THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA IRVINE GIFT OF Mrs. John Williams THEORETICAL ASTRONOMY RELATING TO THE MOTIONS OF THE HEAVENLY BODIES REVOLVING AROUND THE SUN IN ACCORDANCE WITH THE LAW OF UNIVERSAL GRAVITATION EMBRACING SYSTEMATIC DERIVATION OP THE FORMULS FOR THE CALCULATION OF THE GEOCENTRIC AXD HEU CENTRIC PLA7E3, FOR THE DETERMINATION' OF THE ORBITS OF PLANETS AND COMETS, FOE THE CORRECTION OF APPROXIMATE ELEMENTS, ANT FOR THE COMPUTATION- OF NATION OF OBSERVATIONS AND THE METHOD OF LEAST SQUARES. iili numerical (Examples and ^uxili BY JAMES C. WATS OX PHILADELPHIA: J. B. LIPPINCOTT COMPANY. LONDON : 10 HENRIETTA ST., COVENT GARDEN. 1892. Entered, according to Act of Congress, in the year 1868, by J. B. LIPPINCOTT & CO., in the Clerk's Office of the District Court of the United States for the Bustern District of Pennsylvania. PREFACE THE discovery of the great law of nature, the law of gravitation, by NEWTOX, prepared the way for the brilliant achievements which have distinguished the history of astronomical science. A first essential, how- ever, to the solution of those recondite problems which were to exhibit the effect of the mutual attraction of the bodies of our system, was the development of the infinitesimal calculus; and the labors of those who devoted themselves to pure analysis have contributed a most important part in the attainment of the high degree of perfection which character- izes the results of astronomical investigations. Of the earlier efforts to develop the great results following from the law of gravitation, those of EULER stand pre-eminent, and the memoirs which he published have, in reality, furnished the germ of all subsequent investigations in celestial mechanics. In this connection also the names of BERXOUILLI, CLAIRAUT, and D'ALEMBERT deserve the most honorable mention as having contributed also, in a high degree, to give direction to the inves- tigations which were to unfold so many mysteries of nature. By means of the researches thus inaugurated, the great problems of mechanics were successfully solved, many beautiful theorems relating to the planet- ary motions demonstrated, and many useful formulae developed. It is true, however, that in the early stage of the science methods were developed which have since been found to be impracticable, even if not erroneous; still, enough was effected to direct attention in the proper channel, and to prepare the way for the more complete labors of LAGK^XGE and LAPLACE. The genius and the analytical skill of these extraordinary men gave to the progress of Theoretical Astronomy the most rapid strides ; and the intricate investigations which they success- fully performed, served constantly to educe new discoveries, so that of all the problems relating to the mutual attraction of the several planets 4 PREFACE. but litlle more remained to be accomplished by their successors than to develop and simplify the methods which they made known, and to intro- duce such modifications as should be indicated by experience or rendered possible by the latest discoveries in the domain of pure analysis. The problem of determining the elements of the orbit of a comet moving in a parabola, by means of observed places, which had been considered by NEWTON, EULER, BOSCOVICH, LAMBERT, and others, received from LAGKANGE and LAPLACE the most careful consideration in the light of all that had been previously done. The solution given by the former is analytically complete, but far from being practically complete ; that given by the latter is especially simple and practical so far as regards the labor of computation ; but the results obtained by it are so affected by the unavoidable errors of observation as to be often little more than rude approximations. The method which was found to answer best in actual practice, was that proposed by OLBERS in his work entitled Leichteste und bequemste Methode die Bahn eines Cometen zu berechnen, in which, by making use of a beautiful theorem of para- bolic motion demonstrated by EULER and also by LAMBERT, and by adopting a method of trial and error in the numerical solution of certain equations, he was enabled to effect a solution which could be performed with remarkable ease. The accuracy of the results obtained by OLBERS'S method, and the facility of its application, directed the attention of LEGENDRE, IVORY, GAUSS, and ENCKE to this subject, and by them the method was extended and generalized, and rendered appli- cable in the exceptional cases in which the other methods failed. It should be observed, however, that the knowledge of one element, the eccentricity, greatly facilitated the solution; and, although elliptic elements had been computed for some of the comets, the first hypothesis was that of parabolic motion, so that the subsequent process required simply the determination of the corrections to be applied to these ele- ments in order to satisfy the observations. The more difficult problem of determining all the elements of planetary motion directly from three observed places, remained unsolved until the discovery of Ceres by PIAZZI in 1801, by which the attention of GAUSS was directed to this subject, the result of which was the subsequent publication of his Theoria Motus Corporum Ccelestium, a most able work, in which he gave to the world, in a finished form, the results of many years of attention PREFACE. 5 to the subject of which it treats. Ills method for determining all the elements directly from given observed places, as given in the Theorla Mohis, and as subsequently given in a revised form by ENCKE, 1< a yes scarcely any thing to be desired on this topic. In the same work he gave the first explanation of the method of least squares, a method which has been of inestimable service in investigations depending on observed data. The discovery of the minor planets directed attention also to the methods of determining their perturbations, since those applied in the case of the major planets were found to be inapplicable. For a long time astronomers were content simply to compute the special perturba- tions of these bodies from epoch to epoch, and it was not until the com- mencement of the brilliant researches by HANSEN that serious hopes were entertained of being able to compute successfully the general per- turbations of these bodies. By devising an entirely new mode of con- sidering the perturbations, namely, by determining what may be called the perturbations of the time, and thus passing from the undisturbed place to the disturbed place, and by other ingenious analytical and mechanical devices, he succeeded in effecting a solution of this most difficult problem, and his latest works contain all the formulae which are required for the cases actually occurring. The refined and difficult analysis and the laborious calculations involved were such that, even after HANSEN'S methods were made known, astronomers still adhered to the method of special perturbations by the variation of constants as developed by LAGRANGE. The discovery of Astrcea by HENCKE was speedily followed by the discovery of other planets, and fortunately indeed it so happened that the subject of special perturbations was to receive a new improvement. The discovery by BOND and ENCKE of a method by which we determine at once the variations of the rectangular co-ordinates of the disturbed body by integrating the fundamental equations of motion by means of mechanical quadrature, directed the attention of HANSEN to this phase of the problem, and soon after he gave formulae for the determination of the perturbations of the latitude, the mean anomaly, and the loga- rithm of the radius-vector, which are exceedingly convenient in the process of integration, and which have been found to give the most satisfactory results. The formulae for the perturbations of the latitude, 6 PREFACE. true longitude, and radius-vector, to be integrated in the same manner, were afterwards given by BRIJNNOW. Having thus stated briefly a few historical facts relating to the problems of theoretical astronomy, I proceed to a statement of the object of this work. The discovery of so many planets and comets has furnished a wide field for exercise in the calculations relating to their motions, and it has occurred to me that a work which should contain a development of all the formulae required in determining the orbits of the heavenly bodies directly from given observed places, and in correcting these orbits by means of more extended discussions of series of observa- tions, including also the determination of the perturbations, togethei with a complete collection of auxiliary tables, and also such practical directions as might guide the inexperienced computer, might add very materially to the progress of the science by attracting the attention of a greater number of competent computers. Having carefully read the works of the great masters, my plan was to prepare a complete work on this subject, commencing with the fundamental principles of dynamics, and systematically treating, from one point of view, all the problems presented. The scope and the arrangement of the work will be best understood after an examination of its contents ; and let it suffice to add that I have endeavored to keep constantly in view the wants of the computer, providing for the exceptional cases as they occur, and giving all the formulae which appeared to me to be best adapted to the problems under consideration. I have not thought it worth while to trace out the geometrical signification of many of the auxiliary quantities introduced. Those who are curious in such matters may readily derive many beau- tiful theorems from a consideration of the relations of some of these auxiliaries. For convenience, the formulae are numbered consecutively through each chapter, and the references to those of a preceding chapter are defined by adding a subscript figure denoting the number of the chapter. Besides having read the works of those who have given special atten- tion to these problems, I have consulted the Astronomische Nachrichten, the Astronomical Journal, and other astronomical periodicals, in which is to be found much valuable information resulting from the experi- ence of those who have been or are now actively engaged in astro- nomical pursuits. I must also express my obligations to the publishers, PREFACE. 7 Messrs. J. B. LIPPINCOTT & Co., for the generous interest which they have manifested in the publication of the work, and also to Dr. B. A. GOULD, of Cambridge, Mass., and to Dr. OPPOLZER, of Vienna, for valuable suggestions. For the determination of the time from the perihelion and of the true anomaly in very eccentric orbits I have given the method proposed by BESSEL in the Monatliche Correspondenz, vol. xii., the tables for which were subsequently given by BRUNNOW in his Astronomical Notices, and also the method proposed by GAUSS, but in a more convenient form. For obvious reasons, I have given the solution for the special case of parabolic motion before completing the solution of the general problem of finding all of the elements of the orbit by means of three observed places. The differential formulae and the other formulae for correcting approximate elements are given in a form convenient for application, and the formulae for finding the chord or the time of describing the subtended arc of the orbit, in the case of very eccentric orbits, will be found very convenient in practice. I have given a pretty full development of the application of the theory of probabilities to the combination of observations, endeavoring to direct the attention of the reader, as far as possible, to the sources of error to be apprehended and to the most advantageous method of treat- ing the problem so as to eliminate the effects of these errors. For the rejection of doubtful observations, according to theoretical considerations, I have given the simple formula, suggested by CHAUVENET, which fol lows directly from the fundamental equations for the probability of errors, and which will answer for the purposes here required as well as the more complete criterion proposed by PEIRCE. In the chapter devoted to the theory of special perturbations I have taken particular pains to develop the whole subject in a complete and practical form, keeping constantly in view the requirements for accurate and convenient numerical application. The time is adopted as the independent variable in the determination of the perturbations of the elements directly, since experience has established the convenience of this form ; and should it be desired to change the independent variable and to use the differential coefficients with respect to the eccentric anomaly, the equations between this function and the mean motion will enable us to effect readily the required transformation. 8 PREFACE. The numerical examples involve data derived from actual observa- tions, and care has been taken to make them complete in every respect, so as to serve as a guide to the efforts of those not familiar with these calculations ; and when different fundamental planes are spoken of, it is presumed that the reader is familiar with the elements of spherical astronomy, so that it is unnecessary to state, in all cases, whether the centre of the sphere is taken at the centre of the earth, or at any other point in space. The preparation of the Tables has cost me a great amount of labor, logarithms of ten decimals being employed in order to be sure of the last decimal given. Several of those in previous use have been recom- puted and extended, and others here given for the first time have been prepared with special care. The adopted value of the constant of the solar attraction is that given by GAUSS, which, as will appear, is not accurately in accordance with the adoption of the mean distance of the earth from the sun as the unit of space; but until the absolute value of the earth's mean motion is known, it is best, for the sake of uniformity and accuracy, to retain GAUSS'S constant. The preparation of this work has been effected amid many interrup- tions, and with other labors constantly pressing me, by which the progress of its publication has been somewhat delayed, even since the stereo- typing was commenced, so that in some cases I have been anticipated in the publication of formulae which would have here appeared for the first time. I have, however, endeavored to perform conscientiously the self-imposed task, seeking always to secure a logical sequence in the de- velopment of the formulae, to preserve uniformity and elegance in the notation, and to elucidate the successive steps in the analysis, so that the work may be read by those who, possessing a respectable mathematical education, desire to be informed of the means by whicli astronomers are enabled to arrive at so many grand results connected with the motions of the heavenly bodies, and by which the grandeur and sublimity of creation are unveiled. The labor of the preparation of the work will have been fully repaid if it shall be the means of directing a more general attention to the study of the wonderful mechanism of the hea- vens, the contemplation of which must ever serve to impress upon the mind the reality of the perfection of the OMNIPOTENT, the LIVING GOD ! OBSERVATORY, ANN ARBOR, June, 1867. CONTENTS. THEORETICAL ASTRONOMY. CHAPTER I. INVESTIGATION OF THE FUNDAMENTAL EQUATIONS OF MOTION, AND OF THE FOR- MULA FOB DETERMINING, FROM KNOWN ELEMENTS, THE HELIOCENTRIC AND GEOCENTRIC PLACES OF A HEAVENLT BODY, ADAPTED TO NUMERICAL COM- PUTATION FOR CASES OF ANY ECCENTRICITY WHATEVER. PAGE Fundamental Principles 15 Attraction of Spheres 19 Motions of a System of Bodies 23 Invariable Plane of the System 29 Motion of a Solid Body 31 The Units of Space, Time, and Mass 36 Motion of a Body relative to the Sun 38 Equations for Undisturbed Motion 42 Determination of the Attractive Force of the Sun 49 Determination of the Place in an Elliptic Orbit 53 Determination of the Place in a Parabolic Orbit 59 Determination of the Place in a Hyperbolic Orbit 65 Methods for finding the True Anomaly and the Time from the Perihelion in the case of Orbits of Great Eccentricity- 70 Determination of the Position in Space 81 Heliocentric Longitude and Latitude 83 Reduction to the Ecliptic 85 Geocentric Longitude and Latitude 86 Transformation of Spherical Co-ordinates 87 Direct Determination of the Geocentric Right Ascension and Declination 90 .Reduction of the Elements from one Epoch to another 99 Numerical Examples 103 Interpolation 112 Time of Opposition 114 9 10 CONTENTS. CHAPTER II. INVESTIGATION OP THE DIFFERENTIAL FORMULA WHICH EXPRESS THE RELATION BETWEEN THE GEOCENTRIC OR HELIOCENTRIC PLACES OF A HEAVENLY BODY AND THE VARIATIONS OF THE ELEMENTS OF ITS ORBIT. PAOB Variation of the Eight Ascension and Declination 118 Case of Parabolic Motion 125 Case of Hyperbolic Motion 128 Case of Orbits differing but little from the Parabola 130 Numerical Examples 135 Variation of the Longitude and Latitude 143 The Elements referred to the same Fundamental Plane as the Geocentric Places 149 Numerical Example 150 Plane of the Orbit taken as the Fundamental Plane to which the Geocentric Places are referred 153 Numerical Example 159 Variation of the Auxiliaries for the Equator 163 CHAPTER III. INVESTIGATION OF FORMULA FOR COMPUTING THE ORBIT OF A COMET MOVTNO IN A PARABOLA, AND FOR CORRECTING APPROXIMATE ELEMENTS BY THE VARIATION OF THE GEOCENTRIC DISTANCE. Correction of the Observations for Parallax 167 Fundamental Equations 169 Particular Cases 172 Katio of Two Curtate Distances 178 Determination of the Curtate Distances 181 Kelation between Two Radii- Vectores, the Chord joining their Extremities, and the Time of describing the Parabolic Arc 184 Determination of the Node and Inclination 192 Perihelion Distance and Longitude of the Perihelion 194 Time of Perihelion Passage 195 Numerical Example 199 Correction of Approximate Elements by varying the Geocentric Distance 208 Numerical Example 213 CHAPTER IV. DETERMINATION, FROM THREE COMPLETE OBSERVATIONS, OF THE ELEMENTS OF THE ORBIT OF A HEAVENLY BODY, INCLUDING THE ECCENTRICITY OR FORM OF THE CONIC SECTION. Reduction of the Data 220 Corrections for Parallax 223 CONTENTS. 11 PAOt Fundamental Equations 225 Formulae for the Curtate Distances 228 Modification of the Formulae in Particular Cases 231 Determination of the Curtate Distance for the Middle Observation .. 236 Case of a Double Solution 239 Position indicated by the Curvature of the Observed Path of the Body 242 Formulae for a Second Approximation 243 Formulae for finding the Ratio of the Sector to the Triangle ... 247 Final Correction for Aberration 257 Determination of the Elements of the Orbit 259 Numerical Example 264 Correction of the First Hypothesis 278 Approximate Method of finding the Ratio of the Sector to the Triangle ~ 279 CHAPTER V. DETERMINATION OF THE ORBIT OF A HEAVENLY BODY FROM FOUR OBSERVA- TIONS, OF WHICH THE SECOND AND THIRD MUST BE COMPLETE. Fundamental Equations .. 282 Determination of the Curtate Distances 289 Successive Approximations 293 Determination of the Elements of the Orbit 294 Numerical Example 294 Method for the Final Approximation 307 CHAPTER VI. INVESTIGATION OF VARIOUS FORMULA FOR THE CORRECTION OF THE APPROXI- MATE ELEMENTS OF THE ORBIT OF A HEAVENLY BODY. Determination of the Elements of a Circular Orbit 311 Variation of Two Geocentric Distances 313 Differential Formulae 318 Plane of the Orbit taken as the Fundamental Plane 320 Variation of the Node and Inclination 324 Variation of One Geocentric Distance 328 Determination of the Elements of the Orbit by means of the Co-ordinates and Velocities 332 Correction of the Ephemeris 335 Final Correction of the Elements 338 Relation between Two Places in the Orbit 339 Modification when the Semi-Transverse Axis is very large 341 Modification for Hyperbolic Motion 346 Variation of the Semi-Transverse Axis and Ratio of Two Curtate Distances... ~ 349 12 CONTENTS. PAGJI Variation of the Geocentric Distance and of the Eeciprocal of the Semi-Trans- verse Axis 352 Equations of Condition 353 Orbit of a Comet 355 Variation of Two Kadii-Vectores 357 CHAPTER VII. METHOD OF LEAST SQUARES, THEORY OP THE COMBINATION OP OBSERVATIONS, AND DETERMINATION OF THE MOST PROBABLE SYSTEM OF ELEMENTS FROM A SERIES OP OBSERVATIONS. Statement of the Problem 360 Fundamental Equations for the Probability of Errors 362 Determination of the Form of the Function which expresses the Probability ... 363 The Measure of Precision, and the Probable Error 366 Distribution of the Errors x . 367 The Mean Error, and the Mean of the Errors 368 The Probable Error of the Arithmetical Mean 370 Determination of the Mean and Probable Errors of Observations 371 Weights of Observed Values 372 Equations of Condition 376 Normal Equations 378 Method of Elimination 380 Determination of the Weights of the Kesulting Values of the Unknown Quanti- ties 386 Separate Determination of the Unknown Quantities and of their Weights 392 Relation between the Weights and the Determinants 396 Case in which the Problem is nearly Indeterminate 398 Mean and Probable Errors of the Results 399 Combination of Observations 401 Errors peculiar to certain Observations 408 Rejection of Doubtful Observations 410 Correction of the Elements 412 Arrangement of the Numerical Operations 415 Numerical Example 418 Case of very Eccentric Orbits 423 CHAPTER VIII. INVESTIGATION OF VARIOUS FORMULA FOR THE DETERMINATION OF THE SPECIAL PERTURBATIONS OF A HEAVENLY BODY. Fundamental Equations 426 Statement of the Problem 428 Variation of Co-ordinates 42? CONTENTS. 13 PAOK Mechanical Quadrature > 433 The Interval for Quadrature , 443 Mode of effecting the Integration 445 Perturbations depending on the Squares and Higher Powers of the Masses 446 Numerical Example 448 Change of the Equinox and Ecliptic 455 Determination of New Osculating Elements 459 Variation of Polar Co-ordinates 462 Determination of the Components of the Disturbing Force 467 Determination of the Heliocentric or Geocentric Place 471 Numerical Example i74 Change of the Osculating Elements 477 Variation of the Mean Anomaly, the Eadius- Vector, and the Co-ordinate z 480 Fundamental Equations 483 Determination of the Components of the Disturbing Force 489 Case of very Eccentric Orbits 493 Determination of the Place of the Disturbed Body 495 Variation of the Node and Inclination 502 Numerical Example 505' Change of the Osculating Elements 510 Variation of Constants 516 Case of very Eccentric Orbits 523 Variation of the Periodic Time 526 Numerical Example 529 Formulae to be used when the Eccentricity or the Inclination is small 533 Correction of the Assumed Value of the Disturbing Mass 535 Perturbations of Comets 536 Motion about the Common Centre of Gravity of the Sun and Planet 537 Reduction of the Elements to the Common Centre of Gravity of the Sun and Planet 538 Reduction by means of Differential Formulae 540 Near Approach of a Comet to a Planet 546 The Sun may be regarded as the Disturbing Body 548 Determination of the Elements of the Orbit about the Planet 550 Subsequent Motion of the Comet 551 Effect of a Resisting Medium in Space 552 Variation of the Elements on account of the Resisting Medium 554 Method to be applied when no Assumption is made in regard to the Density of the Ether 556 14 CONTENTS. TABLES. PAOl I. Angle of the Vertical and Logarithm of the Earth's Kadius 561 IL For converting Intervals of Mean Solar Time into Equivalent Intervals of Sidereal Time 563 in. For converting Intervals of Sidereal Time into Equivalent Intervals of Mean Solar Time 564 IV. For converting Hours, Minutes, and Seconds into Decimals of a Day... 565 V. For finding the Number of Days from the Beginning of the Year 565 VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit 566 VII. For finding the True Anomaly in a Parabolic Orbit when v is nearly 180 611 VIII. For finding the Time from the Perihelion in a Parabolic Orbit 612 IX. For finding the True Anomaly or the Time from the Perihelion in Orbits of Great Eccentricity 614 X. For finding the True Anomaly or the Time from the Perihelion in El- liptic and Hyperbolic Orbits 618 XL For the Motion in a Parabolic Orbit 619 XII. For the Limits of the Boots of the Equation sin (z' f) = n^ sin 4 z' ... 