John S. Prell
.
THEORETICAL ASTRONOMY
RELATING TO THE
MOTIONS OF THE HEAVENLY BODIES
REVOLVING AROUND THE SUN IN ACCORDANCE WITH
THE LAW OF UNIVERSAL GRAVITATION
EMBRACING
A SYSTEMATIC DERIVATION OF THE FORMULA FOR THE CALCULATION OF THE GEOCENTRIC AND UELIO-
CENTRIC PLACES, FOR THE DETERMINATION OF THE ORBITS OF PLANETS ANi COMETS, FOB
THE CORRECTION OF APPROXIMATE ELEMENTS, AND FOR THE COMPUTATION OF
SPECIAL PERTURBATIONS; TOGETHER WITH THE THEORY Of THE COMBI-
NATION OF OBSERVATIONS AND THE METHOD OF LEAST SQUARES.
Utiih Jtummral femgte and
JOHN S. PRELL
Civil & Mechanical Engineer.
SAN FRANCISCO, CAL.
BY
JAMES C. WATSON
DIRECTOR OF THE OBSERVATORY AT ANN ARBOR, AND PROFESSOR OF ASTRONOMY IN THE
UNIVERSITY OF MICHIGAN
PHILADELPHIA:
J. B. LIPPINCOTT COMPANY.
LONDON: 36 SOUTHAMPTON STREET, COVENT GARDEN.
1900.
Entered, according to Act of Congress, in the year 1868 '-v
J. B. LIPPINCOTT & CO.,
in the Clerk's Office of the District Court of the United States for the Eastern District
of Pennsylvania.
Copyright, 1896, by ANNETTE H. WATSON.
GIFT
555
JOHN S. PRELL
Civil & Mechanical Engineer.
SAN FRANCISCO, CAL.
PREFACE.
THE discovery of the great law of nature, the law of gravitation, by
NEWTON, prepared the way for the brilliant achievements which have
distinguished the history of astronomical science. A first essential, how-
ever, to the solution of those recondite problems which were to exhibit
the effect of the mutual attraction of the bodies of our system, was the
development of the infinitesimal calculus; and the labors of those who
devoted themselves to pure analysis have contributed a most important
part in the attainment of the high degree of perfection which character-
izes the results of astronomical investigations. Of the earlier efforts to
develop the great results following from the law of gravitation, those of
EULER stand pre-eminent, and the memoirs which he published have,
in reality, furnished the germ of all subsequent investigations in
celestial mechanics. In this connection also the names of BERNOTJILLI,
CLAIRATJT, and D'ALEMBERT deserve the most honorable mention as
having contributed also, in a high degree, to give direction to the inves-
tigations which were to unfold so many mysteries of nature. By means
of the researches thus inaugurated, the great problems of mechanics
were successfully solved, many beautiful theorems relating to the planet-
ary motions demonstrated, and many useful formulae developed.
It is true, however, that in the early stage of the science methods
were developed which have since been found to be impracticable, even
if not erroneous; still, enough was effected to direct attention in the
proper channel, and to prepare the way for the more complete labors of
LAGE-O.NGE and LAPLACE. The genius and the analytical skill of these
extraordinary men gave to the progress of Theoretical Astronomy the
most rapid strides ; and the intricate investigations which they success-
fully performed, served constantly to educe new discoveries, so that of
all the problems relating to the mutual attraction of the several planets
3
M718385
4 PREFACE.
but little more remained to be accomplished by their successors than to
develop and simplify the methods which they made known, and to intro-
duce such modifications as should be indicated by experience or rendered
possible by the latest discoveries in the domain of pure analysis.
The problem of determining the elements of the orbit of a comet
moving in a parabola, by means of observed places, which had been
considered by NEWTON, EULER, BOSCOVICH, LAMBERT, and others,
received from LAGRANGE and LAPLACE the most careful consideration
in the light of all that had been previously done. The solution given
by the former is analytically complete, but far from being practically
complete ; that given by the latter is especially simple and practical so
far as regards the labor of computation ; but the results obtained by it
are so affected by the unavoidable errors of observation as to be often
little more than rude approximations. The method which was found to
answer best in actual practice, was that proposed by OLBERS in his
work entitled Leichteste und bequemste Methode die Bahn eines Cometen
zu berechnen, in which, by making use of a beautiful theorem of para-
bolic motion demonstrated by EULER and also by LAMBERT, and by
adopting a method of trial and error in the numerical solution of
certain equations, he was enabled to effect a solution which could be
performed with remarkable ease. The accuracy of the results obtained
by OLBERS'S method, and the facility of its application, directed the
attention of LEGENDRE, IVORY, GAUSS, and ENCKE to this subject, and
by them the method was extended and generalized, and rendered appli
cable in the exceptional cases in which the other methods failed.
It should be observed, however, that the knowledge of one element,
the eccentricity, greatly facilitated the solution ; and, although elliptic
elements had been computed for some of the comets, the first hypothesis
was that of parabolic motion, so that the subsequent process required
simply the determination of the corrections tc be applied to these ele-
ments in order to satisfy the observations. The more difficult problem
of determining all the elements of planetary motion directly from three
observed places, remained unsolved until the discovery of Ceres by
PIAZZI in 1801, by which the attention of GAUSS was directed to this
subject, the result of which was the subsequent publication of his
Theoria Motus Corporum Cwlestium, a most able work, in which he gave
to the world, in a finished form, the results of many years of attention
PRPJFACE. 5
to the subject of which it treats. His method for .determining all the
elements directly from given observed places, as given in the Theoria
Motus, and as subsequently given in a revised form by ENCKE, leaves
scarcely any thing to be desired on this topic. In the same work he
gave the first explanation of the method of least squares, a method
which has been of inestimable service in investigations depending on
observed data.
The discovery of the minor planets directed attention also to the
methods of determining their perturbations, since those applied in the
case of the major planets were found to be inapplicable. For a long
time astronomers were content simply to compute the special perturba-
tions of these bodies from epoch to epoch, and it was not until the com-
mencement of the brilliant researches by HANSEN that serious hopes
were entertained of being able to compute successfully the general per-
turbations of these bodies. By devising an entirely new mode of con-
sidering the perturbations, namely, by determining what may be called
the perturbations of the time, and thus passing from the undisturbed
place to the disturbed place, and by other ingenious analytical and
mechanical devices, he succeeded in effecting a solution of this most
difficult problem, and his latest works contain all the formulse which are
required for the cases actually occurring. The refined and difficult
analysis and the laborious calculations involved were such that, even
after HANSEN'S methods were made known, astronomers still adhered to
the method of special perturbations by the variation of constants as
developed by LAGRANGE.
The discovery of Astrcea by HENCKE was speedily followed by the
discovery of other planets, and fortunately indeed it so happened that
the subject of special perturbations was to receive a new improvement.
The discovery by BOND and ENCKE of a method by which we determine
at once the variations of the rectangular co-ordinates of the disturbed
body by integrating the fundamental equations of motion by means of
mechanical quadrature, directed the attention of HANSEN to this phase
of the problem, and soon after he gave formulse for the determination
of the perturbations of the latitude, the mean anomaly, and the loga-
rithm of the radius-vector, which are exceedingly convenient in the
process of integration, and which have been found to give the most
satisfactory results. The formulse for the perturbations of the latitude,
6 PREFACE.
true longitude, and radius-vector, to be integrated in the same manner,
were afterwards given by BRUNNOW.
Having thus stated briefly a few historical facts relating to the
problems of theoretical astronomy, I proceed to a statement of the
object of this work. The discovery of so many planets and comets has
furnished a wide field for exercise in the calculations relating to their
motions, and it has occurred to me that a work which should contain a
development of all the formulae required in determining the orbits of the
heavenly bodies directly from given observed places, and in correcting
these orbits by means of more extended discussions of series of observa-
tions, including also the determination of the perturbations, together
with a complete collection of auxiliary tables, and also such practical
directions as might guide the inexperienced computer, might add very
materially to the progress of the science by attracting the attention of a
greater number of competent computers. Having carefully read the
works of the great masters, my plan was to prepare a complete work on
this subject, commencing with the fundamental principles of dynamics,
and systematically treating, from one point of view, all the problems
presented. The scope and the arrangement of the work will be best
understood after an examination of its contents ; and let it suffice to add
that I have endeavored to keep constantly in view the wants of the
computer, providing for the exceptional cases as they occur, and giving
all the formulae which appeared to me to be best adapted to the problems
under consideration. I have not thought it worth while to trace out the
geometrical signification of many of the auxiliary quantities introduced.
Those who are curious in such matters may readily derive many beau-
tiful theorems from a consideration of the relations of some of these
auxiliaries. For convenience, the formulae are numbered consecutively
through each chapter, and the references to those of a preceding chapter
are defined by adding a subscript figure denoting the number of the
chapter.
Besides having read the works of those who have given special atten
tion to these problems, I have consulted the Astronomische Nachrichten,
the Astronomical Journal, and other astronomical periodicals, in which
'is to be found much valuable information resulting from the experi-
ence of those w r ho have been or are now actively engaged in astro-
nomical pursuits. I must also express my obligations to the publishers,
PREFACE. 7
Messrs. J. B. LIPPINCOTI & Co., for the generous interest which they
have manifested in the publication of the work, and also to Dr. B. A.
GOULD, of Cambridge, Mass., and to Dr. OPPOLZER, of Vienna, for
valuable suggestions.
For the determination of the time from the perihelion and of the true
anomaly in very eccentric orbits I have given the method proposed by
BESSEL in the Monatliche Correspondent, vol. xii., the tables for which
were subsequently given by BRUNNOW in his Astronomical Notices, and
also the method proposed by GAUSS, but in a more convenient form.
For obvious reasons, I have given the solution for the special case of
parabolic motion before completing the solution of the general problem
of finding all of the elements of the orbit by means of three observed
places. The differential formulae and the other formulae for correcting
approximate elements are given in a form convenient for application,
and the formulas for finding the chord or the time of describing the
subtended arc of the orbit, in the case of very eccentric orbits, will be
found very convenient in practice.
I have given a pretty full development of the application of the
theory of probabilities to the combination of observations, endeavoring
to direct the attention of the reader, as far as possible, to the sources of
error to be apprehended and to the most advantageous method of treat-
ing the problem so as to eliminate the effects of these errors. For the
rejection of doubtful observations, according to theoretical considerations,
I have given the simple formula, suggested by CHAUVENET, which fol
lows directly from the fundamental equations for the probability of
errors, and which will answer for the purposes here required as well as
the more complete criterion proposed by PEIRCE. In the chapter
devoted to the theory of special perturbations I have taken particular
pains to develop the whole subject in a complete and practical form,
keeping constantly in view the requirements for accurate and convenient
numerical application. The time is adopted as the independent variable
in the determination of the perturbations of the elements directly, since
experience has established the convenience of this form ; and should it
be desired to change the independent variable and to use the differential
coefficients with respect to the eccentric anomaly, the equations between
this function and the mean motion will enable us to effect readily the
required transformation.
8 PREFACE.
The numerical examples involve data derived from actual observa-
tions, and care has been taken to make them complete in every respect,
so as to serve as a guide to the efforts of those not familiar with these
calculations ; and when different fundamental planes are spoken of, it is
presumed that the reader is familiar with the elements of spherical
astronomy, so that it is unnecessary to state, in all cases, whether the
centre of the sphere is taken at the centre of the earth, or at any other
point in space.
The preparation of the Tables has cost me a great amount of labor,
logarithms of ten decimals being employed in order to be sure of the
last decimal given. Several of those in previous use have been recom-
puted and extended, and others here given for the first time have been
prepared with special care. The adopted value of the constant of the
solar attraction is that given by GAUSS, which, as will appear, is not
accurately in accordance with the adoption of the mean distance of the
earth from the sun as the unit of space; but until the absolute value of
the earth's mean motion is known, it is best, for the sake of uniformity
and accuracy, to retain GAUSS'S constant.
The preparation of this work has been effected amid many intern; p-
tions, and with other labors constantly pressing me, by which the progress
of its publication has been somewhat delayed, even since the stereo-
typing was commenced, so that in some cases I have been anticipated
in the publication of formulae which would have here appeared for the
first time. I have, however, endeavored to perform conscientiously the
self-imposed task, seeking always to secure a logical sequence in the de-
velopment of the formulae, to preserve uniformity and elegance in the
notation, and to elucidate the successive steps in the analysis, so that the
work may be read by those who, possessing a respectable mathematical
education, desire to be informed of the means by which astronomers are
enabled to arrive at so many grand results connected with the motions
of the heavenly bodies, and by which the grandeur and sublimity of
creation are unveiled. The labor of the preparation of the work will
have been fully repaid if it shall be the means of directing a more
general attention to the study of the wonderful mechanism of the hea-
vens, the contemplation of which must ever serve to impress upon the
mind the reality of the perfection of the OMNIPOTENT, the LIVING GOD !
OBSEBVATOKY, ANN ARBOR, June, 1867.
CONTENTS.
THEORETICAL ASTRONOMY.
CHAPTER I.
INVESTIGATION OP THE FUNDAMENTAL EQUATIONS OF MOTION, AND OF THE FOR-
MULA FOR DETERMINING, FROM KNOWN ELEMENTS, THE HELIOCENTRIC AND
GEOCENTRIC PLACES OF A HEAVENLY BODY, ADAPTED TO NUMERICAL COM-
PUTATION FOR CASES OF ANY ECCENTRICITY WHATEVER.
PAOB
Fundamental Principles 15
Attraction of Spheres 19
Motions of a System of Bodies '. 23
Invariable Plane of the System 29
Motion of a Solid Body 31
The Units of Space, Time, and Mass 36
Motion of a Body relative to the Sun 38
Equations for Undisturbed Motion 42
Determination of the Attractive Force of the Sun 49
Determination of the Place in an Elliptic Orbit 53
Determination of the Place in a Parabolic Orbit 59
Determination of the Place in a Hyperbolic Orbit 65
Methods for finding the True Anomaly and the Time from the Perihelion in the
case of Orbits of Great Eccentricity 70
Determination of the Position in Space 81
Heliocentric Longitude and Latitude 83
Reduction to the Ecliptic 85
Geocentric Longitude and Latitude 86
Transformation of Spherical Co-ordinates 87
Direct Determination of the Geocentric Eight Ascension and Declination 90
.Reduction of the Elements from one Epoch to another 99
Numerical Examples 103
Interpolation < 112
Time of Opposition 114
10 CONTENTS.
CHAPTER II.
INVESTIGATION OF THE DIFFERENTIAL FORMULAE WHICH EXPRESS THE RELATION
BETWEEN THE GEOCENTRIC OR HELIOCENTRIC PLACES OF A HEAVENLY BODY
AND THE VARIATIONS OF THE ELEMENTS OF ITS ORBIT.
PAGB
Variation of the Eight Ascension and Declination 118
Case of Parabolic Motion 125
Case of Hyperbolic Motion 128
Case of Orbits differing but little from the Parabola 130
Numerical Examples 135
Variation of the Longitude and Latitude 143
The Elements referred to the same Fundamental Plane as the Geocentric Places 149
Numerical Example 150
Plane of the Orbit taken as the Fundamental Plane to which the Geocentric
Places are referred 153
Numerical Example 159
Variation of the Auxiliaries for the Equator 163
CHAPTER III.
INVESTIGATION OF FORMULA FOR COMPUTING THE ORBIT OF A COMET MOVING
IN A PARABOLA, AND FOR CORRECTING APPROXIMATE ELEMENTS BY THE
VARIATION OF THE GEOCENTRIC DISTANCE.
Correction of the Observations for Parallax 167
Fundamental Equations 169
Particular Cases 172
Ratio of Two Curtate Distances 178
Determination of the Curtate Distances 181
Relation between Two Radii- Vectores, the Chord joining their Extremities, and
the Time of describing the Parabolic Arc 184
Determination of the Node and Inclination 192
Perihelion Distance and Longitude of the Perihelion 194
Time of Perihelion Passage 195
Numerical Example 199
Correction of Approximate Elements by varying the Geocentric Distance 208
Numerical Example 213
CHAPTER IV.
DETERMINATION, FROM THREE COMPLETE OBSERVATIONS, OF THE ELEMENTS OF
THE ORBIT OF A HEAVENLY BODY, INCLUDING THE ECCENTRICITY OR FORM
OF THE CONIC SECTION.
Reduction of the Data 220
Corrections for Parallax 223
CONTENTS. ll
PAOK
Fundamental Equations 225
Formulae for the Curtate Distances 228
Modification of the Formulae in Particular Cases 231
Determination of the Curtate Distance for the Middle Observation 236
Case of a Double Solution 239
Posjtion indicated by the Curvature of the Observed Path of the Body 242
Formulae for a Second Approximation 243
Formulas for finding the Katio of the Sector to the Triangle 247
Final Correction for Aberration 257
Determination of the Elements of the Orbit 259
Numerical Example 264
Correction of the First Hypothesis 278
Approximate Method of finding the Katio of the Sector to the Triangle 279
CHAPTER V.
DETERMINATION OF THE ORBIT OP A HEAVENLY BODY FROM FOUR OBSERVA-
TIONS, OF WHICH THE SECOND AND THIRD MUST BE COMPLETE.
Fundamental Equations 282
Determination of the Curtate Distances , 289
Successive Approximations 293
Determination of the Elements of the Orbit 294
Numerical Example 294
Method for the Final Approximation 307
CHAPTER VI.
INVESTIGATION OF VARIOUS FORMULAE FOR THE CORRECTION OF THE APPROXI-
MATE ELEMENTS OF THE ORBIT OF A HEAVENLY BODY.
Determination of the Elements of a Circular Orbit 311
Variation of Two Geocentric Distances 313
Differential Formulae 318
Plane of the Orbit taken as the Fundamental Plane 320
Variation of the Node and Inclination 324
Variation of One Geocentric Distance 328
Determination of the Elements of the Orbit by means of the Co-ordinates and
Velocities 332
Correction of the Ephemeris 335
Final Correction of the Elements 338
Kelation between Two Places in the Orbit 339
Modification when the Semi-Transverse Axis is very large 341
Modification for Hyperbolic Motion 346
Variation of the Semi-Transverse Axis and Eatio of Two Curtate Distances 349
12 CONTENTS.
PAQB
Variation of the Geocentric Distance and of the Reciprocal of the Semi-Trans-
verse Axis 352
Equations of Condition 353
Orbit of a Comet 355
Variation of Two Eadii-Vectores 357
CHAPTER VII.
METHOD OF LEAST SQUARES, THEORY OP THE COMBINATION OF OBSERVATIONS,
AND DETERMINATION OF THE MOST PROBABLE SYSTEM OF ELEMENTS FROM
A SERIES OF OBSERVATIONS.
Statement of the Problem 360
Fundamental Equations for the Probability of Errors 362
Determination of the Form of the Function which expresses the Probability ... 363
The Measure of Precision, and the Probable Error 366
Distribution of the Errors 367
The Mean Error, and the Mean of the Errors 368
The Probable Error of the Arithmetical Mean 370
Determination of the Mean and Probable Errors of Observations 371
Weights of Observed Values 372
Equations of Condition 376
Normal Equations 378
Method of Elimination 380
Determination of the Weights of the Resulting Values of the Unknown Quanti-
ties 386
Separate Determination of the Unknown Quantities and of their Weights 392
Eelation between the Weights and the Determinants 396
Case in which the Problem is nearly Indeterminate 398
Mean and Probable Errors of the Results 399
Combination of Observations 401
Errors peculiar to certain Observations 408
Rejection of Doubtful Observations 410
Correction of the Elements 412
Arrangement of the Numerical Operations 415
Numerical Example 418
Case of very Eccentric Orbits 423
CHAPTER VIII.
INVESTIGATION OF VARIOUS FORMULA FOR THE DETERMINATION OF THE SPECIAL
PERTURBATIONS OF A HEAVENLY BODY.
Fundamental Equations 426
Statement of the Problem '. 428
Variation of Co-ordinates 429
CONTENTS. 13
PAQB
Mechanical Quadrature 433
The Interval for Quadrature 443
Mode of effecting the Integration 445
Perturbations depending on the Squares and Higher Powers of the Masses 446
Numerical Example 448
Change of the Equinox and Ecliptic 455
Determination of New Osculating Elements 459
Variation of Polar Co-ordinates 462
Determination of the Components of the Disturbing Force 467
Determination of the Heliocentric or Geocentric Place 471
Numerical Example 474
Change of the Osculating Elements 477
Variation of the Mean Anomaly, the Radius- Vector, and the Co-ordinate 2...... 480
Fundamental Equations 483
Determination of the Components of the Disturbing Force 489
Case of very Eccentric Orbits 493
Determination of the Place of the Disturbed Body 495
Variation of the Node and Inclination 502
Numerical Example 505
Change of the Osculating Elements 510
Variation of Constants 516
Case of very Eccentric Orbits 523
Variation of the Periodic Time 526
Numerical Example 529
Formulae to be used when the Eccentricity or the Inclination is small 533
Correction of the Assumed Value of the Disturbing Mass 535
Perturbations of Comets 536
Motion about the Common Centre of gravity of the Sun and Planet 537
Keduction of the Elements to the Common Centre of Gravity of the Sun and
Planet 538
Keduction by means of Differential Formulae 540
Near Approach of a Comet to a Planet 546
The Sun may be regarded as the Disturbing Body 548
Determination of the Elements of the Orbit about the Planet 550
Subsequent Motion of the Comet 551
Effect of a Resisting Medium in Space 552
Variation of the Elements on account of the Resisting Medium 554
Method to be applied when no Assumption is made in regard to the Density o*
the Ether.... 5*6
14 CONTENTS.
TABLES.
PAOl
I. Angle of the Vertical and Logarithm of the Earth's Radius ~ 561
H. For converting Intervals of Mean Solar Time into Equivalent Intervals
of Sidereal Time 563
III. For converting Intervals of Sidereal Time into Equivalent Intervals
of Mean Solar Time 564
IV. For converting Hours, Minutes, and Seconds into Decimals of a Day... 565
V. For finding the Number of Days from the Beginning of the Year 565
VI. For finding the True Anomaly or the Time from the Perihelion in a
Parabolic Orbit 566
VII. For finding the True Anomaly in a Parabolic Orbit when v is nearly 180 611
VIII. For finding the Time from the Perihelion in a Parabolic Orbit 612
IX. For finding the True Anomaly or the Time from the Perihelion in Orbits
of Great Eccentricity 614
X. For finding the True Anomaly or the Time from the Perihelion in El-
liptic and Hyperbolic Orbits 618
XI. For the Motion in a Parabolic Orbit 619
XII. For the Limits of the Boots of the Equation sin (z 1 )=m sin 4 y! ... 622
XIII. For finding the Eatio of the Sector to the Triangle 624
XIV. For finding the Eatio of the Sector to the Triangle 629
XV. For Elliptic Orbits of Great Eccentricity 632
XVI. For Hyperbolic Orbits 632
XVII. For Special Perturbations 633
XVIII. Elements of the Orbits of the Comets which have been observed 638
XIX. Elements of the Orbits of the Minor Planets 646
XX. Elements of the Orbits of the Major Planets 648
XXI. Constants, &c 649
EXPLANATION OP THE TABLES 651
APPENDIX. Precession 657
Nutation 658
Aberration 659
Intensity of Light 660
Numerical Calculations .. 662
THEORETICAL ASTRONOMY.
CHAPTER I.
INVESTIGATION OF THE FUNDAMENTAL EQUATIONS OF MOTION, AND OF THE FOR-
MULA FOB DETERMINING, FROM KNOWN ELEMENTS, THE HELIOCENTRIC AND
GEOCENTRIC PLACES OF A HEAVENLY BODY, ADAPTED TO NUMERICAL COMPUTA-
TION FOR CASES OF ANY ECCENTRICITY WHATEVER.
1. THE study of the motions of the heavenly bodies does not re-
quire that we should know the ultimate limit of divisibility of the
matter of which they are composed, whether it may be subdivided
indefinitely, or whether the limit is an indivisible, impenetrable atom.
Nor are we concerned with the relations which exist between the
separate atoms or molecules, except so far as they form, in the aggre-
gate, a definite body whose relation to other bodies of the system it
is required to investigate. On the contrary, in considering the ope-
ration of the laws in obedience to which matter is aggregated into
single bodies and systems of bodies, it is sufficient to conceive simply
of its divisibility to a limit which may be regarded as infinitesimal
compared with the finite volume of the body, and to regard the mag-
nitude of the element of matter thus arrived at as a mathematical
point.