622 XIII. For finding the Katio of the Sector to the Triangle 624 XIV. For finding the Eatio of the Sector to the Triangle 629 XV. For Elliptic Orbits of Great Eccentricity 632 XVI. For Hyperbolic Orbits 632 XVII. For Special Perturbations 633 XVIII. Elements of the Orbits of the Comets which have been observed 638 XIX. Elements of the Orbits of the Minor Planets 646 XX. Elements of the Orbits of the Major Planets 648 XXI. Constants, &c 649 EXPLANATION OF THE TABLES 651 APPENDIX. Precession 657 Nutation 658 Aberration 659 Intensity of Light 660 Numerical Calculations.... ... fif.2 THEORETICAL ASTRONOMY. CHAPTER I. INVESTIGATION OF THE FUNDAMENTAL, EQUATIONS OF MOTION, AND OF THE FOR- MULAS FOR DETERMINING, FROM KNOWN ELEMENTS, THE HELIOCENTRIC AND GEOCENTRIC PLACES OF A HEAVENLY BODY, ADAPTED TO NUMERICAL COMPUTA- TION FOR CASES OF ANY ECCENTRICITY WHATEVER. 1. THE study of the motions of the heavenly bodies does not re- quire that we should know the ultimate limit of divisibility of the matter of which they are composed, whether it may be subdivided indefinitely, or whether the limit is an indivisible, impenetrable atom. Nor are we concerned with the relations which exist between the separate atoms or molecules, except so far as they form, in the aggre- gate, a definite body whose relation to other bodies of the system it is required to investigate. On the contrary, in considering the ope- ration of the laws in obedience to which matter is aggregated into single bodies and systems of bodies, it is sufficient to conceive simply of its divisibility to a limit which may be regarded as infinitesimal compared with the finite volume of the body, and to regard the mag- nitude of the element of matter thus arrived at as a mathematical point. An element of matter, or a material body, cannot give itself motion; neither can it alter, in any manner whatever, any motion which may have been communicated to it. This tendency of matter to resist all changes of its existing state of rest or motion is known as inertia, and is the fundamental law of the motion of bodieS. Ex- perience invariably confirms it as a law of nature ; the continuance of motion as resistances are removed, as well as the sensibly unchanged motion of the heavenly bodies during many centuries, affording the 15 16 THEORETICAL ASTRONOI.il'. most convincing proof of its universality. Whenever, therefore, a material point experiences any change of its. state as respects rest or motion, the cause must be attributed to the operation of something external to the element itself, and which we designate by the word force. The nature of forces is generally unknown, and we estimate them by the effects which they produce. They are thus rendered com- parable with some unit, and may be expressed by abstract numbers. 2. If a material point, free to move, receives an impulse by virtue of the action of any force, or if, at any instant, the force by which motion is communicated shall cease to act, the subsequent motion of the point, according to the law of inertia, must be rectilinear and uniform, equal spaces being described in equal times. Thus, if s, v, and t represent, respectively, the space, the velocity, and the time, the measure of v being the space described in a unit of time, we shall have, in this case, s = vt. It is evident, however, that the space described in a unit of time will vary with the intensity of the force to which the motion is due, and, the nature of the force being unknown, we must necessarily compare the velocities communicated to the point by different forces, in order to arrive at the relation of their effects. We are thus led to regard the force as proportional to the velocity; and this also has received the most indubitable proof as being a law of nature. Hence, the principles of the composition and resolution of forces may be applied nlso to the composition and resolution of velocities. If the force acts incessantly, the velocity will be accelerated, and the force which produces this motion is called an accelerating force. In regard to the mode of operation of the force, however, we may consider it as acting absolutely without cessation, or we may regard it as acting instantaneously at successive infinitesimal intervals repre- sented by dt, and hence the motion as uniform during each of these intervals. The latter supposition is that which is best adapted to the requirements of the infinitesimal calculus; and, according to the fundamental principles of this calculus, the finite result will be the same as in the case of a force whose action is absolutely incessant. Therefore, if we represent the element of space by ds, and the ele- ment of time by dt, the instantaneous velocity will be '* ~ dt' which will vary from one instant to another. FUXDAME:N*TAL PRINCIPLES. 1? 3. Since the force is proportional to the velocity, its measure at any instant will be determined by the corresponding velocity. If the accelerating force is constant, the motion will be uniformly accele- rated; and if we designate the acceleration due to the force by/, the unit of/ being the velocity generated in a unit of time, we shall have V =ft. If, however, the force be variable, we shall have, at any instant, the relation the force being regarded as constant in its action during the element of time dt. The instantaneous value of v gives, by differentiation, dv _ d*s ~dt~~dl? and hence we derive /= so that, in varied motion, the acceleration due to the force is mea- sured by the second differential of the space divided by the square of the element of time. 4. By the mass of the body we mean its absolute quantity of mat- ter. The density is the mass of a unit of volume, and hence the entire mass is equal to the volume multiplied by the density. If it is required to compare the forces which act upon different bodies, it is evident tbat the masses must be considered. If equal masses receive impulses by the action of instantaneous forces, the forces acting on each will be to each other as the velocities imparted ; and if we consider as the unit of force that which gives to a unit of mass the unit of velocity, we have for the measure of a force F, denoting the mass by Jf, F = Mv. This is called the quantity of motion of the body, and expresses its capacity to overcome inertia. By virtue of the inert state of matter, there can be no action of a force without an equal and contrary re- action ; for, if the body to which the force is applied is fixed, the equilibrium between the resistance and the force necessarily implies the development of an equal and contrary force ; and, if the body be free to move, in the change of state, its inertia will oppose equal and 2 18 THEORETICAL ASTRONOMY. contrary resistance. Hence, as a necessary consequence of inertia, it follows that action and reaction are simultaneous, equal, and contrary. If the body is acted upon by a force such that the motion is varied, the accelerating force upon each element of its mass is represented by , and the entire motive force F is expressed by = Hi> M being the sum of all the elements, or the mass of the body. Since ds V== W this gives which is the expression for the intensity of the motive force, or of the force of inertia developed. For the unit of mass, the measure of the force is d's d?' and this, therefore, expresses that part of the intensity r>f the motive force which is impressed upon the unit of mass, and is what is usually called the accelerating force. 5. The force in obedience to which the heavenly bodies perform their journey through space, is known as the attraction of gravitation ; and the law of the operation of this force, in itself simple and unique, has been confirmed and generalized by the accumulated Researches of modern science. Not only do we find that it controls the motions of the bodies of our own solar system, but that the revolutions of binary systems of stars in the remotest regions of space proclaim the uni- versality of its operation. It unfailingly explains all the phenomena observed, and, outstripping observation, it has furnished the means of predicting many phenomena subsequently observed. The Jaw of this force is that every particle of matter is attracted by every other particle by a force which varies directly as the mass and inversely as the square of the distance of the attracting particle. This reciprocal action is instantaneous, and is not modified, in any degree, by the interposition of other particles or bodies of matter. It is also absolutely independent of the nature of the molecules them- selves, and of their aggregation. ATTRACTION OF SPHERES. 19 If we consider two bodies the masses of which are m and m f , and whose magnitudes are so small, relatively to their mutual distance /?, that we may regard them as material points, according to the law of gravitation, the action of m on each molecule or unit of m' will be -, and the total force on m' will be P* .m m . ? The action of m' on each molecule of m will be expressed by , and its total action by m' "7* The absolute or moving force with which the masses m and m! tend toward each other is, therefore, the same on each body, which result is a necessary consequence of the equality of action and reaction. The velocities, however, with which these bodies would approach each other must be different, the velocity of the smaller mass exceed- ing that of the greater, and in the ratio of the masses moved. The expression for the velocity of m', which would be generated in a unit of time if the force remained constant, is obtained by dividing the absolute force exerted by m by the mass moved, which gives m 7 and this is, therefore, the measure of the acceleration due to the action of m at the distance p. For the acceleration due to the action of m' we derive, in a similar manner, 6. Observation shows that the heavenly bodies are nearly spherical in form, and we shall therefore, preparatory to finding the equations which express the relative motions of the bodies of the system, de- termine the attraction of a spherical mass of uniform density, or varying from the centre to the surface according to any law, for a point exterior to it. If we suppose a straight line to be drawn through the centre of the sphere and the point attracted, the total action of the sphere on the point will be a force acting along this line, since the mass of the sphere is symmetrical with respect to it. Let dm denote an element 20 THEORETICAL, ASTRONOMY. of the mass of the sphere, and /> its distance from the point attracted; then will dm ~P r express the action of this element on the point attracted. If we sup- pose the density of the sphere to be constant, and equal to unity, the element dm becomes an element of volume, and will be expressed by dm = dx dy dz ; x, y. and z being the co-ordinates of the element referred to a system of rectangular co-ordinates. If we take the origin of co-ordinates at the centre of the sphere, and introduce polar co-ordinates, so that x r cos y cos 0, y = r cos
do df= f- If we suppose the axis of z to be directed to the point attracted, the co-ordinates of this point will be a being the distance of the point from the centre of the sphere, and, since />' = (x - ao- + (y - ) 2 + - O' we shall have P t =a? 2ar sin
2 , with respect to a, gives do a r sin, and 6 are independent of a, and hence
ji* cos we have p = a r, and p = a -f- r ,
and taking the integral with respect to /> between these limits, we
obtain
Integrating, finally, between the limits r = and r = r,, we get
r, being the radius of the sphere, and, if we denote its entire mass by
TO, this becomes
m
Y ==
a
Therefore,
dV m
da
from which it appears that the action of a homogeneous spherical
mass on a point exterior to it, is the same as if the entire mass were
concentrated at its centre. If, in the integration with respect to r,
we take the limits r' and r", we obtain
and, denoting by m the mass of a spherical shell whose radii are r"
and r', this becomes
Consequently, the attraction of a homogeneous spherical shell on a
point exterior to it, is the same as if the entire mass were concentrated
at its centre.
The supposition that the point attracted is situated within a
spherical shell of uniform density, does not change the form of the
FUNDAMENTAL PRINCIPLES.
general equation; but, in the integration with reference to p, the
limits will be p = r -f- a, and p = r a, which give
and this being independent of a, we have
^=-=-
da
Whence it follows that a point placed in the interior of a spherical
shell is equally attracted in all directions, and that, if not subject to
the action of any extraneous force, it will be in equilibrium in every
position.
7. Whatever may be the law of the change of the density of the
heavenly bodies from the surface to the centre, we may regard them
as composed of homogeneous, concentric layers, the density varying
only from one layer to another, and the number of the layers may
be indefinite. The action of each of these will be the same as if its
mass were united at the centre of the shell ; and hence the total action
of the body will be the same as if the entire mass were concentrated
at its centre of gravity. The planets are indeed not exactly spheres,
but oblate spheroids differing but little from spheres ; and the error
of the assumption of an exact spherical form, so far as it relates to
their action upon each other, is extremely small, and is in fact com-
pensated by the magnitude of their distances from each other; for,
whatever may be the form of the body, if its dimensions are small
in comparison with its distance from the body which it attracts, it is
evident that its action will be sensibly the same as if its entire mass
were concentrated at its centre of gravity. If we suppose a system
of bodies to be composed of spherical masses, each unattended with
any satellite, and if we suppose that the dimensions of the bodies
are small in comparison with their mutual distances, the formation
of the equations for the motion of the bodies of the system will be
reduced to the consideration of the motions of simple points endowed
with forces of attraction corresponding to the respective masses. Our
solar system is, in reality, a compound system, the several systems
of primary and satellites corresponding nearly to the case supposed ;
and, before proceeding with the formation of the equations which are
applicable to the general case, we will consider, at first, those for a
simple system of bodies, considered as points and subject to their
mutual actions and the action of the forces which correspond to the
24 THEORETICAL ASTRONOMY.
actual velocities of the different parts of the system for any ins! ant.
It is evident that we cannot consider the motion of any single body
as free, and subject only to the action of the primitive impulsion
which it has received and the accelerating forces which act upon it ;
but, on the contrary, the motion of each body will depend on the
force which acts upon it directly, and also on the reaction due to the
other bodies of the system. The coLsideration, however, of the varia-
tions of the motion of the several bodies of the system is reduced to
the simple case of equilibrium by means of the general principle that,
if we assign to the different bodies of the system motions which are
modified by their mutual action, we may regard these motions as
composed of those which the bodies actually have and of other
motions which are destroyed, and which must therefore necessarily
be such that, if they alone existed, the system would be in equi-
librium. We are thus enabled to form at once the equations for the
motion of a system of bodies. Let m, m', m", &c. be the masses of
the several bodies of the system, and x, y, z, x' } y f , z', &c. their co-
ordinates referred to any system of rectangular axes. Further, let
the components of the total force acting upon a unit of the mass of
m, or of the accelerating force, resolved in directions parallel to the
co-ordinate axes, be denoted by X, Y, and Z y respectively, then will
mX, mY, mZ,
be the forces which act upon the body in the same directions. The
velocities of the body m at any instant, in directions parallel to the
co-ordinate axes, will be
dx_ dy dz
W dP ~dt'
and the corresponding forces are
m m
By virtue of the action of the accelerating force, these forces for the
next instant become
m^+mXeft, m* +
at at
which may be written respectively:
MOTION OF A SYSTEM OF BODIES. 25
The actual velocities for this instant are
dx , dx dy dy dz dz
-dt+ d W ~di + d -di' -dt+ d ^di'
and the corresponding forces are
dx , dx dy . . dy dz . , dz
Comparing these with the preceding expressions for the forces, it
appears that the forces which are destroyed, in directions parallel to
the co-ordinate axes, are
md-j^ + mXdt,
-md + mYdt, tf)
at
In the same manner we find for the forces which will be destroyed
in the case of the body m! :
m'd--
at
and similarly for the other bodies of the system. According to the
general principle above enunciated, the system under the action of
these forces alone, will be in equilibrium. The conditions of equi-
librium for a system of points of invariable but arbitrary form, and
subject to the action of forces directed in any manner whatever, are
IX, = 0, IY, = 0, IZ, = 0,
0, Z (Z,y - YJ = 0,
in which X,, Y,, Z,, denote the components, resolved parallel to the
26 THEORETICAL ASTRONOMY.
co-ordinate axes, of the forces acting on any point, and , rr, y, z, the
co-ordinates of the point. These equations are equally applicable to
the case of the equilibrium at any instant of a system of variable
form ; and substituting in them the expressions (3) for the force <=> dU
Btroyed in the case of a system of bodies, we shall have
at
which are the general equations for the motions of a system of bodies.
8. Let x,, y,, z t) be the co-ordinates of the centre of gravity of the
system, and, by differentiation of the equations for the co-ordinates
of the centre of gravity, which are
_ Imx _ Imy _ Zmz
X >-~2^' y '--Zm~' Z '2^>
we get
v &* v d _ m (y yO , _ m(z jO
~7~~~' / ' ~?
Hence we derive
m (Yx - Xy~) + m' (Y'x' X'^j = 0,
and generally
28 THEORETICAL ASTRONOMY.
In a similar manner, we find
Zm(Xz Zx) = Q, (7)
Im (Zy Yz) = 0.
These relations will not be altered if, in addition to their reciprocal
action, the bodies of the system are acted upon by forces directed to
the origin of co-ordinates. Thus, in the case of a force acting upon
m, and directed to the origin of co-ordinates, we have, for its action
alone,
Yx = Xy t Xz = Zx, Zy=Yz,
and similarly for the other bodies. Hence these forces disappear
from the equations, and, therefore, when the several bodies of the
system are subject only to their reciprocal action and to forces directed
to the origin of co-ordinates, the last three of equations (4) become
*(-)-*
the integration of which gives
Im (xdy ydx) = cat,
Im(zdxxdz}=c'dt, (8)
Im (ydz zdy} = c"dt,
c, c', and c" being the constants of integration. Now, xdy ydx is
double the area described about the origin of co-ordinates by the pro-
jection of the radius- vector, or line joining m with the origin of co-ordi-
nates, on the plane of xy during the element of time dt ; and, further,
zdx xdz and ydz zdy are respectively double the areas described,
during the same time, by the projection of the radius-vector on the
planes of xz and yz. The constant c, therefore, expresses double the
sum of the products formed by multiplying the areal velocity of each
body, in the direction of the co-ordinate plane xy, by its mass; and
c', c", express the same sum with reference to the co-ordinate planes
xz and yz respectively. Hence the sum of the areal velocities of the
several bodies of the system about the origin of co-ordinates, each
multiplied by the corresponding mass, is constant; and the sum of
the areas traced, each multiplied by the corresponding mass, is pro-
portional to the time. If the only forces which operate, are those
INVARIABLE PLANE. 29
resulting from the mutual action of the bodies which compose the
system, this result is correct whatever may be the point in space
taken as the origin of co-ordinates.
The areas described by the projection of the radius-vector of each
body on the co-ordinate planes, are the projections, on these planes, of
the areas actually described in space. We may, therefore, conceive of
a resultant, or principal plane of projection, such that the sum of the
areas traced by the projection of each radius-vector on this plane,
when projected on the three co-ordinate planes, each being multiplied
by the corresponding mass, will be respectively equal to the first
members of the equations (8). Let a, /?, and f be the angles which
this principal plane makes with the co-ordinate planes xy, xz, and yz,
respectively; and let 8 denote the sum of the areas traced on this
plane, in a unit of time, by the projection of the radius-vector of
each of the bodies of the system, each area being multiplied by the
corresponding mass. The sum S will be found to be a maximum,
and its projections on the co-ordinate planes, corresponding to the
element of time dt, are
S cos a dt, S cos /5 dt, S cos f dt.
Therefore, by means of equations (8), we have
c = S cos a, c' = S cos /3, c" = S cos r,
and, since cos 2 a + cos 2 /? + cos 2 f = 1,
Hence we derive
COS a = , . .. =r, COS /? =
VV + c" 1 + c" 2 ' l/V + c ' 2 + c">
c"
COS Y =
These angles, being therefore constant and independent of the time,
show that this principal plane of projection remains constantly par-
allel to itself during the motion of the system in space, whatever
may be the relative positions of the several bodies ; and for this
reason it is called the invariable plane of the system. Its position
with reference to any known plane is easily determined when the
velocities, in directions parallel to the co-ordinate axes, and the
masses and co-ordinates of the several bodies of the system, are
known. The values of c, c', c" are given by equations (8), and
30 THEORETICAL ASTRONOMY.
hence the values of a, /?, and f y which determine the position of the
invariable plane.
Since the positions of the co-ordinate planes are arbitrary, we may
suppose that of xy to coincide with the invariable plane, which gives
cos /9 = and cos f = 0, and, therefore, c' = and c" = 0. Further,
since the positions of the axes of x and y in this plane are arbitrary,
it follows that for every plane perpendicular to the invariable plane,
the sum of the areas traced by the projections of the radii- vectores
of the several bodies of the system, each multiplied by the corre-
sponding mass, is zero. It may also be observed that the value of S
is constant whatever may be the position of the co-ordinate planes,
and that its value is necessarily greater than that of either of the
quantities in the second member of the equation,
except when two of them are each equal to zero. It is, therefore, a
maximum, and the invariable plane is also the plane of maximum
areas.
10. If we suppose the origin of co-ordinates itself to move with
uniform and rectilinear motion in space, the relations expressed by
equations (8) will remain unchanged. Thus, let x,, y,, z, be the co-
ordinates of the movable origin of co-ordinates, referred to a fixed
point in space taken as the origin; and let x , y , z , x f , y ', z f , &c.
be the co-ordinates of the several bodies referred to the movable
origin. Then, since the co-ordinate planes in one system remain
always parallel to those of the other system of co-ordinates, we shall
have
* = x, + ar , y = y, -f y , z = z,+ z ,
nnd similarly for the other bodies of the system. Introducing these
values of x, y, and z into the first three of equations (4), they become
The condition of uniform rectilinear motion of the movable origin
gives
MOTION OF A SOLID BODY. 3]
and the preceding equations beo>me
(9)
-
dr
Substituting the same values in the last three of equations (4), ob-
serving that the co-ordinates x,, y,, z, are the same for all the bodies
of the system, and reducing the resulting equations by means of
equations (9), we get
C10)
Hence it appears that the form of the equations for the motion of the
system of bodies, remains unchanged when we suppose the origin of
co-ordinates to move in space with a uniform and rectilinear motion.