An element of matter, or a material body, cannot give itself
motion; neither can it alter, in any manner whatever, any motion
which may have been communicated to it. This tendency of matter
to resist all changes of its existing state of rest or motion is known
as inertia, and is the fundamental law of the motion of bodies. Ex-
perience invariably confirms it as a law of nature; the continuance of
motion as resistances are removed, as well as the sensibly unchanged
motion of the heavenly bodies during many centuries, affording the
16
16 THEORETICAL ASTRONOMY.
most convincing proof of its universality. Whenever, therefore, a
material point experiences any change of its state as respects rest or
motion, the cause must be attributed to the operation of something
external to the element itself, and which we designate by the word
force. The nature of forces is generally unknown, and we estimate
them by the effects which they produce. They are thus rendered com-
parable with some unit, and may be expressed by abstract numbers.
2. If a material point, free to move, receives an impulse by virtue
of the action of any force, or if, at any instant, the force by which
motion is communicated shall cease to act, the subsequent motion of
the point, according to the law of inertia, must be rectilinear and
uniform, equal spaces being described in equal times. Thus, if s, v,
and t represent, respectively, the space, the velocity, and the time, the
measure of v being the space described in a unit of time, we shall
have, in this case,
s = vt.
It is evident, however, that the space described in a unit of time will
vary with the intensity of the force to which the motion is due, and,
the nature of the force being unknown, we must necessarily compare
the velocities communicated to the point by different forces, in order
to arrive at the relation of their effects. We are thus led to regard
the force as proportional to the velocity; and this also has received
the most indubitable proof as being a law of nature. Hence, the
principles of the composition and resolution of forces may be applied
also to the composition and resolution of velocities.
If the force acts incessantly, the velocity will be accelerated, and
the force which produces this motion is called an accelerating force.
In regard to the mode of operation of the force, however, we may
consider it as acting absolutely without cessation, or we may regard
it as acting instantaneously at successive infinitesimal intervals repre-
sented by dt, and hence the motion as uniform during each of these
intervals. The latter supposition is that which is best adapted to
the requirements of the infinitesimal calculus; and, according to the
fundamental principles of this calculus, the finite result will be the
same as in the case of a force whose action is absolutely incessant.
Therefore, if we represent the element of space by ds, and the ele-
ment of time by dt, the instantaneous velocity will be
ds
- V = ~dt>
which will vary from one instant to another.
FUNDAMENTAL PRINCIPLES. 17
3. Since the force is proportional to the velocity, its measure at
any instant will be determined by the corresponding velocity. If
the accelerating force is constant, the motion will be uniformly accele-
rated; and if we designate the acceleration due to the force by/, the
unit of/ being the velocity generated in a unit of time, we shall have
If, however, the force be variable, we shall have, at any instant,
the relation
efc
/ ~~ dt'
the force being regarded as constant in its action during the element
of time dt. The instantaneous value of v gives, by differentiation,
dv _ d*s
~dt~~W
and hence we derive
so that, in varied motion, the acceleration due to the force is mea-
sured by the second differential of the space divided by the square
of the element of time.
4. By the mass of the body we mean its absolute quantity of mat-
ter. The density is the mass of a unit of volume, and hence the
entire mass is equal to the volume multiplied by the density. If it
is required to compare the forces which act upon different bodies, it
is evident that the masses must be considered. If equal masses
receive impulses by the action of instantaneous forces, the forces
acting on each will be to each other as the velocities imparted; and
if we consider as the unit of force that which gives to a unit of mass
the unit of velocity, we have for the measure of a force F, denoting
the mass by Jf,
F = Mv.
This is called the quantity of motion of the body, and expresses its
capacity to overcome inertia. By virtue of the inert state of matter,
there can be no action of a force without an equal and contrary re-
action ; for, if the body to which the force is applied is fixed, the
equilibrium between the resistance and the force necessarily implies
the development of an equal and contrary force ; and, if the body be
free to move, in the change of state, its inertia will oppose equal and
18 THEORETICAL ASTRONOMY.
contrary resistance. Hence, as a necessary consequence of inertia, it
follows that action and reaction are simultaneous, equal, and contrary.
If the body is acted upon by a force such that the motion is varied,
the accelerating force upon each element of its mass is represented by
"7- , and the entire motive force F is expressed by
M being the sum of all the elements, or the mass of the body. Since
_ds_
this gives
which ^is the expression for the intensity of the motive force, or of
the force of inertia developed. For the unit of mass, the measure
of the force is
d?s
and this, therefore, expresses that part of the intensity of the motive
force which is impressed upon the unit of mass, and is what is usually
called the accelerating force.
5. The force in obedience to which the heavenly bodies perform
their journey through space, is known as the attraction of gravitation;
and the law of the operation of this force, in itself simple and unique,
has been confirmed and generalized by the accumulated researches of
modern science. Not only do we find that it controls the motions of
the bodies of our own solar system, but that the revolutions of binary
systems of stars in the remotest regions of space proclaim the uni-
versality of its operation. It unfailingly explains all the phenomena
observed, and, outstripping observation, it has furnished the means
of predicting many phenomena subsequently observed. The law of
this force is that every particle of matter is attracted by every other
particle by a force which varies directly as the mass and inversely as
the square of the distance of the attracting particle.
This reciprocal action is instantaneous, and is not modified, in any
degree, by the interposition of other particles or bodies of matter. It
is also absolutely independent of the nature of the molecules them-
selves, and of their aggregation.
ATTRACTION OF SPHERES. 19
If we consider two bodies the masses of which are m and m', and
whose magnitudes are so small, relatively to their mutual distance />,
that we may regard them as material points, according to the law of
gravitation, the action of m on each molecule or unit of m' will be
, and the total force on m! will be
m'.
/>'
The action of m f on each molecule of m will be expressed by , and
its total action by
m'
-r-
The absolute or moving force with which the masses m and m f tend
toward each other is, therefore, the same on each body, which result
is a necessary consequence of the equality of action and reaction.
The velocities, however, with which these bodies would approach
each other must be different, the velocity of the smaller mass exceed-
ing that of the greater, and in the ratio of the masses moved. The
expression for the velocity of m', which would be generated in a unit
of time if the force remained constant, is obtained by dividing the
absolute force exerted by m by the mass moved, which gives
. 7
and this is, therefore, the measure of the acceleration due to the
action of m at the distance p. For the acceleration due to the
action of m r we derive, in a similar manner,
6. Observation shows that the heavenly bodies are nearly spherical
in form, and we shall therefore, preparatory to finding the equations
which express the relative motions of the bodies of the system, de-
termine the attraction of a spherical mass of uniform density, or
varying from the centre to the surface according to any law, for a
point exterior to it.
If we suppose a straight line to be drawn through the centre of the
sphere and the point attracted, the total action of the sphere on the
point will be a force acting along this line, since the mass of the
sphere is symmetrical with respect to it. Let dm denote an element
20 THEORETICAL ASTRONOMY.
of the mass of the sphere, and p its distance from the point attracted ;
then will
dm
express the action of this element on the point attracted. If we sup-
pose the density of the sphere to be constant, and equal to unity, the
element dm becomes an element of volume, and will be expressed by
dm = dx dy dz ;
x y y. and z being the co-ordinates of the element referred to a system
of rectangular co-ordinates. If we take the origin of co-ordinates
at the centre of the sphere, and introduce polar co-ordinates, so that
x = r cos *, with respect to a, gives
dp a r sin dr d
P
Thia must be integrated between the limits tp = -f- fa an( ^ ^ =
22 THEORETICAL ASTRONOMY.
but since p is a function of , we have
7 P 7
r cos *.
a J
Integrating, finally, between the limits r and r = r,, we get
r, being the radius of the sphere, and, if we denote its entire mass by
m, this becomes
F=-.
a'
Therefore,
dV m
da a?
from which it appears that the action of a homogeneous spherical
mass on a point exterior to it, is the same as if the entire mass were
concentrated at its centre. If, in the integration with respect to r,
we take the limits r' and r", we obtain
and, denoting by m the mass of a spherical shell whose radii are r"
and r f , this becomes
Consequently, the attraction of a homogeneous spherical shell on a
point exterior to it, is the same as if the entire mass were concentrated
at its centre.
The supposition that the point attracted is situated within a
spherical shell of uniform density, does not change the form of the
FUNDAMENTAL PRINCIPLES.
general equation; but, in the integration with reference to />, the
limits will be p = r + a, and p = r a, which give
V=
and this being independent of a, we have
A- - d -?=0
^i- - 7 - \Jm
da
Whence it follows that a point placed in the interior of a spherical
shell is equally attracted in all directions, and that, if not subject to
the action of any extraneous force, it will be in equilibrium in every
position.
7. Whatever may be the law of the change of the density of the
heavenly bodies from the surface to the centre, we may regard them
as composed of homogeneous, concentric layers, the density varying
only from one layer to another, and the number of the layers may
be indefinite. The action of each of these will be the same as if its
mass were united at the centre of the shell ; and hence the total action
of the body will be the same as if the entire mass were concentrated
at its centre of gravity. The planets are indeed not exactly spheres,
but oblate spheroids differing but little from spheres ; and the error
of the assumption of an exact spherical form, so far as it relates to
their action upon each other, is extremely small, and is in fact com-
pensated by the magnitude of their distances from each other ; for,
whatever may be the form of the body, if its dimensions are small
in comparison with its distance from the body which it attracts, it is
evident that its action will be sensibly the same as if its entire mass
were concentrated at its centre of gravity. If we suppose a system
of bodies to be composed of spherical masses, each unattended with
any satellite, and if we suppose that the dimensions of the bodies
are small in comparison with their mutual distances, the formation
of the equations for the motion of the bodies of the system will be
reduced to the consideration of the motions of simple points endowed
with forces of attraction corresponding to the respective masses. Our
solar system is, in reality, a compound system, the several systems
of primary and satellites corresponding nearly to the case supposed ;
and, before proceeding with the formation of the equations which are
applicable to the general case, we will consider, at first, those for a
simple system of bodies, considered as points and subject to their
mutual actions and the action of the forces which correspond to the
24 THEORETICAL ASTRONOMY.
actual velocities of the different parts of the system for any instant.
It is evident that we cannot consider the motion of any single body
as free, and subject only to the action of the primitive impulsion
which it has received and the accelerating forces which act upon it ;
but, on the contrary, the motion of each body will depend on the
force which acts upon it directly, and also on the reaction due to the
other bodies of the system. The coLsideration, however, of the varia-
tions of the motion of the several bodies of the system is reduced to
the simple case of equilibrium by means of the general principle that,
if we assign to the different bodies of the system motions which are
modified by their mutual action, we may regard these motions as
composed of those which the bodies actually have and of other
motions which are destroyed, and which must therefore necessarily
be such that, if they alone existed, the system would be in equi-
librium. We are thus enabled to form at once the equations for the
motion of a system of bodies. Let m, m', m", &c. be the masses of
the several bodies of the system, and x, y, z, x' t y', z f , &c. their co-
ordinates referred to any system of rectangular axes. Further, let
the components of the total force acting upon a unit of the mass of
m, or of the accelerating force, resolved in directions parallel to the
co-ordinate axes, be denoted by X, F, and Z, respectively, then will
mX, m Y, mZ,
be the forces which act upon the body in the same directions. The
velocities of the body m at any instant, in directions parallel to the
co-ordinate axes, will be
\
dx dy dz
~di' W dt'
and the corresponding forces are
dx dy dz
m ~j7> m ~jl* m ~jT'
dt dt dt
By virtue of the action of the accelerating force, these forces for the
next instant become
*j + mXdt, .m+mYdt, m j + mZdt >
which may be written respectively:
MOTION OF A SYSTEM OF BODIES. 25
dz , dz .. dz
The actual velocities for this instant are
dx , dx dy dy dz dz
and the corresponding forces are
dx , dx dy dy dz . . dz
- -
Comparing these with the preceding expressions for the forces, it
appears that the forces which are destroyed, in directions parallel to
the co-ordinate axes, are
dx
md -j- -|- mXdt,
-md^ + mYdt, (3)
md-j--{-mZdt.
In the same manner we find for the forces which will be destroyed
in the case of the body m' :
-m'd + m'X'dt,
at
and similarly for the other bodies of the system. According to the
general principle above enunciated, the system under the action of
these forces alone, will be in equilibrium. The conditions of equi-
librium for a system of points of invariable but arbitrary form, and
subject to the action of forces directed in any manner whatever, are
ZX, = 0, ZY, = 0, ZZ, = 0,
S ( T> - X,y) = 0, Z (X t z - Z,x) = 0, Z (Z,y - I = ,
which X,, Y h Z,, denote the components, resolved parallel to the
n
26 THEORETICAL ASTRONOMY.
co-ordinate axes, of the forces acting on any point, and r, y, z, the
co-ordinates of the point. These equations are equally applicable to
the case of the equilibrium at any instant of a system of variable
form ; and substituting in them the expressions (3) for the forces de-
stroyed in the case of a system of bodies, we shall have
2m r-5 2mX = 0,
aP
(4)
which are the general equations for the motions of a system of bodies.
8. Let x,, y,, z,, be the co-ordinates of the centre of gravity of the
system, and, by differentiation of the equations for the co-ordinates
of the centre of gravity, which are
2mx 2my 2mz
x, = ^ ) y. = ^ t z. = ^ >
2m 2m 2m
we get
d* 2m dP 2m
Introducing these values into the first three of equations (4), they
become
d*x f 2mX d*y, 2m Y d*z f 2mZ t .
"dP = '' ~^ y ~dP ~ ' ~2m' ~dP ~ ~2m '
from which it appears that the centre of gravity of the system moves
in space as if the masses of the different bodies of which it is com-
posed, were united in that point, and the forces directly applied to it.
If we suppose that the only accelerating forces which act on the
bodies of the system, are those which result from their mutual action,
we have the obvious relation :
= m'X',
MOTION OF A SYSTEM OF BODIES. 27
and similarly for ^any two bodies ; and, consequently,
so that equations (5) become
Integrating these once, and denoting the constants of integration by
c, c', c", we find, by combining the results,
and hence the absolute motion of the centre of gravity of the system,
when subject only to the mutual action of the bodies which compose
it, must be uniform and rectilinear. Whatever, therefore, may be
the relative motions of the different bodies of the system, the motion
of its centre of gravity is not thereby affected.
9. Let us now consider the last three of equations (4), and suppose
the system to be submitted only to the mutual action of the bodies
which compose it, and to a force directed toward the origin of co-
ordinates. The action of m' on ra, according to the law of gravita-
tion, is expressed by , in which p denotes the distance of m from m'.
To resolve this force in directions parallel to the three rectangular
axes, we must multiply it by the cosine of the angle which the line
joining the two bodies makes with the co-ordinate axes respectively,
which gives
m'(z'-z)
~~
Further, for the components of the accelerating force of m on m', we
have
m(x x') , m(y y f ) , m(g /)
""?"' ~7 ~7~
Hence we derive
m ( Yx Xy} + m' ( TV Xy ) = 0,
and generally
28 THEORETICAL ASTRONOMY.
In a similar manner, we find
Zm (Xi ZOD) = 0, (7)
Im (Zq F) = 0.
These relations will not be altered if, in addition to their reciprocal
action, the bodies of the system are acted upon by forces directed to
the origin of co-ordinates. Thus, in the case of a force acting upon
m, and directed to the origin of co-ordinates, we have, for its action
alone,
Yx = Xy, Xz = Zx, %y=Yz,
and similarly for the other bodies. Hence these forces disappear
from the equations, and, therefore, when the several bodies of the
system are subject only to their reciprocal action and to forces directed
to the origin of co-ordinates, the last three of equations (4) become
d*z
^
d*y
the integration of which gives
2m {xdy ydx) = cat,
Im (zdx xdz) = c'dt, (8;
2m (ydz zdy] = c"dt,
c, c', and G" being the constants of integration. Now, xdy ydx is
double the area described about the origin of co-ordinates by the pro-
jection of the radius- vector, or line joining m with the origin of co-ordi-
nates, on the plane of xy during the element of time dt ; and, further,
zdx xdz and ydz zdy are respectively double the areas described,
during the same time, by the projection of the radius-vector on the
planes of xz and yz. The constant c, therefore, expresses double the
sum of the products formed by multiplying the areal velocity of each
body, in the direction of the co-ordinate plane xy, by its mass; and
c r , G fr , express the same sum with reference to the co-ordinate planes
xz and yz respectively. Hence the sum of the areal velocities of the
several bodies of the system about the origin of co-ordinates, each
multiplied by the corresponding mass, is constant; and the sum of
the areas traced, each multiplied by the corresponding mass, is pro-
portional to the time. If the only forces which operate, are those
INVARIABLE PLANE. 29
resulting from the mutual action of the bodies which compose the
system, this result is correct whatever may be the point in space
taken as the origin of co-ordinates.
The areas described by the projection of the radius-vector of each
body on the co-ordinate planes, are the projections, on these planes, of
the areas actually described in space. We may, therefore, conceive of
a resultant, or principal plane of projection, such that the sum of the
areas traced by the projection of each radius-vector on this plane,
when projected on the three co-ordinate planes, each being multiplied
by the corresponding mass, will be respectively equal to the first
members of the equations (8). Let , /9, and 7- be the angles which
this principal plane makes with the co-ordinate planes xy, xz, and yz,
respectively; and let S denote the sum of the areas traced on this
plane, in a unit of time, by the projection of the radius-vector of
each of the bodies of the system, each area being multiplied by the
corresponding mass. The sum S will be found to be a maximum,
and its projections on the co-ordinate planes, corresponding to the
element of time dt, are
S cos a dt, S cos p dt, S cos Y dt.
Therefore, by means of equations (8), we have
c = S cos a, c f S cos /?, c" = S cos p,
and, since cos 2 a -f- cos 2 /? + cos 2 f = 1,
2 = c 2 -f c' 2 -f c" 2 .
Hence we derive
c'
COS a =
1/V + c' 2 + c" 2 1/c 2 + c' 2 -j- c" 2
c"
cos y = -
These angles, being therefore constant and independent of the time,
show that this principal plane of projection remains constantly par-
allel to itself during the motion of the system in space, whatever
may be the relative positions of the several bodies; and for this
reason it is called the invariable plane of the system. Its position
with reference to any known plane is easily determined when the
velocities, in directions parallel to the co-ordinate axes, and the
masses and co-ordinates of the several bodies of the system, are
known. The values of c, c', c" are given by equations (8), and
THEORETICAL ASTRONOMY.
hence the values of , /?, and ?-, which determine the position of the
invariable plane.
Since the positions of the co-ordinate planes are arbitrary, we may
suppose that of xy to coincide with the invariable plane, which gives
cos /9 = and cos p = 0, and, therefore, c' and c" = 0. Further,
since the positions of the axes of x and y in this plane are arbitrary,
it follows that for every plane perpendicular to the invariable plane,
the sum of the areas traced by the projections of the radii-vectores
of the several bodies of the system, each multiplied by the corre-
sponding mass, is zero. It may also be observed that the value of 8
is constant whatever may be the position of the co-ordinate planes,
and that its value is necessarily greater than that of either of the
quantities in the second member of the equation,
2 = c 2 -f c' 2 -f c" 2 ,
except when two of them are each equal to zero. It is, therefore, a
maximum, and the invariable plane is also the plane of maximum
areas.
10. If we suppose the origin of co-ordinates itself to move with
uniform and rectilinear motion in space, the relations expressed by
equations (8) will remain unchanged. Thus, let x,, y n z, be the co-
ordinates of the movable origin of co-ordinates, referred to a fixed
point in space taken as the origin ; and let X Q , y , z , x f , yj, z/, &c.
be the co-ordinates of the several bodies referred to the movable
origin. Then, since the co-ordinate planes in one system remain
always parallel to those of the other system of co-ordinates, we shall
have
and similarly for the other bodies of the system. Introducing these
values of x, y, and z into the first three of equations (4), they become
The condition of uniform rectilinear motion of the movable origin
gives
MOTION OF A SOLID BODY. 31
and the preceding equations become
0, (9)
ZmZ=Q.
Substituting the same values in the last three of equations (4), ob-
serving that the co-ordinates x t , y,, z, are the same for all the bodies
of the system, and reducing the resulting equations by means of
equations (9), we get
(/7 2 r f]i~ \
*. ^jr ~ *~jjr } - a* (A - 2O = 0, (10)
- r =0.
Hence it appears that the form of the equations for the motion of the
system of bodies, remains unchanged when we suppose the origin of
co-ordinates to move in space with a uniform and rectilinear motion.
11. The equations already derived for the motions of a system of
bodies, considered as reduced to material points, enable us to form at
once those for the motion of a solid body. The mutual distances of
the parts of the system are, in this case, invariable, and the masses
of the several bodies become the elements of the mass of the solid
body. If we denote an element of the mass by c?m, the equations (5)
for the motion of the centre of gravity of the body become
-=zdm, (11)
the summation, or integration with reference to dm, being taken so as
to include the entire mass of the body, from which it appears that
the centre of gravity of the body moves in space as if the entire mass
were concentrated in that point, and the forces applied to it directly.
If we take the origin of co-ordinates at the centre of gravity of
the body, and suppose it to have a rectilinear, uniform motion in
space, and denote the co-ordinates of the element dm, in reference to
this origin, by x , y Q) z w we have, by means of the equations (10),
THEORETICAL ASTRONOMY.
&X R
d*y d\ \ . /-,__ _ N , . v
^ ~dl y ~di? } ~~J ' ~~ y *) ~ '
d' 2 x n d*z \ r
z -^-~ x o-^-JdmJ(Xz Zx )dm = 0,
d\\, Crr
-%-$- )dm-J(Zy ~Yz )dm =0,
dt 2
the integration with respect to dm being taken so as to include the
entire mass of the body. These equations, therefore, determine the
motion of rotation of the body around its centre of gravity regarded
as fixed, or as having a uniform rectilinear motion in space. Equa-
tions (11) determine the position of the centre of gravity for any
instant, and hence for the successive instants at intervals equal to dt;
and we may consider the motion of the body during the element of
time dt as rectilinear and uniform, whatever may be the form of its
trajectory. Hence, equations (11) and (12) completely determine the
position of the body in space, the former relating to the motion of
translation of the centre of gravity, and the latter to the motion of
rotation about this point. It follows, therefore, that for any forces
which act upon a body we can always decompose the actual motion
into those of the translation of the centre of gravity in space, and of
the motion of rotation around this point; and these two motions may
be considered independently of each other, the motion of the centre
of gravity being independent of the form and position of the body
about this point.
If the only forces which act upon the body are the reciprocal action
0f the elements of its mass and forces directed to the origin of co-
ordinates, the second terms of equations (12) become each equal to
zero, and the results indicated by equations (8) apply in this case
also. The parts of the system being invariably connected, the plane
of maximum areas, or invariable plane, is evidently that which is
perpendicular to the axis of rotation passing through the centre of
gravity, and therefore, in the motion of translation of the centre of
gravity in space, the axis of rotation remains constantly parallel to
itself. Any extraneous force which tends to disturb this relation
will necessarily develop a contrary reaction, and hence a rotating body
resists any change of its plane of rotation not parallel to itself. Wo
may observe, also, that on account of the invariability of the mutual
distances of the elements of the mass, according to equations (8), the
motion of rotation must be uniform.