11. The equations already derived for the motions of a system of
bodies, considered as reduced to material points, enable us to form at
once those for the motion of a solid body. The mutual distances of
the parts of the system are, in this case, invariable, and the masses
of the several bodies become the elements of the mass of the solid
body. If we denote an element of the mass by dm, the equations (5)
for the motion of the centre of gravity of the body become
m^=fzdm, (11)
the summation, or integration with reference to dm, being taken so as
to include the entire mass of the body, from which it appears that
the centre of gravity of the body moves in space as if the entire mass
were concentrated in that point, and the forces applied to it directly.
If we take the origin of co-ordinates at the centre of gravity of
the body, and suppose it to have a rectilinear, uniform motion in
space, and denote the co-ordinates of the element dm, in reference to
this origin, by x , y w z w we have, by means of the equations (10),
THEORETICAL ASTRONOMY.
the integration with respect to c?m being taken so as to include the
entire mass of the body. These equations, therefore, determine the
motion of rotation of the body around its centre of gravity regarded
as fixed, or as having a uniform rectilinear motion in space. Equa-
tions (11) determine the position of the centre of gravity for any
instant, and hence for. the successive instants at intervals equal to dt;
and we may consider the motion of the body during the element of
time dt as rectilinear and uniform, whatever may be the form of its
trajectory. Hence, equations (11) and (12) completely determine the
position of the body in space, the former relating to the motion of
translation of the centre of gravity, and the latter to the motion of
rotation about this point. It follows, therefore, that for any forces
which act upon a body we can always decompose the actual motion
into those of the translation of the centre of gravity in space, and of
the motion of rotation around this point ; and these two motions may
be considered independently of each other, the motion of the centre
of gravity being independent of the form and position of the body
about this point.
If the only forces which act upon the body are the reciprocal action
of the elements of its mass and forces directed to the origin of co-
ordinates, the second terms of equations (12) become each equal to
zero, and the results indicated by equations (8) apply in this case
also. The parts of the system being invariably connected, the plane
of maximum areas, or invariable plane, is evidently that which is
perpendicular to the axis of rotation passing through the centre of
gravity, and therefore, in the motion of translation of the centre of
gravity in space, the axis of rotation remains constantly parallel to
itself. Any extraneous force which tends to disturb this relation
will necessarily develop a contrary reaction, and hence a rotating body
resists any change of its plane of rotation not parallel to itself. "Wo
may observe, also, that on account of the invariability of the mutual
distances of the elements of the mass, according to equations (8), the
motion of rotation must be uniform.
12. We shall now consider the action of a system of bodies on a
MOTION OF A SOLID BODY. 33
distant mass, which we will denote by M. Let X Q , y , z , x ', y ' } z f ,
&c. be the co-ordinates of the several bodies of the system referred
to its centre of gravity as the origin of co-ordinates; x n y n and z,
the co-ordinates of the centre of gravity of the system referred to
the centre of gravity of the body M. The co-ordinates of the body
m, of the system, referred to this origin, will therefore be
x = x, -f x ot y = y f + y * = */ + %>
and similarly for the other bodies of the system. If we denote by
r the distance of the centre of gravity of m from that of M, the
accelerating force of the former on an element of mass at the centre
of gravity of the latter, resolved parallel to the axis of x, will be
mx
-pr,
and, therefore, that of the entire system on the element of Jf, resolved
; n the same direction, will be
S-.
We have also
r = (x, + x Y + (y, + y Y + 0, + *)',
and, if we denote by r, the distance of the centre of gravity of the
system from M,
r? = x? + y? + z?.
Therefore
- = (X, + X Q ~) (r* + 2
We shall now suppose the mutual distances of the bodies of the
system to be so small in comparison with the distance r, of its centre
of gravity from that of M, that terms of the order r 2 may be neglected ;
a condition which is actually satisfied in the case of the secondary
systems belonging to the solar system. Hence, developing the second
factor of the second member of the last equation, and neglecting terms
of the order r 2 , we shall have
_ > , ^o _ 3s, (a?,a? + y,y + g,s )
r" r/ "T r , r *
and
^mx -m . Imx 3#, ,
s -^ = * r * + 3 TT (
3
34 THEORETICAL ASTRONOMY.
But, since x , y , z , are the co-ordinates in reference to the centre of
gravity of the system as the origin, we have
Imx u = 0, Zmy Q = 0,
and the preceding equation reduces to
y^ x
"~^- x> r
In a similar manner, we find
The second members of these equations are the expressions for the
total accelerating force due to the action of the bodies of the system
on M, resolved parallel to the co-ordinate axes respectively, when we
consider the several masses to be collected at the centre of gravity
of the system. Hence we conclude that when an element of mass
is attracted by a system of bodies so remote from it that terms of the
order of the squares of the co-ordinates of the several bodies, referred
to the centre of gravity of the system as the origin of co-ordinates,
may be neglected in comparison with the distance of the system from
the point attracted, the action of the system will be the same as if
the masses were all united at its centre of gravity.
If we suppose the masses m, m', m", &c. to be the elements of the
mass of a single body, the form of the equations remains unchanged;
and hence it follows that the mass M is acted upon by another mass,
or by a system of bodies, as if the entire mass of the body, or of tho
system, were collected at its centre of gravity. It is evident, also,
that reciprocally in the case of two systems of bodies, in which the
mutual distances of the bodies are small in comparison with the
distance between the centres of gravity of the two systems, their
mutual action is the same as if all the several masses in each system
were collected at the common centre of gravity of that system ; and
the two centres of gravity will move as if the masses were thus
united.
13. The results already obtained are sufficient to enable us to form
the equations for the motions of the several bodies which compose the
solar system. If these bodies were exact spheres, which could be
considered as composed of homogeneous concentric spherical shells,
the density varying only from one layer to another, the action of
MOTION OF A SYSTEM OF BODIES. 35
each on an element of the mass of another would be the same as if
the entire mass of the attracting body were concentrated at its centre
of gravity. The slight deviation from this law, arising from the
ellipsoidal form of the heavenly bodies, is compensated by the mag-
nitude of their mutual distances; and, besides, these mutual distances
are so great that the action of the attracting body on the entire mass
of the body attracted, is the same as if the latter were concentrated
at its centre of gravity. Hence the consideration of the reciprocal
action of the single bodies of the system, is reduced to that of material
points corresponding to their respective centres of gravity, the masses
of which, however, are equivalent to those of the corresponding
bodies. The mutual distances of the bodies composing the secondary
systems of planets attended with satellites are so small, in comparison
with the distances of the different systems from each other and from
the other planets, that they act upon these, and are reciprocally acted
upon, in nearly the same manner as if the masses of the secondary
systems were united at their common centres of gravity, respectively.
The motion of the centre of gravity of a system consisting of a
planet and its satellites is not affected by the reciprocal action of the
bodies of that system, and hence it may be considered independently
of this action. The difference of the action of the other planets on
a planet and its satellites will simply produce inequalities in the
relative motions of the latter bodies as determined by their mutual
action alone, and will not affect the motion of their common centre
of gravity. Hence, in the formation of the equations for the motion
of translation of the centres of gravity of the several planets or
secondary systems which compose the solar system, we have simply
to consider them as points endowed with attractive forces correspond-
ing to the several single or aggregated masses. The investigation
of the motion of the satellites of each of the planets thus attended,
forms a problem entirely distinct from that of the motion of the
common centre of gravity of such a system. The consideration of
the motion of rotation of the several bodies of the solar system about
their respective centres of gravity, is also independent of the motion
of translation. If the resultant of all the forces which act upon a
planet passed through the centre of gravity, the motion of rotation
would be undisturbed; and, since this resultant in all cases very
nearly satisfies this condition, the disturbance of the motion of rota-
tion is very slight. The inequalities thus produced in the motion
of rotation are, in fact, sensible, and capable of being indicated by
observation, only in the case of the earth and moon. It has, indeed,
30 THEORETICAL ASTRONOMY.
been rigidly demonstrated that the axis of rotation of the earth rela-
tive to the body itself is fixed, so that the poles of rotation and the
terrestrial equator preserve constantly the same position in reference
to the surface; and that also the velocity of rotation is constant.
This assures us of the permanency of geographical positions, and,
in connection with the fact that the change of the length of the
mean solar day arising from the variation of the obliquity of the
ecliptic and in the length of the tropical year, due to the action of
the sun, moon, and planets upon the earth, is absolutely insensible,
amounting to only a small fraction of a second in a million of
years, assures us also of the permanence of the interval which we
adopt as the unit of time in astronomical investigations.
14. Placed, as we are, on one of the bodies of the system, it in
only possible to deduce from observation the relative motions of the
different heavenly bodies. These relative motions in the case of the
comets and primary planets are referred to the centre of the sun,
since the centre of gravity of this body is near the centre of gravity
of the system, and its preponderant mass facilitates the integration 1
of the equations thus obtained. In the case, however, of the secondary
systems, the motions of the satellites are considered in reference to
the centre of gravity of their primaries. We shall, therefore, form
the equations for the motion of the planets relative to the centre of
gravity of the sun ; for which it becomes necessary to consider more
particularly the relation between the heterogeneous quantities, space,
time, and mass, which are involved in them. Each denomination,
being divided by the unit of its kind, is expressed by an abstract
number; and hence it offers no difficulty by its presence in an equa-
tion. For the unit of space we may arbitrarily take the mean dis-
tance of the earth from the sun. and the mean solar day may be
taken as the unit of time. But, in order that when the space is
expressed by 1, and the time by 1, the force or velocity may also be
expressed by 1, if the unit of space is first adopted, the relation of
the time and the mass which determines the measure of the force
wi". be such that the units of both cannot be arbitrarily chosen.
Thus, if we denote by / the acceleration due to the action of the
mass m on a material point at the distance a, and by/ the accelera-
tion corresponding to another mass m' acting at the same distance,
we have the relation
MOTION RELATIVE TO THE SOX. 37
and hence, since the acceleration is proportional to the mass, it may
be taken as the measure of the latter. But we have, for the measure
of/,
dfc
J ~ df
Integrating this, regarding /as constant, and the point to move from
a state of rest, we get
= $&. (13)
The acceleration in the case of a variable force is, at any instant.
measured by the velocity which the force acting at that instant would
generate, if supposed to remain constant in its action, during a unit
of time. The last equation gives, when 1,
/=2;
and hence the acceleration is also measured by double the space which
would be described by a material point, from a state of rest, during
a unit of time, the force being supposed constant in its action during
this time. In each case the duration of the unit of time is involved
in the measure of the acceleration, and hence in that of the mass on
which the acceleration depends; and the unit of mass, or of the force,
will depend on the duration which is chosen for the unit of time. In
general, therefore, we regard as the unit of mass that which, acting
constantly at a distance equal to unity on a material point free to
move, .will give to this point, in a unit of time, a velocity which,
if the force ceased to act, would cause it to describe the unit of dis-
tance in the unit of time.
Let the unit of time be a mean solar day; W the acceleration due
to the force exerted by the mass of the sun at the unit of distance;
and /the acceleration corresponding to the distance r; then will
and 1? becomes the measure of the mass of the sun. The unit of
mass is, therefore, equal to the mass of the sun taken as many times
as 12 is contained in unity. Hence, when we take the mean solar
day as the unit of time, the mass of the sun is measured by F; by
which we are to understand that if the sun acted during a mean solar
day, on a material point free to move, at a distance constantly equal
to the mean distance of the earth from the sun, it would, at the end
of that time, have communicated to the point a velocity which, if
38 THEORETICAL ASTRONOMY.
the force did not thereafter act, would cause it to describe, in a unu
of time, the space expressed by F.
The acceleration due to the action of the sun at the unit of distance
is designated by F, since the square root of this quantity appears
frequently in the formulae which will be derived.
If we take arbitrarily the mass of the sun as the unit of mass, the
unit of time must be determined. Let t denote the number of mean
solar days which must be taken for the unit of time when the unit
of mass is the mass of the sun. The space which the force due to
this mass, acting constantly on a material point at a distance equal to
the mean distance of the earth from the sun, would cause the point
to describe in the time t, is, according to equation (13),
But, since t expresses the number of mean solar days in the unit oi
time, the measure of the acceleration corresponding to this unit is 2s,
and this being the unit of force, we have
JW = 1 ;
and hence
Therefore, if the mass of the sun is regarded as the unit of mass, the
number of mean solar days in the unit of time will be equal to unity
divided by the square root of the acceleration due to the force exerted
by this mass at the unit of distance. The numerical value of k will
be subsequently found to be 0.0172021, which gives 58.13244 mean
solar days for the unit of time, when the mass of the sun is taken as
the unit of mass.
15. Let x, y, z be the co-ordinates of a heavenly body referred to
the centre of gravity of the sun as the origin of co-ordinates; r its
radius-vector, or distance from this origin; and let m denote the
quotient obtained by dividing its mass by that of the sun; then,
taking the mean solar day as the unit of time, the mass of the sun is
expressed by k 2 , and that of the planet or comet by mk 2 . For a
second body let the co-ordinates be x*, y', z'; the distance from the
sun, r'j and the mass, m'& 2 ; and similarly for the other bodies of the
system. Let the co-ordinates of the centre of gravity of the sun
referred to any fixed point in space be , 7, , the co-ordinate planes
being parallel to those of x, y, and z, respectively; then will the
MOTIOX RELATIVE TO THE SUN.
acceleration due to the action of TO on the sun be expressed by .
and the three components of this force in directions parallel to the
co-ordinate axes, respectively, will be
.
The action of m' on the sun will be expressed by
and hence the acceleration due to the combined and simultaneous
action of the several bodies of the system on the sun, resolved par-
allel to the co-ordinate axes, will be
,, v wur my mz
? ?' -pr
The motion of the centre of gravity of tjie sun, relative to the fixed
origin, will, therefore, be determined by the equations
~dT-
If we multiply these three equations respectively by z, y, anct x,
and add the products, we obtain
cz c'y + c"x = 0.
This, being the equation of a plane passing through the origin of
co-ordinates, shows that the path of the body relative to the sun is a
plane curve, and that the plane of the orbit passes through the centre
of the sun.
Again, if we multiply the first of equations (19) by 2dx, the second
by 2dy, and the third by 2dz, take the sum and integrate, we shall
find
>
But, since r 2 = a 2 + f + z 2 , we shall have, by differentiation,
rdr = xdx -f- ydy -\- zdz.
Therefore, introducing this value into the preceding equation, we obtain
^+^f+if_af(l + m) (2Q)
ar r
h being an arbitrary constant.
44 THEORETICAL ASTRONOMY.
If we add together the squares of the expressions for c, c', and c",
and put c 2 + c' 2 + c" 2 = 4/ 2 , we shall have
(x* -\- y* -f a 2 ) (cfo 2 -f eft/ 2 -f c?2 2 ) (a:c?a; -}- ycZy -f z ^ 2 ) 2 Af*.
M+df+d* r*dr*_
eft 2 df J '
If we represent by dv the infinitely small angle contained between
tw consecutive radii- vectores r and r + dr, since dar 2 + d^/ 2 + d# is
the square of the element of path described by the body, we shall
have
da? + df + dz* = dr* + rW.
Substituting this value in the preceding equation, it becomes
i*dv = 2fdt. (22)
The quantity r*dv is double the area included by the element of path
described in the element of time dt, and by the radii-vectores r and
r + dr; and/, therefore, represents the areal velocity, which, being a
constant, shows that the radius-vector of a planet or comet describes
equal areas in equal intervals of time.
From the equations (20) and (21) we find, by elimination,
dt = - -^L- (23)
1/2r& 2 (1 + m) Ar 2 4/ 2
Substituting this value of dt in equation (22), we get
*=- -3*- (24)
V2r& 2 (1 + TO) Ar 2 4/'
which gives, in order to find the maximum and minimum values of r,
dr rl/2rtf (1 + m) Ar 2 4/ 2
~dv= -2T~
or
2r& 2 (1 -f m) Ar 2 4/ 2 = 0.
Therefore
P(l-fm) , I 4/ 2 ~
"A +V-1T
and
P(l+m)_ / 4/ 2
h \
are, respectively, the maximum and minimum values of r. The
MOTION EELATIVE TO THE SUN.
points of the orbit, or trajectory of the body relative to the sun, cor-
responding to these values of r, are called the apsides; the former,
the aphelion, and the latter, the perihelion. If we represent these
values, respectively, by a(l + e) and d(l e\ we shall have
in which p = a (1 e 2 }. Introducing these values into the equation
(24), it becomes
=
r-
\ a
the integral of which gives
(o being an arbitrary constant. Therefore we shall have
I 1 I = cos (v w\
e \ r
from which we derive
p
1 -f- e cos (y w)
which is the polar equation of a conic section, the pole being at the
focus, p being the semi-parameter, e the eccentricity, and v to the
angle at the focus between the radius-vector and a fixed line, in the
plane of the orbit, making the angle to with the semi-transverse
axis a.
If the angle v ft) is counted from the perihelion, we have (o = 0,
and
P
-\-e cos v
(25)
The angle v is called the true anomaly.
Hence we conclude that the orbit of a heavenly body revolving
around the sun is a conic section with the sun in one of the foci.
Observation shows that the planets revolve around the sun in ellipses,
usually of small eccentricity, while the comets revolve either in
ellipses of great eccentricity, in parabolas, or in hyperbolas, a cir-
cumstance which, as we shall have occasion to notice hereafter, greatly
46 THEORETICAL ASTRONOMY.
lessens the amount of labor in many computations respecting their
motion.
Introducing into equation (23) the values of h and 4/ 2 already
found, we obtain
I/a rdr
= k^/Y^m ' v/aV (a ^rf 2 '
which may be written
dt==
the integration of which gives
J
In the perihelion, r a (1 e), and the integral reduces to i' = C;
therefore, if we denote the time from the perihelion by t 0) we shall
have
_.i _?=r. (27)
In the aphelion, r a (1 + e) ; and therefore we shall have, for the
time in which the body passes from the perihelion to the aphelion,
< = or
r being the periodic time, or time of one revolution of the planet
around the sun, a the semi-transverse axis of the orbit, or mean dis-
tance from the sun, and re the semi-circumference of a circle whose
radius is unity. Therefore we shall have
<28)
MOTION RELATIVE TO THE SUN. 47
For a second planet, we shall have
and, consequently, between the mean distances and periodic times of
any two planets, we have the relation
f =jr ( 29 >
If the masses of the two planets TO and TO' are very nearly the
same, we may take 1 + TO = 1 -j- TO'; and hence, in this case, it follows
that the squares of the periodic times are to each other as the cubes of
the mean distances from the sun. The same result may be stated in
another form, which is sometimes more convenient. Thus, since ~ab
is the area of the ellipse, a and b representing the semi-axes, we
shall have
~ab , 11.
- =/= areal velocity;
and, since b 2 = a? (1 e*), we have
y
which becomes, by substituting the value of r already found,
-m). (30)
In like manner, for a second planet, we have
and, if the masses are such that we may take 1 -f m sensibly equal
to 1 -H TO', it follows that, in this case, the areas described in equal
times, in different orbits, are proportional to the square roots of their
parameters.
17. We shall now consider the signification of some of the con-
stants of integration already introduced. Let t denote the inclination
of the orbit of TO to the plane of xy, which is thus taken as the plane
of reference, and let & be the angle formed by the axis of x and the
line of intersection of the plane of the orbit with the plane of xy;
then will the angles i and SI determine the position of the plane of
48 THEORETICAL ASTRONOMY.
the orbit in space. The constants c, c', and c", involved in the
equation
cz _ c 'y -J- C " X = 0,
are, respectively, double the projections, on the co-ordinate planes,
xy, xz, and yz, of the areal velocity // and hence we shall have
The projection of 2/ on a plane passing through the intersection of
the plane of the orbit with the plane of xy, and perpendicular to the
latter, is
2/sini;
and the projection of this on the plane of xz, to which it is inclined
at an angle equal to ft, gives
c' 2/sinicos ft.
Its projection on the plane of yz gives
c" 2/smtsin SI.
Hence we derive
z cos i y sin i cos ft -f- x sin i sin SI = 0, (31)
which is the equation of the plane of the orbit; and, by means of
the value of / in terras of p, and the values of c, c', c", we derive,
also,
m) cos ft sini, (32;
* dt y dt~
dz_ z dx__
y~ ~ z -Ji = ki/p(l-\-m) sin ft sin t.