12. We shall now consider the action of a svstem of bodies on a
MOTION OF A SOLID BODY. 33
distant mass, which we will denote by M. Let x 0) y w z w x f , 2/ ', z Q ',
&c. be the co-ordinates of the several bodies of the system referred
to its centre of gravity as the origin of co-ordinates; x lt y,, and z,
the co-ordinates of the centre of gravity of the system referred to
the centre of gravity of the body M. The co-ordinates of the body
m, of the system, referred to this origin, will therefore be
and similarly for the other bodies of the system. If we denote by
r the distance of the centre of gravity of m from that of M y the
accelerating force of the former on an element of mass at the centre
of gravity of the latter, resolved parallel to the axis of x, will be
mx
and, therefore, that of the entire system on the element of M t resolved
; n the same direction, will be
v mx
r 5 "'
We have also
r* = (x, + x Q Y + (y, + 2/ ) 2 + 0, + *)',
and, if we denote by r t the distance of the centre of gravity of the
system from Jf,
r, = *, + ?, + *,*,
Therefore
| = O, + *o) (r? + 2 (x, x + y, y + z, $ + r 2 )
We shall now suppose the mutual distances of the bodies of the
system to be so small in comparison with the distance r, of its centre
of gravity from that of Jf, that terms of the order r 2 may be neglected ;
a condition which is actually satisfied in the case of the secondary
systems belonging to the solar system. Hence, developing the second
factor of the second member of the last equation, and neglecting terms
of the order r 2 , we shall have
_ _/ , ^o _ 3s, (Xa?o + y/
r 3 3 ' , 3 r 5
r, r f r t
and
-mx -m . 2mx n 3rc. ,
r == x , - + 3 -i (x,Smx Q + y,Zmy Q
T r, r f i,
I
34 THEORETICAL ASTRONOMY.
But, since # , y , z , are the co-ordinates in reference to the centre of
gravity of the system as the origin, we have
2mx = 0, Zmy Q = 0, 2wz == 0,
and the preceding equation reduces to
v w# 2m
s -?=*'^f-
In a similar manner, we find
v my Im mz Im
V =J V r 7 = V
The second members of these equations are the expressions for the
total accelerating force due to the action of the bodies of the system
on Mj resolved parallel to the co-ordinate axes respectively, when we
consider the several masses to be collected at the centre of gravity
of the system. Hence we conclude that when an element of mass
is attracted by a system of bodies so remote from it that terms of the
order of the squares of the co-ordinates of the several bodies, referred
to the centre of gravity of the system as the origin of co-ordinates,
may be neglected in comparison with the distance of the system from
the point attracted, the action of the system will be the same as if
the masses were all united at its centre of gravity.
If we suppose the masses m, m', m", &c. to be the elements of the
mass of a single body, the form of the equations remains unchanged;
and hence it follows that the mass M is acted upon by another mass,
or by a system of bodies, as if the entire mass of the body, or of the
system, were collected at its centre of gravity. It is evident, also,
that reciprocally in the case of two systems of bodies, in which the
mutual distances of the bodies are small in comparison with the
distance between the centres of gravity of the two systems, their
mutual action is the same as if all the several masses in each system
were collected at the common centre of gravity of that system ; and
the two centres of gravity will move as if the masses were thus
united.
13. The results already obtained are sufficient to enable us to form
the equations for the motions of the several bodies which compose the
solar system. If these bodies were exact spheres, which could be
considered as composed of homogeneous concentric spherical shells,
the density varying only from one layer to another, the action of
MOTION OF A SYSTEM OF BODIES. 35
each on an element of the mass of another would be the same as if
the entire mass of the attracting body were concentrated at its centre
of gravity. The slight deviation from this law, arising from the
ellipsoidal form of the heavenly bodies, is compensated by the mag-
nitude of their mutual distances; and, besides, these mutual distances
are so great that the action of the attracting body on the entire mass
of the body attracted, is the same as if the latter were concentrated
at its centre of gravity. Hence the consideration of the reciprocal
action of the single bodies of the system, is reduced to that of material
points corresponding to their respective centres of gravity, the masses
of which, however, are equivalent to those of the corresponding
bodies. The mutual distances of the bodies composing the secondary
systems of planets attended with satellites are so small, in comparison
with the distances of the diiferent systems from each other and from
the other planets, that they act upon these, and are reciprocally acted
upon, in nearly the same manner as if the masses of the secondary
systems were united at their common centres of gravity, respectively.
The motion of the centre of gravity of a system consisting of a
planet and its satellites is not affected by the reciprocal action of the
bodies of that system, and hence it may be considered independently
of this action. The difference of the action of the other planets on
a planet and its satellites will simply produce inequalities in the
relative motions of the latter bodies as determined by their mutual
action alone, and will not affect the motion of their common centre
of gravity. Hence, in the formation of the equations for the motion
of translation of the centres of gravity of the several planets or
secondary systems which compose the solar system, we have simply
to consider them as points endowed with attractive forces correspond-
ing to the several single or aggregated masses. The investigation
of the motion of the satellites of each of the planets thus attended,
forms a problem entirely distinct from that of the motion of the
common centre of gravity of such a system. The COL ^deration of
the motion of rotation of the several bodies of the solar system about
their respective centres of gravity, is also independent of the motion
of translation. If the resultant of all the forces which act upon a
planet passed through the centre of gravity, the motion of rotation
would be undisturbed; and, since this resultant in all cases very
nearly satisfies this condition, the disturbance of the motion of rota-
tion is very slight. The inequalities thus produced in the motion
of rotation are, in fact, sensible, and capable of being indicated by
observation, only in the case of the earth and moon. It has, indeed,
36 THEORETICAL ASTRONOMY.
been rigidly demonstrated that the axis of rotation of the earth rela-
tive to the body itself is fixed, so that the poles of rotation and the
terrestrial equator preserve constantly the same position in reference
to the surface; and that also the velocity of rotation is constant.
This assures us of the permanency of geographical positions, and,
in connection with the fact that the change of the length of the
mean solar day arising from the variation of the obliquity of the
ecliptic and in the length of the tropical year, due to the action of
the sun, moon, and planets upon the earth, is absolutely insensible,
amounting to only a small fraction of a second in a million of
years, assures us also of the permanence of the interval which we
adopt as the unit of time in astronomical investigations.
14. Placed, as we are, on one of the bodies of the system, it is
only possible to deduce from observation the relative motions of the
different heavenly bodies. These relative motions in the case of the
comets and primary planets are referred to the centre of the sun,
since the centre of gravity of this body is near the centre of gravity
of the system, and its preponderant mass facilitates the integration
of the equations thus obtained. In the case, however, of the secondary
systems, the motions of the satellites are considered in reference to
the centre of gravity of their primaries. We shall, therefore, form
the equations for the motion of the planets relative to the centre of
gravity of the sun ; for which it becomes necessary to consider more
particularly the relation between the heterogeneous quantities, space,
time, and mass, which are involved in them. Each denomination,
being divided by the unit of its kind, is expressed by an abstract
number; and hence it offers no .difficulty by its presence in an equa-
tion. For the unit of space we may arbitrarily take the mean dis-
tance of the earth from the sun. and the mean solar day may be
taken as the unit of time. But, in order that when the space is
expressed by 1, and the time by 1, the force or velocity may also be
expressed by 1, if the unit of space is first adopted, the relation of
the time and the mass which determines the measure of the force -
will be such that the units of both cannot be arbitrarily chosen.
Thus, if we denote by / the acceleration due to the action of the
mass m on a material point at the distance a, and by/' the accelera-
tion corresponding to another mass m' acting at the same distance,
we have the relation
MOTION EELATIVE TO THE SUN. 37
and hence, since the acceleration is proportional to the mass, it may
be taken as the measure of the latter. But we have, for the measure
of/,
Integrating this, regarding /as constant, and the point to move from
a state of rest, we get
= i/P. (13)
The acceleration in the case of a variable force is, at any instant,
measured by the velocity which the force acting at that instant would
generate, if supposed to remain constant in its action, during a unit
of time. The last equation gives, when t = 1,
/=2;
and hence the acceleration is also measured by double the space which
would be described by a material point, from a state of rest, during
a unit of time, the force being supposed constant in its action during
this time. In each case the duration of the unit of time is involved
in the measure of the acceleration, and hence in that of the mass on
which the acceleration depends; and the unit of mass, or of the force,
will depend on the duration which is chosen for the unit of time. In
general, therefore, we regard as the unit of mass that which, acting
constantly at a distance equal to unity on a material point free to
move, will give to this point, in a unit of time, a velocity which,
if the force ceased to act, would cause it to describe the unit of dis-
tance in the unit of time.
Let the unit of time be a mean solar day; F the acceleration due
to the force exerted by the mass of the sun at the unit of distance;
and /the acceleration corresponding to the distance r; then will
and k 2 becomes the measure of the mass of the sun. The unit of
mass is, therefore, equal to the mass of the sun taken as many times*
as k 2 is contained in unity. Hence, when we take the mean solar
day as the unit of time, the mass of the sun is measured by #*; by
which we are to understand that if the sun acted during a mean solar
day, on a material point free to move, at a distance constantly equal
to the mean distance of the earth from the sun, it would, at the end
of that time, have communicated to the point a velocity which, if
38 THEORETICAL ASTRONOMY.
the force did not thereafter act, would cause it to describe, in a unu
of time, the space expressed by ft?.
The acceleration due to the action of the sun at the unit of distance
is designated by F, since the square root of this quantity appears
frequently in the formulae which will be derived.
If we take arbitrarily the mass of the sun as the unit of mass, the
unit of time must be determined. Let t denote the number of mean
solar days which must be taken for the unit of time when the unit
of mass is the mass of the sun. The space which the force due to
this mass, acting constantly on a material point at a distance equal to
the mean distance of the earth from the sun, would cause the point
to describe in the time t, is, according to equation (13),
But, since t expresses the number of mean solar days in the unit of
time, the measure of the acceleration corresponding to this unit is 2s,
and this being the unit of force, we have
*V=1;
and hence
Therefore, if the mass of the sun is regarded as the unit of mass, the
number of mean solar days in the unit of time will be equal to unity
divided by the square root of the acceleration due to the force exerted
by this mass at the unit of distance. The numerical value of k will
be subsequently found to be 0.0172021, which gives 58.13244 mean
solar days for the unit of time, when the mass of the sun is taken as
the unit of mass.
15. Let x, y, z be the co-ordinates of a heavenly body referred to
the centre of gravity of the sun as the origin of co-ordinates; r its
radius-vector, or distance from this origin; and let m denote the
quotient obtained by dividing its mass by that of the sun; then,
taking the mean solar day as the unit of time, the mass of the sun is
expressed by F, and that of the planet or comet by W& 2 . For a
second body let the co-ordinates be x f , y f , z f ; the distance from the
sun, r'; and the mass, ra'F; and similarly for the other bodies of the
system. Let the co-ordinates of the centre of gravity of the sun
referred to any fixed point in space be , 77, , the co-ordinate planes
being parallel to those of x, y, and z, respectively; then will the
MOTION RELATIVE TO THE SUN. 39
mk*
acceleration due to the action of m on the sun be expressed by .
and the three components of this force in directions parallel to the
co-ordinate axes, respectively, will be
, ,
r 3 r 3 r 3
The action of ra' on the sun will be expressed by
and hence the acceleration due to the combined and simultaneous
action of the several bodies of the system on the sun. resolved par-
allel to the co-ordinate axes, will be
The motion of the centre of gravity of the sun, relative to the fixed
origin, will, therefore, be determined by the equations
=
= , = (14)
Let p denote the distance of m from m f ; p r its distance from w",
adding an accent for each successive body considered; then will the
action of the bodies m', m", &c. on m be
of which the three components parallel to the co-ordinate axes, re-
spectively, are
The action of the sun on m, resolved in the same manner, is expressed
by
which are negative, since the force tends to diminish the co-ordinates
x, y, and z. The three components of the total action of the othei
bodies of the system on m are, therefore,
40 THEOKETICAL ASTRONOMY.
Vx , wm'tf x)
"T 5 " ~7 '
k *y ; X ^ + y, c + 2,
the equations which determine the absolute motion are
eP . (?a? ^ __ m' (20)
h being an arbitrary constant.
44 THEORETICAL ASTRONOMY.
If we add together the squares of the expressions for c, c', and c"
and put c 2 + c' 2 + c" 2 = 4/ 2 , we shall have
(x* -f f + * 2 ) (cfe* + dy* + <** 2 ) 0*k + ydy -f
dz 2 + cos I
being an arbitrary constant. Therefore we shall have
I 1 1 = cos (v w),
e \ r /
from which we derive
r ^^^
e cos (1;
which is the polar equation of a conic section, the pole being at the
focus, p being the semi -para meter, e the eccentricity, and v o the
angle at the focus between the radius-vector and a fixed line, in the
plane of the orbit, making the angle co with the semi-transverse
axis a.
If the angle v co is counted from the perihelion, we have a) = 0,
and
(25)
P
COS V
The angle v is called the true anomaly.
Hence we conclude that the orbit of a heavenly body revolving
around the sun is a conic section with the sun in one of the foci.
Observation shows that the planets revolve around the sun in ellipses,
usually of small eccentricity, while the comets revolve either in
ellipses of great eccentricity, in parabolas, or in hyperbolas, a cir-
cumstance which, as we shall have occasion to notice hereafter, greatly
46 THEORETICAL ASTRONOMY.
lessens the amount of labor in many computations respecting their
motion.
Introducing into equation (23) the values of h and 4f already
found, we obtain
dt =
-
m j/aV (a r) 2 '
which may be written
dt =
or
^-(V)/
the integration of which gives
J
ae
(26)
In the perihelion, r = a (1 e), and the integral reduces to ' = C/
therefore, if we denote the time from the perihelion by t Q , we shall
have
o / -i/a r\ /~ (a r\*\
7 / ^r == cos - ) e \l --
jfe|/l-f-w\ \ ae j \ \ ae ] ]
(27)
In the aphelion, r = a (1 -f- e) ; and therefore we shall have, for the
time in which the body passes from the perihelion to the aphelion,
t, = Jr, or
1 T-
"
r being the periodic time, or time of one revolution of the planet
around the sun, a the semi-transverse axis of the orbit, or mean dis-
tance from the sun, and n the semi-circumference of a circle whose
radius is unity. Therefore we shall have
(28)
MOTION RELATIVE TO THE SUN. 47
For a second planet, we shall have
and, consequently, between the mean distances and periodic times of
any two planets, we have the relation
fft^ = - (29)
If the masses of the two planets m and m r are very nearly the
same, we may take 1 -f- m = 1 -f- m r and hence, in this case, it follows
that the squares of the periodic times are to each other as the cubes of
the mean distances from the sun. The same result may be stated in
another form, which is sometimes more convenient. Thus, since rcab
is the area of the ellipse, a and b representing the semi-axes, we
shall have
=f= areal velocity;
and, since b 2 = a 2 (1 e 2 ), we have
T
which becomes, by substituting the value of r already found,
In like manner, for a second planet, we have
and, if the masses are such that we may take 1 -f- m sensibly equal
to 1 -f m', it follows that, in this case, the areas described in equal
times, in different orbits, are proportional to the square roots of their
parameters.
17. We shall now consider the signification of some of the con-
stants of integration already introduced. Let i denote the inclination
of the orbit of m to the plane of xy, which is thus taken as the plane
of reference, and let be the angle formed by the axis of x and the
line of intersection of the plane of the orbit with the plane of xy ;
then will the angles i and & determine the position of the plane of
48 THEORETICAL ASTRONOMY.
the orbit in space. The constants c, c', and c", involved in the
equation
cz c'y -\- c"x = 0,
are, respectively, double the projections, on the co-ordinate planes,
xy, xz, and yz, of the areal velocity// and hence we shall have
e = 2/cosi.
The projection of 2f on a plane passing through the intersection ot
the plane of the orbit with the plane of XT/, and perpendicular to the
latter, is
2/sini;
and the projection of this on the plane of xz y to which it is inclined
at an angle equal to &, gives
c' = 2/ sin i cos 1.
Its projection on the plane of yz gives
Hence we derive
z cos i y sin i cos Q> -f- x sin i sin = 0, (31)
which is the equation of the plane of the orbit; and, by means of
the value of f in terms of p, and the values of c, c', c", we derive,
also,
dz dx , 73 ; T . . ,,,
'-j z -j- = Jc i/p (1 + *) cos & sm l > (.&%)
at at
__ = cvpLm sn
Cvv Ctv
These equations will enable us to determine & , i, and p, when, for
any instant, the mass and co-ordinates of m, and the components of
its velocity, in directions parallel to the co-ordinate axes, are known.
The constants a and e are involved in the value of p, and hence four
constants, or elements, are introduced into these equations, two of
which, a and e, relate to the form of the orbit, and two, > and i, to
the position of its plane in space. If we measure the angle v a)
from the point in which the orbit intersects the plane of xy, the con-
stant co will determine the position of the orbit in its own plane.
Finally, the constant of integration C, in equation (26), is the time
MOTION RELATIVE TO THE SUN. 4S
of passage through the perihelion; and this determines the position
of the body in its orbit. When these six constants are known, the
undisturbed orbit of the body is completely determined.
Let V denote the velocity of the body in its orbit; then will
equation (20) become
At the perihelion, r is a minimum, and hence, according to this
equation, the corresponding value of V is a maximum. At the
aphelion, V is a minimum.
In the parabola, a = oo*, and hence
which will determine the velocity at any instant, when r is known.
It will be observed that the velocity, corresponding to the same value
of r, in an elliptic orbit is less than in a parabolic orbit, and that,
since a is negative in the hyperbola, the velocity in a hyperbolic
orbit is still greater than in the case of the parabola. Further, since
the velocity is thus found to be independent of the eccentricity, the
direction of the motion has no influence on the species of conic section
described.
If the position of a heavenly body at any instant, and the direction
and magnitude of its velocity, are given, the relations already derived
will enable us to determine the six constant elements of its orbit.
But since we cannot know in advance the magnitude and direction
of the primitive impulse communicated to the body, it is only by
the aid of observation that these elements can be derived; and
therefore, before considering the formulae necessary to determine
unknown elements by means of observed positions, we will investi-
gate those which are necessary for the determination of the helio-
centric and geocentric places of the body, assuming the elements to
be known. The results thus obtained will facilitate the solution of
the problem of finding the unknown elements from the data furnished
by observation.
18. To determine the value of k, which is a constant for the solar
system, we have, from equation (28),
50 THEORETICAL ASTRONOMY.
In the case of the earth, a = 1, and therefore
*=
fl/1 4-
m
In reducing this formula to numbers we should properly use, for r,
the absolute length of the sidereal year, which is invariable. The
effect of the action of the other bodies of the system on the earth is
to produce a very small secular change in its mean longitude correr
spending to any fixed date taken as the epoch of the elements; and
a correction corresponding to this secular variation should be applied
to the value of r derived from observation. The effect of this cor-
rection is slightly to increase the observed value of r; but to deter-
mine it with precision requires an exact knowledge of the masses of
all the bodies of the system, and a complete theory of their relative
motions, a problem which is yet incompletely solved. Astronomical
usage has, therefore, sanctioned the employment of the value of k
found by means of the length of the sidereal year derived directly
from observation. This is virtually adopting as the unit of space a
distance which is very little less than the absolute, invariable mean
distance of the earth from the sun; but, since this unit may be arbi-
trarily chosen, the accuracy of the results is not thereby affected.
The value of r from which the adopted value of k has been com-
puted, is 365.2563835 mean solar days; and the value of the com-
bined mass of the earth and moon is
~~ 354710*
Hence we have log V = 2.5625978148; log j/1 + m = 0.0000006122;
log 2x = 0.7981798684; and, consequently,
log k = 8.2355814414.
If we multiply this value of k by 206264.81, the number of seconds
of arc corresponding to the radius of a circle, we shall obtain its
value expressed in seconds of arc in a circle whose radius is unity, or
on the orbit of the earth supposed to be circular. The value of k in
seconds is, therefore,
log k = 3.5500065746.
o
The quantity expresses the mean angular motion of a planet
in a mean solar day, and is usually designated by //. "We shall,
therefore, have
MOTION RELATIVE TO THE SUN. 51
^, '" ' | C33)
for the expression for the mean daily motion of a planet.
Since, in the case of the earth, 1/1 -f- m differs very little from 1,
it will be observed that k very nearly expresses the mean angular
motion of the earth in a mean solar day.
In the case of a small planet or of a comet, the mass m is so small
that it may, without sensible error, be neglected ; and then we shall
have
^ = 4 (M)
a
For the old planets whose masses are considerable, the rigorous ex-
pression (33) must be used.
19. Let us now resume the polar equation of the ellipse, the pole
being at the focus, which is
1 -f e cos v
If we represent by = e;
and, since a(l e 2 ) is half the parameter of the transverse axis,
which we have designated by p, we have .
r
1 -}- sin (f> cos v
The angle
) tan |v. (45)
Again.
V\-\- e = 1/1 -f sin y = 1/1 -|- 2 sin p cos fop,
which may be written
1/1 -}- e I/sin 2 -|^ -f c s 2 i^ + 2 sin Jp cos Jp,
or _
1/1 -|- e = sin ^ -f cos ?
In a similar manner we find
1/1 e = sin \
, and 6 = a cos ^, hence
r sin v cos ^ sin v
. _,
sin E =
1 -f- e GOBV
Equation (41) gives
r cos v + ae p cos v
or
_, ^) cos v + ae ag2 cos v
and, putting a cos 2 (p instead of _p, and sin
(sin 8 Jf(l e cos IT) )
dM v
Expanding these terms, and performing the operations indicated, we
get
^ = 1 + 2e cos M + - (6 xjos 2 M 4 sin 2 Jf)
-f | (16 cos 8 M 36 sin 2 Jf cos Jf) -f- . . . ,
PLACE IN THE ORBIT. 57
which reduces to
, (51)
i
Equation (22) gives
. 2fdt
=
and, since /= }&|/p(l + m), we have
dv = ^j- dt, (52)
or ___
, &V 1 -f- Wl Cl 2 /
But g = fa and therefore
a?
2 2 CX
By expanding the factor j/1 e 2 , we obtain
and hence
Substituting for its value from equation (51), and integrating, we
get, since v = when M=0,
vM=2e sin M +je 2 sin2Jf + -^- (13 sin 33f 3 sin Jf ) +. . . (53)
which is the expression for the equation of the centre to terms involving
e 3 . In the same manner, this series may be extended to higher powers
of e.
When the eccentricity is very small, this series converges very
rapidly ; and the value of v M for any planet may be arranged in
a table with the argument M.
For the purpose, however, of computing the places of a heavenly
body from the elements of its orbit, it is preferable to solve the
equations which give v and E directly ; and when the eccentricity is
58 THEOEETICAL ASTRONOMY.
very great, this mode is indispensable, since the series will not in
that case be sufficiently convergent.
It will be observed that the formula which must be used in obtain-
ing the eccentric anomaly from the mean anomaly is transcendental,
and hence it can only be solved either by series or by trial. But
fortunately, indeed, it so happens that the circumstances of the celes-
tial motions render these approximations very rapid, the orbits being
usually either nearly circular, or else very eccentric.
If, in equation (50), we put F(E] = E, and consequently F(M]
= M, we shall have, performing the operations indicated and reducing,
E = M -f e sin M + ^ sin 2 M + &c. (54)
Let us now denote the approximate value of E computed from this
equation by E ot then will
in which &E Q is the correction to be applied to the assumed value of E.
Substituting this in equation (39), we get
M= E -f &E Q e sin E Q e cos E *E ;
and, denoting by M Q the value of M corresponding to E w we shall
also have
M Q = E Q esmE .
Subtracting this equation from the preceding one, we obtain
M-M,
- -- = = A A..
1 e cos -c/
It remains, therefore, only to add the value of &E Q found from this
formula to the first assumed value of E, or to E w and then, using
this for a new value of E , to proceed in precisely the same manner
for a second approximation, and so on, until the correct value of E is
obtained. When the values of E for a succession of dates, at equal
intervals, are to be computed, the assumed values of E may be ob-
tained so closely by interpolation that the first approximation, in the
manner just explained, will give the correct value; and in nearly
every case two or three approximations in this manner will suffice.
Having thus obtained the value of E corresponding to M for any
instant of time, we may readily deduce from it, by the formulse
already investigated, the corresponding values of r and v.
In the case of an ellipse of very great eccentricity, corresponding
to the orbits of many of the comets, the most convenient method of
PLACE IN THE CEBIT. 59
computing r and v, for any instant, is somewhat different. The
manner of proceeding in the computation in such cases we shall con-
sider hereafter; and we will now proceed to investigate the formulae
for determining r and v, when the orbit is a parabola, the formulse
for elliptic motion not being applicable, since, in the parabola, a = ,
and e 1.
22. Observation shows that the masses of the comets are insensible
in comparison with that of the sun; and, consequently, in this case,
m and equation (52), putting for p its value 2g, becomes
k V2q dt = r*dv,
or
which may be written
kdt
3-^ = 2 (1 -f- tan 2 J-y) sec 2 l>vdv = (!-}- tan 2 Jv) d tan 0.