These equations will enable us to determine ft, i, and p, when, for
any instant, the mass and co-ordinates of m, and the components of
its velocity, in directions parallel to the co-ordinate axes, are known.
The constants a and e are involved in the value of p, and hence four
constants, or eleinents, are introduced into these equations, two of
which, a and e, relate to the form of the orbit, and two, ft and i, to
the position of its plane in space. If we measure the angle v to
from the point in which the orbit intersects the plane of xy, the con-
stant to will determine the position of the orbit in its own plane.
Finally, the constant of integration C, in equation (26), is the time
MOTION RELATIVE TO THE SUN. 4S
of passage through the perihelion; and this determines the position
of the body in its orbit. When these six constants are known, the
undisturbed orbit of the body is completely determined.
Let F denote the velocity of the body in its orbit; then will
equation (20) become
At the perihelion, r is a minimum, and hence, according to this
equation, the corresponding value of F is a maximum. At the
aphelion, F is a minimum.
In the parabola, a = oo, and hence
which will determine the velocity at any instant, when r is known.
It will be observed that the velocity, corresponding to the same value
of r, in an elliptic orbit is less than in a parabolic orbit, and that,
since a is negative in the hyperbola, the velocity in a hyperbolic
orbit is still greater than in the case of the parabola. Further, since
the velocity is thus found to be independent of the eccentricity, the
direction of the motion has no influence on the species of conic section
described.
If the position of a heavenly body at any instant, and the direction
and magnitude of its velocity, are given, the relations already derived
will enable us to determine the six constant elements of its orbit.
But since we cannot know in advance the magnitude and direction
of the primitive impulse communicated to the body, it is only by
the aid of observation that these elements can be derived; and
therefore, before considering the formula necessary to determine
unknown elements by means of observed positions, we will investi-
gate those which are necessary for the determination of the helio-
centric and geocentric places of the body, assuming the elements to
be known. The results thus obtained will facilitate the solution of
the problem of finding the unknown elements from the data furnished
by observation.
18. To determine the value of Ic, which is a constant for the solar
system, we have, from equation (28),
60 THEORETICAL ASTRONOMY.
Tn the case of the earth, a = 1, and therefore
In reducing this formula to numbers we should properly use, for r,
the absolute length of the sidereal year, which is invariable. The
effect of the action of the other bodies of the system on the earth is
to produce a very small secular change in its mean longitude corre-
sponding to any fixed date taken as the epoch of the elements; and
a correction corresponding to this secular variation should be applied
to the value of T derived from observation. The effect of this cor-
rection is slightly to increase the observed value of r; but to deter-
mine it with precision requires an exact knowledge of the masses of
all the bodies of the system, and a complete theory of their relative
motions, a problem which is yet incompletely solved. Astronomical
usage has, therefore, sanctioned the employment of the value of k
found by means of the length of the sidereal year derived directly
from observation. This is virtually adopting as the unit of space a
distance which is very little less than the absolute, invariable mean
distance of the earth from the sun; but, since this unit may be arbi-
trarily chosen, the accuracy of the results is not thereby affected.
The value of r from which the adopted value of k has been com-
puted, is 365.2563835 mean solar days; and the value of the com-
bined mass of the earth and moon is
Hence we have log T = 2.5625978148; log j/l+m = 0.0000006122;
log 2n = 0.7981798684; and, consequently,
log k = 8.2355814414.
If we multiply this value of k by 206264.81, the number of seconds
of arc corresponding to the radius of a circle, we shall obtain its
value expressed in seconds of arc in a circle whose radius is unity, or
on the orbit of the earth supposed to be circular. The value of k in
seconds is, therefore,
log k = 3.5500065746.
The quantity - expresses the mean angular motion of a planet
in a mean solar day, and is usually designated by p. We shall.
therefore, have
MOTION RELATIVE TO THE SUN. 51
n = kVl + m , (33)
a*
for the expression for the mean daily motion of a planet.
Since, in the case of the earth, V\ + m differs very little from 1,
it will be observed that k very nearly expresses the mean angular
motion of the earth in a mean solar day.
In the case of a small planet or of a comet, the mass m is so small
that it may, without sensible error, be neglected; and then we shall
-
or
(34)
For the old planets whose masses are considerable, the rigorous ex-
pression (33) must be used.
19. Let us now resume the polar equation of the ellipse, the pole
being at the focus, which is
r= (1 -e^
1 + e cos v
If we represent by = e;
and, since a(l e 2 ) is half the parameter of the transverse axis,
which we have designated by p, we have
1 + sin + 40 cos i ( v ~~ 40* tn ^ s S^ ves
^+y (37)
It appears from this formula that r increases with v, and becomes in-
finite when 1 + e cosv = 0, or cost; = cos 4, in which case v = 180
4 : consequently, the maximum positive value of v is represented
by 180 4> an d the maximum negative value by (180 40-
Further, it is evident that the orbit will be that branch of the hyper-
bola which corresponds to the focus in which the sun is placed, since,
under the operation of an attractive force, the path of the body must
be concave toward the centre of attraction. A body subject to a
force of repulsion of the same intensity, and varying according to
the same law, would describe the other branch of the curve.
The problem of finding the position of a heavenly body as seen
from any point of reference, consists of two parts: first, the deter-
mination of the place of the body in its orbit; and then, by means
>f this and of the elements which fix the position of the plane of the
PLACE IN THE ORBIT. 53
orbit, and that of the orbit in its own plane, the determination ot
the position in space.
In deriving the formulae for finding the place of the body in its
orbit, we will consider each species of conic section separately, com-
mencing with the ellipse.
20. Since the value of a r can never exceed the limits ae and
-j- ae, we may introduce an auxiliary angle such that we shall have
a r
= cos E.
This auxiliary angle E is called the eccentrie anomaly; and its geo
metrical signification may be easily known from its relation to the
true anomaly. Introducing this value of - - into the equation
(27) and writing t T in place of t , T being the time of perihelion
passage, and t the time for which the place of the planet in its orbit
is to be computed, we obtain
(38)
But -- j - = mean daily motion of the planet = p. ; therefore
The quantity [t(t T) represents what would be the angular distance
from the perihelion if the planet had moved uniformly in a circular
orbit whose radius is a, its mean distance from the sun. It is called
the mean anomaly, and is usually designated by M. We shall, there-
fore, have
M=(t-T),
M=EesmE. (39)
When the planet or comet is in its perihelion, the true anomaly,
mean anomaly, and eccentric anomaly are each equal to zero. All
three of these increase from the perihelion to the aphelion, where
they are each equal to 180, and decrease from the aphelion to the peri-
helion, provided that they are considered negative. From the peri-
helion to the aphelion v is greater than E, and E is greater than M.
The same relation holds true from the aphelion to the perihelion, if
ve regard, in this case, the values of v, E, and M as negative.
As soon as the auxiliary angle E is obtained by means of the mean
motion and eccentricity, the values of r and v may be derived. For
54 THEORETICAL ASTRONOMY.
this purpose there are various formulae which may be applied in
practice, and which we will now develop.
The equation
^-'-00.*
gives
-). (40)
This also gives
a r ae = a cos E ae,
or
=. a cos E ae,
which, by means of equation (25), reduces to
r cos v = a cos E ae. (41^
If we square both members of equations (40) and (41), and subtract
the latter result from the former, we get
r sin v = oj/1 e* sin E = b sin E. (42)
By means of the equations (41) and (42) it may be easily shown
that the auxiliary angle E, or eccentric anomaly, is the angle at the
centre of the ellipse between the semi-transverse axis, and a line
drawn from the centre to the point where the prolongation of the
ordinate perpendicular to this axis, and drawn through the place of
the body, meets the circumference of the circumscribed circle.
Equations (40) and (41) give
r(l=pcost;) = a(l:te) (1 + coaE).
By using first the upper sign, and then the lower sign, we obtain, by
reduction,
Vr sin \v = i/o(l + e) sin $E,
Vr cos & = Va(l e) cos $E, (43)
which are convenient for the calculation of r and v, and especially so
when several places are required. By division, these equations give
PLACE IN THE ORBIT. 55
Since e sin y>, we have
1 -f e 1
Consequently,
tan $E = tan (45 ^) tan $>o. (45)
Again.
VI -f- e = 1/1 + sin -f cos 2?>.
In a similar manner we find
1/1 e = sin \ , and 6 = a cos instead of p, and sin (p for e, we get
(48)
If wo multiply the first of equations (43) by cos 2?, and the
56 THEORETICAL ASTRONOMY.
second by sin^E, successively add and subtract the products, and
reduce by means of the preceding equations, we obtain
sin \ (v + E} */- cos ^ and 1? directly; and when the eccentricity is
58 THEORETICAL ASTRONOMY.
very great, this mode is indispensable, since the series will not in
that ease be sufficiently convergent.
It will be observed that the formula which must be used in obtain-
ing the eccentric anomaly from the mean anomaly is transcendental,
and hence it can only be solved either by series or by trial. But
fortunately, indeed, it so happens that the circumstances of the celes-
tial motions render these approximations very rapid, the orbits being
usually either nearly circular, or else very eccentric.
If, in equation (50), we put F(E) = E, and consequently F(M]
= M, we shall have, performing the operations indicated and reducing.
E = M + e sin M + ^ sin 2 M + &c. (54)
Let us now denote the approximate value of E computed from this
equation by E , then will
in which A-E" is the correction to be applied to the assumed value of E.
Substituting this in equation (39), we get
M = E + A-E" e sin E e cos E bE ;
and, denoting by M the value of M corresponding to E , we shall
also have
M = E e sin E .
Subtracting this equation from the preceding one, we obtain
_. - _^.
1 ecos E
It remains, therefore, only to add the value of &E found from this
formula to the first assumed value of E, or to E , and then, using
this for a new value of E w to proceed in precisely the same manner
for a second approximation, and so on, until the correct value of E is
obtained. When the values of E for a succession of dates, at equal
intervals, are to be computed, the assumed values of E may be ob-
tained so closely by interpolation that the first approximation, in the
manner just explained, will give the correct value; and in nearly
every case two or three approximations in this manner will suffice.
Having thus obtained the value of E corresponding to M for any
instant of time, we may readily deduce from it, by the formulae
already investigated, the corresponding values of r and v.
In the case of an ellipse of very great eccentricity, corresponding
to the orbits of many of the comets, the most convenient method of
PLACE IS THE ORBIT. 59
computing r and v, for any instant, is somewhat different. The
manner of proceeding in the computation in such cases we shall 3011-
sider hereafter; and we will now proceed to investigate the formula
for determining r and t?, when the orbit Ls a parabola, the formulae
for elliptic motion not being applicable, since, in the parabola, a = co ,
and e = 1.
22. Observation shows that the masses of the comets are insensible
in comparison with that of the sun; and, consequently, in this case,
m = and equation (52), putting for p its value 2q, becomes
or
which may be written
-^-j = (1 + tan 1 v) sec 1 $vdv = (1 + tan 1 $v) d tan t?.
1/2 3*
Integrating this expression between the limits T and t, we obtain
k( t T J = tan Aw + { tan 8 v, (55)
1/2 9*
which is the expression for the relation between the true anomaly
and the time from the perihelion, in a parabolic orbit.
Let us now represent by r the time of describing the arc of a
parabola corresponding to t; = 90 ; then we shall have
3k
V*
tyt
Now, - is constant, and its logarithm is 8.5621876983; and if we
take <7 = 1, which is equivalent to supposing the comet to move in
a parabola whose perihelion distance is equal to the semi-transverse
axis of the earth's orbit, we find
log r ** = 2.03987229, or r, = 109.61558 days ;
that is, a comet moving in a parabola whose perihelion distance
60 THEORETICAL ASTRONOMY.
is equal to the mean distance of the earth from the sun, requires
109.61558 days to describe an arc corresponding to v = 90.
Equation (55) contains only such quantities as are comparable with
each other, and by it t T, the time from the perihelion, may be
readily found when the remaining terms are known; but, in order
to find v from this formula, it will be necessary to solve the equation
of the third degree, tan \v being the unknown quantity. If we put
x = tan \v, this equation becomes
a* 4. 3 X a = 0,
in which a is the known quantity, and is negative before, and positive
after, the perihelion passage. According to the general principle in
the theory of equations that in every equation, whether complete or
incomplete, the number of positive roots cannot exceed the number
of variations of sign, and that the number of negative roots cannot
exceed the number of variations of sign, when the signs of the terms
containing the odd powers of the unknown quantity are changed, it
follows that when a is positive, there is one positive root and no
negative root. When a is negative, there is one negative root and
no positive root; and hence we conclude that equation (55) can have
but one real root.
We may dispense with the direct solution of this equation by
forming a table of the values of v corresponding to those of t T
in a parabola whose perihelion distance is equal to the mean distance
of the earth from the sun. This table will give the time correspond-
ing to the anomaly v in any parabola, whose perihelion distance is
q, by multiplying by V
By means of these formula, the co-ordinates are found directly
from the eccentric anomaly, when the constants A x , J y , J,, i x , i y , i,,
y x , v y , and v z , have been computed from those already found, or from
a, b, c, A, B, and C. This method is very convenient when a groat
POSITION IX SPACE. 95
number of geocentric places are to be computed ; but, when only a
few places are required, the additional labor of computing so many
auxiliary quantities will not be compensated by the facility afforded
in the numerical calculation, when these constants have been deter-
mined. Further, when the ephemeris is intended for the comparison
of a series of observations in order to determine the corrections to be
applied to the elements by means of the differential formula? which
we shall investigate in the following chapter, it will always be ad-
visable to compute the co-ordinates by means of the radius-vector
and true anomaly, since both of these quantities will be required in
finding the differential coefficients.
38. In the case of a hyperbolic orbit, the co-ordinates may be com-
puted directly from F, since we have
r cos v a (e sec _F),
r sin v = a tan 4 tan F;
and, consequently,
x = ae sin a' sin A' a sec F sin a' sin A' -j- a tan 4 tan F sin a' cos A ',
y = ae sin b' sin I? a sec F sin b' sin B 1 -\- a tan 4 tan F sin b' cos B,
z =. ae sin c' sin C" a sec F sin c' sin C' -\- a tan 4 tan F sin c' cos C".
Let us now put
ae sin a' sin A' = *
a sin a' sin A = yu x ,
a tan 4. sin a' cos A = v x ;
ae sin b' sin B = A,,
a sin b' sins' = n,,
a tan 4. sin b' cos B' = v y ;
ae sin c' sin C' = >*
a sin c' sin C' = /*
a tan 4 sin c' cos C' = v t .
Then we shall have
x ^ + n f sec F -f v,. tan .F,
y = ^ + j^ sec .F + *, tan J 1 , (106)
z = i, + JM, sec F + v t tan F.
In a similar manner we may derive expressions for the co-ordinates,
in the case of a hyperbolic orbit, when the auxiliary quantity a is
used instead of F.
39. If we denote by ~', &', and V the elements which determine
the position of the orbit in space when referred to the equator as the
96 THEORETICAL ASTRONOMY.
fundamental plane, and by a) the angular distance between the
ascending node of the orbit on the ecliptic and its ascending node on
the equator, being measured positively from the equator in the
direction of the motion, we shall have
To find Sl f and i f , we have, from the spherical triangle formed by
the intersection of the planes of the orbit, ecliptic, and equator with
the celestial vault,
cos i' = cos i cos e sin i sin e cos SI >
sin i' sin &' = sin i sin SI,
sin i' cos &' = cos i sin e -f- sin i cos e cos SI .
Let us now put
n sin N== cost,
n cosN= sin i cos &,
and these equations reduce to
cos i' = n sin (N e),
sin i' sin &' = sin i sin SI,
sin t' cos ' = n cos (-AT e) ;
from which we find
cot i' = tan (.tf- e) cos & '. ( 107 )
Since sin i is always positive, cos N and cos & must have the
signs. To prove the numerical calculation, we have
sin i cos cos N
sin i cos &' cos (JV e)
the value of the second member of which must agree with that used
in computing & '.
In order to find = tan ^.Etan (45 -f ?),
and differentiate, we find
dv dE _ _ d cos E} dip. (IB")
124 THEORETICAL ASTKONOMY.
sin v cos? , -r, cosv+e , ,
Isow, since sm E= , and cos K=^, - we shall have
1 + e cos v I -f- e cos v
_, ae cos sin 2 v a cos . (l4t)
a
Further, we have
M=
T being the epoch for which the mean anomaly is M w and
_
Differentiating these expressions, we get
and substituting these values in the expressions for dr and dv, we
have, finally,
dr = a tan an d ~r> which are
rfT dg de dT dq de
required in finding the differential coefficients of the heliocentric co-
ordinates with respect to the elements T, q, and e, these quantities
being substituted for M , p, and S + / 5 O (1 - )' + Ac. (29)
If it is required to find the expression for -7- in the case of the
variation of the elements of parabolic motion, or when 1 e is very
small, we may regard the coefficient of 1 e as constant, and neglect
terms multiplied by the square and higher powers of 1 *- e. By
differentiating the equation (29) according to these conditions, and
regarding u and e as variable, we get
= (1 + O du Q X iw 6 ) de;
and, since du = |(1 + w 2 ) dv, this gives
The values of the second member, corresponding to different values
of Vy may be tabulated with the argument v; but a table of this kind
is by no means indispensable, since the expression for -j- may be
changed to another form which furnishes a direct solution with the
same facility. Thus, by division, we have
and since, in the case of parabolic motion,
this becomes
(31)
If we differentiate the equation
^
1 -{- e cos i/
regarding r, v, and e as variables, we shall have
dr 2r* sinVv r'esint; dv
132 THEORETICAL ASTRONOMY.
In the case of parabolic motion, e = l, and this equation is easily
transformed into
7 / 7. \
(33)
Substituting for -- its value from (SI), and reducing, we get
(34)
The equations (31) and (34) furnish the values of and to be
used in forming the expressions for the variation of the place of the
body when the parabolic eccentricity is changed to the value 1 -f de.
When the eccentricity to which the increment is assigned differs but
little from unity, we may compute the value of -7- directly from
equation (30). A still closer approximation would be obtained by
using an additional term of (29) in finding the expression for -=- ; but
a more convenient formula may be derived, of which the numerical
application is facilitated by the use of Table IX. Thus, if we differ-
entiate the equation
v = V+ A (lOOi) + B (1000 2 + (7(100*7,
regarding the coefficients A, B, and C as constant, and introducing
the value of i in terms of e, we have
dt> = dF__200J_ ._ 4 iA_ (100i) _ 600(7
in which s . 206264.8, the values of A, B, and C, as derived from
the table, being expressed in seconds. To find , we have
de
k(i-T)VT+l
which gives, by differentiation,
k(t p de dV
and if we introduce the expression for the value of M used as the
argument in finding V by means of Table VI., the result is
DIFFERENTIAL FORMULA. 133
Hence we have
dv JfcoafjF 200,1 400J? 6000
(1 ~- (10(>l/ ' (
by means of which the value of is readily found.
When the eccentricity differs so much from that of the parabola
that the terms of the last equation are not sufficiently convergent,
the expression for , which will furnish the required accuracy, may
be derived from the equations (75) 1 and (76) r If we differentiate the
first of these equations with respect to e, since B may evidently be
regarded as constant, we get
dw__ 9
de~ '
If we take the logarithms of both members of equation (76) n and
differentiate, we get
dv _ dC . dw _ 4de _ ,g-^
smv ~" ~C + slnw ~~ (1 + e) (I -f- 9e)'
To find the differential coefficient of C with respect to e, it will be
sufficient to take
which gives
The equation
gives
d
= + 507.264, -|L = - 179.315.
Therefore, according to (1), we shall have
coeaA=-J-1.42345Aff0.03845A 0.27641 At +1.99400^^
+ 1.13004A3/ + 507.264A.a,
A
log ~ = 0.186517., log ^ = 0.186517..
If we wish to obtain the differential coefficients of v and r with
respect to log 3 instead of q, we have
dv q dv dr q d r
dlogq A O dq dlogq ^ dq
in which ). is the modulus of the system of logarithms.
Then we compute the value of -j- by means of the equation (30).
(35), (39), or (40). The correct value as derived from (39) is
= 0.24289.
de
The values derived from (35), omitting the last term, from (40) and
from (30), are, respectively, 0.24440, 0.24291, and 0.23531.
The close agreement of the value derived from (40) with the correct
value is accidental, and arises from the particular value of v, which
is here such as to make the assumptions, according to which equation
(40) is derived from (39), almost exact.
Finally, the value of -^- may be found by means of (32), whicJi
gives
J = + 0.70855.
ae
When, in addition to the differential coefficients which depend on
the elements T, q, and e, those which depend on the position of the
orbit in space have been found, the expressions for the variation of
the geocentric right ascension and declination become
NUMERICAL EXAMPLES. 141
COS <5 Aa = COS S -? A* + COS 3 ~ A & 4- COS 5 -? Ai + COS 5 A I 7
a- d& dt dT
_ c?a c?o
-f- COS A , we must put
dr dv
de' de
dr __ * (63)
dr j dv
-r-, hcosH=r-j-,
dq' dq
and the equations become
In the numerical application of these formulae, the values of the
second members of the equations (63) are found as already exem-
plified for the cases of parabolic orbits and of elliptic and hyperbolic
orbits in which the eccentricity differs but little from unity. In the
same manner, the differential coefficients of ^ and /9 with respect to
any other elements which determine the form of the orbit may be
computed.
NUMERICAL EXAMPLES. 148
In the case of a parabolic orbit, if the parabolic eccentricity is
supposed to be invariable, the terms involving e vanish. Further,
in the case of parabolic elements, we have
. ~ dr k sin v , dv
g sin Cr = ~j~m == - = T tan ~>v ,
d J? ~Y 2o ~ d T
dv
which give
tan G = tan ^v.
Hence there results G = 18Q%v, and g = k\~, which is the
expression for the linear velocity of a comet moving in a parabola.
Therefore,
fcl/2 ,
For the case in which the motion is considered as being retrograde,
180 i must be used instead of i in computing the values of A w
A, n, N, C , and (7, and the equations (55), (56), and the first two
of (58), remain unchanged. But, for the differential coefficients with
respect to i, the values of D and D must be found from the last two
of equations (57), using the given value of i directly ; and then we
shall have
cos /9 -TT = sin t sin u cos (A & ),
55. EXAMPLES. The equations thus derived for the differential
coefficients of A and ft with respect to the elements of the orbit,
referred to the ecliptic as the fundamental plane, are applicable when
any other plane is taken as the fundamental plane, if we consider ^
and /9 as having the same signification in reference to the new plane
that they have in reference to the ecliptic, the longitudes, however,
being measured from the place of the descending node of this plane
on the ecliptic. To illustrate their numerical application, let it be
required to find the differential coefficients of the geocentric right
ascension and declination of Eurynome with respect to the ele-
ments of its orbit referred to the equator, for the date 1865 February
24.5 mean time at Washington, using the data given in Art. 41.
150 THEOEETICAL ASTRONOMY
In the first place, the elements which are referred to the ecliptic
must be referred to the equator as the fundamental plane ; and, by
means of the equations (109)^ we obtain
' = 353 45' 35".87, **= 19 26' 25".76, > 9 = 212 32' 17".71,
and
<' = + = 50 I dr d f n TT-N dlj h . , -f-f^.
cos ij ; = -7 cos (0 u H), r- = -T sin w sm (0 u H ).
d/j- a o/t J
If we express r and v in terms of the elements T, q, and e, the
values of the auxiliaries /, g, h, F, &c. must be found by means of
(64); and, in the same manner, any other elements which determine
the form of the orbit and the position of the body in its orbit, may
be introduced.
The partial differential coefficients with respect to the elements
having been found, we have
dO do dO do
COS 1) A0 = COS i) -j- *X + COS 1? -r- A? -f COS 1} AM. + COS TJ A/,
dx df dM dfj.
DIFFERENTIAL FORMULA. 157
from which it appears that, by the introduction of as one of the
elements of the orbit, when the geocentric places are referred directly
to the plane of the unchanged orbit as the fundamental plane, the
variation of the geocentric longitude in reference to this plane depends
on only four elements.
57. It remains now to derive the formulae for finding the values
of r t and d from those of X and ,3. Let x 0f y w z be the geocentric co-
ordinates of the body referred to a system in which the ecliptic is
the plane of xy, the positive axis of x being directed to the point
whose longitude is ft ; and let x r , y f , z f be the geocentric co-ordi-
nates of the body referred to a system in which the axis of x remains
the same, but in which the plane of the unchanged orbit is the plane
of xy; then we shall have
a- = J cos p cos 0* ft ), ' = -1 cos r t cos 9,
y,= d cos ,3 sin (<* ft ), y ' = A cos r t sin 0,
z, = J sin ,3, z,' = J sin r h
and also
z ' = y sin i + z cos i.
Hence we obtain
COST) COS = COS, 3 COS 0* ft),
cos 7 sin = cos /5 sin (^ ft ) cos i + sin ,3 sin ?', (80 ;
sin ij = cos ,3 sin (J ft ) sin i + sin ,3 cos i.
These equations correspond to the relations between the parts of a
spherical triangle of which the sides are i, 90 7, and 90 /9,
the angles opposite to 90 r t and 90 being respectively
90 -f (J ft) and 90 d. Let the other angle of the triangle be
denoted by 7-, and we have
cos T) sin r = sin * cos (J ft), ,gj
cosi7cosr = i3inisin(>l ft ) sin p -f cos i cos p.
The equations thus obtained enable us to determine 57, 6, and f from
i and /?. Their numerical application is facilitated by the intro-
duction of auxiliary angles. Thus, if we put
-ax < 82)
158 THEORETICAL ASTRONOMY.
in which n is always positive, we get
cos Jj cos e = cos 13 cos (A ft),
cos i) sin 6 = n cos ( N t), (8 3)
=nsin(Ni\
from which ^ and may be readily found. If we also put
n' sin N' = cost, ,,,
n' cos ^T = sin i sin (A ft ),
we shall have
cot N' = tan i sin 0* ft),
If f is small, it may be found from the equation
The quadrants in which the angles sought must be taken, are easily
determined by the relations of the quantities involved ; and the
accuracy of the numerical calculation may be checked as already
illustrated for similar cases.
If we apply Gauss's analogies to the same spherical triangle, we get
rin (45 - &) sin (45 - $ (0 + r )) =
cos(45 H- i (i - ft)) sin (45 - W + i);,
sin (45 - ,)' cos (45 - (0 + r )) =
sin (45 + (* ft)) sin (45 - (j9 t)),
cos (45 - ,) sin (45 - (* r)) = (87)
cos (45 + { (X - ft )) cos (45 $(fi + i)),
cos (45 - ^) cos (45 $ (0 r )) =
sin (45 + ^C; - ft )) cos (45 - (0 - 0),
from which we may derive 37, 0, and f.
When the problem is to determine the corrections to be applied to
the elements of the orbit of a heavenly body, in order to satisfy
givei. observed places, it is necessary to find the expressions for
cosjy A0 and &q in terms of cos/9 A^ and A/9. If we differentiate the
first and second of equations (80), regarding ft and i (which here
determine the position of the fundamental plane adopted) as con-
stant, eliminate the terms containing dy from the resulting equations,
and reduce by means of the relations of the parts of the spherical
triangle, we get
NUMERICAL EXAMPLE. 151
cos TJ dd = cos f cos ,9 dl -\- sin 7 d/9.
Differentiating the last of equations (80), and reducing, we find
dy = sin f cos /9 dl -f cos f dp.
The equations thus derived give the values of the differential co-
efficients of 6 and y with respect to ^ and p ; and if the differences
A/ and & t 3 are small, we shall have
cos TJ A0 = cos ^ cos ft AA -f sm 7" A &
Aiy = sin 7- cos ,9 AA -(- cos f A,9.
The value of f required in the application of numbers to these
equations may generally be derived with sufficient accuracy from
(86), the algebraic sign of cosf being indicated by the second of
equations (81) ; and the values of r t and 6 required in the calculation
of the differential coefficients of these quantities with respect to the
elements of the orbit, need not be determined with extreme accuracy.
58. EXAMPLE. Since the spherical co-ordinates which are fur-
nished directly by observation are the right ascension and declina-
tion, the formulae will be most frequently required in the form for
finding 37 and 6 from a and S. For this purpose, it is only necessary
to write a and 3 in place of A and ,?, respectively, and also Q, ', ',
ft/, %', and u' in place of , i, a, %, and u, in the equations which
have been derived for the determination of ^ and 0, and for the
differential coefficients of these quantities with respect to the elements
of the orbit.
To illustrate this clearly, let it be required to find the expressions
for cos 37 A# and &y in terms of the variations of the elements in the
case of the example already given; for which we have
/ = 5010'7".29, ' = 35345'35".87, t*=19 26' 25".76.
These are the elements which determine the position of the orbit of
Eurynome . referred to the mean equinox and equator of 1865.0.
We have, further,
log/= 0.62946, Iog0 = 0.34593, log h = 2.97759,
F=33914'0", G = 350 11' 16", H = 14 30' 48",
u' = 179 13' 58".
In the first place, we compute >;, 6, and f by means of the formulae
160 THEORETICAL ASTRONOMY.
(83j and (85), or by means of (87), writing a, d, ft', and i' instead
of )>, , ft, and i, respectively. Hence we obtain
* = 188 31' 9", i) = 1 59' 28", r = 19 17' 7".
Since the equator is here considered as the fundamental plane, the
longitude d is measured on the equator from the place of the ascend-
ing node of the orbit on this plane. The values of the differential
r>oefficients are then found by means of the formulas
d6 d-n r ... .
7 = 0, -7^7 = cos TI sin i cos u ,
d0 o ^ I -
di' di' A
do r f ,, di] r
(tit a f , .^T. di)
008?-^ = * cos (* ' GO, -r~ = -
CLJtl A uM
which give
do di)
cos i) -- = 0, -^- = 4- 0.0204,
cos i) ~ = 4- 1.5051, -^- = 4- 0.0086,
cos >j - = + 2.0978, -^- = 4- 0.0422.
cos 7 - = + 1.1922, -^- = 4- 0.0143,
cos i} -^- -f 538.00, -&. = 1.71.
Therefore, the equations for cos TJ A^ and A^ become
cos 13 A0 = 4- 1.5051 A/ + 2.0978 A? 4- 1.1922 Ajtf + 538.00 AJ
AT = 4- 0.0086 A/ 4- 0.0422 A? 4- 0.0143 *M 9 1.71 A/I
4- 0.6072 Aft' 4- 0.0204 Ai'.
If we assign to the elements of the orbit the variations
DIFFERENTIAL FORMULA. 161
A/ = -6".64, Aft' = -14".12, At*:=-8".86,
Af = + 10", Ajf. = -f 10", A/* = + 0.01,
we have
A/ = Ao*' + cos H A ft' = 19".96 ;
and the preceding equations give
cos i) A0 = + 8".24, A, = 6".96.
With the same values of AO/, Aft', &c., we have already found
cos 3 Aa = + 5".47, A<5 = 9".29,
which, by means of the equations (88), writing a and 3 in place of
>l and /9, give
cos 7 A0 = -f 8".23, A 7 = 6".96.
59. In special cases, in which the differences between the calcu-
lated and the observed values of two spherical co-ordinates are given,
and the corrections to be applied to the assumed elements are sought,
it may become necessary, on account of difficulties to be encountered
in the solution of the equations of condition, to introduce other ele-
ments of the orbit of the body. The relation of the elements chosen
to those commonly used will serve, without presenting any difficulty,
for the transformation of the equations into a form adapted to the
special case. Thus, in the case of the elements which determine the
form of the orbit, we may use a or log a instead of p, and the
equation
kVl -f m
^ '~^~
gives
(89)
in which 1 is the modulus of the system of logarithms. Therefore,
the coefficient of A^ is transformed into that of A log a by multiply-
ing it by f j; and if the unit of the mth decimal place of the loga-
rithms is taken as the unit of A log a, the coefficient must be also
multiplied by 10~ m . The homogeneity of the equation is not disturbed,
since p. is here supposed to be expressed in seconds.
If we introduce logp as one of the elements, from the equation
p = a cos" f
11
162 THEORETICAL ASTRONOMY.
we get ^
ogp- i- P- o an? y>,
or
dfj. = | d logp 3/x tan , the coefficients of A^>
are changed ; and if we denote by cos 3 \ -^- } and I ~ } the values of
\ a I s dfj.
in which s = 206264".8. If the values of the differential coefficients
with respect to // and and -p: by
d l and p,
respectively, will give the values of the differential coefficients of
the heliocentric longitude and latitude with respect to x, y, and 2.
Combining these with the expressions for the differential coefficients
of the heliocentric co-ordinates with respect to the elements of the
orbit, we obtain the values of cos 6 A and A& in terms of the varia-
tions of the elements.
The equations for dx, dy, and dz in terms of du, eZft, and di, may
also be used to determine the corrections to be applied to the co-or-
dinates in order to reduce them from the ecliptic and mean equinox
of one epoch to those of another, or to the apparent equinox of the
date. In this case, we have
du = d* dft.
When the auxiliary constants A, B, a, 6, &c. are introduced, to
find the variations of these arising from the variations assigned to
the elements, we have, from the equations (99) u
cot A = tan ft cos i,
cot B = cot ft cos i sin i cosec ft tan e,
cot C = cot ft cos i -f- sin i cosec ft cot e,
in which i may have any value from to 180. If we differentiate
these, regarding all the quantities involved as variable, and reduce
by means of the values of sin a, sin b, and sin c, we get
cosi ,_ sin .4 . ,. . . ,.
dA = . , c?ft . sin ft sm i di,
sin a sma
dB = . , . (cos i cos e sin i sin e cos ft ) dft
sin 2 6
. sinl? , ^ ,. . sinisinft ,
- (cos ft sin ^ cos e -j- cos ^ sin e) di -\ r-^-r - as,
dC= . , (cos i sin e + sin i cos e cos ft ) rfft
sm 2 c
sin C , , j. . sin i sin
H -- -. - (cos ft sin i sm e cos i cos e) d^ -\ -- t
sin c
and these, by means of (101)^ reduce to
104 THEORETICAL ASTRONOMY.
= ^~ d& sin A cot a di,
sin* a
. (91)
Let us now differentiate the equations (101) w using only the upper
sign, and the result is
da = sin i sin A d& + cos A di,
db = sin i sin JB d& -f- cos B di -f cos c cosec b ds,
dc = sin i sin C d& -f- cos C di cos b cosec c ds.
If we multiply the first of these equations by cot a, the second by
cot 6, and the third by cote, and denote by ^ the modulus of the
system of logarithms, we get
dlogsina= A sin i cot a sin A d& -f <* cot a cos A di,
d log sin b = ^ sin i cot b sin dQ -{- -* cot b cos B di -f A - r-^r de,
d log sin c = A sin i cot c sin C dQ + ^o c t c cos Gdi A - r- 2 - de.
(92)
The equations (91) and (92) furnish the differential coefficients of
A, B, C, log sin a, &c. with respect to , *, and e; and if the varia-
tions assigned to SI, i, and e are so small that their squares may be
neglecitd, the same equations, writing &A, A&, At, &c. instead of
the differentials, give the variations of the auxiliary constants. In
the case of equations (92), if the variations of &, i, and e are ex-
pressed in seconds, each term of the second member must be divided
by 206264.8, and if the variations of log sin a, log sin 6, and log sine
are required in units of the mth decimal place of the logarithms, each
term of the second member must also be divided by 10.
If we differentiate the equations (81) 1} and reduce by means of the
same equations, we easily find
cos b dl = cosi sec b du -f cos b d& sin b cos (I &) di, /QOX
db = sini cos(l
which determine the relations between the variations of the elements
of the orbit and those of the heliocentric longitude and latitude.
By differentiating the equations (88) w neglecting the latitude of
DIFFERENTIAL FORMULAE. 165
the sun, and considering X, /?, J, and O as variables, we derive, after
reduction,
cosfidl = ^ cos (A O) dQ,
R (94)
dp = j sin sin (A Q ) d0 ,
which determine the variation of the geocentric latitude and longitude
arising from an increment assigned to the longitude of the sun. Ii
appears, therefore, that an error in the longitude of the sun will
produce the greatest error in the computed geocentric longitude of a
heavenly body when the body is in opposition.
166 THEORETICAL ASTRONOMY.
CHAPTER III.
INVESTIGATION OF FORMTJUE FOB COMPUTING THE ORBIT OF A COMET MOVING IN
A PARABOLA, AND FOR CORRECTING APPROXIMATE ELEMENTS BY THE VARIATION
OF THE GEOCENTRIC DISTANCE.
61. THE observed spherical co-ordinates of the place of a heavenly
body furnish each one equation of condition for the correction of the
elements of its orbit approximately known, and similarly for the
determination of the elements in the case of an orbit wholly unknown ;
and since there are six elements, neglecting the mass, which must
always be done in the first approximation, the perturbations not
being considered, three complete observations will furnish the six
equations necessary for finding these unknown quantities. Hence,
the data required for the determination of the orbit of a heavenly
body are three complete observations, namely, three observed longi-
tudes and the corresponding latitudes, or any other spherical co-
ordinates which completely determine three places of the body as
seen from the earth. Since these observations are given as made at
some point or at different points on the earth's surface, it becomes
necessary in the first place to apply the corrections for parallax. In
the case of a body whose orbit is wholly unknown, it i? impossible
to apply the correction for parallax directly to the place of the body ;
but an equivalent correction may be applied to the places of the
earth, according to the formula which will be given in the next
chapter. However, in the first determination of approximate ele-
ments of the orbit of a comet, it will be sufficient to neglect entirely
the correction for parallax. The uncertainty of the observed places
of these bodies is so much greater than in the case of well-defined
objects like the planets, and the intervals between the observations
which will be generally employed in the first determination of the
orbit will be so small, that an attempt to represent the observed places
with extreme accuracy will be superfluous.
When approximate elements have been derived, we may find the
distances of the comet from the earth corresponding to the three
observed places, and hence determine the parallax in right ascension
DETERMINATION OF AN ORBIT. 167
and in declination for each observation by means of the usual formulae.
Thus, we have
_ xp cos ?>' sin (o 0)
A cos d
tan p =Mp,
give
log p = 9.480952, log p" = 9.310779.
NUMERICAL EXAMPLE. 203
From these values of p and //', it appears that the comet was very
near the earth at the time of the observations.
The heliocentric places are then found by means of the equations
(71) and (72). Thus we obtain
I = 106 40' 50".5, b = + 33 1' 10".6, logr = 9.912082,
r=112 31 9.9, b"= + 23 55 5.8, logr" =9.935116.
The agreement of these values of r and r" with those previously
found, checks the accuracy of the calculation. Further, since the
heliocentric longitudes are increasing, the motion is direct.
The longitude of the ascending node and the inclination of the
orbit may now be found by means of the equations (74), (75), or (70);
and we get
& == 304 43' 11".5, i = 64 31' 21". 7.
The values of u and u" are given by the formulae
tan(J &) tail (7" &)
tan u = -- - r-^A tan u' ^^-,
cos i cos i
u and I & being in the same quadrant in the case of direct motion
Thus we obtain
u = 142 52' 12".4, ' u"= 153 18' 49".4.
Then the equation
x s = (r" r cos (tt" u))* + i 3 sin' (u" u)
gives
log x = 9.201423,
and the agreement of this value of X with that previously found,
proves the calculation of SI, i, u, and u".
From the equations
7=r sin i (i (u" + u) a>) = */?}'
Vq sm \ (u" u) /rr"
we get
tf = 22' 47".4, a, = 115 40' 6".3, log g = 9.88737B.
Hence we have
TT = w + ^ = 60 23' 17".8,
204 THEORETICAL ASTRONOMY.
and
i = u * = 27 12' 6".l, t/ 7 = u" < = 37 38' 43 M.
Then we obtain
log m = 9.9601277 f log q = 0.129061,
and, corresponding to the values of v and t>", Table VI. gives
log Jf = 1.267163, log M" = 1.424152.
Therefore, for the time of perihelion passage, we have
T=t M- = t 13.74364,
m
and
M"
T=if'=f 19.72836.
m
The first value gives T= 1863 Dec. 27.56473, and the second gives
T Dec. 27.56463. The agreement between these results is the final
proof of the calculation of the elements from the adopted value of
Jf=
P
If we find T by means of Table VIII., we have
log N = 0.021616, log N" = 0.018210,
and the equation
2 2
T= t Nr* sin v = if' N"il sin v",
in which log ^= 1.5883273, gives for T the values Dec. 27.56473
and Dec. 27.56469.
Collecting together the several results obtained, we have the fol-
lowing elements :
T = 1863 Dec. 27.56471 Washington mean time.
= 6023'17".8^
and Mean
log q =-9.887378.
Motion Direct.