1/2 g 1
Integrating this expression between the limits T and t, we obtain
^ = tan ^v + I tan 8 -^ v, (55)
1/2 g f
which is the expression for the relation between the true anomaly
and the time from the perihelion, in a parabolic orbit.
Let us now represent by r the time of describing the arc of a
parabola corresponding to v = 90 ; then we shall have
2
or
T/2- 1 " 3 '
3k __ 4 f
~
Now, ^ is constant, and its logarithm is 8.5621876983; and if we
take q = l, which is equivalent to supposing the comet to move in
a parabola whose perihelion distance is equal to the semi-transverse
axis of the earth's orbit, we find
log r days = 2.03987229, or r = 109.61558 days ;
that is, a comet moving in a parabola whose perihelion distance
60 THEORETICAL ASTRONOMY.
is equal to the mean distance of the earth from the sun, requires
109.61558 days to describe an arc corresponding to v = 90.
Equation (55) contains only such quantities as are comparable with
each other, and by it t T, the time from the perihelion, may be
readily found when the remaining terms are known; but, in order
to find v from this formula, it will be necessary to solve the equation
of the third degree, tan \v being the unknown quantity. If we put
x = tan Ji?, this equation becomes
ar + 3z a = 0,
in which a is the known quantity, and is negative before, and positive
after, the perihelion passage. According to the general principle in
the theory of equations that in every equation, whether complete or
incomplete, the number of positive roots cannot exceed the number
of variations of sign, and that the number of negative roots cannot
exceed the number of variations of sign, when the signs of the terms
containing the odd powers of the unknown quantity are changed, it
follows that when a is positive, there is one positive root and no
negative root. When a is negative, there is one negative root and
no positive root; and hence we conclude that equation (55) can have
but one real root.
We may dispense with the direct solution of this equation by
forming a table of the values of v corresponding to those of t T
in a parabola whose perihelion distance is equal to the mean distance
of the earth from the sun. This table will give the time correspond-
ing to the anomaly v in any parabola, whose perihelion distance is
q, by multiplying by q 2 , the time which corresponds to the same
anomaly in the table. We shall have the anomaly v corresponding
to the time t T by dividing t T by (f y and seeking in the table
the anomaly corresponding to the time resulting from this division.
A more convenient method, however, of finding the true anomaly
from the time, and the reverse, is to use a table of the form gene-
rally known as Barker's Table. The following will explain its con-
struction :
Multiplying equation (55) by 75, we obtain
(t T) = 75 tan Jv + 25 tan 8 >.
M=75 tan v -f 25 tan* |v,
1/2 <
Let us now put
PLACE IN THE ORBIT. 61
75k
and C = - =r, which is a constant quantity ; then will
v 2
The value of C is
log <7 = 9.9601277069.
Again. let us take
-=?
which is called the mean daily motion in the parabola; then will
M = m (t T) == 75 tan v + 25
If we now compute the values of M corresponding to successive
values of v from v = to v = 180, and arrange them in a table
with the argument v, we may derive at once, from this table, for the
time (t T) either M when v is known, or v when M m (t T)
is known. It may also be observed that when t T is negative, the
value of v is considered as being negative, and hence it is not neces-
sary to pay any further attention to the algebraic sign of.t T than
to give the same sign to the value of v obtained from the table.
Table VI. gives the values of If for values of v from to 180,
with differences for interpolation, the application of which will be
easily understood.
23. When v approaches near to 180, this table will be extremely
inconvenient, since, in this case, the differences between the values of
M for a difference of one minute in the value of v increase very
rapidly ; and it will be very troublesome to obtain the value of v
from the table with the requisite degree of accuracy. To obviate
the necessity of extending this table, we proceed in the following
manner:
Equation (55) may be written
1/2 q
(1+3 cot 2
and, multiplying and dividing the second member by (1 + cot 2 Jt>)*,
we shall ha\e
F) ,. .,, '
_ = i tan- 4, (1 + cot'
62 THEORETICAL ASTRONOMY.
2
But 1 + cot 2 Jt; = - - and consequently
sin v tan v
1/2 5* ~3sin 8 v (1 + cot 2 ^) 3 '
Now, when t? approaches near to 180, cot^v will be very small, and
the second factor of the second member of this equation will nearly
= 1. Let us therefore denote by w the value of v on the supposition
that this factor is equal to unity, which will be strictly true when
v = 180, and we shall have, for the correct value of v, the following
equation :
V = W + A ,
A O being a very small quantity. We shall therefore have
and, putting tan %w = 0, and tan \ A O = x, we get, from this equation,
"""
6 s 1 6x """ (1 dx} y
Multiplying this through by # 3 (1 0#) 3 , expanding and reducing,
there results the following equation :
+ 0* (2 + W + W +
Dividing through by the coefficient of x, we obtain
1 + 30 _ 2 . 2 (2 + 60* +
04 1 /5fi N v "U/ o /- i A aZ T
Let us now put
1 + 30*
then, substituting this in the preceding equation, inverting the series
and reducing, we obtain finally
But tan ^A O = x, therefore
PLACE IN THE ORBIT. 63
Substituting in this the value of x above found, and reducing, we
obtain
2 -f 320* + 160 8 + 100 8 , ,
&C '
For all the cases in which this equation is to be applied, the third
term of the second member will be insensible, and we shall have, to
a sufficient degree of approximation,
Table VII. gives the values of A O , expressed in seconds of arc,
corresponding to consecutive values of w from w = 155 to w = 180
In the application of this table, we have only to compute the vaiue
of M precisely as for the case in which Table VI. is to be used,
namely,
M=m(t T};
then will w be given by the formula
[206
-v-jT
since we have already found
or
Having computed the value of w from this equation, Table VII.
will furnish the corresponding value of A ; and then we shall have,
for the correct value of the true anomaly,
v w -f- A O ,
which will be precisely the same as that obtained directly from Table
VI., when the second and higher orders of differences are taken into
account.
If v is given and the time t T is required, the table will give,
by inspection, an approximate value of A , using v as argument, and
then w is given by
w = v A.
64 THEORETICAL ASTRONOMY.
The exact value of A is then found from the table, and hence we
derive that of w and finally t T from
t-T- J?
V J- - yy *
GO Bin* to
24. The problem of finding the time t T when the true anomaly
is given, may also be solved conveniently, and especially so when v is
small, by the following process:
Equation (55) is easily transformed into
^- =
3v
from which we obtain, since q = r cos 2 |v,
~2^~ \ 1/2 / ~ \ 1/2 / *
Let us now put
sin^v
sm x = 7==,
1/2
and we have
- - - = 3 sin x 4 sin's = sin 3#.
2r*
Consequently,
which admits of an accurate and convenient numerical solution. To
facilitate the calculation we put
~- _ sin 3x
J\
sin v
the values of which may be tabulated with the argument v. "When
v 0, we shall have N= fl/2, and when v = 90, we have N= 1 ;
from which it appears that the value of ^V changes slowly for values
of v from to 90. But when v = 180, we shall have N= *
and hence, when v exceeds 90, it becomes necessary to introduce au
auxiliarv different from N. We shall, therefore, put in this case,
W = JV sin v = sin 3#;
PLACE IN THE ORBIT. 65
from which it appears that N f = 1 when v = 90, and that JV' = Jv'iB
when v = 180. Therefore we have, finally, when v is less than 90,
2 f
t T=- 7 c 1 -Nr sinv,
OK
and, when v is greater than 90,
o
in which log = 1.5883272995, from which t T is easily derived
oA/
when v is known.
Table VIII. gives the values of N, with diiferences for interpola-
tion, for values of v from v = to v = 90, and the values of N r
for those of v from v = 90 to v = 180.
25. We shall now consider the case of the hyperbola, which differs
from the ellipse only that e is greater than 1 ; and, consequently, the
formulae for elliptic and hyperbolic motion will differ from each other
only that certain quantities which are positive in the ellipse are nega-
tive or imaginary in the hyperbola. We may, however, introduce
auxiliary quantities which will serve to preserve the analogy between
the two, and yet to mark the necessary distinctions.
For this purpose, let us resume the equation
T =
2 cos J- (v + 4) cos J (v 4)'
When v 0, the factors cos | fa + ^ an ^ cos(v ^) in the de-
nominator will be equal ; and since the limits of the values of v are
180 4> and (180 $), it follows that the first factor will vanish
for the maximum positive value of v, and that the second factor will
vanish for the maximum negative value of v, and, therefore, that, in
either case, r = oo.
In the hyperbola, the semi-transverse axis is negative, and, conse-
quenlly, we have, in this case,
p ==a (e 2 1), or a p cot* 4.
We have, also, for the perihelion distance,
q = a(e 1).
Let us now put
tan IF = tan ' (56)
66 THEOKETICAL ASTRONOMY.
which is analogous to the formula for the eccentric anomaly E in an
ellipse: and. since e= - , we shall have
cos 4*
_
e+l~l + COS+~
and, consequently,
tan \F = tan ^v tan ^. (57)
We shall now introduce an auxiliary quantity
2tan|^ , ,, ,
Since tan F= TTTr' we shall nave
1 tan' %F
a l\
i 1 \
(60)
- 1^=1]
Wi/
Squaring this equation, adding 1 to both members, and reducing we
obtain
cos
Replacing
cosF ~~ 2cosi(^H- *) cos i (v 4) "~ 2 cos (v -f- 4) cos (v
which reduces to
1 ~. f ~ I ^/vc,A
(62)
PLACE IN THE ORBIT. 67
If we add ip 1 to both members of this equation, we shall havp
coaF p
Taking first the upper sign, and then the lower sign, and reducing,
we get
VcosF
1/r cos %> = _--) cog , ^ (63)
VcosF
These equations for finding r and v, it will be observed, are analogous
to those previously investigated for an elliptic orbit. These equations
give, by division,
ten v =
which is identical with the equation (56), and may be employed to
verify the computation of r and v.
Multiplying the last of equations (63) by the first, putting for
1 its value tan 2 ij/, and reducing, we obtain
r sin v = a tan 4/ tan F=a tan 4 1 ' + X) + &c -
In the case of the parabola, e = 1 and i 0, and this equation becomes
identical with (55).
Let us now put
72 THEORETICAL ASTRONOMY.
and also
then the angle V will not be the true anomaly in the parabola, but
an angle derived from the solution of a cubic equation of the same
form as that for finding the parabolic anomaly; and its value may
be found by means of Table VI., if we use for M the value com-
puted from
75K^ _T) \T2
- 3 * \ 9
1/2 f
Let U be expanded into a series of the form
which is evidently admissible, a, /?, f y ____ being functions of u and
independent of i. It remains now to determine the values of the
coefficients a, /?, f, &c., and, in doing so, it will only be necessary to
consider terms of the third order, or those involving i 3 , since, for
nearly all of those cases in which the eccentricity is such that terms
of the order i 4 will sensibly affect the result, the general formulae
already derived, with the ordinary means of solution, will give the
required accuracy. We shall, therefore, have
U + I U 3 = U 4- ai + & 4 rP 4 l(u 4 ai 4- & + rW,
or, again neglecting terms of the order i*,
But we have already found, (70),
k(tT)i/l + e = n + ^ = u + _ 2 . u , + ^
2g f
+ 3? (iw 5 + > 7 ) 4i 3 (> 7 + > 9 ).
Since the first members of these equations are identical, it follows, by
the principle of indeterminate coefficients, that the coefficients of the
like powers of i are equal, and we shall, therefore, have
ua * + (i _|_ w .) p = 4. 3 (>< -f 4r T ),
3 4- 2wa/9 4- (1 + w 2 ) r = 4 (> 7 4- ^U 9 ).
From the first of these equations we find
PLACE IN THE CEBIT. 73
The second equation gives
3(K + ju*)-n.
l + u
or, substituting for a its value just found, and reducing,
g = 3 0* + f-H" 7 + jft * + jftu")
(1 + **)
We have also
_ 4 (i^ 7 + X) j 3 2a^ti
r " 1 + u 2
and hence, substituting the values of oc and ft already found, and
reducing, we obtain finally
(I -I- <
Again, we have
tan U tan (u { oil
Developing this, and neglecting terms of the order i 4 , we get
tan^U^f
Now, since w = tan Ji? and U= tan \ F, we shall have
or
2a .
,, (72)
Substituting in this equation the values of a, ft, and ? already found,
and reducing, we obtain finally
(!+')'
THEORETICAL ASTRONOMY.
This equation can be used whenever the true anomaly in the
ellipse or hyperbola is given, and the time from the perihelion is to
be determined. Having found the value of F, we enter Table VI.
with the argument F and take out the corresponding value of M;
and l,hen we derive t T from
t _ T= M<
in which log C = 9.96012771.
For the converse of this, in which the time from the perihelion is
given and the true anomaly is required, it is necessary to express the
difference v F in a series of ascending powers of i, in which the
coefficients are functions of U. Let us, therefore, put
u = U 4- o'i -f /3'i 2 -f r '$ + & c .
Substituting this value of u in equation (70), and neglecting terms
multiplied by i 4 and higher powers of i, we get
_ 4 [73^2 _ 2 Ua' 2 4 U 7 1 17') t".
But, since the first member of this equation is equal to U + J Z7 S , we
shall have, by the principle of indeterminate coefficients,
From these equations, we find
s _ Iff W + ill! u 9 + VftV ff" + f If u 13 + fill + i-VV
(i + tf 2 ) 6
If we interchange v and F in equation (72), it becomes, writing a ; ,
0', f' for a, /9, ^,
_ T7 ,
*
PLACE IN THE ORBIT. 75
2/3' 2a' 2 i7
+ /_^ ___ 4a^^7_ 2(t7^j) \
Ml + tf 2 (1 + tf 2 ) 2 (1 + tf 2 ) 3 /
Substituting in this equation the above values of cc/, /?', and p', and
reducing, we obtain, finally,
_ F[ . ,
22
. If I ^ T + JIM ^ 9 + f If f I ^ n + If If ^ 18
2
by means of which v may be determined, the angle V being taken
from Table VI., so as to correspond with the value of M derived
from
Equations (73) and (74) are applicable, without any modification,
to the case of a hyperbolic orbit which differs but little from the
parabola. In this case, however, e is greater than unity, and, conse-
quently, i is negative.
28. In order to render these formulae convenient in practice, tables
may be constructed in the following manner:
Let x = v or F, and tan Ja? = 6, and let us put
<0*+|0s
100(1 -f- 2 ) 2 '
^
10000 (1
ii
10000 (1 + 2 ) 4
1000000(1 + 2 ) 6
+ iiii^ 9 + flff^ 11 + If ^ 13 + jiff*" +
1000000 (1 + ^ 2 ) 6
wherein s expresses the number of seconds corresponding to the
length of arc equal to the radius of a circle, or log s== 5.31442513.
We shall, therefore, have :
v=V+A (1000 + 5(10017+
76 THEORETICAL ASTRONOMY.
and, when x = v,
V=v-A (lOOi) + & (lOOi) 2 - C' (lOOi) 8 .
Table IX. gives the values of A, B, B', <7, and C' for consecu-
tive values of x from x = to x = 149, with differences for inter-
polation.
When the value of v has been found, that of r may be derived
from the formula
r= g(l-M) ^
1 -j- e cos v
Similar expressions arranged in reference to the ascending powers
of (1 e) or of I I ^ - 1 II may be derived, but they do not con-
2 \^ \
I 1 I is
1 -f~ & ' i
than i, yet the coefficients are, in each case, so much greater than
those of the corresponding powers of i, that three terms will not
afford the same degree of accuracy as the same number of terms in
the expressions involving i.
29. Equations (73) and (74) will serve to determine v or t T in
nearly all cases in which, with the ordinary logarithmic tables, the
general methods fail. However, when the orbit differs considerably
from a parabola, and when v is of considerable magnitude, the results
obtained by means of these equations will not be sufficiently exact,
and we must employ other methods of approximation in the case that
the accurate numerical solution of the general formulae is still impos-
sible. It may be observed that when E or F exceeds 50 or 60, the
equations (39) and (69) will furnish accurate results, even when e
differs but little from unity. Still, a case may occur in which the
perihelion distance is very small and in which v may be very great
before the disappearance of the comet, such that neither the general
method, nor the special method already given, will enable us to de-
termine v or t T with accuracy; and we shall, therefore, investigate
another method, which will, in all cases, be sufficiently exact when
the general formulae are inapplicable directly. For this purpose, let
us resume the equation
= E e sin E t
PLACE IN THE ORBIT. 77
which, since q = a(l e), may be written
If we put
we shall have
^e 201/1 \ 1 l + 9e
" *
2
Let us now put
9E -f sin .E
201/2
and
then we have
When jB is known, the value of w may, according to this equation,
be derived directly from Table VI. with the argument
and then from w we may find the value of A. It remains, therefore,
to find the value of B; and then that of v from the resulting value
of A.
Now, we have
2 tan
and if we put tan 2 \E = r, we get
Sn = == * -^ r
We have, also,
E = 2 tan" 1 r^= 2r^ (1 JT + Jt - ^r 8 + Ac.)-
78 THEOKETICAL ASTRONOMY.
Therefore,
15 (E sin E) = 2t* (10r
and
9E + sin E = 2r^ (10 ^ T + y r y r 8 + if i* &c.).
Hence, by division,
and, inverting this series, we get
which converges rapidly, and from which the value of may be
found.
Let us now put
A_ l_
r~0 2 '
then the values of O may be tabulated with the argument A; and,
besides, it is evident that as long as A is small C 2 will not differ
much from 1 -f- \A.
Next, to find B, we have
- 40.),
and hence
_ 62 . 9
~~~ "" T ^ ~ 2525^+ 33eST6 T
from which we easily find
S = 1 + T | s ^ 2 + ,|^ s + riW,^ 4 + &c.
If we compare equations (44) and (56), we get
tan E = T tan J.P.
Hence, in the case of a hyperbolic orbit, if we put tan 2 |.F r', we
must write r' in place of r in the formulae already derived ; and,
from the series which gives A in terms of r, it appears that A is in
this case negative. Therefore, if we distinguish the equations for
PLACE IN THE CEBIT. 79
Hyperbolic motion from those for elliptic motion by writing A', B f t
and C f in place of A, B, and C, respectively, we shall have
" - Ac.
Table X. contains the values of log B and log C for the ellipse
and the hyperbola, with the argument A, from A = to A = 0.3,
For every case in which A exceeds 0.3, the general formulae (39)
and (69) may be conveniently applied, as already stated.
The equation
Hi Q
tan |v = -v/ T 5 -- tan \E
gives
JL. &
or, substituting the value of A in terms of w,
tan^=(7tan>^^l^. (76)
The last of equations (43) gives
~l+tan f j-tf
Hence we derive
The equation for v in a hyperbolic orbit is of precisely the same form
as (76), the accents being omitted, and the value of A being computed
from
For the radius-vector in a hyperbolic orbit, we find, by means of the
last of equations (63),
T== (l
When t T is given and r and v are required, we first assume
B = 1 , and enter Table VI. with the argument
i 1 B
80 THEORETICAL ASTRONOMY.
in which log C Q = 9.96012771, and take out the corresponding value
of w. Then we derive A from the equation
in the case of the ellipse, and from (78) in the case of a hyperbolic
orbit. With the resulting value of A, we find from Table X. the
corresponding value of log JB, and then, using this in the expression
for My we repeat the operation. The second result for A will not
require any further correction, since the error of the first assumption
of B = 1 is very small ; and, with this as argument, we derive the
value of log C from the table, and then v and r by means of the
equations (76) and (77) or (79).
When the true anomaly is given, and the time t T is required,
we first compute T from
in the case of the ellipse, or from
in the case of the hyperbola. Then, with the value of r as argu-
ment, we enter the second part of Table X. and take out an approxi-
mate value of A, and, with this as argument, we find logJ? and log C.
The equation
will show whether the approximate value of A used in finding
log C is sufficiently exact, and, hence, whether the latter requires any
correction. Next, to find w, we have
5(1 -H)'
and, with w as argument, we derive M from Table VI. Finally, we
have
(80)
by means of which the time from the perihelion may be accurately
determined.
POSITION IN SPACE. 81
30. We have thus far treated of the motion of the heavenly bodies,
relative to the sun, without considering the positions of their orbits
in space ; and the elements which we have employed are the eccen-
tricity and semi-transverse axis of the orbit, and the mean anomalv
at a given epoch, or, what is equivalent, the time of passing thft
perihelion. These are the elements which determine the position of
the body in its orbit at any given time. It remains now to fix its
position in space in reference to some other point in space from which
we conceive it to be seen. To accomplish this, the position of its
orbit in reference to a known plane must be given ; and the elements
which determine this position are the longitude of the perihelion, the
longitude of the ascending node, and the inclination of the plane of
the orbit to the known plane, for which the plane of the ecliptic is
usually taken. These three elements will enable us to determine the
co-ordinates of the body in space, when its position in its orbit has
been found by means of the formula already investigated.
The longitude of the ascending node, or longitude of the point
through which the body passes from the south to the north side of
the ecliptic, which we will denote by & , is the angular distance of
this point from the vernal equinox. The line of intersection of the
plane of the orbit with the fundamental plane is called the line of
nodes.
The angle which the plane of the orbit makes with the plane of
the ecliptic, which we will denote by i, is called the inclination of
the orbit. It will readily be seen that, if we suppose the plane of
the orbit to revolve about the line of nodes, when the angle i exceeds
180, & will no longer be the longitude of the ascending node, but
will become the longitude of the descending node, or of the point
through which the planet passes from the north to the south side of
the ecliptic, which is denoted by t5, and which is measured, as in the
case of & ? from the vernal equinox.
It will easily be understood that, when seen from the sun, so long
as the inclination of the orbit is less than 90, the motion of the
body will be in the same direction as that of the earth, and it is then
said to be direct. When the inclination is 90, the motion will be at
right angles to that of the earth ; and when i exceeds 90, the motion
in longitude will be in a direction opposite to that of the earth, and
it is then called retrograde. It is customary, therefore, to extend the
inclination of the orbit only to 90, and if this angle exceeds a right
angle, to regard its supplement as the inclination of the orbit, noting
simply the distinction that the motion is retrograde.
82 THEORETICAL ASTRONOMY.
The I'jngitude of the perihelion, which is denoted by x, fixes the
position of the orbit in its own plane, and is, in the case of direct
motion, the sum of the longitude of the ascending node and the
angular distance, measured in the direction of the motion, of the
perihelion from this node. It is, therefore, the angular distance of
the perihelion from a point in the orbit whose angular distance back
from the ascending node is equal to the longitude of this node; or
it may be measured on the ecliptic from the vernal equinox to the
ascending node, then on the plane of the orbit from the node to the
place of the perihelion.
In the case of retrograde motion, the longitudes of the successive
points in the orbit, in the direction of the motion, decrease, and the
point in the orbit from which these longitudes in the orbit are
measured is taken at an angular distance from the ascending node
equal to the longitude of that node, but taken, from the node, in the
same direction as the motion. Hence, in this case, the longitude of
the perihelion is equal to the longitude of the ascending node dimi-
nished by the angular distance of the perihelion from this node.
It may, perhaps, seem desirable that the distinctions, direct and
retrograde motion, should be abandoned, and that the inclination of
the orbit should be measured from to 180, since in this case
one set of formulae would be sufficient, while in the common form
two sets are in part required. However, the custom of astronomers
seems to have sanctioned these distinctions, and they may be per-
petuated or not, as may seem advantageous.
Further, we may remark that in the case of direct motion the sum
of the true anomaly and longitude of the perihelion is called the
true longitude in the orbit; and that the sum of the mean anomaly
and longitude of the perihelion is called the mean longitude, an ex-
pression which can occur only in the case of elliptic orbits.
In the case of retrograde motion the longitude in the orbit is equal
to the longitude of the perihelion minus the true anomaly.
31. We will now proceed to derive the formulae for determining
the co-ordinates of a heavenly body in space, when its position in its
orbit is known.
For the co-ordinates of the position of the body at the time t s we
have
x = r cos v,
3 = r sin v,
POSITION IN SPACE. 83
the line of apsides being taken as the axis of x, and the origin being
taken at the centre of the sun.