73. The elements thus derived will, in all cases, exactly represent
the extreme places of the comet, since these only have been used in
finding the elements after p and p" have been found. If, by means
NUMERICAL EXAMPLES. 205
of these elements, we compute n and n ff , and correct the value of Jf,
the elements which will then be obtained will approximate nearer
the true values; and each successive correction will furnish more
accurate results. When the adopted value of M is exact, the result-
ing elements must by calculation reproduce this value, and since the
computed values of x 1 , x", /9, and ft" will be the same as the observed
values, the computed values of // and ft' must be such that when
substituted in the equation for J/, the same result will be obtained
as whon the observed values of X' and ft' are used. But, according
to the equations (13) and (14), the value of M depends only on the
inclination to the ecliptic of a great circle passing through the places
of the sun and comet for the time t', and is independent of the angle
at the earth between the sun and comet. Hence, the spherical co-
ordinates of any point of the great circle joining these places of the
sun and comet will, in connection with those of the extreme places,
give the same value of J/, and when the exact value of M has been
used in deriving the elements, the computed values of x' and ft' must
give the same value for w' as that which is obtained from observa-
tion. But if we represent by ij/ the angle at the earth between the
sun and comet at the time t', the values of i// derived by observation
and by computation from the elements will differ, unless the middle
place is exactly represented. In general, this difference will be small,
and since w' is constant, the equations
cos V = cos $ cos (/' O')
sin 4/ cos w' = cos ,? sin (A' '), (93)
sin V sin id = sin $,
give, by differentiation,
cos ft dX = cos w' sec ft d$', . .
d? = sin w' cos (A' ') d*'.
From these we get
cosj5'dA'__tan(A'' ')
d? sin ff
which expresses the ratio of the residual errors in longitude and
latitude, for the middle place, when the correct value of M has been
used.
Whenever these conditions are satisfied, the elements will be
correct on the hypothesis of parabolic motion, and the magnitude
of the final residuals in the middle place will depend on the deviation
of the actual orbit of the comet from the parabolic form. Further,
206 THEORETICAL ASTRONOMY.
when elements have been derived from a value of M which has not
been finally corrected, if we compute X' and /9' by means of these
elements, and then
the comparison of this value of tan w' with that given by observa-
tion will show whether any further correction of M is necessary, and
if the difference is not greater than what may be due to unavoidable
errors of calculation, we may regard M as exact.
To compare the elements obtained in the case of the example
given with the middle place, we find
t/ = 32 31' 13".5, u' = 148 11' 19".8, log / = 9.922836.
Then from the equations
tan (/ Q, ) = cos i tan u',
tan b' = tan i sin (T SI ),
we derive
t = 109 46' 48".3, b' = 28 24' 56".0.
By means of these and the values of O' and R f , we obtain
;' = 302 57' 41".l, p = 57 39' 37".0 ;
and, comparing these results with the observed values of X and /?',
the residuals for the middle place are found to be
Comp. Obs.
cos p*X= + 3".6, A/5 = + I'M.
The ratio of these remaining errors, after making due allowance for
unavoidable errors of calculation, shows that the adopted value of
M is not exact, since the error of the longitude should be less than
that of the latitude.
The value of w' given by observation is
log tan /== 0.966314,
and that given by the computed values of /(' and ft' is
log tan w' = 0.966247.
The difference being greater than what can be attributed to errors of
calculation, it appears that the value of M requires further cor-
NUMERICAL EXAMPLES. 207
rectiou. Since the difference is small, we may derive the correct
value of 31 by using the same assumed value of ,, and, instead of
the value of tan w' derived from observation, a value differing as
much from this in a contrary direction as the computed value differs.
Thus, in the present example, the computed value of log tan w' is
0.000067 less than the observed value, and, in finding the new value
of M, we must use
log tan uf = 0.966381
in computing $, and /9 " involved in the first of equations (14). If
the first of equations (10) is employed, we must use, instead of tan^S'
as derived from observation,
tan jf = tan u/ sin (/' O')>
or
log tan ft = 0.966381 -f- log sin (// ') = 0.198559,
the observed value of /' being retained. Thus we derive
log M= 9.829586,
and if the elements of the orbit are computed by means of this
value, they will represent the middle place in accordance with the
condition that the difference between the computed and the observed
value of tan w' shall be zero.
A system of elements computed with the same data from
log M 9.822906 gives for the error of the middle place,
C.-O.
cos jf A// = 1' 26".2, A,? = 40".l.
If we interpolate by means of the residuals thus found for two values
of M, it appears that a system of elements computed from
log M= 9.829586
will almost exactly represent the middle place, so that the data are
completely satisfied by the hypothesis of parabolic motion.
The equations (34) and (32) give
log ~ = 0.006955, log -jp = 0.006831,
and from (10) we get
log M' = 9.822906, log M" = 9.663729..
208 THEORETICAL ASTRONOMY.
Then by means of the equation (33) we derive, for the corrected
value of M.
log M =9.829582,
which differs only in the sixth decimal place from the result obtained
by varying tanw' and retaining the approximate values p = ^/ == w/'
74. When the approximate elements of the orbit of a comet are
known, they may be corrected by using observations which include
a longer interval of time. The most convenient method of effecting
this correction is by the variation of the geocentric distance for the
time of one of the extreme observations, and the formulae which
may be derived for this purpose are applicable, without modification,
to any case in which it is possible to determine the elements of the
orbit of a comet on the supposition of motion in a parabola. Since
there are only five elements to be determined in the case of parabolic
motion, if the distance of the comet from the earth corresponding to
the time of one complete observation is known, one additional com-
plete observation will enable us to find the elements of the orbit.
Therefore, if the elements are computed which result from two or
more assumed values of A differing but little from the correct value,
by comparison of intermediate observations with these different sys-
tems of elements, we may derive that value of the geocentric distance
of the comet for which the resulting elements will best represent the
observations.
In order that the formulae may be applicable to the case of any
fundamental plane, let us consider the equator as this plane, and,
supposing the data to be three complete observations, let A, A', A"
be the right ascensions, and _D, D' } D" the declinations of the sun
for the times t, t', t". The co-ordinates of the first place of the earth
referred to the third are
x =. R" cos D" cos A" R cos D cos A,
y R" cos D" sin A" R cos D sin A,
z = R" sin D" R sin D.
If we represent by g the chord of the earth's orbit between the places
for the first and third observations, and by G and K, respectively,
the right ascension and declination of the first place of the earth as
seen from the third, we shall have
x = g cosK cos G,
y = g cosK sin G,
z = g sin K,
VARIATION OF THE GEOCENTRIC DISTANCE. 209
and, consequently,
g cos K cos ( G A) = R' cos D" cos (A" A) R cos D,
g cos K sin ( G A} = R' cos W sin (A" A\ (96)
gsmK = K' sin Z>" R sin D,
from which ^, IT, and G may be found.
If we designate by x n y,, z, the co-ordinates of the first place of
the comet referred to the third place of the earth, we shall have
x, = J cos 8 cos a -}- g cos K cos O t
y, = J cos 5 sin a + g cos JT sin G,
z, = J sin 3 -f- <7 sm -K,
Let us now put
x, = h' cos * cos H',
y t = h' cos ? sin IT,
z, h' sin C 7 ,
and we get
h' cos f cosCET GT) = A cost cos(o GO + ? cos^",
A' cos r sin (^ G) = ^J cos 3 sin (a (?), (97)
A' sin C' = J sin 3 -|- ^ sin K,
from which to determine H f , ', and A'.
If we represent by ^ ' the angle at the third place of the earth
between the actual first and third places of the comet in space, we
obtain
cos ?'= cos ? cos H' cos 5" cos o"+ cos *' sin H' cos 3" sin o" -f sin C' sin o",
or
cos / = cos r cos d" cos (a" "') + sin r sin 3" ; (98)
and if we put
e sin/ sin <5",
e cos/= cos 3" cos (a" H')
this becomes
cos ? ' = ecos(r-/). (99)
Then we shall have
x 1 = h'* + J'" 2A' J" cos ?'
x' = ( J" A' cos ?')* + A" sin 1 /, (100)
in which 4" is the distance of the comet from the earth correspond-
ing to the last observation. We have, also, from equations (44) and
(45),
r =(J tfcos*) 1 + 7? sin 1 4-, QOV,
/' = ( J" #' cos V') 1
u
210 THEORETICAL ASTRONOMY.
in which ^ is the angle at the earth between the sun and comet at
the time t, and 4/' the same angle at the time t". To find their
values, we have
cos 4- = cos D cos S cos (a A) + sin D sin <$,
cos *"= cos D" cos S" cos (a" 4") + sin H' sin d",
which may be still further reduced by the introduction of auxiliary
angles as in the case of equation (98).
Let us now put
h'- sin f a, a', and a" will be such that the third differences may be neg-
lected, this formula may be assumed to express exactly any value of
the function corresponding to a value of the argument not differing
212 THEORETICAL ASTRONOMY.
much from J, or within the limits x dJ and x + 84, the as-
sumed values A <5J, J, and J + d A being so taken that the correct
value of J shall be either within these limits or very nearly so.
To find the coefficients m, n, and o, we have
m n + o = a. m = a', m-\-n + o = a",
whence
m = a', n =l(a"a), o = $(a"+ a) a'.
Now, in order that the middle place may be exactly represented in
right ascension, we must have
()'+"()+-='>
*Vora which we find
a: 9< =
In the same manner, the condition that the middle place shall be
exactly represented in declination, gives
In order that the orbit shall exactly represent the middle place, both
conditions must be satisfied simultaneously; but it will rarely happen
that this can be effected, and the correct value of x must be found
from those obtained by the separate conditions. The arithmetical
mean of the two values of x will not make the sum of the squares
of the residuals a minimum, and, therefore, give the most probable
value, unless the variation of cos d f A', for a given increment as-
signed to J, is the same as that of A<5'. But if we denote the value
of x for which the error in a' is reduced to zero by x', and that for
which A<5' = 0, by x", the most probable value of x will be
in which n = \ (a" a) and n r = \ (d" d). It should be observed
that, in order that the differences in right ascension and declination
shall have equal influence in determining the value of x, the values
of a, a', and a" must be multiplied by cos d'. The value of d J in
most conveniently expressed in units of the last decimal place of the
logarithms employed.
NUMERICAL EXAMPLE. 213
If the elements are already known so approximately that the first
assumed value of J differs so little from the true value that the
second differences of the residuals may be neglected, two assumptions
in regard to the value of A will suffice. Then we shall have o = 0,
and hence
m = a', n = a" a'.
The condition that the middle place shall be exactly represented,
gives the two equations
(a" a') x -f a' 3 J = 0,
(d"-d')x + d'dJ = Q.
The combination of these equations according to the method of least
squares will give the most probable value of x, namely, that for
which the sum of the squares of the residuals will be a minimum.
Having thus determined the most probable value of x, a final
system of elements computed with the geocentric distance J -f- .r,
corresponding to the time t, will represent the extreme places exactly,
and will give the least residuals in the middle place consistent with
the supposition of parabolic motion. It is further evident that we
may use any number of intermediate places to correct the assumed
value of J, each of which will furnish two equations of condition
for the determination of x, and thus the elements may be found
which will represent a series of observations.
76. EXAMPLE. The formula? thus derived for the correction of
approximate parabolic elements by varying the geocentric distance,
are applicable to the case of any fundamental plane, provided that
a, 8, A, D, &c. have the same signification with respect to this plane
that they have in reference to the equator. To illustrate their
numerical application, let us take the following normal places of
the Great Comet of 1858, which were derived by comparing an
epheineris with several observations made during a few days before
and after the date of each normal, and finding the mean difference
between computation and observation :
Washington M. T. a 6
1858 June 11.0 141 18' 30".9 + 24 46' 25".4,
July 13.0 144 32 49 .7 27 48 .8,
Aug. 14.0 152 14 12 .0 + 31 21 47 .9,
which are referred to the apparent equinox of the date. These
places are free from aberration.
214 THEORETICAL ASTRONOMY.
Wo shall take the ecliptic for the fundamental plane, and con-
verting these right ascensions and declinations into longitudes and
latitudes, and reducing to the ecliptic and mean equinox of 1858.0,
the times of observation being expressed in days from the beginning
of the yenr, we get
t =162.0, vl = 135 51' 44".2, ft = + 9 6' 57".8,
t' = 194.0, / = 137 39 41 .2. p = 12 55 9 .0,
f!" = 226.0, A" = 142 51 31 .8, ft' = + 18 36 28 .7.
From the American Nautical Almanac we obtain, for the true places
of the sun,
= 80 24' 32".4, log E = 0.006774,
0' =110 55 51 .2, Iog.R' =0.007101,
"=141 33 2.0, log #' = 0.005405,
the longitudes being referred to the mean equinox 1858.0.
When the ecliptic is the fundamental plane, we have, neglecting
the sun's latitude, D = 0, and we must write ^ and /9 in place of ex
and , and in place of A, in the equations which have been derived
for the equator as the fundamental plane. Therefore, we have
sin (-)=#' sin ("-);
cos * = cos /? cos (J 0), cos V = cos ft" cos (A" 0")
E sin 4 = B, K' sin 4" = ",
from which to find G, g, 6, B, b ff , and B", all of which remain
unchanged in the successive trials with assumed values of J. Thus
we obtain
G = 201 T 57".4, log B = 9.925092, b = -f 0.568719,
log^ = 0.013500, log B" = 9.510309, b" = + 0.959342.
Then we assume, by means of approximate elements already
known,
log A = 0.397800,
and from
A' cos C' cos (H r (?) = A cos /? cos (A GO + g,
h' cos ? sin (IT (?) = J cos /? sin (Jl G).
A' sin C' = J sin /?,
we find IT 7 , ', and h f . These give
5"' = 153 46' 20".5, :' = + 7 24' 16".4, log h' = 0.487484.
NUMERICAL EXAMPLE.
Next, from
cos and p",
respectively. Thus we obtain
/ = 159 43' 14".2, b = + 10 50' 14".0, logr = 0.323447,
J" = 144 1747.8, b" = + 35 1428.7, log r" = 0.052347.
The agreement of these results for r and r" with those already
obtained, proves the accuracy of the calculation. Since the helio-
centric longitudes are diminishing, the motion is retrograde.
Then from (74) we get
ft = 165 17' 30".3, i = 63 6' 32".5 ;
and from
tan u = _ tan u n = _ tan (r-
we obtain
u = 12 10' 12".6, u" = 40 18' 51". 2,
the values of u and I ft being in the same quadrant when the
motion is retrograde. The equation (79) gives log x = 0.090630,
which agrees with the value already found.
The formulae (81) give
w = 129 6' 46".3, log q = 9.760326,
and hence we have
v = u u> = 116 56' 33".7, v" = u" u> = 88 47' 55".l,
from which we get
T= 1858 Sept. 29.4274.
From these elements we find
log r 1 = 0.212844, v' = 107 7' 34".0, u' = ?1 59' 12".3,
and from
tan (JF ft) = cos i tan u',
tan b' = tan i sin (F ft ),
we get
r = 154 56' 33".4, V= + 19 30' 22".l.
NUMERICAL EXAMPLE. 217
By means of these and the values of O' and R', we obtain
A' = 137 39' 13".3, & = -f 12 54' 45".3,
and comparing these results with observation, we have, for the error
of the middle place,
C.-O.
cos p AA' = 27".2, A^ = 23".7.
From the relative positions of the sun, earth, and ccmet at the
time t" it is easily seen that, in order to diminish these residuals, the
geocentric distance must be increased, and therefore we assume, for
a second value of A,
log A = 0.398500,
from which we derive
H' = 153 44' 57".6, C' = + 7 24' 26".l, log h' = 0.488026,
log C= 9.912587, logc = 0.472115, logr = 0.324207,
log A" = 0.311054, logr" = 0.054824, log x = 0.089922.
Then we find the heliocentric places
I = 159 40' 33".8, b = + 10 50' 8".6, logr =0.324207,
T=144 1712.1, b" = + 35 837.8, log r" = 0.054825,
and from these,
= 165 15' 41".l, i = 63 2' 49".2,
u = 12 10 30 .8, u" = 40 13 26 .0,
a, = 128 54 44 .4, log q = 9.763620,
T= 1858 Sept. 29.8245, log r' = 0.214116,
v ' = 106 55' 43".8, u'= 21 59' 0".6,
V = 154 53 32 .3, b' = + 19 29 31 .9,
A' = 137 3939.7, /3' = + 12 55 2.9.
Therefore, for the second assumed value of J, we have
C.-O.
cos ft AA' = 1".5, A,9' = 6".l.
Since these residuals are very small, it will not be necessary to
make a third assumption in regard to J, but we may at once derive
the correction to be applied to the last assumed value by means of
the equations (109). Thus we have
' = 1.5, a '=-27.2, d' = 6.1, cos C
E sin O (4 J ) cos ^ sin A = R cos i' sin O P sin ^ cos b sin /,
( A, J ) sin /5 = B sin J p sin ?: sin i .
If we suppose the axis of x to be directed to the point whose longi-
tude is O , these become
DETERMINATION OF AN OEBIT. 223
H cos (O GO) (4 4.) cos ft cos (/I ) =
E cos 2; Po sin TT O cos & cos (7 .),
# sin (G - ) (J, - 4,) cos sin (A ) = (2;
f> ? sin TT O cos 6 sin (1 9 Q ),
( J, J ) sin P = R sin ^2/^^qpr,
we have
_o
_ p _ 2pcot2/
and from equations (70),
sin i (-u" ) tan 6?' . , cos /
cot 2/ = - -r - , sin 2y' = - j-f-v - r.
cos/ cos^Cw" u)
Therefore the formulae (87) reduce to
* sin ( - i (w" + )) = -; tan
"
cos (88)
e cos o> tt w = - / sec 2 w w ^
cos Y Vn'
from which also e and to may be derived. Then
sin <{> = e,
and the agreement of cos y as derived from this value of y with that
given by (86) will serve as a further proof of the calculation. The
longitude of the perihelion will be given by
or, when i exceeds 90, and the distinction of retrograde motion is
adopted, by n = & w.
DETERMINATION OF AN ORBIT. 261
To find a, we have
p (a cos ^p) 1
cos*?' p '
or it may be computed directly from the equation
T"
4s *rr" cos 1 \ (u" u) sin^ (E" E)'
(89)
which results from the substitution, in the last term of the preceding
equation, of the expressions for a cos l)
we compute K, I, ,3 , a , 6, c, c/, /, and A. The angle I must be less
than 90, and the value of ^9 must be determined with the greatest
possible accuracy, since on this the accuracy of the resulting elements
principally depends. Thus we obtain
K= 4 47' 29".48, log tan 1= 9.3884640,
,e o = 2 52' 59"f if, log a = 6.8013583,,,
log b = 2.5456342., log c = 2.2328550., '
log d = 1.2437914, log/= 1.3587437., log h = 3.9247G9L
The formulae
__ 8in(x"
, /
+/
sin(/"-/)
_ sin (V 1} E sin (A - Q)
1 ~sin(A" x) J b
give
log MI = 9.8946712, log J//' = 9.6690383,
log M a = 1.9404111, log J/," = 0.7306625..
The quantities thus far obtained remain unchanged in the suc-
cessive approximations to the values of P and Q.
For the first hypothesis, from
ij sin C = R sin 4.',
1, cos C == *, J^ cos Vt
268 THEORETICAL ASTRONOMY.
we obtain
logr =9.0782249, log T" = 9.0645575,
log P = 9.9863326, log Q = 8.1427824,
log c = 2.2298567 B , log k = 0.0704470,
log / == 0.0716091, log i? = 0.3326925,
C = 8 24' 49".74, Iogm = 1.2449136.
The quadrant in which must be situated is determined by the con-
dition that % shall have the same sign as ^.
The value of z' must now be found by trial from the equation
sin (/ C) = m sin 4 if.
Table XII. shows that of the four roots of this equation one exceeds
180, and is therefore excluded by the condition that sin z' must be
positive, and that two of these roots give z r greater than 180 i]/,
and are excluded by the condition that z' must be less than 180 ty.
The remaining root is that which belongs to the orbit of the planet,
and it is shown to be approximately 10 40' ; but the correct value
is found from the last equation by a few trials to be
z' = Q V 22".96.
The root which corresponds to the orbit of the earth is 18 20' 41",
and differs very little from 180 ij/.
Next, from
:
(i + -2-
1+P\ ' 2r
we derive
log r' = 0.3025672, log / = 0.01 23991,
log n = 9.7061 229, log n" = 9.6924555,
log P = 0.0254823, log p" = 0.0028859.