If we take the line of nodes as the axis of x, we shall have
x = r cos (v -f w),
y = r sin (v -f- w),
to being the arc of the orbit intercepted between the place of the
perihelion and of the node, or the angular distance of the perihelion
from the node.
Ndw, we have co = 7r ft in the case of direct motion, and a) -
ft TT in the case of retrograde motion ; and hence the last equations
become
x = r cos (v TT qp ft)
y = r sin (v db ?r qp ft)
the upper and lower signs being taken, respectively, according as
the motion is direct or retrograde. The arc v7rq=ft=wis called
the argument of the latitude.
Let us now refer the position of the body to three co-ordinate
planes, the origin being at the centre of the sun, the ecliptic being
taken as the plane of xy, and the axis of #, in the line of nodes.
Then we shall have
x' = r cos u,
y f = db r sin u cos i,
z' r sin u sin i.
If we denote the heliocentric latitude and longitude of the body, at
the time t, by b and I, respectively, we shall have
x' = r cos b cos (I ft ),
y f = r cos b sin (I ft ),
/ = r sin b,
and, consequently,
cos u = cos b cos (I &),
sin u cos i = cos b sin (7 ft), (81)
sin w sin i = sin 6.
From these we derive
tan (I ft ) tan w cos i,
tan 6 = db tan i sin (f ft ), (82)
which serve to determine I and 6, when ft, w, and t are given. Since
84 THEORETICAL ASTRONOMY.
cos b is always positive, it follows that I & and u must lie in the
same quadrant when i is less than 90 ; but if i is greater than 90,
or the motion is retrograde, I ft and 360 u will belong to the
same quadrant. Hence the ambiguity which the determination of
/ ft by means of its tangent involves, is wholly avoided.
If we use the distinction of retrograde motion, and consider i
always less than 90, I ft and u will lie in the same quadrant.
32. By multiplying the first of the equations (81) by sin u, and
the second by cos u, and combining the results, considering only the
upper sign, we derive
cos b sin (u I -f- ft ) = 2 sin u cos u sin' ^*,
or
cos b sin (u I + ft ) = sin 2u sin 8 \i.
In a similar manner, we find
cos b cos (u l-\- ft ) = cos'it -J- sin 2 w cos i,
which may be written
cos b cos (u 1-\- ft ) = \ (1 -f- cos 2-w) + J (1 cos 2w) cos i,
or
cos b cos (u l-\- ft ) = (1 + cost) -j- i (1 cos i) cos 2u;
and hence
cos b cos (M / -f~ ft ) = cos 7 i -f- sin* Ji cos 2u.
If we divide this equation by the value of cos b sin (u I + ft )
already found, we shall have
f 7 * ^\ tan*J*sin2ti , QO ,
tan (u I + ft) = r L 2 , . . JT-. (83)
1 -f- tan 2 i cos 2it
The angle u 1+ ft is called the reduction to the ecliptic; and the
expression for it may be arranged in a series which converges rapidly
when i is small, as in the case of the planets. In order to effect this
development, let us first take the equation
n sin. a;
tan v = j :
1 -f- n cos x
Differentiating this, regarding y and n as variables, and reducing, we
find
dy sin a;
~dn 1 -J- 2n cos x -f- ^
POSITION IN SPACE. 85
which gives, by division, or by the method of indeterminate coefficients,
-~ = sin x n sin 2x + n 2 sin 3# n 3 sin 4# -4- &c.
dn
Integrating this expression, we get, since y = when x = 0,
y = n sin x \r? sin 2# -f- |n 3 sin 3# \n^ sin 4# -f- ____ , (84)
which is the general form of the development of the above expression
for tan y. The assumed expression for tan y corresponds exactly with
the formula for the reduction to the ecliptic by making n = tan 2 Jt
and x = 2u; and hence we obtain
u I -j- & = tan 2 \i sin 2u ^ tan 4 i sin 4^ -f~ i tan 6 Ji sin Qu
&c. ' (85)
When the value of i does not exceed 10 or 12, the first two terms
of this development will be sufficient. To express u I -f- & in
seconds of arc, the value derived from the second member of this
equation must be multiplied by 206264.81, the number of seconds
corresponding to the radius of a circle.
If we denote by jR e the reduction to the ecliptic, we shall have
But we have v = M -f the equation of the centre ; hence
1 = M -f TT -f- equation of the centre reduction to the ecliptic,
and, putting L = M -f- TT = mean longitude, we get
I L -f- equation of centre reduction to ecliptic. (86)
In the tables of the motion of the planets, the equation of the
centre (53) is given in a table with M as the argument ; and the
reduction to the ecliptic is given in a table in which i and u are the
arguments.
33. In determining the place of a heavenly body directly from
the elements of its orbit, there will be no necessity for computing the
reduction to the ecliptic, since the heliocentric longitude and latitude
may be readily found by the formulae (82). When the heliocentric
place has been found, we can easily deduce the corresponding geo-
centric place.
Let x, y, z be the rectangular co-ordinates of the planet or comet
referred to the centre of the sun, the plane of xy being in the ecliptic,
86 THEORETICAL ASTRONOMY.
the positive axis of x being directed to the vernal equinox, and the
positive axis of z to the north pole of the ecliptic. Then we shall
have
x = r cos b cos I,
y = r cos b sin I,
2 =r sin b.
Again, let X y F, Z be the co-ordinates of the centre of the sun re-
ferred to the centre of the earth, the plane of XY being in the eclip-
tic, and the axis of X being directed to the vernal equinox ; and let
denote the geocentric longitude of the sun, R its distance from
the earth, and I its latitude. Then we shall have
X=R cos S cos O,
Y= R cos 2 sin O,
Z = R sin 2.
Let x'j y f , z' be the co-ordinates of the body referred to the centre of
the earth ; and let X and ft denote, respectively, the geocentric longi-
tude and latitude, and J, the distance of the planet or comet from the
earth. Then we obtain
x f = A cos ft cos A,
y f =Jco8^Bin^ (87)
z' = A sin p.
But, evidently, we also have
and, consequently,
A cos p cos A = r cos b cos l-{- R cos 2 cos O >
A cos p sin A = r cos 6 sin I -J- .R cos 2" sin O > (88)
J sin /? = r sin b -\- R sin 2".
If we multiply the first of these equations by cos Q, and the second
by sin Q, and add the products; then multiply the first by sin Q,
and the second by cos O , and subtract the first product from the
second, we get
A cos p cos (A Q) = r cos b cos (I Q) -J- R cos I,
A cos/3sm(A 0) = rcos6sin(J O), (89)
A sin p = r sin b -j- R sin 2".
It will be observed that this transformation is equivalent to the sup-
position that the axis of x, in each of the co-ordinate systems, is
POSITION IN SPACE. 87
directed to a point whose longitude is > or that the system has been
revolved about the axis of z to a new position for which the axis of
abscissas makes the angle Q with that of the primitive system. We
may, therefore, in general, in order to effect such a transformation in
systems of equations thus derived, simply diminish the longitudes by
the given angle.
The equations (89) will determine A, /9, and A when r. 6. and I have
been derived from the elements of the orbit, the quantities J?, , and
Z being furnished by the solar tables; or, when J, /9, and A are given,
these equations determine /, 6, and r. The latitude 2 1 of the sun
never exceeds 0".9, and, therefore, it may in most cases be neg-
lected, so that cos 1 and sin = 0, and the last equations become
A cos ft cos 0* Q ) = r cos b cos (I ) -j- R,
A cos/5 sin (A 0) =r cosb sin (I ), (90)
A sin /? =r sin b.
If we suppose the axis of x to be directed to a point whose longi-
tude is &, or to the ascending node of the planet or comet, the equa-
tions (88) become
A cos/? cos (A &) =r cos it -f- ^ cos -T cos (O &),
A cos/? sin (A Q) = r smu cosi + R cos 2 sin (0 &), (91)
A sin /? = r sin u sin i -f- R sin ,
by means of which /9 and ^ may be found directly from & , i, r, and u.
If it be required to determine the geocentric right ascension and
declination, denoted respectively by a and d, we may convert the
values of f) and A into those of a and 8. To effect this transforma-
tion, denoting by e the obliquity of the ecliptic, we have
cos 8 cos a = cos /? cos A,
cos 5 sin a = cos /? sin A cos e sin sin e,
sin d = cos /5 sin A sin e -f- sin /5 cos e.
Let us now take
n sin JV= sin/?,
w cos JV= cos /? sin A,
and we shall have
COS d COS tt = COS /? COS A,
cos 5 sin a = n cos (JV+ 0>
sin d =n sin ( JV -j- )
88 THEORETICAL, ASTRONOMY.
Therefore, we obtain
(92)
cos N
tan 5 = tan (N + e) sin a.
We also have
cos (N -f- e ) _ cos <5 sin a
cos N cos /? sin A '
which will serve to check the calculation of a and d. Since ccs d ana
cos /? are always positive, cos a and cos A must have the same sign,
and thus the quadrant in which oc is to be taken, is determined.
For the solution of the inverse problem, in which cc and d are
given and the values of X and /? are required, it is only necessary to
interchange, in these equations, a and ^, 3 and /?, and to write s in
place of e.
34. Instead of pursuing the tedious process, when several places
are required, of computing first the heliocentric place, then the geo-
centric place referred to the ecliptic, and, finally, the geocentric right
ascension and declination, we may derive formulae which, when cer-
tain constant auxiliaries have once been computed, enable us to derive
the geocentric place directly, referred either to the ecliptic or to the
equator.
We will first consider the case in which the ecliptic is taken as the
fundamental plane. Let us, therefore, resume the equations
x' = r cos u,
y' = d= r sin u cos i,
z' = r sin u sin i,
in which the axis of x is supposed to be directed to the ascending node
of the orbit of the body. If we now pass to a new system x, y, z,
the origin and the axis of z remaining the same, in which the axis
of x is directed to the vernal equinox, we shall move it back, in a
negative direction, equal to the angle &, and, consequently,
x = x f cos R> if sin
y = x' sin Q -}- y' cos
Therefore, we obtain
fc = r (cos u cos & =P sin'it cos i sin & ),
y = r ( sin u cos i cos & -}- cos u sin & ), (93)
z = r sin u sin i,
POSITION IN SPACE. 89
which are the expressions for the heliocentric co-ordinates of a planet
or comet referred to the ecliptic, the positive axis of x being directed
to the vernal equinox. The upper sign is to be used when the
motion is direct, and the lower sign when it is retrograde.
Let us now put
cos Q = sin a sin A,
^F cos i sin & = sin a cos A,
sin a = smbsmB,
cos i cos & = sin b cos B,
in which sin a and sin 6 are positive, and the expressions for the co-
ordinates become
x = r sin a sin (J. -f- u) t
y = rsmb sin (B -f w), (95)
z =r sin i sin it.
The auxiliary quantities a, 6, J., and .B, it will be observed, are
functions of & and i 9 and, in computing an ephemeris, are constant
so long as these elements are regarded as constant. They are called
the constants for the ecliptic.
To determine them, we have, from equations (94),
cot A = q= tan & cos i, cot B cot Q> cos i,
cos & . , sin &
sma= -p sm0 = ^-;
sinJ. sin li
the upper sign being used when the motion is direct, and the lower
sign when it is retrograde.
The auxiliaries sin a and sin b are always positive, and, therefore,
sin A and cos & , sin B and sin & , respectively, must have the same
signs. The quadrants in which A and B are situated, are thus deter-
mine^.
From the equations (94) we easily find
cos a = sin i sin & >
cos b = sin t cos & . (96)
If we add to the heliocentric co-ordinates of the body the co-ordi-
nates of the sun referred to the earth, for which the equations have
already been given, we shall have
x -j- X= A cos ft cos A,
y + Y= J cos /9 sin A, (97)
2 + Z = A sin /?,
90 THEORETICAL ASTRONOMY.
which suffice to determine A, /?, and J. The values of a and 8 may
be derived from these by means of the equations (92).
35. "We shall now derive the formulae for determining a and d
directly. For this purpose, let #, y, z be the heliocentric co-ordinates
of the body referred to the equator, the positive axis of x being
directed to the vernal equinox. To pass from the system of co-
ordinates referred to the ecliptic to those referred to the equator as
the fundamental plane, we must revolve the system negatively around
the axis of x, so that the axes of z and y in the new system make
the angle e with those of the primitive system, e being the obliquity
of the ecliptic. In this case, we have
~" ~
X X,
2/" = y cos s z sin e,
z" = y sin e -j- z cos e.
Substituting for x, y, and z their values from equations (93), and
omitting the accents, we get
x = r cos u cos & qc r sin u cos i sin & ,
y r cos u sin & cos s-\-r sin u ( cos i cos & cos e sin i sin e), (98)
z = r cos u sin & sin e-\-r sin u (db cos i cos & sin e -j- sin i cos e).
These are the expressions for the heliocentric co-ordinates of the
planet or comet referred to the equator. To reduce them to a con-
venient form for numerical calculation, let us put
cos & = sin a sin A,
qp cos i sin & = sin a cos A,
sin & cos e = sin b sin i?,
cos i cos & cos e sin i sin e = sin b cos -B,
sin 2 sin e = sin c sin 0,
cos i cos & sin e -f sin i cos e = sin c cos (7;
and the expressions for the co-ordinates reduce to
x = r sin a sin (A -f- u),
y = rsmb sin (. -f u), (100)
2 = r sin c sin ( (7 -f- u).
The auxiliary quantities, a, 6, c, JL, jB, and C, are constant so long
as & and i remain unchanged, and are called constants for th( equator.
It will be observed that the equations involving a and A, regard-
ing the motion as direct, correspond to the relations between the
parts of a quadrantal triangle of which the sides are i and a, the
POSITION IN SPACE. 91
angle included between these sides being that which we designate by
A, and the angle opposite the side a being 90 Q> . In the case
of b and jB, the relations are those of the parts of a spherical triangle
of which the sides are 6, i, and 90 -f- e, B being the angle included
by i and 6, and 180 & the angle opposite the side b. Further,
in the case of c and (7, the relations are those of the parts of a
spherical triangle of which the sides are c, i, and e, the angle C being
that included by the sides i and c, and 180 & that included by
the sides i and e. We have, therefore, the following additional
equations :
cos a = sin i sin & ,
cos b = cos & sin i cos s cos i sin e, (101)
cos e = cos Q sin i sin s -f- cos i cos e.
In the case of retrograde motion, we must substitute in these
180 i in place of i.
The geometrical signification of the auxiliary constants for the
equator is thus made apparent. The angles a, b, and c are those
which a line drawn from the origin of co-ordinates perpendicular to
the plane of the orbit on the north side, makes with the positive co-
ordinate axes, respectively ; and A, B, and C are the angles which
the three planes, passing through this line and the co-ordinate axes,
make with a plane passing through this line and perpendicular to the
line of nodes.
In order to facilitate the computation of the constants for the
equator, let us introduce another auxiliary quantity E , such that
sin i = e Q sin E Q ,
cos i cos Q = e Q cos E w
e (} being always positive. We shall, therefore, have
tani
~ cos Q,'
Since both e Q and sini are positive, the angle E cannot exceed 180;
and the algebraic sign of tan E Q will show whether this angle is to
be taken in the first or second quadrant.
The first two of equations (99) give
cot A = HP tan & cosi;
and the first gives
cos &
sm a = T .
sin A
92 THEORETICAL ASTRONOMY.
From the fourth of equations (99), introducing e Q and E ti , we get
sin b cos B = e cos JE cos e e sin E Q sin e = e cos (J^ -{- e )
But
sin 6 sin B = sin & cos e ;
therefore
sin & cos e tan & cos E Q ' cos e
We have, also,
. , sin cos
sin b = . p .
sm.B
In a similar manner, we find
cot C= C si sin (^Q + e )
~ tan & cos EQ ' sin e
and
sin O sin e
sine
sin
The auxiliaries sin a, sin 6, and sin c are always positive, and, there-
fore, sin A and cos &, sini? and sin &, and also sin Q and sin &,
must have the same signs, which will determine the quadrant in
which each of the angles A, B, and C is situated.
If we multiply the last of equations (99) by the third, and the
fifth of these equations by the fourth, and subtract the first product
from the last, we get, by reduction,
sin b sin c sin ( C B) = sin i sin &.
But
sin a cos A = =F cos i sin & J
and hence we derive
sin b sin c sin ( C B}
sin a cos A
= tan i,
which serves to check the accuracy of the numerical computation of
the constants, since the value of tan i obtained from this formula
must agree exactly with that used in the calculation of the values of
these constants.
If we put A' = A n T &, B' = B n q= a, and C' = C n
^F & , the upper or lower sign being used according as the motion is
direct or retrograde, we shall have
POSITION IN SPACE. 93
= r sin a sin (A' -f- v),
y = r sin 5 sin (' -f *), (102)
z = r sin c sin ( C' + v),
a transformation which is perhaps unnecessary, but which is con-
venient when a series of places is to be computed.
It will be observed that the formula for computing the constants
a, 6, c, A, By and (7, in the case of direct motion, are converted into
those for the case in which the distinction of retrograde motion id
adopted, by simply using 180 i instead of i.
36. When the heliocentric co-ordinates of the body have been
found, referred to the equator as the fundamental plane, if we add to
these the geocentric co-ordinates of the sun referred to the same
fundamental plane, the sum will be the geocentric co-ordinates of
the body refeired also to the equator.
For the co ordinates of the sun referred to the centre of the earth,
we have, neglecting the latitude of the sun,
X= RcosQ,
Y=- .Rsin O cose,
Z = R sin O sin e = Ftan e,
in which R represents the radius-vector of the earth, O the sun's
longitude, and e the obliquity of the ecliptic.
We shall, therefore, have
x -}- JT = A cos cos a,
y -f Y= A cos 8 sin a, (103)
z -f Z= Jsind,
which suffice to determine a, d, and J.
If we have regard to the latitude of the sun in computing its geo-
centric co-ordinates, the formulae will evidently become
JT Ii cos O cos S,
Y= R sin Q cos S cos e R sin 2 sine, (104)
Z = H sin O cos sin e -\- JK sin S cos e,
in which, since S can never exceed it 0".9, cos 2 is very nearly
equal to 1, and sin I = I.
The longitudes and latitudes of the sun may be derived from a
solar ephemeris, or from the solar tables. The principal astronomical
ephemerides, such as the Berliner Astronomisches Jahrbuch, the
Nautical Almanac, and the American Ephemeris and Nautical Al-
94 THEOKETICAL ASTRONOMY.
manac, contaiD, for each year for which they are published, the
equatorial co-ordinates of the sun, referred both to the mean equinox
and equator of the beginning of the year, and to the apparent equinox
of the date, taking into account the latitude of the sun.
37. In the case of an elliptic orbit, we may determine the co-
ordinates directly from the eccentric anomaly in the following
manner :
The equations (102) give, accenting the letters a, 6, and c,
x = r cos v sin a' sin A' -j- r sin v sin a' cos A',
y =r cos v sin b r sin B r -f- r sin v sin b' cos B',
z = r cos v sin c r sin C' -[-r sin v sin c' cos C'.
Now, since r cos v = a cos E ae, and r sin v = a cos ' sin B' = A y ,
a sin b f sin _B' = ^t y ,
a tan 4> sin b' cos B f = v y ;
ae sin c' sin C" = >*
a sin c' sin G r = /*
a tan 4 sin c' cos C" = v z .
Then we shall have
x = A x -f fj. K sec .F + v x tan jP,
v = A y 4- ^ sec jF + v y tan F, (106;
2 = A s -j- /7. z sec F -f- v z tan jP.
In a similar manner we may derive expressions for the co-ordinates,
in the case of a hyperbolic orbit, when the auxiliary quantity a is
used instead of F.
39. If we denote by ;:', ', and V the elements which determine
the position of the orbit in space when referred to the equator as the
96 THEORETICAL ASTRONOMY.
fundamental plane, and by o) the angular distance between the
ascending node of the orbit on the ecliptic and its ascending node on
the equator, being measured positively from the equator in the
direction of the motion, we shall have
To find Q,' and i', we have, from the spherical triangle formed by
the intersection of the planes of the orbit, ecliptic, and equator with
the celestial vault,
cos if = cos i cos e sin i sin e cos & ,
sin i r sin &' = sin i sin ^ ,
sin i' cos &' = cos t sin e -f- sin i cos e cos SI
Let us now put
n sin JV = cos i t
n cos JV= sin i cos &,
and these equations reduce to
cos i' = n sin ( JV e),
sin i' sin &' = sin i sin & ,
sin i' cos &' = n cos (JV e) ;
from which we find
HT cot i cos JV
tan JV= , tan ' = ^ ^ tan &,
cos 1 cos (JV e)
cot i' = tan (JV e) cos &'. ( 107 )
Since sin i is always positive, cos JV and cos & must have the same
signs. To prove the numerical calculation, we have
sin i cos Q _ cos JV
sin i' cos ' cos (JV e)'
the value of the second member of which must agree with that used
in computing & '.
In order to find CD O , we have, from the same triangle,
sin CM O sin i' = sin & sin e,
cos a> n sin i' = cos e sin i -{- sin e cos i cos & .
Liet us now take
m sin Jlf = cos e,
m cos Jlf = sin e cos & ;
and we obtain
POSITION IN SPACE. 97
cot M = tan e cos &,
and, also, to check the calculation,
sin e cos & cos Jf
sin i' cos % cos (M i)'
If we apply Gauss's analogies to the same spherical triangle, we
get
cos^' cos^ (' + > ) == cosi cosi (*' + s \ MAQ x
sin It sin I (&' ) sin J& sin i (t e),
The quadrant in which \ (&' + ^ ) or K^ ^o) ^ s situated, must be
so taken that sin \i f and cos \V shall be positive ; and the agreement
of the values of the latter two quantities, computed by means of the
value of \i f derived from tan Ji', will serve to check the accuracy of
the numerical calculation.
For the case in which the motion is regarded as retrograde, we
must use 180 i instead of i in these equations, and we have, also,
We may thus find the elements TT', & ', and i f y in reference to the
equator, from the elements referred to the ecliptic; and using the
elements so found instead of it, &, and i, and using also the places
of the sun referred to the equator, we may derive the heliocentric
and geocentric places with respect to the equator by means of the
formulae already given for the ecliptic as the fundamental plane.
If the position of the orbit with respect to the equator is given,
and its position in reference to the ecliptic is required, it is only
necessary to interchange & and &', as well as i and 180 i', e
remaining unchanged, in these equations. These formulae may
also be used to determine the position of the orbit in reference to
any plane in space; but the longitude & must then be measured
from the place of the descending node of this plane on the ecliptic.
The value of &, therefore, which must be used in the solution of the
equations is, in this case, equal to the longitude of the ascending
node of the orbit on the ecliptic diminished by the longitude of the
descending node of the new plane of reference on the ecliptic. The
quantities & ', i', and w will have the same signification in reference
7
98 THEORETICAL ASTRONOMY.
to this plane that they have in reference to the equator, with this dis-
tinction, however, that & ' is measured from the descending node of
this new plane of reference on the ecliptic ; and e will in this case
denote the inclination of the ecliptic to this plane.
40. We have now derived all the formulae which can be required
in the case of undisturbed motion, for the computation of the helio-
centric or geocentric place of a heavenly body, referred either to the
ecliptic or equator, or to any other known plane, when the elements
of its orbit are known ; and the formulae which have been derived
are applicable to every variety of conic section, thus including all
possible forms of undisturbed orbits consistent with the law of uni-
versal gravitation. The circle is an ellipse of which the eccentricity
is zero, and, consequently, M=v = u, and r = a, for every point of
the orbit. There is no instance of a circular orbit yet known ; but
in the case of the discovery of the asteroid planets between Mars
and Jupiter it is sometimes thought advisable, in order to facilitate
the identification of comparison stars for a few days succeeding the
discovery, to compute circular elements, and from these an ephemeris.