The values of the curtate distances having thus been found, the
heliocentric places for the three observations are now computed from
NUMERICAL EXAMPLE. 269
r cos b cos (I O) p cos (A Q) R,
r cos 6 sin (7 O) = />sm(A Q),
r sin 6 = /> tan /? ;
/ cos 6' cos (? O ') = /' cos (/ O') ',
/ cos b' sin (f O') = /o'sm(/l' O')>
/ sin 6' = p' tan /S' ;
r" cos 6" cos (r O") = P" cos (A" Q") ",
r" cos 6" sin (" O") = p" sin (X" Q"),
r"sin&" = />"tan/S",
which give
J = 5 14' 39".53, log tan b =8.4615572, logr =0.3040994,
I' = 1 45 11 .28, log tan b' = 8.4107555, log /= 0.3025673,
I" = 10 21 34 .57, log tan b" =8.3497911, log r" = 0.3011010.
The agreement of the value of logr 7 thus obtained with that already
found, is a proof of part of the calculation. Then, from
firm i 7\ ^-v tan 6" 4- tan 6
tan , sm (} (I" + - 8) = 2co3 ,^_,y
i j\ ^\ tan 6" tan b
we get
& = 207 2' 38".16, i = 4 27' 23".84,
u = 158 8' 25".78, u' = 160 39' 18".13, u" = 163 16' 4".42.
The equation
tan b' = tan i sin (^' ^ )
gives log tan 6' = 8.4107514, which differs 0.0000041 from the value
already found directly from p'. This difference, however, amounts
to only O'^OS in the value of the heliocentric latitude, and is due to
errors of calculation. If we compute n and n" from the equations
_ r'r" sin (u" u') _ rr' sin (u f u)
~ rr" sin (u" u)' ~ rr" sin (u" u) '
the results should agree with the values of these quantities previously
computed directly from P and Q. Using the values of u, u', and
u" just found, we obtain
log n = 9.7061158, log n" = 9.6924683,
270 THEORETICAL ASTEONOMY.
which differ in the last decimal places from the values used in finding
p and p". According to the equations
d log n = 21.055 cot (ti" ') du',
d log n" = -f 21.055 cot (u' it) du',
the differences of logn and logn" being expressed in units of the
seventh decimal place, the correction to u' necessary to make the two
values of logw agree is 0".15; but for the agreement of the two
values of log n", u' must be diminished by 0".26, so that it appears
that this proof is not complete, although near enough for the first
approximation. It should be observed, however, that a great circle
passing through the extreme observed places of the planet passes
very nearly through the third place of the sun, and hence the values
of p and p" as determined by means of the last two of equations (18)
are somewhat uncertain. In this case it would be advisable to com-
pute p and p", as soon as p' has been found, by means of the equa-
tions (22) and (23). Thus, from these equations we obtain
log p = 0.025491 8, log p" = 0.0028874,
and hence
I = 514'40".05, log tan b =8.4615619, log r = 0.3041042,
J"= 10 21 34.19, log tan b" =8.3497919, log r"= 0.3011017,
& = 207 2' 32".97, i = 4 27' 25".13,
u = 158 8' 31".47, u' = 160 39' 23".31, u" = 163 16' 9".22.
The value of log tan b f derived from X' and these values of & and i,
is 8.4107555, agreeing exactly with that derived from p' directly.
The values of n and n" given by these last results for u, u' and u",
are
log n = 9.7061144, log n" = 9.6924640 ;
and this proof will be complete if we apply the correction du'= 0".18
to the value of u', so that we have
u" u' = 2 36' 46".09, u' u = 2 30' 51".66.
The results which have thus been obtained enable us to proceed to
a second approximation to the correct values of P and Q, and we
may also correct the times of observation for the time of aberration
by means of the formulae
t = t C P sec ft if = t ' C P ' sec p, i" = f " Cp" sec /?",
wherein log C= 7.760523, expressed in parts of a day. Thus we gfit
t = 257.67467, t = 264.41976, if' = 271.38044,
NUMERICAL EXAMPLE. 271
and hence
log T = 9.0782331, log T> = 9.3724848, log r" = 9.0645692.
Then, to find the ratios denoted by s and s", we have
sin f cos G = sin $ (u" '),
sin f sin G = cos (w" w') cos 2/,
cos p = cos \ (u" u') sin 2/ ;
tan/'^l,
sin Y" cos G" = sin \ (u' u),
sin Y" sin G" = cos ^ (u r u) cos 2/',
cos /' = cos A (u' it) sin 2/' ;
from which we obtain
/ = 44 57' 6".00, x " = 44 56' 57".50,
r= 1 18 35 .90, r " =1 15 40 .69,
log m = 6.3482114, log m" = 6.3163548,
logy = 6.1163135, log/' = 6.0834230.
From these, by means of the equations
using Tables XIII. and XIV., we compute s and s". First, in the
case of s, we assume
ij = y-^-: = 0.0002675,
s ~r J
and, with this as the argument, Table XIII. gives log s 2 = 0.0002581 .
Hence we obtain x' = 0.000092, and, with this as the argument,
Table XIV. gives = 0.00000001 ; and, therefore, it appears that a
repetition of the calculation is unnecessary. Thus we obtain
logs =0.0001290, logs" =0.0001 200.
When the intervals are small, it is not necessary to use the formulas
272 THEOEETICAL ASTKONOMY.
in the complete form here given, since these ratios may then be found
by a simpler process, as will appear in the sequel. Then, from
_ _
~~ n" cos (u" u') cos (u" u) cos (' w)'
\ve find
log P = 9.9863451, log Q = 8.1431341,
with which the second approximation may be completed. We now
compute c , k , 1 , z', &c. precisely as in the first approximation ; but
we shall prefer, for the reason already stated, the values of f) and p"
computed by means of the equations (22) and (23) instead of those
obtained from the last two of the formulse (18). The results thus
derived are as follows :
log c = 2.2298499 n , log k = 0.0714280,
log 1 = 0.0719540, log 7] = 0.3332233,
C = 8 24' 12".48, log m = 1.2447277,
z' = 9 0' 30".84,
log / == 0.3032587, log p' = 0.0137621,
log n = 9.7061153, log n"= 9.6924604,
lo sf > = 0.0269143, logp"= 0.0041748,
I = 515'57".26, log tan b =8.4622524, logr =0.3048368,
I' = 7 46 2.76, log tan b' =8.4114276, log /= 0.303258.7,
J" = 10 22 0.91, log tan b" = 8.3504332, log i" 0.3017481,
ft = 207 0' 0".72, i = 4 28' 35".20,
u = 158 12' 19".54, u' = 160 42' 45".82, u" = 163 19' 7".14.
The agreement of the two values of log r' is complete, and the value
of log tan b f computed from
tan b' = tan i sin (V Q ),
is log tan b f = 8.41 14279, agreeing with the result derived directly
from //. The values of n and n" obtained from the equations (54)
are
log n = 9.7061156, log n" = 9.6924603,
which agree with the values already used in computing p and p", and
the proof of the calculation is complete. We have, therefore,
u" u' = 2 36' 21".32, u' u = 2 30' 26".28, u" u = 5& 47".60.
From these values of u"u' and u' u, we obtain
log s = 0.0001284, log s" = 0.0001193,
NUMERICAL, EXAMPLE. 273
and, recomputing P and Q, we get
log P = 9.9863452, log Q = 8.1431359,
which differ so little from the preceding values of these quantities
that another approximation is unnecessary. We may, therefore, from
the results already derived, complete the determination of the elements
of the orbit.
The equations
sin / cos G' = sin -^ (u" w),
sin / sin G' = cos ^ {u" u) cos 2/',
cos y' = cos \ (u" w) sin 2/,
give
x ' = 44 53' 53".25, / = 2 33' 52".97, log tan G' = 8.9011435,
log m' = 6.9332999, log/ = 6.7001345.
From these, by means of the formulae
and Tables XIII. and XIV., we obtain
logs' 2 = 0.0009908, log a/ = 6.5494116.
Then from
_/s'r/'sin(tt" tf)\'
p ~\ * r
we get
log j9 = 0.3691818.
The values of log p given by
s/r" sin (u" u'} \ 2 / s'W sin (V w)
are 0.3691824 and 0.3691814, the mean of which agrees with the
result obtained from u" u, and the differences between the separate
results are so small that the approximation to P and Q is sufficient.
The equations
-E) = V7,
rini (?)
a cos , M" = E' e sin JE".
from which we get
M= 338 8' 36".71, IT = 339 54' 10".61, M" = 341 43* 6".97 ;
and if M denotes the mean anomaly for the date T=1863 Sept. 21.5
Washington mean time, from the formulae
= tf !*(? T)
= M" r*(tr T\
we obtain the three values 339 55' 25".97, 339 55' 25".96, and
339 55' 25".96, the mean of which gives
J/ = 339 55' 25".96.
The agreement of the three results for Jf is a final proof of the
accuracy of the entire calculation of the elements.
Collecting together the separate results obtained, we have the fol-
lowing elements :
Epoch = 1863 Sept. 21.5 Washington mean time.
M = 339 55' 25".96
-= 37 15 40 .29)
Q = 207 7*> > Ech P tlc and Mean
i= 4 28 35:20J Eq-ox 1863.0.
(99)
in which log fa = 8.8596330. We have, also, to the same degree of
approximation,
ioy=i.^ log " = (ioo)
For the values
log r = 9.0782331, 1(^=9.3724848, log r"= 9.0645692,
log/ = 0.3032587,
these formulae give
log* = 0.0001277, logs' = 0.0004953, log *" = 0.0001199,
which differ but little from the correct values 0.0001284, 0.0004954,
and 0.0001193 previously obtained.
Since
secV = 1 + 6 sm 2 tf -f &c.,
the second of equations (65) gives
Substituting this value in the first of equations (66), we get
If we neglect terms of the fourth order with respect to the time, it
will be sufficient in this equation to put y' = f , according to (71), and
hence we have
and, since s' l is of the second order with respect to r', we have,
to terms of the fourth order,
280 THEORETICAL ASTRONOMY.
Therefore,
which, when the intervals are small, may be used to find s' from r
and r". In the same manner, we obtain
(102j
For logarithmic calculation, when addition and subtraction loga-
rithms are not used, it is more convenient to introduce the auxiliary
angles , r , and %" } by means of which these formulae become
. , ,. T' 2 cos 6 / . ., r" 2 cosV'
log * = & - T*-
in which log |A = 9.7627230. For the first approximation these
equations will be sufficient, even when the intervals are considerable,
to determine the values of s and s" required in correcting P and Q.
The values of r, r', r", and r" above given, in connection with
log r = 0.3048368, log r" = 0.3017481,
give
log s = 0.0001284, log s' = 0.0004951, log s" = 0.0001193.
These results for log 8 and log s" are correct, and that for log s' differs
only 3 in the seventh decimal place from the correct value.
CEBIT FROM FOUR OBSERVATIONS. 281
CHAPTER V.
DETERMINATION OF THE OBBIT OF A HEAVENLY BODY FROM FOUB OBSERVATIONS,
OF WHICH THE SECOND AND THIRD MUST BE COMPLETE.
95. THE formulae given in the preceding chapter are not sufficient
to determine the elements of the orbit of a heavenly body when its
apparent path is in the plane of the ecliptic. In this case, however,
the position of the plane of the orbit being known, only four ele-
ments remain to be determined, and four observed longitudes will
furnish the necessary equations. There is no instance of an orbit
whose inclination is zero ; but, although no such case may occur, it may
happen that the inclination is very small, and that the elements
derived from three observations w r ill on this account be uncertain,
and especially so, if the observations are not very exact. The diffi-
culty thus encountered may be remedied by using for the data in the
determination of the elements one or more additional observations,
and neglecting those latitudes which are regarded as most uncertain.
The formulae, however, are most convenient, and lead most expe-
ditiously to a knowledge of the elements of an orbit wholly unknown,
when they are made to depend on four observations, the second and
third of which must be complete ; but of the extreme observations
only the longitudes are absolutely required.
The preliminary reductions to be applied to the data are derived
precisely as explained in the preceding chapter, preparatory to a de-
termination of the elements of the orbit from three observations.
Let t, t', t", t'" be the times of observation, r, r', r", r"' the radii-
vectores of the body, u, u f , u", u'" the corresponding arguments of
the latitude, R, R', R", R'" the distances of the earth from the sun,
and O, O', O", O'" the longitudes of the sun corresponding to
these times. Let us also put
[rV"] = rV" sin ('" ")
and
' m=
282 THEORETICAL ASTRONOMY.
Then, according to the equations (5) 3 , we shall have
nx x' + ri'x" = 0,
ny -tf +n"f = 0,
W V x" ri"x'" = Q,
, X', X", I'" be the observed longitudes, ft ', ", /?'" the ob-
served latitudes corresponding to the times t, t f , t", t"', respectively,
and A, A', A", A'" the distances of the body from the earth. Further,
let
A'" cos ?" = it" t
and for the last place we have
x'" == //" cos A'" R" cos "',
y"' = /'" sin x"' #" sin 0'".
Introducing these values of x'" and y"' t and the corresponding values
of x, x' } x", y, y f , y" into the equations (2), they become
= n (p cos A R cos Q) (/ cos A' R cos 0')
+ n" G>" cos A" tf'cosQ"),
= n OB sin A .R sin ) (p 1 sin A' R sin ')
+ w" (p" sin A" J2" sin "),
= w,' (/>' cos /' ft cos ') OB" cos /" J?" cos ") (3)
+ '" (p"' cos A'" #" cos '").
= n' (p' sin A' ^' sin ') (/>" sin A" 12" sin ")
+ n'" (p'" sin A'" - R" sin "').
If we multiply the first of these equations by sin I, and the second
by cos ^, and add the products, we get
= nR sin (x ) (/ sin (A' A) + R sin (A '))
-f n" ( P " sin (A" - A) + R' sin (A - ")) ; (4)
and in a similar manner, from the third and fourth equations, we
find
= n' (p' sin (A'" A') R sin (A'" ')) (5)
(p" sin (A'" A") R" sin (A'" ")) ri"R" sin (A'" '")
Whenever the values of n, n r , n", and n'" are known, or may be
determined in functions of the time so as to satisfy the conditions of
motion in a conic section, these equations become distinct or inde-
Dendent of each other ; and, since only two unknown quantities p 1
ORBIT FROM FOUR OBSERVATIONS. 283
and p" are involved in them, they will enable us to determine these
curtate distances.
Let us now put
cos p sin (X A) =A, cos p' sin (A" A) =B,
cos p' sin (A'" A") = C, cos /? sin (X" A') = D, ( 6 )
and the preceding equations give
Ap' sec ,? Bri'p" sec ,3" = nR sin (A ) R f sin (A Q')
H- ri'R" sin (JL "),
sec /? CP" sec j9"= w'Jr sin (/" ') R" sin (/" 0") (7)
If we assume for n and w" their values in the case of the orbit of
the earth, which is equivalent to neglecting terms of the second order
in the equations (26) 3 , the second member of the first of these equa-
tions reduces rigorously to zero ; and in the same manner it can be
shown that when similar terms of the second order in the corre-
sponding expressions for n' and n" are neglected, the second member
of the last equation reduces to zero. Hence the second member of
each of these equations will generally differ from zero by a quantity
which is of at least the second order with respect to the intervals of
time between the observations. The coefficients of p f and p" are of
the first order, and it is easily seen that if we eliminate p" from
these equations, the resulting equation for p' is such that an error of
the second order in the values of n and n" may produce an error of
the order zero in the result for p', so that it will not be even an
approximation to the correct value ; and the same is true in the case
of p". It is necessary, therefore, to retain terms of the second order
in the first assumed values for w, w', n", and n'"; and, since the
terms of the second order involve r r and r", we thus introduce two
additional unknown quantities. Hence two additional equations in-
volving r', r", p', p" and quantities derived from observation, must
be obtained, so that by elimination the values of the quantities sought
may be found.
From equation (34) 4 we have
p' sec p = K cos V vV* R'* sin 2 *', (8)
which is one of the equations required ; and similarly we find, for
the other eauation.
P" sec P' = R" cos V Vi"* R"* sin* 4". (9)
284 THEORETICAL ASTRONOMY.
Introducing these values into the equations (7), and putting
*'=Vr'*-R"^,
x ! '=V / r" i -R" 2 sin 2 4",
we get
Ax' - Bn"x" = nR sin (A - ) R' sin (A ')
+ ri'R" sin (A ") AR 1 cos 4' + n"BR" cos 4",
ZtoY OB" = n'R sin (A'" ') R" sin (A'" ")
+ n'"R'" sin (A'" '") n'lXR' cos 4' + CR" cos 4".
Let us now put
or
,_cos/5"sin(A" /I) cos/3'sinQl'" A')
~~ cos yS' sin (/ A) ' A ~ cos /9" sin (A'" /')'
. .. _. . R' sin (A'" ')
h R cos 4 -7^ = e ,
R sin (A _ Q) , jR"' sin (A'" '")
1 = i ?^ = d,
-a. O
and we have
x" = h"n!x' + ri"d" a" + n'c". ^
These equations will serve to determine x' and x", and hence r' and
r", as soon as the values of n, n f , n", and n'" are known.
96. In order to include terms of the second order in the values of
n and n", we have, from the equations (26) 3 ,
and, putting
these give
ORBIT FROM FOUR OBSERVATIONS. 285
e=w'.
Lot us now put
t"' = *("'-"), r '=k(r-t'\ (is\
and, making the necessary changes in the notation in equations (26) s ,
we obtain
, i* , ^r- ,
= 1 + i "
-r^ dr"
From these we get, including terms of the second order,
and hence, if we put
P" = ^ "=(n' + n'"-l)r" 8 , (17)
we shall have, since r ' = r + r r// ,
When the intervals are equal, we have
and these expressions may be used, in the case of an unknown orbit,
tor the first approximation to the values of these quantities.
The equations (13) and (17) give
= n"P';
i / of\ aw
J-TTJ-V+W*
and, introducing these values, the equations (12) become
286 THEORETICAL ASTRONOMY.
' a$ '
the value of k being expressed in seconds of arc, or logfc = 3.5500066,
we get
log a = 0.3881359, log ft 2.9678027.
For the eccentric anomalies we have
w) tan (45 ?),
E f = tan(u' ) tan (45 ?),
tan ^E" = tan (M" o) tan (45 p),
tan J ,E'" = tan (tt w ) tan (45 ?),
from which the results are
E = 329 11' 46".01, " = 12 5' 33".63,
' = 354 29 11 .84, '" = 39 34 34 .65.
The value of J (E'" E} thus derived differs only 0".03 from that
obtained directly from x .
For the mean anomalies, we have
M =E esinE, M" =E"
M' = E'e sin E', M'" = E'" e sin E'",
which give
M = 334 55' 39".32, M" = 9 44' 52".82,
M' = 355 33 42 .97, .'" = 32 26 44 .74.
finally, if M Q denotes the mean anomaly for the epoch T= 1864
Jan. 1.0 mean time at Greenwich, from
M = M p.(t T) =M' fi(1fT)
= M"-n (If' T~) = M'" - ft ({" - T),
we obtain the four values
M = l 29'39".40
39 .49
39 .40
39 .40,
the agreement of which completely proves the entire calculation of
the elements from the data. Collecting together the several results,
we have the following elements :
NUMERICAL EXAMPLE. 307
Epoch = 1864 Jan. 1.0 Greenwich mean time.
M = 1 29' 39".42
= 4 36 47. 20
V = 11 15 52 .22
log a = 0.3881 359
log j = 2.9678027
n = 928".54447.
102. The elements thus derived completely represent the four ob-
served longitudes and the latitudes for the second and third places,
which are the actual data of the problem ; but for the extreme lati-
tudes the residuals are, computation minus observation,
A/9 = 4".47, Ay9'" = -f 1".23.
These remaining errors arise chiefly from the circumstance that the
position of the plane of the orbit cannot be determined from the
second and third places with the same degree of precision as from
the extreme places. It would be advisable, therefore, in the final
approximation, as soon as p', p", n, n", n f , and n" f are obtained, to
compute from these and the data furnished directly by observation
the curtate distances for the extreme places. The corresponding
heliocentric places may then be found, and hence the position of the
plane of the orbit as determined by the first and fourth observations.
Thus, by means of the equations (37) and (38), we obtain
log p = 0.0256953, log p'" = 0.4492542.
With these values of p and p'", the following heliocentric places are
obtained :
I = 7 16' 51".54, log tan b =8.4289064, logr =0.3083732,
r = 91 37 40 .96, logtan&'" = 8.8638549 n , log r"' = 0.3172678.
Then from
tan.t sin ( (?" + f) ) = \ (tan b'" + tan 6) sec (f" I),
tan i cos ( (V" + ) = 2 (tan b"' tan 6) cosec^ (f" I),
we get
= 206 42' 45".23, i = 4 36' 49".76.
For the arguments of the latitude the results are
u = 160 30' 35".99, u'" = 244 59' 12".53.