The elements which determine the form of the orbit remain con-
stant so long as the system of elements is regarded as unchanged ;
but those which determine the position of the orbit in space, TT, & ,
and i, vary from one epoch to another on account of the change of
the relative position of the planes to which they are referred. Thus
the inclination of the orbit will vary slowly, on account of the change
of the position of the ecliptic in space, arising from the perturbations
of the earth by the other planets ; while the longitude of the peri-
helion and the longitude of the ascending node will vary, both on
account of this change of the position of the plane of the ecliptic,
and also on account of precession and nutation. If TT, &, and i are
referred to the true equinox and ecliptic of any date, the resulting
heliocentric places will be referred to the same equinox and ecliptic ;
and, further, in the computation of the geocentric places, the longi-
tudes of the sun must be referred to the same equinox, so that the
resulting geocentric longitudes or right ascensions will also be re-
ferred to that equinox. It will appear, therefore, that, on account
of these changes in the values of n, &, and i, the auxiliaries sin a,
sin 6, sin c, A, J5, and C, introduced into the formulae for the co-
ordinates, will not be constants in the computation of the places for
a series of dates, unless the elements are referred constantly, in the
calculation, to a fixed equinox and ecliptic. It is customary, there-
POSITION IN SPACE. 99
fore, to reduce the elements to the ecliptic and mean equinox of the
beginning of the year for which the ephemeris is required, and then
to compute the places of the planet or comet referred to this equinox,
using, in the case of the right ascension and declination, the mean
obliquity of the ecliptic for the date of the fixed equinox adopted, in
the computation of the auxiliary constants and of the co-ordinatey
of the sun. The places thus found may be reduced to the true
equinox of the date by the well-known formulae for precession and
nutation. Thus, for the reduction of the right ascension and declina-
tion from the mean equinox and equator of the beginning of the
year to the apparent or true equinox and equator of any date, usually
the date to which the co-ordinates of the body belong, we have
A<* =/+ g sin (G + a) tan d,
for which the quantities /, g, and G are derived from the data given
either in the solar and lunar tables, or in astronomical ephemerides,
such as have already been mentioned.
The problem of reducing the elements from the ecliptic of one
date t to that of another date t f may be solved by means of equations
(109), making, however, the necessary distinction in regard to the
point from which Q> and &>' are measured. Let d denote the longi-
tude of the descending node of the ecliptic of t' on that of t y and
let T} denote the angle which the planes of the two ecliptics make
with each other, then, in the equations (109), instead of & we must
write & 0, and, in order that Q, ' shall be measured from the
vernal equinox, we must also write & ' 6 in place of Q, ' . Finally,
we must write T] instead of e, and A
os (' + Aw) = cosi (& 0) cos (i + 17),
sin Jt* sin J (' Aw) = sin J (& 0) sin J (i 9 ),
sin $i! cosi (' AW) = C os (& 0) sin J (t -f r,).
These equations enable us to determine accurately the values of & ',
i', and AW, which give the position of the orbit in reference to the
ecliptic corresponding to the time t f , when d and y are known. The
longitudes, however, will still be referred to the same mean equinox
as before, which we suppose to be that of t; and, in order to refer
100 THEORETICAL ASTRONOMY.
them to the mean equinox of the epoch t', the amount of the pre-
cession in longitude during the interval t f t must also be applied.
If the changes in the values of the elements are not of consider-
able magnitude, it will be unnecessary to apply these rigorous formulae,
and -we may derive others sufficiently exact, and much more con-
venient in application. Thus, from the spherical triangle formed by
the intersection of the plane of the orbit and of the planes of the
two ecliptics with the celestial vault, we get
sin 7] cos (ft 0) = cos i' sin i -f- sin if cos i cos Aw,
from which we easily derive
sin (i r i) = sin TJ cos (ft 0) -f- 2 sin if cos i sin 2 JAW. (112)
We have, further,
sin Aw sin i' = sin f) sin (ft 0),
or
Shl(ft - 0) r+ + n\
sin AW = sm -TJ - ^-3 -. (113)
sin*
We have, also, from the same triangle,
sin AW cos i' = cos (ft 0) sin (ft' 0)
-f sh (ft 0) cos (ft' 0) cos 17,
which givec
sin (ft' ft) = sin Aw cos if 2 sin (ft 0) cos (ft' 0) sin 2 ^,
or
sin (ft' ft) = sin >? sin (ft 0)coti'
2 sin (ft 0) cos (ft' 0) sin 1 7. (114)
Finally, we have
TT' TT= ft f ft + Aw.
Since ^ is very small, these equations give, if we apply also the pre-
cession in longitude so as to reduce the longitudes to the mean equinox
of the date *'
smi
i f = i_-^cos(ft 0) -
8
2 -0), (115)
POSITION IN SPACE. 101
in which is the annual precession in longitude, and in which
s = 206264". 8. In most cases, the last terms of the expressions for
i f , Q'y and TT', being of the second order, may be neglected.
For the case in which the motion is regarded as retrograde, we
must put 180 i and 180 i', instead of i and i', respectively, in
the equations for AO>, i f , and & ' ; and for TT', in this case, we have
which gives
*' = * + (*' 0-^7 7 sin (& 0) tan ' ^s
Civ S
If we adopt BesseFs determination of the luni-solar precession and
of the variation of the mean obliquity of the ecliptic, we have, at the
time 1750 + r,
-^ = 50".21129 + O."0002442966r,
at
^ = 0".48892 0/'000006143r,
at
and, consequently,
73 = (0."48892 O."000006143r) (f f) ;
and in the computation of the values of these quantities we must put
T \(t r 4" 1750, t and t 1 being expressed in years.
The longitude of the descending node of the ecliptic of the time t
on the ecliptic of 1750.0 is also found to be
351 36' 10" 5".21 (t 1750),
which is measured from the mean equinox of the beginning of the year
1750.
The longitude of the descending node of the ecliptic of t f on that
of t f measured from the same mean equinox, is equal to this value
diminished by the angular distance between the descending node of
the ecliptic of t on that of 1750 and the descending node of the
ecliptic of t' on that of t, which distance is, neglecting terms of the
second order,
5"31(f 1750);
and the result is
351 36' 10" 5".21 (t 1750) 5".21 (if 1750),
351 36' 10" -10".42( 1750) 5".21(/ *).
102 THEORETICAL ASTRONOMY.
To reduce this longitude to the mean equinox at the time t, we must
add the general precession during the interval t 1750, or
50".21 (t 1750),
so that we have, finally,
e = 351 36' 10" + 39".79 (t 1750) 5".21 (if <).
When the elements TT, &, and i have been thus reduced from the
ecliptic and mean equinox to which they are referred, to those of the
date for which the heliocentric or geocentric place is required, they
may be referred to the apparent equinox of the date by applying the
nutation in longitude. Then, in the case of the determination of the
right ascension and declination, using the apparent obliquity of the
ecliptic in the computation of the co-ordinates, we directly obtain the
place of the body referred to the apparent equinox. But, in com-
^uting a series of places, the changes which thus take place in the
elements themselves from date to date induce corresponding changes
in the auxiliary quantities a, 6, c, A, B, and CJ so that these are no
longer to be considered as constants, but as continually changing their
values by small differences. The differential formulae for the com-
putation of these changes, which are easily derived from the equations
(99), will be given in the next chapter; but they are perhaps unneces-
sary, since it is generally most convenient, in the cases which occur, to
compute the auxiliaries for the extreme dates for which the ephemeris
is required, and to interpolate their values for intermediate dates.
It is advisable, however, to reduce the elements to the ecliptic and
mean equinox of the beginning of the year for which the ephemeris
is required, and using the mean obliquity of the ecliptic for that
epoch, in the computation of the auxiliary constants for the equator,
the resulting geocentric right ascensions and declinations will be
referred to the same equinox, and they may then be reduced to the
apparent equinox of the date by applying the corrections for preces-
sion and nutation.
The places which thus result are free from parallax and aberration.
In comparing observations with an ephemeris, the correction for par-
allax is applied directly to the observed apparent places, since this
correction varies for different places on the earth's surface. The cor-
rection for aberration may be applied in two different modes. We
may subtract from the time of observation the time in which the
light from the planet or comet reaches the earth, and the true place
for this reduced time is identical with the apparent place for the time
NUMERICAL EXAMPLES. 103
of observation ; or, in case we know the daily or hourly motion of
the body in right ascension and declination, we may compute the
motion during the interval which is required for the light to pass
from the body to the earth, which, being applied to the observed
place, gives the true place for the time of observation.
We may also include the aberration directly in the ephemeris by
using the time t 497 S .78^ in computing the geocentric places for
the time t, or by subtracting from the place free from aberration, com-
puted for the time t, the motion in a and d during the interval
497*.78 J, in which expression A is the distance of the body from tlu
earth, and 497.78 the number of seconds in which light traverses the
mean distance of the earth from the sun.
It is customary, however, to compute the ephemeris free from
aberration and to subtract the time of aberration, 497*.78 J, from the
time of observation when comparing observations with an ephemeris,
according to the first method above mentioned. The places of the
sun used in computing its co-ordinates must also be free from aberra-
tion; and if the longitudes derived from the solar tables include
aberration, the proper correction must be applied, in order to obtain
the true longitude required.
41. EXAMPLES. We will now collect together, in the proper
order for numerical calculation, some of the principal formulae which
have been derived, and illustrate them by numerical examples, com-
mencing with the case of an elliptic orbit. Let it be required to find
the geocentric right ascension and declination of the planet Eurynome
, for mean midnight at Washington, for the date 1865 February
24, the elements of the orbit being as follows :
Epoch = 1864 Jan. 1.0 Greenwich mean time.
29' 40".21
20 33 .09^ Ecliptic and Mean
Equinox, 1864.0.
log a = 0.3881319
log fJL = 2.9678088
P = 928".55746
When a series of places is to be computed, the first thing to oe
done is to compute the auxiliary constants used in the expressions for
the co-ordinates, and although but a single pi ice is required in the
problem proposed, yet we will proceed in this manner, in order to
104 THEORETICAL ASTRONOMY.
exhibit the application of the formulae. Since the elements it, ,
and i are referred to the ecliptic and mean equinox of 1864.0, we will
first reduce them to the ecliptic and mean equinox of 1865.0. For
this reduction we have t 1864.0, and t f = 1865.0, which give
rll
5f = 50".239, 9 = 352 51' 41", 7 = 0".4882.
at
Substituting these values in the equations (115), we obtain
i f i = At = 0".40, A& = + 53".61, ATT ^ + 50".23 ;
and hence the elements which determine the position of the orbit in
reference to the ecliptic of 1865.0 are
TT = 44 21' 23".32, = 206 43' 33".74, i = 4 36' 50".ll.
For the same instant we derive, from the American Ephemeris and
Nautical Almanac, the value -of the mean obliquity of the ecliptic,
which is
e = 23 27' 24".03.
The auxiliary constants for the equator are then found by means oi
the formulas
cot A = tan & cos i, tan E Q = -- ,
COS do
cosi c
tan & cos E Q cos e
Coi0=
tan & cos Q sin e
cos Q> . , sin & cos e . sin & sm e
sm a = r , sm b = - : ^ , sm c - : ~ .
sm A SIHJD sm C
The angle E is always less than 180, and the quadrant in which it is
to be taken, is indicated directly by the algebraic sign of tan E Q . The
values of sin a, sin 6, and sin c are always positive, and, therefore, the
angles A, B, and C must be so taken, with respect to the quadrant in
which each is situated, that sin A and cos &, sin B and sin &, and also
sin C and sin & , shall have the same signs. From these we derive
A = 296 39' 5".07, log sin a = 9.9997156,
B = 205 55 27 .14, log sin b = 9.9748254,
C = 212 32 17 .74, - log sin c = 9.5222192.
Finally, the calculation of these constants is proved by means of the
formula
NUMERICAL EXAMPLES. 105
. sin b sin c sin ( C -B)
tan i = :
sin a cos A
whic;i gives log tan i = 8.9068875, agreeing with the value 8.9068876
derived directly from i.
Next, to find r and u. The date 1865 February 24.5 mean time
at Washington reduced to the meridian of Greenwich by applying
the difference of longitude, 5* 8 m 1P.2, becomes 1865 February
24.714018 mean time at Greenwich. The interval, therefore, from
the epoch for which the mean anomaly is given and the date for
which the geocentric place is required, is 420.714018 days; and mul-
tiplying the mean daily motion, 928". 55745, by this number, and
adding the result to the given value of M, we get the mean anomaly
for the required place, or
M = 1 29' 40".21 + 108 30' 57".14 = 110 0' 37".35.
The eccentric anomaly E is then computed by means of the equation
M=E esmE,
the value of e being expressed in seconds of arc. For Eurynome we
have log sin = 37 35' 0".0, or log 6 = 0.1010188; and lug a
= 0.6020600, to find r and v. First, we compute ^Vfrom
" at
in which log A = 9.6377843, and we obtain
logN= 8.7859356; N= 0.06108514.
The value of F must now be found from the equation
N= el tan F log tan (45 + F).
NUMERICAL EXAMPLES. 109
If we assume F= 30, a more approximate value may be derived
from
which gives F, = 28 40' 23", and hence N, = 0.072678. Then we
compute the correction to be applied to this value of F, by means of
the equation
(N-N,^o S *F
' l(e cosF,-) *'
wherein s = 206264".8; and the result is
*F, = 4.6097 (N N f ~) s = 3 3' 43".0.
Hence, for a second approximation to the value of F } we have
.F, = 25 36'40".0.
The corresponding value of Nis N, 0.0617653, and hence
AJ F, = 5.199 (N NJ s = 12' 9".4.
The third approximation, therefore, gives F, = 25 24' 30".6, and,
repeating the operation, we get
J F T =2524'27".74.
which requires no further correction.
To find r, we have
which gives
log r = 0.2008544.
Then, v is derived from
tan v cot ^4 tan F,
and we find
v = 67 3' 0".0.
When several places are required, it is convenient to compute
and r by means of the equations
110 THEORETICAL ASTRONOMY.
For the given values of a and e we have log V a(e + 1) = 0.4782649,
logl / a(e 1) = 0.0100829, and hence we derive
v = 67 2' 59".92, log r = 0.2008545.
It remains yet to illustrate the calculation of v and r for elliptic
and hyperbolic orbits in which the eccentricity differs but little from
unity. First, in the case of elliptic motion, let t T= 68.25 days;
e = 0.9675212; and log q = 9.7668134. We compute M from
wherein log C = 9.9601277, which gives
log 3f= 2.1404550.
\Yith this as argument we get, from Table VI.,
V= 101 38' 3".74,
and then with this value of V as argument we find, from Table IX.,
A --= 1540".08, B = 9".506, C= 0".062.
^ _ e
Then we have log i = log = 8.217680, and from the equation
1 -f- e
v = V+ -4(1000
we get
v = V+ 42' 22".28 -f 25".90 -f 0".28 = 102 20' 52".20.
The value of r is then found from
-f e cos v '
namely,
log r = 0.1614051.
We may also determine r and v by means of Table X. Thus, we
first compute M from
Assuming B = 1, we get log M= 2.13757, and, entering Table VI.
with this as argument, we find w = 101 25'. Then we compute A
from
5(l--e
A ~ * 1 9e
NUMERICAL EXAMPLES. Ill
which gives A = 0.024985. With this value of A as argument, we
find, from Table X.,
log B = 0.0000047.
The exact value of M is then found to be
log M= 2.1375635,
which, by means of Table VI., gives
w = 101 24' 36".26.
By means of this we derive
A = 0.02497944,
and hence, from Table X.,
log C =0.0043771.
Then we have
which gives
v = 102 20' 52".20,
agreeing exactly with the value already found. Finally, r is given by
" (1 -{-AC 2 ) cos 1 }*'
from which we get
log r = 0.1614052.
Before the time of perihelion passage, t T is negative ; but the
value of v is computed as if this were positive, and is then considered
as negative.
In the case of hyperbolic motion, i is negative, and, with this dis-
tinction, the process when Table IX. is used is precisely the same
as for elliptic motion; but when table X. is used, the value of A
must be found from
5(e 1)
^(f+
and that of r from
(1 .AC 2 ) cos 8 in'
the values of log B and log C being taken from the columns of the
table which belong to hyperbolic motion.
In the calculation of the position of a comet in space, if the motion
112 THEORETICAL ASTRONOMY.
is retrograde and the inclination is regarded as less than 90, the dis-
tinctions indicated in the formulae must be carefully noted.
42. When we have thus computed the places of a planet or comet
for a series of dates equidistant, we may readily interpolate the places
for intermediate dates by the usual formulae for interpolation. The
interval between the dates for which the direct computation is made
should also be small enough to permit us to neglect the effect of the
fourth differences in the process of interpolation. This, however, is
not absolutely necessary, provided that a very extended series of
places is to be computed, so that the higher orders of differences may
be taken into account. To find a convenient formula for this inter-
polation, let us denote any date, or argument of the function, by
a + na>> and the corresponding value of the co-ordinate, or of the
function, for which the interpolation is to be made, by / (a -f nco).
If we have computed the values of the function for the dates, or
arguments, a to, a, a -f a>, a -f- 2w, &c., we may assume that an
expression for the function which exactly satisfies these values will
also give the exact values corresponding to any intermediate value
of the argument. If we regard n as variable, we mav expand the
function into the series
f(a + no>) =/() + An + JSn 9 + On 9 -f &c. (116)
and if we regard the fourth differences as vanishing, it is only neces-
sary to consider terms involving n s in the determination of the
unknown coefficients A, B, and C. If we put n successively equal
to 1, 0, 1, and 2, and then take the successive differences of these
values, we get
I. Diff. II. Diff. III. Diff.
/(<,-) =f(a)-A + B -C
f(a + 2>) =/(o) + 2A + 4B + 8 C
If we symbolize, generally, the difference f(a + nco) f(a -\-(n 1 ) ), the difference / (a + (n-+ J) a>) /( a + (n J) a)
by /" (a -f- wa>), and similarly for the successive orders of differences^
these may be arranged as follows :
Argument. Function. I. Diff. II. Diff. III. Diff.
a a> /( a _a/)
a /(a) /(-iK> f (a )
INTERPOLATION. 113
Comparing these expressions for the differences with the above, we
get
C=J/"(a + i), J = i/"(a),
A =f (a + J) - if' (a) - J/ (a + ,
which, from the manner in which the differences are formed, give
C= $ (/" (a 4- ,) -/" (a)), B = if" (a),
A =/ (a + ) -/(a) - i/" (a) - J (/" (a + ,) -/" (a)).
To find the value of the function corresponding to the argument
a + Jw, we have n = , and, from (116),
/(a + ) =/(a) + 14 + ^ + 4 a
Substituting in this the values of A, B, and (7, last found, and re-
ducing, we get
/(a -H0 = i (/(a + +/()) ~ I (i (/" ( + ) +/" W)),
in which only fourth differences are neglected, and, since the place
of the argument for n = is arbitrary, we have, therefore, generally,
/(a + (n + ) ) = i (/( + (* + !) ") +/(a + no,))
- 4 (i (/' (+( + 1) +/' ( + ))) (1.17)
Hence, to interpolate the value of the function corresponding to a
date midway between two dates, or values of the argument, for which
the values are known, we take the arithmetical mean of these two
known values, and from this we subtract one-eighth of the arith-
metical mean of the second differences which are found on the same
horizontal line as the two given values of the function.
By extending the analytical process here indicated so as to include
the fourth and fifth differences, the additional term to be added to
equation (117) is found to be
+ no,))),
and the correction corresponding to this being applied, only sixth
differences will be neglected.
It is customary in the case of the comets which do not move too
rapidly, to adopt an interval of four days, and in the case of the
asteroid planets, either four or eight days, between the dates for which
the direct calculation is made. Then, by interpolating, in the case of
an interval co, equal to four days, for the intermediate dates, we
obtain a series of places at intervals of two days ; and, finally, inter-
8
114 THEORETICAL ASTRONOMY.
polating for the dates intermediate to these, we derive the places at
intervals of one day. When a series of places has been computed,
the use of differences will serve as a check upon the accuracy of the
calculation, and will serve to detect at once the place which is not
correct, when any discrepancy is apparent. The greatest discordance
will be shown in the differences on the same horizontal line as the
erroneous value of the function ; and the discordance will be greater
and greater as we proceed successively to take higher orders of dif-
ferences. In order to provide against the contingency of systematic
error, duplicate calculation should be made of those quantities in
which such an error is likely to occur.
The ephemerides of the planets, to be used for the comparison of
observations, are usually computed for a period of a few weeks before
and after the time of opposition to the sun ; and the time of the
opposition may be found in advance of the calculation of the entire
ephemeris. Thus, we find first the date for which the mean longitude
of the planet is equal to the longitude of the sun increased by 180 ;
then we compute the equation of the centre at this time by means of
the equation (53), using, in most cases, only the first term of the
development, or
v M =
e being expressed in seconds. Next, regarding this value as con-
stant, we find the date for which
L -J- equation of the centre
is equal to the longitude of the sun increased by 180 ; and for this
date, and also for another at an interval of a few days, we compute
u, and hence the heliocentric longitudes by means of the equation
tan (I Q ) = tan u cos i.
Let these longitudes be denoted by I and l f , the times to which they
correspond by t and t' 9 and the longitudes of the sun for the same
times by O and Q ' ; then for the time t , for which the heliocentric
longitudes of the planet and the earth are the same, we have
or (118)
the first of these equations being used when I 180 O is less
TIME OF OPPOSITION. 115
than V 180 O'. If the time t differs considerably from t or
t', it may be necessary, in order to obtain an accurate result, to repeat
the latter part of the calculation, using t Q for t. and taking t' at a
small interval from this, and so that the true time of opposition shall
fall between t and t' . The longitudes of the planet and of the sun
must be measured from the same equinox.
When the eccentricity is considerable, it will facilitate the calcula-
tion to use two terms of equation (53) in finding the equation of the
centre, and, if e is expressed in seconds, this gives
_ M = 2e sin Jf + 5 . t. s i n 2M 9
4 S
8 being the number of seconds corresponding to a length of arc equal
to the radius, or 206264".8 ; and the value of v M will then be
expressed in seconds of arc. In all cases in which circular arcs are
involved in an equation, great care must be taken, in the numerical
application, in reference to the homogeneity of the different terms.
If the arcs are expressed by an abstract number, or by the length of
arc expressed in parts of the radius taken as the unit, to express them
in seconds we must multiply by the number 206264.8 ; but if the
arcs are expressed in seconds, each term of the equation must contain
only one concrete factor, the other concrete factors, if there be any,
being reduced to abstract numbers by dividing each by s the number
of seconds in an arc equal to the radius.
43. It is unnecessary to illustrate further the numerical application
of the various formulae which have been derived, since by reference
to the formulae themselves the course of procedure is obvious. It
may be remarked, however, that in many cases in which auxiliary
angles have been introduced so as to render the equations convenient
for logarithmic calculation, by the use of tables which determine the
logarithms of the sum or difference of two numbers when the loga-
rithms of these numbers are given, the calculation is abbreviated,
and is often even more accurately performed than by the aid of the
auxiliary angles.
The logarithm of the sum of two numbers may be found by meana
of the tables of common logarithms. Thus, we have
If we put
log tan x = J (log b log a),
116 THEORETICAL ASTRONOMY.
we shall have
log (a -f 6) = log a 2 log cos x,
or
log (a -f- 6) = log b 2 log sin x.
The first form is used when cos x is greater than sin x, and the second
form when cos a; is less than sin a;.
It should also be observed that in the solution of equations of the
form of (89), after tan (A 0) using the notation of this particular
case has been found by dividing the second equation by the first,
the second members of these equations being divided by cos (A 0)
and sin (^ ), respectively, give two values of J cos /?, which should
agree within the limits of the unavoidable errors of the logarithmic
tables ; but, in order that the errors of these tables shall have the
least influence, the value derived from the first equation is to be pre-
ferred when cos (A ) is greater than sin (^ 0), and that derived
from the second equation when cos (A 0) is less than sin (^ ).
The value of J, if the greatest accuracy possible is required, should
be derived from J cos /9 when /9 is less than 45, and from J sin /?
when /? is greater than 45.
In the application of numbers to equations (109), when the values
of the second members have been computed, we first, by division,
find tan|(&' -f- ^ ) an( ^ tanj(&' fti ); then, if sin|(&' -f w ) is
greater than cos|(&' + w ), we find cos^' from the first equation;
but if sin f (&>' -f = sin e sin & .
In all cases, care should be taken in determining the quadrant in
which the angles sought are situated, the criteria for which are fixed
either by the nature of the problem directly, or by the relation of the
algebraic signs of the trigonometrical functions involved.