308 THEORETICAL ASTRONOMY.
The equations
tan b' = tan i sin (/ SI )>
tan b" = tan i sin (I" r- ft),
give
log tan 5' = 8.1827129 n , log tan b" = 8.6342104,,
and the comparison of these results with those derived directly from
ft' and p" exhibits a difference of + 1".04 in b', and of 0".06 in
b". Hence, the position of the plane of the orbit as determined from
the extreme places very nearly satisfies the intermediate latitudes.
If \ve compute the remaining elements by means of these values
of r, r'", and u, u" f , the separate results are :
log tan G = 8.0522282 n , log m = 9.7179026,
log = 0.2917731, log x = 8.9608397,
logp = 0.3712405, i (E" E) = 17 35' 42".12,
log (a cos = 197 37' 47".72, log e 9.2907906,
dr dM dr dp
~dA~~dj'dA + ~dM ' ~dA ~r~dH"dA'
and
dv _ dv d . dv (Of, . rfv d;j. 1 . , fl
'+' + ' = ' ' (
_ _ =
dJ + ^ ' d A "*" dlf ' rf J "*" ^ ' d J
dr" '.
If AJ and AJ" are expressed in seconds of arc, the corresponding
values of Ar and Ar" must be divided by 206264.8. The corrected
results thus obtained should agree with the values of r and r" com-
puted directly from the corrected values of , v", p, and e by means
of the polar equation of the conic section. Finally, we have
dz = sin T) dA,
and similarly for dz" and the last of equations (73) 2 gives
rsinwAi' rcoswsint' A&' =sini?AJ,
r" sin u" Ai' r" cos u" sin i' A & ' = sin r t " A J",
(18)
from which to find AI V and A & ', u and w" being the arguments of
the latitude in reference to the equator. We have also, according to
(72),
Aw = A/ COSt' A&',
A-' = A/ -f- 2 sin* ^ A ',
from which to find the corrections to be applied to a> f and ~ r . The
elements which refer to the equator may then be converted into those
for the ecliptic by means of the formulae which may be derived from
(109^ by interchanging SI and &' and 180 i' and t.
The final residuals of the longitudes may be obtained by substi-
tuting the adopted values of A J and AJ" in the several equations of
condition, or, which affords a complete proof of the accuracy of the
entire calculation, by direct calculation from the corrected elements ;
and the determination of the remaining errors in the values of ^ will
show how nearly the position of the plane of the orbit corresponding
to the corrected distances satisfies the intermediate latitudes.
Instead of , respect-
ively, in the equations (13), (14), and (15), and the partial differential
coefficients of r, r", v, and v" with respect to these elements must be
computed by means of the various differential formulae which have
already been investigated. Further, in all these cases, the homo-
geneity of the formulae must be carefully attended to.
108. The approximate elements of the orbit of a heavenly body
may also be corrected by varying the elements which fix the position
of the plane of the orbit. Thus, if the observed longitude and lati-
tude and the values of SI and i are given, the three equations (91),
will contain only three unknown quantities, namely, J, r, and u, and
the values of these may be found by elimination. When the observed
latitude /9 is corrected by means of the formula (6) 4 , the latitudes of
the sun disappear from these equations, and if we multiply the first
by sin (O SI) sin/9, the second (using only the upper sign) by
cos ( O SI ) sin /9, and the third by sin (^ O) cos /9, and add
the products, we get
~~ cos i sin /3 cos (O SI ) sin i cos ft sin (>i O)'
from which u may be found. If we multiply the second of these
equations by sin /9, and the third by cos /9 sin (A SI), and add the
products, we find
(20)
sin u (sin i cot /3 sin U Q ) cos i)
The expression for r in terms of the known quantities may also be
found by combining the first and second, or by combining the first
and third, of equations (91) r If we put
n cosN= sin /? cos (O SI),
nsmN= cos /3 sin 0* O).
the formula for u becomes
cos N
The last of equal ions (91) t shows that sin u and sin/9 must have the
same sign, and thus the quadrant in which u must be taken is deter-
mined. Putting, also,
m cos M = sin u,
msin H ;= sin w cot /3 sin (>l SI),
VARIATION OF THE NODE AND INCLINATION. 25
we have
__cos^_ JZejn_(0--jB)
cos (If +0" sinu
When any other plane is taken as the fundamental plane, the
latitude of the sun (which will then refer to this plane) will be re-
tained in the equations (91)! and in the resulting expressions for u
and r.
The value of u may also be obtained by first computing w and -^
by means of the equations (42) 3 , and then, if z denotes the angle at
the planet or comet between the earth and sun, the values of u and
z, as may be readily seen, will be determined by means of the rela-
tions of the parts of a spherical triangle of which the sides are
180 (z + ^), 180 -f O SI, and u, the angle opposite to the
side u being that which we designate by tr, and the side 180 -f O &
being included by this and the inclination i. Let S= 180 (z
and, according to Napier's analogies, this spherical triangle gives
_
'
from which S and u are readily found. Then we have
z = 180 4 S,
snz
to find r.
If we assume approximate values of R and /, as given by a system
of elements already known, the equations here given enable us to find
r, U, r", and u" from A, /9 and /", /3", corresponding to the dates t
and t" of the fundamental places selected, and from these results for
two radii-vectores and arguments of the latitude, the remaining
elements may be derived. From these the geocentric place of the
body may be found for the date t' of any intermediate or additional
observed place, and the difference between the computed and the
observed place will indicate the degree of precision of the assumed
values of ft and L Then we assign to ft the increment , q = 2a sin 1 (45 ^?) ;
and the value of to may be found by means of the equations (87) 4 or
(88) 4 .
116. The process here indicated will be applied chiefly in the de-
termination of the orbits of comets, and generally for cases in which
a is large. In such cases the angles and d will be small, so that
the slightest errors will have considerable influence in vitiating the
value of t" t as determined by equation (61); but if we transform
this equation so as to eliminate the divisor a* in the first member, the
uncertainty of the solution may be overcome. The difference sine
'342 THEORETICAL ASTRONOMY.
may be expressed by a series which converges rapidly when e is small,
Thus, let us put
e sin = y sin* ^e, a? = sin 2 -je >
and we have
-~ = 2 cosec \s | y cot ^e,
-j = 4 cosec -^e.
Therefore
dy _ 8 6y cos ^e _ 4 3y (1 2a?)
_ _
da; ~ sin^e 2*(1 x)
If we suppose y to be expanded into a series of the form
we get, by differentiation,
and substituting for -~ the value already obtained, the result is
2#e + (4 r 2/3) x 1 + (65 4 r ) a? + &c. = 4 3a + (60 3,9) x
+ (6/9 - 3 r ) ^ l + (6r - 35) * + &c.
Therefore we have
4 3a:=0, 60 3^9=2/9,
6 3r = 4r 2/9, 6r 35 = 65 4r,
from which we get
4.6 4.6.8 4.6.8.10 .
>&C "
- 37ff 3T5T7' 3.5.7.9
Hence we obtain
and, in like manner,
which, for brevity, may be written
e sin e = | Q sin* e,
RELATION BETWEEN TWO PLACES IN THE ORBIT. 343
Combining these expressions with (61), and substituting for sin s and
sin %d their values given by the equations (60), there results
8t'=g(r + f" + x)t=F #0 + r "_x)f, (65)
the upper sign being used when the heliocentric motion of the body
is less than 180, and the lower sign when it is greater than 180.
The coefficients Q and Q' represent, respectively, the series of terms
enclosed in the parentheses in the second members of the equations
(62) and (63), and it is evident that their values may be tabulated
with the argument e or d, as the case may be. It will be observed,
however, that the first two terms of the value of Q are identical with
the first two terms of the expansion of (co$\s)~ into a series of
ascending powers of sin Je, while the difference is very small between
the coefficients of the third terms. Thus, we have
(cos Je)- = (1 sin 2 JO'* =1 + 1 sin 2 ^ + g--~ sin* $e
6 . 11 . 16 .
+ 5Tl ( J') dA'
Let 7i, n', n", &c. be the observed values of x, and m the number of
observations ; then we have
A = n x, A' = ri x, A" = n" x, &c.,
and hence
dA _ dA' dA"
j -j -j = 1.
ax ax ax
Therefore the equation (4) becomes
Cn x) dlog?(n' x)
d( n -x)
,
This equation will serve to determine the value of x as soon as the
form of the function symbolized by
m Vm
which agrees with the first of equations (28).
Let P denote the weight of the sum X, p f the weight of x f , and p"
that of x" \ then we shall have
374 THEORETICAL ASTRONOMY.
from which we get
Since the unit of weight is arbitrary, we may take
P f = ^ P" = JV & c -5
nnd hence we have, for the weight of the algebraic sum of any
number of values,
P = ~W = r' 1 + r"* + r'"* + &c.'
or. whatever may be the unit of weight adopted,
P = (39)
In the case of a series of observed values of a quantity, if we
designate by r r the probable error of a residual found by comparing
the arithmetical mean with an observed value, by r the probable
error of the observation, by x the arithmetical mean, and by n any
observed value, the probable error of
n = x + v,
according to (36), will be
r Q being the probable error of the arithmetical mean. Hence we derive
7* """";
and if we adopt the value
r' = 0.8453^,
m
the expression for the probable error of an observation becomes
r = 0.8453 - J2L=, (40)
in which [v] denotes the sum of the residuals regarded as positive,
and m the number of observations.
133. Let n, n', n" } &c. denote the observed values of x, and let^p,
p',p", &c. be their respective weights; then, according to the defi-
METHOD OF LEAST SQUARES. 375
nition of the weight, the value n may be regarded as the arithmetical
mean of p observations whose weight is unity, and the same is true
in the case of n', n", &c. We thus resolve the given values into
p+p f +p" + .... observations of the weight unity, and the arith-
metical mean of all these gives, for the most probable value of x,
_pn+ p'n' + p"n" + &c. _ [ pn\
"~ ' " ~ '
The unit of weight being entirely arbitrary, it is evident that the
relation given by this equation is correct as well when the quantities
p, p', p", &c. are fractional as when they are whole numbers. The
weight of x as determined by (41) is expressed by the sum
P +P' +p"+P m + &c-
and the probable error of ar is given by
(42)
when r, denotes the probable error of an observation whose weight
is unity. The value of r, must be found by means of the observa-
tions themselves. Thus, there will be p residuals expressed by
n x w p' residuals expressed by n' x w and similarly in the case of
n", n'", &c. Hence, according to equation (31), we shall have
,, = 0.8745^
(43)
in which m denotes the number of values to be combined, or the
number of quantities n, n', n", &c. For the mean error of x , we
have the equations
t .
I/O] n* -DO]'
If different determinations of the quantity x are given, for which
the probable errors are r, r 7 , r", &c., the reciprocals of the squares
of these probable errors may be taken as the weights of the respective
values n, n ; , n", &c., and we shall have
376 THEORETICAL ASTRONOMY.
with the probable error
f =^ + 4,+j, + ...; (46)
The mean errors may be used in these equations instead of the pro-
bable errors.
134. The results thus obtained for the case of the direct observa-
tion of the quantity sought, are applicable to the determination of
the conditions for finding the most probable values of several un-
known quantities when only a certain function of these quantities is
directly observed. In the actual application of the formulas it will
always be possible to reduce the problem to the case in which the
quantity observed is a linear function of the quantities sought. Thus,
let V be the quantity observed, and , ^, , &c. the unknown quan-
tities to be determined, so that we have
Let , %, , &c. be approximate values of these quantities supposed
to be already known by means of previous calculation, and let x, y,
z, &c. denote, respectively, the corrections which must be applied to
these approximate values in order to obtain their true values. Then,
if we suppose that the previous approximation is so close that the
squares and products of the several corrections may be neglected, we
have
T . __ dV , dV . dV
F - F '=**+-3i*+*' + -"
.
and thus the equation is reduced to a linear form. Hence, in general,
if we denote by n the difference between the computed and the ob-
served value of the function, and similarly in the case of each obser-
vation employed, the equations to be solved are of the following
form :
ax -\- by -f- cz -\- du -f- ew -\- ft -f- n = 0,
a'x + b'y + c'z + d'u + e'w +/' -f n' = 0, (47)
a"x + b"y + c"z + d"u + e"w +ft + n"= 0,
&c. &c.
which may be extended so as to include any number of unknown
quantities. If the number of equations is the same as the number
of unknown quantities, the resulting values of these will exactly
satisfy the several equations ; but if the number of equations exceeds
the number of unknown quantities, there will not be any system of
METHOD OF LEAST SQUARES. 377
values for these wLich will reduce the second members absolutely to
zero, and we can only determine the values for which the errors for
the several equations, which may be denoted by v, v', v", &c., will be
those which we may regard as belonging to the most probable values
of the unknown quantities.
Let J, J', J", &c. be the actual errors of the observed quantities;
then the probability that these occur in the case of the observations
used in forming the equations of condition, will be expressed by
and the most probable values of the unknown quantities will be those
which make P a maximum. The form of the function /]+ [on] = 0,
[06] + [M] y + M 2 + [MJ u + [6e] to + [&/] < + [6] = 0,
[ac>-f [6c]y + [cc] + [cci] + [ce] w + [c/] < + [en] = 0,
[ad] re + [ W] y 4- [cd] 2 + [dd] M + [de] 10 + [d/] + [d] 0, ^ O1 ;
M x + [be] y + [_ce] z + [de] u + [ee~] w + [e/] < + [en~] = 0,
in which
[aa] = aa -{- a'a' -j- a"a" -{"
[a&]=a& + a'6' + a"6" + ....
[oc] = ac + oV + a"c" + . . . .
[66] = 66 4- 6'6' 4- b"b" 4- ....
&c. &c.
The equations of condition are thus reduced to the same number as
the number of the unknown quantities, and the solution of these
will give the values for which the sum of the squares of the residuals
will be a minimum. These final equations are called normal equations.
When the observations are not equally precise, in accordance with
the condition that AV + A'V 2 + h m v" 2 + Ac. shall be u minimum,
METHOD OF LEAST SQUARES. 379
each equation of condition must be multiplied by the measure of
precision of the observation; or, since the weight is proportional to
the square of the measure of precision, each equation of condition
must be multiplied by the square root of the weight of the observa-
tion, and the several equations of condition, being thus reduced to
the same unit of weight, must be combined as indicated by the equa-
tions (51).
135. It will be observed that the formation of the first normal
equation is effected by multiplying each equation of condition by
the coefficient of x in that equation and then taking the sum of all
the equations thus formed. The second normal equation is obtained
in the same manner by multiplying by the coefficient of y; and thus
by multiplying by the coefficient of each of the unknown quantities
the several normal equations are formed. These equations will gene-
rally give, by elimination, a system of determinate values of the
unknown quantities x, y, z, &c. But if one of the normal equations
may be derived from one of the others by multiplying it by a con-
stant, or if one of the equations may be derived by a combination of
two or more of the remaining equations, the number of distinct rela-
tions will be less than the number of unknown quantities, and the
problem will thus become indeterminate. In this case an unknown
quantity may be expressed in the form of a linear function of one or
more of the other unknown quantities. Thus, if the number of
independent equations is one less than the number of unknown
quantities, the final expressions for all of these quantities except one,
will' be of the form
X = a + Pt, y = a'+l?t, z = a" + fi't, &C. (53)
The coefficients a, , a', ft', &c. depend on the known terms and co-
efficients in the normal equations, and if by any means t can be de-
termined independently, the values of x, y, z, &c. become determinate.
It is evident, further, that when two of the normal equations may be
rendered nearly identical by the introduction of a constant factor, the
problem becomes so nearly indeterminate that in the numerical appli-
cation the resulting values of the unknown quantities will be very
uncertain, so that it will be necessary to express them as in the equa-
tions (53).
The indeterrnination in the case of the normal equations results
necessarily from a similarity in the original equations of condition,
and when the problem becomes nearly indeterminate, the identity of
380 THEORETICAL ASTRONOMY.
the equations will be closer in the normal equations than in the equa-
tions of condition from which they are derived. It should be observed,
also, that when we express x, y, z, &c. in terms of t, as in (53), the
normal equation in t, which is the one formed by multiplying by the
coefficient of t in each of the equations of condition, is not required.
136. The elimination in the solution of the equations (51) is most
conveniently effected by the method of substitution. Thus, the first
of these equations gives
[06] [ac] [ad] [ae] [of] [an],
[aa] f [aa] [aa] [aa] [aa] [aa] '
and if we substitute this for x in each of the remaining normal equa-
tions, and put
(54)
fcflrn=
[i ^
[ee] B=t [oe] = [ ee .l], [ e /] _ t^i [ /] :
M ' (57)
- [an] = [en .l], (58)
we obtain
METHOD OF LEAST SQUARES. 381
[16.1] y + [6c.l] z + [6rf.l] u + [6e.l] w + [6/.l]< + [6n.l] = 0,
[6c.l] y + [cc.l] 2 + [crf.l] u + [ce.l] w + [c/.l] t + [cn.l] = 0,
[6d.l] y + [cd.l] 2 + [dd.l] w + [cfe.l] 10 + [d/.l] * + [d.l] = 0, (59)
[6.l] y + [?.!] + [ sin (p = e e the value given bv
the first of equations (183), the result is
2^ sin (192)
Substituting in this the values of a ft and i i given by (190),
we get
a = ft, + -^_ - r , - l ~? sm l l .p'q's + 1\ (193)
1 sin i t cos i '^^
CHANGE OF THE OSCULATING ELEMENTS. 51 5
T being expressed in seconds of arc. Finally, for the longitude of
the perihelion, we have
*=*+&*, (194)
and the elements of the instantaneous orbit are completely deter-
mined. When we neglect terms of the third order, this equation,
substituting the values given by (190) and (192), becomes
It should also be observed that the inclination i which appears in
these formulae is supposed to be susceptible of any value from to
180, and hence when i exceeds 90 and the elements are given in
accordance with the distinction of retrograde motion, they are to be
changed to the general form by using 180 i instead of ?, and
2& x instead of TT.
The accuracy of the numerical process may be checked by com-
puting the heliocentric place of the body for the date to which the
new elements belong by means of these elements, and comparing the
results with those obtained directly by means of the equations (155).
We may remark, also, that when the inclination does not differ much
from 90, the reduction of the longitudes to the fundamental plane
becomes uncertain, and F may be very large, and hence, instead of
the ecliptic, the equator must be taken as the fundamental plane to
which the elements and the longitudes are referred.
192. Although, by means of the formulae which have been given,
the complete perturbations may be determined for a very long period
of time, using constantly the same osculating elements, yet, on
account of the ease with which new elements may be found from 3M,
.. d$M dv , dftz. ,
v, oz,, JT~> -ji> and 57-' and on account of the facility afforded in
the calculation of the indirect terms in the equations for the differen-
tial coefficients so long as the values of the perturbations are small,
it is evident that the most advantageous process will be to compute
8M, v, and dz, only with respect to the first power of the disturbing
force, and determine new osculating elements whenever the terms of
the second order must be considered. Then the integration will
Hgain commence with zero, and will be continued until, on account
of the terms of the second order, another change of the elements is
required. The frequency of this transformation will necessarily de-
5io THEORETICAL ASTRONOMY.
pend on the magnitude of the disturbing force; and if the disturbed
body is so near the disturbing body that a very frequent change of
the elements becomes necessary, it may be more convenient either to
include the terms of the second order directly in the computation
of the values of dM, v, and Sz,, or to adopt one of the other methods
which have been given for the determination of the perturbations of
a heavenly body. In the case of the asteroid planets, the consider-
ation of the terms of the second order in this manner will only
require a change of the osculating elements after an interval of seve-
ral years, and whenever this transformation shall be required, the
equations for
and hence
= (e sin v R -\- ), (200)
dt kvp (1 + w)
for the determination of dp.
The first of the equations (97) gives
and hence we obtain
Sr
Jtl/l + ro
^ = ^^=S. (201)
dt ytl/1 4- m
The equation p = a (I e 2 ) gives
Equating these values of -^ and introducing the value of -^-
already found, we get
de 1
dt kVp(l+m)
(202)
520 THEORETICAL ASTRONOMY.
and since
= 1 4- e cos v, = 1 e cos E,
r a
E being the eccentric anomaly in the instantaneous orbit, this becomes
~ = * (p sin S)
,
-(*-4)^ 0208)
The equation (205) gives
522 THEORETICAL ASTRONOMY.
1
P cot P cos v -fr
kVp(l + m)
~~dt'
by means of which (208) reduces to
_ d* 2r COS y I? /, , N
cos Mean Equinox 1864.0
t = 4 37 .51 J
, and the differential coefficients of and
q, with respect to the elements of the orbit, need not be determined
with great accuracy.
Next, we compute -^ and ^ J * from equations (12), and from
(16), the values of * ^' * $C =, Ac., by means of which,
dy d