DIFFERENTIAL FORMULAE. 117
CHAPTER II.
INVESTIGATION OF THE DIFFERENTIAL FORMULAE WHICH EXPRESS THE RELATION
BETWEEN THE GEOCENTRIC OR HELIOCENTRIC PLACES OF A HEAVENLY BODY
AND THE VARIATION OF THE ELEMENTS OF ITS ORBIT.
44. IN many calculations relating to the motion of a heavenly
body, it becomes necessary to determine the variations which small
increments applied to the values of the elements of its orbit will pro-
duce in its geocentric or heliocentric place. The form, however, in
which the problem most frequently presents itself is that in which
approximate elements are to be corrected by means of the differences
between the places derived from computation and those derived from
observation. In this case it is required to find the variations of the
elements such that they will cause the differences between calculation
and observation to vanish ; and, since there are six elements, it follows
that six separate equations, involving the variations of the elements
as the unknown quantities, must be formed. Each longitude or right
ascension, and each latitude or declination, derived from observation,
will furnish one equation ; and hence at least three complete observa-
tions will be required for the solution of the problem. When more
than three observations are employed, and the number of equations
exceeds the number of unknown quantities, the equations of condi-
tion which are obtained must be reduced to six final equations, from
which, by elimination, the corrections to be applied to the elements
may be determined.
If we suppose the corrections which must be applied to the ele-
ments, in order to satisfy the data furnished by observation, to be so
small that their squares and higher powers may be neglected, the
variations of those elements which involve angular measure being
expressed in parts of the radius as unity, the relations sought may
be determined by differentiating the various formulae which determine
the position of the body. Thus, if we represent by 6 any co-ordi-
nate of the place of the body computed from the assumed elements
of the orbit, we shall have, in the case of an elliptic orbit,
118 THEORETICAL ASTRONOMY.
M Q being the mean anomaly at the epoch T. Let 6 f denote the value
of this co-ordinate as derived directly or indirectly from observation ;
then, if we represent the variations of the elements by ATI, A&, Ai,
&c., and if we suppose these variations to be so small that their
squares and higher powers may be neglected, we shall have
do do , do do
d A *+5a A a+-3r A< +27
do .... dO
.
The differential coefficients - 1 , ^ , &c. must now be derived from
d* a$>
the equations which determine the place of the body when the ele-
ments are known.
We shall first take the equator as the plane to which the positions
of the body are referred, and find the differential coefficients of the
geocentric right ascension and declination with respect to the elements
of the orbit, these elements being referred to the ecliptic as the fun-
damental plane. Let x, y, z be the heliocentric co-ordinates of the
body in reference to the equator, and we have
9 =/(*, y, z),
or
do . , do , , do ,
dO = -j- dx -4- -j- dy 4- -,- dz.
dx r dy y dz
Hence we obtain
dO _ d0_ dx dO_ dy_ dO_ dz ;
~dn dx ' d-K dy'dTtdz'dn*
and similarly for the differential coefficients of with respect to the
other elements. We must, therefore, find the partial differential co-
efficients of d with respect to #, y, and z, and then the partial differen-
tial coefficients of these co-ordinates with respect to the elements. In
the case of the right ascension we put = oc, and in the case of the
declination we put 6 = 3.
45. If we differentiate the equations
x + X= A cos 8 cos a,
y -{- Y= A cos d sin a,
z -\- Z = A sin d,
regarding X, F, and Z as constant, we find
DIFFERENTIAL FORMULAE. 119
dx = cos u cos 8 d 4 A sin a cos 8 da, A cos a sin d dd,
dy == sin a cos d d A -\- A cos a cos $ da, A sin a sin d dd,
dz = sin d d A -j- A cos d dd.
From these equations, by elimination, we obtain
.. , sin a , cos a , /rk>l
cos d da, = az -| -p cfy, (3)
,, cos a sin d sin a sin d cos<5
<** = dx ~
Therefore, the partial differential coefficients of a and d with respect
to the heliocentric co-ordinates are
da, sin a dS cos a sin <5
dx ~ A ' dx ~ A
do, cos a a*<5 sin a sin 5 ...
cos # -j- = -j , T- = -. , (4)
dy A dy A
da. dd cos#
j = 0. j == ^
dz dz A
Next, to find the partial differential coefficients of the co-ordinates
x, y, Zj with respect to the elements, if we differentiate the equations
(100)!, observing that sin a, sin 6, sin c, J., B, C, are functions of ft
and i y we get
dx = - dr -f # cot ( J. -|- w) efot -J- -j^- rfft -f -^r c^i,
df2 = ? dr + 2 cot (0+ u) du + ^ rf^ + -jjjr di.
To find the expressions for ~, -^, &c., we have the equations
d/oo di'
x = r cos w cos & r sin M sin & cos i,
y r cos -w sin & cos e -j- r sin w cos ^ cos i cos e r sin u sin ^ sin e,
3 = r cos u sin & sin s -j- r sin u cos & cos i sin e -f r sin u sin i cos e.
which give, by differentiation,
-=^- = r cos u sin & r sin u cos & cos i,
d&
_ J^ = r cos w cos & cos e r sin w sin ft cos i cos e,
120 THEORETICAL ASTRONOMY.
dz
- = r cos u cos & sin e r sin u sin & cos i sin e.
&
dte
- = r sm it sin $6 sin *>
ew
and remain unchanged; and we
have, also,
da; dy . dz , rtN
T^- = rsmttcosa. -^n- = rsmwcoso. -77- = rsmttcosc. (9)
di di di
It is advisable, in order to avoid the use of two sets of formulae, in
part, to regard the motion as direct and the inclination as susceptible
of any value from to 180. If the elements which are given are
for retrograde motion, we take the supplement of i instead of i; and
if we designate the longitude of the perihelion, when the motion is
considered as being retrograde, by (TT), we shall have
If we introduce, as one of the elements of the orbit, the distance
of the perihelion from the ascending node, we have
du = dv -f~ d* *
and, hence,
dx dx
122 THEORETICAL ASTKONOMY.
The values of > - , and must, in this case, be found by means
of the equations (5).
By means of these expressions for the differential coefficients of the
co-ordinates x, y, z, with respect to the various elements, and those
given by (4), we may derive the differential coefficients of the geo-
centric right ascension and declination with respect to the elements
&, ij and n or o>, and also with respect to r and v, by writing suc-
cessively a and d in place of 6, and &, i, &c., in place of TT in the
equation (2). The quantities r and v, however, are functions of the
remaining elements y>, M Q , and /*; and we have
, dr , dr ,,.. dr ,
, dv j dv , f dv ,
dv =^ -j- d(p -\- - J1 r F dM Q -f- -j dp.
d, Jf , and //, are
dx
d
and have been
aM d/j. a/j.
found, the partial differential coefficients of the heliocentric co-ordj
nates with respect to the elements tp, M w and // will be completely
determined, and hence, by means of (2), making the necessary
changes, the differential coefficients of a and d with respect to these
elements.
46. If we differentiate the equation
M=E
DIFFERENTIAL FORMULA. 123
we shall have
dM= dE(l e cos E) cos
we get
J a cosy
a? cos y j , , . sin v /
dv = - - ---
sn v a cos
2 ^ . \ ,
- + \\d =PJ and - = 1 + sin cos v, this becomes
QJ COS (P ^ \ ? /< r*N
dv = dM -4- 1 h tan y cos v] sm v c?^. (12)
r 2
If we differentiate the equation
r = a (1 e co .E),
we shall have
dr = - da -f ae sin .E , the result
is
.__ r l ^ a _}- a tan ^ sin-y dM-\- (ae sin ^J sin v a cos p cos E) dy.
124 THEORETICAL ASTRONOMY.
X T . -n sin v cos sin v dM a cos
. (14)
Further, we have
T being the epoch for which the mean anomaly is M w and
A; 1/1 + m
^= - 3 --
a^
Differentiating these expressions, we get
), and q = r cos 2 Jv, we have
1 3& (* T) tan Jv 1 0-21 i
x i_ =-(14- tan 2 \v 3 sm 2 \v sm 2 \v tan 2
q r*V 2q r
__ cosv
r
We also have
tan 4v
Therefore, equation (19) reduces to
(20)
, ^ N
(21)
-.
I/ 2q
If we introduce cZ log q instead of dg, this equation becomes
From the equations (17), (18), (20), and (21), we derive
dr ksinv dv IcV 2q
dT. ~ V5q 5? r 2
dr dv 3& (t T)
V^V/O /. 7 y > V^^y
dq dq
dr_ _qcosv dv 3^ (t T} 1/2?
" " "
and then we have, for the differential coefficients of x with respect to
T and q or log 7,
DIFFEEENTIAL FOKMUL^. 127
dx _ dx dr dx dv dx dx dr dx dv
dT == ~d^"dT + ~d^"dT' ~dq = ~dr ' ~efy + ~dv "dq
dx dx dr dx dv
'
d log q dr d log q dv ' d log q'
and similarly for the differential coefficients of y and z with respect
to these elements. The expressions for the partial differential co-
efficients of x, y, and z, respectively, with respect to r and v are the
same as already found in the case of elliptic motion. We shall thus
obtain the equations which express the relation between the variations
of the geocentric places of a comet and the variation of the parabolic
elements of its orbit, and which may be employed either to correct
the approximate elements by means of equations of condition fur-
nished by comparison of the computed place with the observed place,
or to determine the change in the geocentric right ascension and
declination corresponding to given increments assigned to the ele-
ments.
48. We may also, in the case of an elliptic orbit, introduce T, q,
and e instead of the elements T / a sin v tan 4 sin v \ ,
ai? = ^,dJ\ n I 1 h : I 4'
r tan F \ r cos ^ sin ^ /
DIFFERENTIAL FORMULA. 129
Bufc, since r sin v = a tan $ tan F, and p a tan 2 ^ this reduces to
* = ^v5i-(| + 1 )^d*. (25)
If we differentiate the equation
we get
r , , T, d-F 7 a tan 4* T
dr = - da -f ae tan* F ^ H -- ^ -- cfy.
a smF cosF cos4-
Trrr
Substituting in this equation the value of ^ we obtain
7 r , , a?e tan F 7 . _ / a 2 e tan 2 jP a \ tan 4*
dr = - da -\ -- dN tt I ---- =f )
a r \ r cos jP/ cos 4
which is easily reduced to
r 7 sinv 7 , T . pi r ae
dr = a da+a ^ dN + -r(^-^
But, since
r ae a
cos F ~~ cos 2 F ~~ cos F'
this reduces to
r
or
7 r , sin v , , r . cos v , , n
'ia^- Si:
4sinw
which, by means of (76)^ reduces to
dv lctT
' '
If we introduce the quantity M which is used as the argument in
finding w by means of Table VI., this equation becomes
dv 9 M cos 2 ^w . 8 tan Av f
\6y)
de ~ 2 (1 + 9e) " 75 tan^w v (1 + e) (1 + 9e)'
This equation remains unchanged in the case of hyperbolic motion,
the value of C being taken from the column of the table which cor-
responds to this case-; and it will furnish the correct value of -=- in
de
all cases in which the last term of equation (23) is not conveniently
C?7*
applicable. The value of -- is then given by the equation (32).
When the eccentricity differs very little from unity, we may put
B = 1, and _
tan \w = tan v j/^ (i + g e ),
cos 2 ^w = O 2 cos 2 %v.
Then we shall have
^ 2 2k (t T)
C 2 sm v .- , cos 4 4w.
; 1/2 g*
The equation
= (1 + A C") cos 2 Jw = (1 + i-^) cos 2 >,
gives
| l = (1 + p) cos 4 Jw = Ccos 4 w.
Hence we derive
T)
NUMERICAL EXAMPLES. 135
[f we substitute this value in equation (39), and put C 2 (1 + e) 2,
we get
im
r 2 (1 + e) (1 + 9e)'
and when e 1, this becomes identical with equation (31).
51. EXAMPLES. We will now illustrate, by numerical examples,
the formulae for the calculation of the variations of the geocentric
right ascension and declination arising from small increments assigned
to the elements. Let it be required to find for the date 1865 Feb-
ruary 24.5 mean time at Washington, the differential coefficients of
the right ascension and declination of the planet Eurynome with
respect to the elements of its orbit, using the data and results given
in Art. 41. Thus we have
a = 181 8' 29".29, d = 4 42' 21".56, log J = 0.2450054,
log r = 0.428285, v = 129 3' 50".5, u = 326 41' 40".l,
A = 296 39' 5".0, B = 205 55' 27".l, C= 212 32' 17".7,
log sin a = 9.999716, log sin b = 9.974825, log sin c = 9.522219,
log x = 0.425066 n , log y = 9.511920, log z = 8.077315,
e = 23 27' 24' 7 .0, tT= 420.714018.
First, by means of the equations (4), we compute the following
values:
log cos d ~ = 8.054308, log -^ = 8.668959 W ,
log C os d ^ = 9.754919 , log -^ = 6.968348 ,
dy dy
log ^ 9.753529.
Then we find the differential coefficients of the heliocentric co-ordi-
nates, with respect to TT, &, i, v, and r, from the formulae (7), which
give
io s = io = - 399496 -
log -- = 7.876553, log -- = 8.830941, log -- = 9.222898.,
log -T- = 8.726364, log -- = 9.687577, log -- = 0.142443
,,,
log ~ =-- 9.996780 n , log -^- = 9.083635, log -^ = 7.649030.
136 THEORETICAL ASTRONOMY.
, dx dy .. dz .
In computing the values of -TT-, -^p and -TT, those of cos a, cos 6,
and' cos c may generally be obtained with sufficient accuracy from
Bin a, sin 6, and sine. Their algebraic signs, however, must be
strictly attended to. The quantities sin a, sin 6, and sin c are always
positive ; and the algebraic signs of cos a, cos 6, and cos c are indicated
at once by the equations (101)i, from which, also, their numerical
values may be derived. In the case of the example proposed, it will
be observed that cos a and cos b are negative, and that cos c is positive.
To find the values of cos d -j- and -7-, we have, according to equa-
(ITC CfrTT
tion (2),
. da, ^ da dx , .dady , . ., N
cos<5 = cos<5- -fcosd --, (41)
dn dx dn dy d*
which give
dd _ dd dx dd dy dd dz
dn ~ dx dn dy dn dz dn'
= + 1.42345, ==_ 0.48900.
n dv dn dv
In the case of &, i, and r, we write these quantities successively in
place of Tt in the equations (41), and hence we derive
cos 8 4^ = - 0-03845, -^- = - 0.09533,
dQ> "66
cos d -^- = 0.27641, -2L = 0.78993,
ai
cos d ~ = 0.08020, -^- = + 0.04873
Next, from (16), we compute the following values:
log ^ = 0.179155, log ^- = 9.577453, log ~ = 2.376581,,
G dy> dM dp
log ^ = 0.171999, log -^r = 9.911247, log ~ = 2.535234.
& d dM Q dfji
j$y (j y
\Ve may now find ^ , -^r^, &c. by means of the equations (11),
and thence the values of cos d -= , -3, &c. : but it is most convenient
d -7-, and -7-
dr dv dr dv
in connection with the numerical values last found, according to the
NUMERICAL, EXAMPLES. 137
equations which result from the analytical substitution of the expres-
sions for -j-, -j > -y-, &c., in equation (2), writing successively
and fJL in place of TT. Thus, we have
. da, .da,dr. .da, dv
cos <5 - = cos d . -- k cos <5 -r- -r ,
a?' ar* a^ av a?>
d<5 __ d dr dd dv
d
+ 1.13004A3f -}- 507.264A/.,
A5 = 0.48900 ATT 0.09533 A & 0.78993Ai 0.65307 A
0.38023A^f 179.315A/Z.
To prove the calculation of the coefficients in these equations, we
assign to the elements the increments
Ajf = + 10", ATT == - 20", A ft = - 10", Ai = + 10",
A^ = + 10", A/ = + 0".01,
BO that they become
Epoch = 1864 Jan. 1.0 Greenwich mean time.
M = 1 29' 50".21
TT = 44 20 13 .09 ^
ft = 206 42 30 .13 > Mean Equinox 1864.0
t = 4 37 .51 J
V = 11 16 1 .02
log a = 0.3881288
L = 928".56745
With these elements we compute the geocentric place for 1865 Feb-
ruary 24.5 mean time at Washington ; and the result is
a = 181 8' 34".81, d = - 4 42' 30".58, log A = 0.2450284,
138 THEORETICAL ASTRONOMY.
which are referred to the mean equinox and equator of 1865.0. The
difference between these values of a and d and those already given, as
derived from the unchanged elements, gives
Aa = + 5".52, COS * Aa = + 5".50, A sin v,
dv ~ a 2 cos
--
T
-- = h sinH=
9r \
smv(t T) y- 206264.8 J,
cy , ,-.. a* cos , we must put
=r-,, (63)
-- -j-, - 5 -
aq aq
and the equations become
= sn
S 9 (64)
ooi^-^iiaU + Jr+ii),
4^ = \ B sin (jB H- H+ u).
In the numerical application of these formulae, the values of the
second members of the equations (63) are found as already exem-
plified for the cases of parabolic orbits and of elliptic and hyperbolic
orbits in which the eccentricity differs but little from unity. In the
same manner, the differential coefficients of A and ft with respect to
any other elements which determine the form of the orbit may be
computed.
NUMERICAL EXAMPLES. 149
In the case of a parabolic orbit, if the parabolic eccentricity is
supposed to be invariable, the terms involving e vanish. Further,
in the case of parabolic elements, we have
. ~ dr ksmv dv
= --
which give
tan G = tan
V2
-, which is the
expression for the linear velocity of a comet moving in a parabola.
Therefore,
= 212 32' 17".71,
and
a/ == w + a> = 50 10' 7".29,
which are the elements which determine the position of the orbit in
space when the equator is taken as the fundamental plane. These
elements are referred to the mean equinox and equator of 1865.0.
Writing a and d in place of ^ and /9, and &', i f , w f in place of &, i,
and co, respectively, we have
A Q sin A cos (<*&') cos i r , A cos A = sin (a &') ;
n sin N= sin i f , n cosN= cost' sin (a &');
J5 sin B = n sin (N + #) B Q cos -B = sin <5 cos (a &') ;
<7 sin 0= cos (a &'), ^o c <> s C^sin (a &') cosi';
A sin Z> = cos i', D cos D = sin i' sin (a & ') ;
/ sin F=a cos ^ cos v,
/ 2 \
/ cos F= 1 -- h tan ? cos v } r sin v;
Vcos?' /
gsmG = a tan f ,
sin i sin & = sin i' sin & f >
sin i sin w = sin & ' sin e,
sin i cos w = cos sin i' sin e cos i' cos & ',
152 THEORETICAL ASTRONOMY.
from which the values of &, i, and co may be found from those of
& ' and i r . If we differentiate the first of these equations, regarding
e as constant, and reduce by means of the other given relations, we
get
di = cos to di' -f- sin a> sin i' d & '. (68)
Interchanging i and 180 i', and also & and &', we obtain
di' = cos tt> di sin w sin idQ.
Eliminating di from these equations, and introducing the value
sin i' sin &
sini sin&y'
the result is
sin Q sin to K
Biny ami
If we differentiate the expression for cos co derived from the same
spherical triangle, and reduce, we find
da> cos i dl cos i' dQ'.
Substituting for dQ, its value given by the preceding equation, and
reducing by means of
sin & ' cos i' = sin & cos w cos i cos Q, sin w ,
we get
sin tn ein m
(70)
The equations (68), (69), and (70) give the partial differential co-
efficients of & , i, and w with respect to & ; and i', and if we sup-
pose the variations of the elements, expressed in parts of the radius
as unity, to be so small that their squares may be neglected, we shall
have
sin oj n
*^=Esf'"
sin
At = sin ta sin i' A Q f -f- cos % Ai r ,
If we apply these formulae to the case of Eurynome, the result is
W O = 4.420A&' +
& = _ 3.488A Q' + 6.686A/,
Ai = 0.179A^' 0.843 AI* ;
DIFFERENTIAL FORMULAE. 153
and if we assign the values
A' = 14".12, Ai' = 8".86, AC// = 6".64,
we get
AOI O = + 3".36, A ^ = 10".0, Ai = + 10".0, A> = 10".0,
and, hence, the elements which determine the position of the orbit in
reference to the ecliptic.
The elements a/, & ', and i f may also be changed into those for
which the ecliptic is the fundamental plane, by means of equations
which may be derived from (109)! by interchanging & and &' and
180 i'andt.
56. If we refer the geocentric places of the body to a plane whose
inclination to the plane of the ecliptic is i y and the longitude of whose
ascending node on the ecliptic is &, which is equivalent to taking
the plane of the orbit corresponding to the unchanged elements as
the fundamental plane, the equations are still further simplified.
Let x'j y'j z f be the heliocentric co-ordinates of the body referred to
a system of co-ordinates for which the plane of the unchanged orbit
is the plane of xy, the positive axis of x being directed to the as-
cending node of this plane on the ecliptic; and let x, y, z be the
heliocentric co-ordinates referred to a system in which the plane of
xy is the plane of the ecliptic, the positive axis of x being directed
*o the point whose longitude is Q . Then we shall have
dtf = dy cos i -f- dz sin i t
dz' = dy sin i -f- dz cos i.
Substituting for dx, dy y and dz their values given by the equations
(47), we get
x r
dx f = dr r sin u du r sin u cos i dQ,
dy' = dr -f- r cos u du -\- r cos u cos i d& y
dz' = dr r cos u sin i dQ> + r sul u di*
It will be observed that we have, so long as the elements remain
unchanged,
a/ = r cos u, y f = r sin u, z' = 0,
154 THEOKETICAL ASTRONOMY.
and hi.'nce, omitting the accents, so that x 9 y, z will refer to the plane
of the unchanged orbit as the plane of xy, the preceding equations
give
dx = cos u dr r sin u du r sin u cos i dQ ,
dy = sin u dr -\- r cos u du -j- r cos u cos i dQ ,
dz r cos u sin ^ dQ> -f- r sin i* cfo*.
The value of ^> is subject to two distinct changes, the one arising
fiom the variation of the position of the orbit in its own plane, and
the other, from the variation of the position of the plane of the orbit.
Let us take a fixed line in the plane of the orbit and directed from
the centre of the sun to a point the angular distance of which, back
from the place of the ascending node on the ecliptic, we shall desig-
nate by ff- and let the angle between this fixed line and the semi-
transverse axis be designated by . Then we have
X = a> -f ff.
The fixed line thus taken is supposed to be so situated that, so long
as the position of the plane of the orbit remains unchanged, we have
But if the elements which fix the position of the plane of the orbit
are supposed to vary, we have the relations
dff = cosi dl,
da> = d% cos i d& , (72)
dn = dx + (1 cosi) d& = d x + 2 sm*i d&.
Now, since u = v -f- to, we have
u=v+X ff *
and
du dv -f- dx dff = dv -f- d% cos i d &
Substituting this value of du in the equations for dx, dy, dz, they
reduce to
dx = cos u dr r sin u dv r sin u dx,
dy = sin u dr -f r cos u dv + r cos u d%, (73)
dz = r cos u sin i dQ, -j- r sin u di.
The inclination is here supposed to be susceptible of any value from
to 180, and if the elements are given with the distinction of
retrograde motion we must use 180 i instead of i.
Let us now denote by 6 the geocentric longitude of the body mea-
sured in the plane of the unchanged orbit (which is here taken as the
DIFFERENTIAL FORMULA.
155
fundamental plane) from the ascending node of this plane on the
ecliptic, and let the geocentric latitude in reference to the same plane
be denoted by y. Then we shall have
x -f- X A cos y cos 0,
y + Y= A cos y sin 0,
z + Z = A sin y,
in which X, F, Z are the geocentric co-ordinates of the sun referred
to the same system of co-ordinates as x, y } and z. These equations
give, by differentiation,
dx = cos y cos 6 d A A sin y cos dy A cos y sin dO t
dy = cos y sin 6 d A A sin y sin dfy -f~ ^ cos ^ cos ^ ^>
efe = sin y dA -\- A cos ^ cfy ;
and hence we obtain
7 . sin , . cos
cos y do = dx -\ -. dy,
, _ sin y cos , __ sin 17 sin .
(74)
These give
dO sin
dO cos
cos y -j- = . ,
eta ^
cos^ , = -. ,
dy A
cfy sinvy cos 6
dy siny sin0
dx A
dy A
dd
cos y -=- = ;
c?>? cosi?
and from (73) we get
dx
~ = COS U,
dr
dx _ dx _
dv dy
dz
'dr
-0:
" = 0>
dx .
dy
-^- = sin,
c?y c?v dz dz A
-/- = -/- = r cos w, - j - ; -y- = ;
(2v c?/ dfv ^/
^2
-^ = r cos it SIP x ;
ds
- - = r sin w.
d
Substituting the values thus found, in the equations
dO dO dx . dO dy
cos y 3- = cos y -=- -j h cos 7 j~ ' j >
dv ax av ay av
dy dy dx dy dy dy dz
dv dx dv dy dv dz d^
156 THEORETICAL ASTRONOMY.
we get
do do r fn
cos >? -=- = cos y -j- = -, cos (0 u) t
dv d% A
(76)
In a similar manner, we derive
do 1 . fa d^ 1
cos *] 3 = sin (0 u), ~ = sm y cos (0 u) t
dO df) r
cos V) -= = 0, -- = cos yj sm i cos u, (77 )
do di) . r
COS Tj TV- = 0, yf- = -f- COS I] Sin W.
CM Cw d
If we introduce the elements ^, Jtf" , and //, which determine r and v,
we have, from
de dO dr , rf^ c?v
cos i? -j = cos T? -j- -j h cos i? -j- -j ,
1 d [ ' dv dy
df) _ df) dr df) dv
d
? -- = ?cos(0 u G),
,, , i I (79)
dO h , dr) h . f _.
cos r) = cos (0 u L ). ^ = -j- sm TJ sm (0 u M ).
d/j- A dp A
If we express r and v in terms of the elements T, g, and e, the
values of the auxiliaries /, g, A, F, &c. must be found by means of
(64) ; and, in the same manner, any other elements which determine
the form of the orbit and the position of the body in its orbit, may
be introduced.
The partial differential coefficients with respect to the elements
having been found, we have
do do do dO
COS JJ A0 = COS r) -- Ay 4- COS f] -7 ACP -f- COS r] -rTT A M n + COS Tt -7- &fJ.
d% dy dM Q dp
DIFFERENTIAL FORMULA. 157
from which it appears that, by the introduction of as one of the
elements of the orbit, when the geocentric places are referred directly
to the plane of the unchanged orbit as the fundamental plane, the
variation of the geocentric longitude in reference to this plane depends
on only four elements.
57. It remains now to derive the formulae for finding the values
of T) and 6 from those of X and /?. Let X Q , y , z be the geocentric co-
ordinates of the body referred to a system in which the ecliptic is
the plane of xy, the positive axis of x being directed to the point
whose longitude is & ; and let #/, y ', z ' be the geocentric co-ordi-
nates of the body referred to a system in which the axis of x remains
the same, but in which the plane of the unchanged orbit is the plane
of xy; then we shall have
x A cos /? cos (A ft ), x ' = A cos i? cos 0,
y o = J cos /? sin (A Q ), y ' = A cos T? sin 0,
Z Q = J sin , z ' = A sin iy,
and also
< = *0
yo = y cos * -f z o sin *
z ' = y Q sin i + z cos i.
Hence we obtain
COS r) COS = COS /? COS (A & ),
cos T) sin = cos /? sin (A & ) cos i + sin sin t, (80 )
sin 17 = cos sin (A & ) sin i -\- sin /? cos i.
These equations correspond to the relations between the parts of a
spherical triangle of which the sides are i, 90 57, and 90 ft
the angles opposite to 90 ^ and 90 ft being respectively
90 -f- (X ^) and 90 0. Let the other angle of the triangle be
denoted by f, and we have
cos 17 sin Y = sin * cos (A & ),
cos T? cos r = sin i sin (A & ) sin ft -f- cos i cos /?.
(81)
The equations thus obtained enable us to determine y, 0, and f from
I and /9. Their numerical application is facilitated by the intro-
duction of auxiliary angles. Thus, if we put
, ,
n cos ^= cos j9 sin (A ft),
158 THEORETICAL ASTRONOMY.
in which n is always positive, we get
cos TI cos 6 = cos /? cos (A & ),
cos rj sin 6 = n cos (N t), (83)
sin 7] = ra sin (JV i\
from which ^ and may be readily found. If we also put
n' sin N' = cos i,
n' cos JV = sin i sin (A ft ),
we shall have
cot N r - tan i sin (A ^ ),
If f is small, it may be found from the equation
sin* cos (A -S
cos??
The quadrants in which the angles sought must be taken, are easily
determined by the relations of the quantities involved ; and the
accuracy of the numerical calculation may be checked as already
illustrated for similar cases.
If we apply Gauss's analogies to the same spherical triangle, we get
fiin (45 - j,) sin (45 - J (0 + r )) =
cos (45 -f i (A - )) sin (45 J ( + i)),
sin (45 - j?) cos (45 - J (* + r )) =
sin (45 + i (A - &)) sin (45 - J (0 0),
cos (45 J?) sin (45 j (0 r)) = (87)
cos (45 + i (A - 8)) cos (45 - i (p + t)),
cos (45 - ^) cos (45 - J (? = sin Y cos /5 AA -f cos y A/1
The value of p required in the application of numbers to these
equations may generally be derived with sufficient accuracy from
(86), the algebraic sign of cosf being indicated by the second of
equations (81) ; and the values of rj and 6 required in the calculation
of the differential coefficients of these quantities with respect to the
elements of the orbit, need not be determined with extreme accuracy.
58. EXAMPLE. Since the spherical co-ordinates which are fur-
nished directly by observation are the right ascension and declina-
tion, the formulae will be most frequently required in the form for
finding rj and from a and d. For this purpose, it is only necessary
to write a and d in place of A and /9, respectively, and also & ', V,
co f , %', and u' in place of &, i, a), %, and u, in the equations which
have been derived for the determination of rj and 0, and for the
differential coefficients of these quantities with respect to the elements
of the orbit.
To illustrate this clearly, let it be required to find the expressions
for cos rj A# and &rj in terms of the variations of the elements in the
case of the example already given ; for which we have
d \ d i s a/j.
dd I dd\ nf i dd
in which s = 206264".8. If the values of the differential coefficients
with respect to fjt and -7 ~n - > and -7-= - by
d
) du -f- sin (I Q> ) di,
which determine the relations between the variations of the elements
of the orbit and those of the heliocentric longitude and latitude.
By differentiating the equations (88) w neglecting the latitude of
DIFFERENTIAL FORMULA. 165
the sun, and considering 1, /9, J, and O as variables, we derive, after
reduction,
r>
cos /? dX = _ cos (A O)dO,
R (94)
dp = -- sin sin A O
which determine the variation of the geocentric latitude and longitude
arising from an increment assigned to the longitude of the sun. It
appears, therefore, that an error in the longitude of the sun will
produce the greatest error in the computed geocentric longitude of a
heavenly body when the body is in opposition.
166 THEORETICAL ASTRONOMY.
CHAPTER III.
INVESTIGATION OP FORMULAE FOR COMPUTING THE ORBIT OF A COMET MOVING IN
A PARABOLA, AND FOR CORRECTING APPROXIMATE ELEMENTS BY THE VARIATION
OF THE GEOCENTRIC DISTANCE.
61. THE observed spherical co-ordinates of the place of a heavenly
body furnish each one equation of condition for the correction of the
elements of its orbit approximately known, and similarly for the
determination of the elements in the case of an orbit wholly unknown ;
and since there are six elements, neglecting the mass, which must
alw r ays be done in the first approximation, the perturbations not
being considered, three complete observations will furnish the six
equations necessary for finding these unknown quantities. Hence,
the data required for the determination of the orbit of a heavenly
body are three complete observations, namely, three observed longi-
tudes and the corresponding latitudes, or any other spherical co-
ordinates which completely determine three places of the body as
seen from the earth. Since these observations are given as made at
some point or at different points on the earth's surface, it becomes
necessary in the first place to apply the corrections for parallax. In
the case of a body whose orbit is wholly unknown, it i? impossible
to apply the correction for parallax directly to the place of the body ;
but an equivalent correction may be applied to the places of the
earth, according to the formula? which will be given in the next
chapter. However, in the first determination of approximate ele-
ments of the orbit of a comet, it will be sufficient to neglect entirely
the correction for parallax. The uncertainty of the observed places
of these bodies is so much greater than in the case of well-defined
objects like the planets, and the intervals between the observations
which will be generally employed in the first determination of the
orbit will be so small, that an attempt to represent the observed places
with extreme accuracy will be superfluous.
When approximate elements have been derived, we may find the
distances of the comet from the earth corresponding to the three
observed places, and hence determine the parallax in right ascension
DETERMINATION OF AN ORBIT. 167
and in declination for each observation by means of the usual formulae.
Thus, we have
?r/> cos ' sin (o 0)
A cos d
tanr =
COS (a 0)'
sin and i, and these elements may refer to any fundamental
plane whatever. If we multiply the first of these equations by
sin (it" it'), the second by sin (u" it), and the third by
sin (u f it), and add the products, we find, after reduction,
- sin (u" u'} ~ sin (u" u) -f ^ sin (u' u) = 0,
which, by introducing the values of [rr'], [rr"], and [r'r"], becomes
[//'] x [rr"] x' + [rr'] x" = 0.
If we put
[r'r"] [rr'] , Q ,
n== D^J n FT
weget > " "_ r^
In precisely the same manner, we find
ny */' + n"f = 0, _
>' i ~"~"__n W
DETERMINATION OF AN ORBIT. 169
~ince the coefficients in these equations are independent of the posi-
tions of the co-ordinate planes, except that the origin is at the centre
of the sun, it is evident that the three equations are identical, ana
express simply the condition that the plane of the orbit passes through
the centre of the sun ; and the last two might have been derived
from the first by writing successively y and z in place of x.
Let ^, A', X" be the three observed longitudes, /9, /?', ft" the corre-
sponding latitudes, and A, J', A" the distances of the body from the
earth ; and let
A cos p = P , A' cos p = f>', A" cos p' = //',
which are called curtate distances. Then we shall have
x = p cos A R cos O , d = p' cos A' R' cos Q ',
y = p sin A R sin O , y' = p' sin X' R' sin O',
z p tan ft, z' = p' tan p,
of' = P " cos I" .fl"cosQ",
in which the latitude of the sun is neglected. The data may be so
transformed that the latitude of the sun becomes 0, as will be ex-
plained in the next chapter ; but in the computation of the orbit of
a comet, in which this preliminary reduction has not been made, it
will be unnecessary to consider this latitude which never exceeds 1",
while its introduction into the formulae would unnecessarily com-
plicate some of those which will be derived. If we substitute these
values of x, x> ', &c. in the equations (4) and (5), they become
= n (p cos A R cos O ) (p r cos / R r cos O ')
+ n" (p" cos A" R" cos O"),
= n (p sin A R sin O ) (p sin X' R' sin O') (6)
+ n" (p" sin A" R" sin "),
= np tan /9 p' tan p -f ri'p" tan ft".
These equations simply satisfy the condition that the plane of the
orbit passes through the centre of the sun, and they only become
distinct or independent of each other when n and n" are expressed
in functions of the time, so as to satisfy the conditions of undisturbed
motion in accordance with the law of gravitation. Further, they
involve five unknown quantities in the case of an orbit wholly
unknown, namely, w, n", p y p', and p" ; and if the values of n and
n" are first found, they will be sufficient to determine p, p', and p".
170 THEOEETICAL ASTRONOMY.
The determination, however, of n and n" to a sufficient degree of
accuracy, by means of the intervals of time between the obsei vations,
requires that p f should be approximately known, and hence, in
general, it will become necessary to derive first the values of n, n" y
and p r after which those of p and p tf may be found from equations
(6) by elimination. But since the number of equations will then
exceed the number of unknown quantities, we may combine them in
such a manner as will diminish, in the greatest degree possible, tho
effect of the errors of the observations. In special cases in which
the conditions of the problem are such that when the ratio of two
curtate distances is known, the distances themselves may be deter-
mined, the elimination must be so performed as to give this ratio
with the greatest accuracy practicable.
63. If, in the first and second of equations (6), we change the
direction of the axis of x from the vernal equinox to the place of the
sun at the time t' t and again in the second, from the equinox to the
second place of the body, we must diminish the longitudes in these
equations by the angle through which the axis of x has been moved,
and we shall have
= n(p cos(A 0') jRcos(O' O)) 0' cos (A' 0') #)
+ n" ( P " cos (A"_ 00 - R" cos (O" - 00),
= 7i 0> sin (A 0')+jRsm(0' O)) /sin(A' 0')
+ n" (," sin (A" _ 00 -R" Bin(0"- ')), (7)
= nO>sin(A' A) +jRsin(0 /)) R sin(O' A')
- n" (p" sin (A" A') R" sin (O " *')),
= np tan p p' tan p -f ri'p" tan p'.
If we multiply the second of these equations by tan/3', and the
fourth by sin (A' 0'), and add the products, we get
= n" P " (tan p' sin (A" O tan 0" sin (A' O 0)
n".R"sin(0" tan jfiH- n/ (tan ^ sin (A 0') tan/?sin(A' OO)
+ nJ2sin(' )tan/3 / . (8)
Let us now denote double the area of the triangle formed by the
sun and two places of the earth corresponding to R and R f fry
and we shall have
[#]= KR f sm(Q' O),
and similarly
[ JH2"] = RR" sin ( O " O ),
'] = R'R" sin(O" 00-
ORBIT OF A HEAVENLY BODY. 17l
Then, if we put
_ ,
[#']' ~ IEK"Y
we obtain
Substituting this in the equation (8), and dividing by the coefficient
of p", the result is
_ Q _
~ p n" ' tan ft" sin (A' Q ') tan ft' sin (A" Q ')
V tan ft' sin (A Q ') tan ft sin (A' Q ')
n (A' Q ') tan ft' sin (A" Q '
_!L _^M _ J?sin(0' Q)tanff' _
n" N" J tan /3" sin (A' - Q') tan /3' sin (A" 0')'
Let us also put
M' = tan ^ sin ( ; ~ Q ') ~ tan ^ sin (*' Q')
~~ tan ft" sin (A' 0') tan ft' sin (A" 0')'
j^ __ _ sin(0'-0)tanf __
tan/5" sin (A' 0') tan/3' sin (A" 0')'
and the preceding equation reduces to
-*, JTR (11)
"We may transform the values of 3/ r and M " so as to be better
adapted to logarithmic calculation with the ordinary tables. Thus,
if w' denotes the inclination to the ecliptic of a great circle passing
through the second place of the comet and the second place of the
sun, the longitude of its ascending node will be O', and we shall
have
sin (A' O') tan w' = tan ft'. (12)
Let /9 , ft Q " be the latitudes of the points of this circle corresponding
to the longitudes A and A", and we have, also,
tan ft = sin (A 0') tan w',
tan ft" = sin (A" Q') tan w'.
Substituting these values for tan/9', sin (A O 7 ) and sin (A" O ; )
in the expressions for M ' and M ", and reducing, they become
sin(/y- /?) cos p' cos ft "
sin (ft" /3 ") ' cos ft cos ft '
=^ S in( '-
172 THEORETICAL ASTRONOMY.
When the value of -77 has been found, equation (11) will give the
relation between p and p" in terms of known quantities. It is evi-
dent, however, from equations (14), that when the apparent path of
the comet is in a plane passing through the second place of the
sun, since, in this case, ft /? and ft" = /? ", we shall have M'= ^
and M fr = oo. In this case, theiefore, and also when /9 ft and
ft" ft Q " are very nearly 0, we must have recourse to some other
equation which may be derived from the equations (7), and which
does not involve this indetermination.
It will be observed, also, that if, at the time of the middle obser-
vation, the comet is in opposition or conjunction with the sun, the
values of M f and M " as given by equation (14) will be indeter-
minate in form, but that the original equations (10) will give the
values of these quantities provided that the apparent path of the
comet is not in a great circle passing through the second place of the
sun. These values are
,_ sin (A 0') _ s in(Q' Q)
" sin (A"')' " sm(A"_ 0') '
Hence it appears that whenever the apparent path of the body is
nearly in a plane passing through the place of the sun at the time of
the middle observation, the errors of observation will have great
influence in vitiating the resulting values of M f and M" and to
obviate the difficulties thus encountered, we obtain from the third of
equations (7) the following value of p" :
_ n sin (A' A)
p ==p f
^E sin (G - A') - ^ # sin (' - AO + 12" sin (" - X)
(15)
We may also eliminate p between the first and fourth of equa-
tions (7). If we multiply the first by tan/3', and the second by
cos (A' '), and add the products, we obtain
= n"p" (tan f cos (A" 0') tan /?" cos (A' '))
ri'R" tan p cos(O" 0') + w/> (tan /3' cos (A Q') tan/?cos(A' 00)
nR tan fi cos (O' O) + R' tan /3',
from which we derive
ORBIT OF A HEAVENLY BODY. 173
"_ JL tan ^ cos (A Q ') tan ff cos (A' Q')
p ~~ P n" ' tan /?" cos (A' 0') tan /5' cos (A" Q ') (16)
R" tan/?' cos(O" 00 + 7 12 tan ^ COB (0' 0) R'tan/?
tan /S" cos (A' Q ') tan $ cos (A" Q')
Let us now denote by I' the inclination to the ecliptic of a great
circle passing through the second place of the comet and that point
of the ecliptic whose longitude is 0' 90, which will therefore be
the longitude of its ascending node, and we shall have
cos (A' 0') tan I' = tan p ; (17)
and, if we designate by /9, and /? the latitudes of the points of this
circle corresponding to the longitudes ^ and X", we shall also have
tan /?, = cos (A 0')tan.T',
tan /? = cos (A" 0') tan I'.
Introducing these values into equation (16), it reduces to
_ n sin (/9, /9) cos /3" cos /?
~ p '
sin (/5" /?) cos /5 cos /?, (19)
tan r cos /5" cos/?,, / f ,. . n f . R' \
rinC9"- A( ) (^"^(e'-GO + ^^cosCG'-Q)-^).
from which it appears that this equation becomes indeterminate when
the apparent path of the body is in a plane passing through that
point of the ecliptic whose longitude is equal to the longitude of the
second place of the sun diminished by 90. In this case we may use
equation (11) provided that the path of the comet is not nearly in
the ecliptic. When the comet, at the time of the second observation,
is in quadrature with the sun, equation (19) becomes indeterminate
in form, and we must have recourse to the original equation (16),
which does not necessarily fail in this case.
When both equations (11) and (16) are simultaneously nearly in-
determinate, so as to be greatly affected by errors of observation, the
relation between p and p" must be determined by means of equation
(15), which fails only when the motion of the comet in longitude is
very small. It will rarely happen that all three equations, (14),
(15), and (16), are inapplicable, and when such a case does occur it
will indicate that the data are not sufficient for the determination of
the elements of the orbit. In general, equation (16) or (19) is to be
used when the motion of the comet in latitude is considerable, and
equation (15) when the motion in longitude is greater than in latitude.
174 THEORETICAL ASTRONOMY.
64. The formulae already derived are sufficient to determine the
relation between //' and p when the values of n and n" are known,
and it remains, therefore, to derive the expressions for these quan-
tities.
If we put
t(f -f) = r",
k (" - O = r, (20)
and express the values of x, y, z, x lf y y", z" in terms of re', y f , z r by
expansion into series, we have
f _dx^ T" J_ dV r^_ __ 1_ dV r" 8
x ~ Substituting these values of the differential coefficients in equa-
tions (21), and the corresponding expressions for y and y", and
putting
ORBIT OF A HEAVENLY BODY. 175
-i__i^_i_^L ^'+i/l 12/rf/y 3 dy\
2>' 24- + 2'6-3 + ^~ *'
T" r" 3 T"* dr r
'~'
_ 1 __, 1 __ , , 3 d
*' 3 ~ t " 2 ' + ^''~ 5 " +4 " "'
___ _ .
~yfc 6 ^r' 3 nV* dt " "
we obtaiD
From these equations we easily derive
>
Civ
' ' (23)
. .
The first members of these equations are double the areas of the
triangles formed by the radii-vectores and the chords of the orbit
between the places of the comet or planet. Thus,
y'z-s'y =|W], sV-a/y^ry], y"x - J'y = [n"], (24)
and x'dy' y'dx' is double the area described by the radius-vector
during the element of time dt> and, consequently, , - ia
double the areal velocity. Therefore we shall have, neglecting the
mass of the body,
iii which p is the semi-parameter of the orbit. The equations (23 j,
therefore, become
[r/] = bk V>, [//'] = b"k V~p, [rr"] - (aW
Substituting for a, 6, a", b" their values from (22), we find, since
176 THEORETICAL ASTRONOMY.
( 25)
From these equations the values of n = ^ J an( ^ n " == r "n ma 7
be derived ; and the results are
r" ? A;r'* ' dt '"
which values are exact to the third powers of the time, inclusive.
In the case of the orbit of the earth, the term of the third order.
dR'
being multiplied by the very small quantity j- > is reduced to a
dt
superior order, and, therefore, it may be neglected, so that in this
case we shall have, to the same degree of approximation as in (26),
n [r'r"~\
From the equations (26) or from (25), since , = TTTT, we find
i 4 _ i i
e -^T~
Since this equation involves r' and -3-, it is evident that the value
dt
of ~, in the case of an orbit wholly unknown, can be determined
Tl
only by successive approximations. In the first approximation to
the elements of the orbit of a heavenly body, the intervals between
the observations will usually be small, and the series of terms of (28)
will converge rap'dly, so that we may take
n r
n" ~ T"'
ORBIT OF A HEAVENLY BODY, 177
and similarly
W f = ^'
Hence the equation (11) reduces to
l>" = ~M'p. (29)
It will be observed, further, that if the intervals between the observa-
tions are equal, the term of the second order in equation (28)
vanishes, and the supposition that , = , is correct to terms of the
71 T
third order. It will be advantageous, therefore, to select observa-
tions whose intervals approach nearest to equality. But if the
observations available do not admit of the selection of those which
give nearly equal intervals, and these intervals are necessarily very
unequal, it will be more accurate to assume
JL- -JL
n" ~~ N'"
and compute the values of N and N ff by means of equations (9),
since, according to (27) and (28), if r' does not differ much from R',
the error of this assumption will only involve terms of the third
order, even when the values of r and r fr differ very much.
Whenever the values of p and p" can be found when that of their
ratio is given, we may at once derive the corresponding values of r
and r", as will be subsequently explained.
The values of r and r" may also be expressed in terms of r ' by
means of series, and we have
j +* dy r " 2 _ &
, dr' r dV r a
from which we derive
k w
neglecting terms of the third order. Therefore
dr'_k(r"-r) t
~dt~ ~T+V'~'
12
178 THEORETICAL ASTEONOMY.
and when the intervals are equal, this value is exact to terms of the
fourth order. We have, also,
which gives
r )!_ r . (31)
Therefore, when r and r" have been determined by a first approxi-
mation, the approximate values of r' and -^- are obtained from these
equations, by means of which the value of may be recomputed
from equation (28). We also compute
N _#.B"sin(0"-0') (32)
N" ' K'sm(C)' 0)
and substitute in equation (11) the values of and -^ thus found.
n" N
designate by M the ratio of the cui
we have
If we designate by M the ratio of the curtate distances t o and //',
NR
In the numerical application of this, the approximate value of p will
be used in computing the last term of the second member.
In the case of the determination of an orbit when the approximate
elements are already known, the value of -77 may be computed from
n __rV'sin(0" iQ ,,
"~~ ''
and that of -^ from (32); and the value of M derived by means of
these from (33) will not require any further correction.
65. When the apparent path of the body is such that the value
of M f , as derived from the first of equations (10), is either indeter-
minate or greatly aifected by errors of observation, the equations (15)
and (16) must be employed. The last terms of these equations may
be changed to a form which is more convenient in the approximations
to the value of the ratio of p" to p.
Let Y t Y f , Y" be the ordinates of the sun when the axis of
ORBIT OF A HEAVENLY BODY. 179
abscissas is directed to that point in the ecliptic whose longitude is
^', and we have
Y =E sin(O *'),
Y' =E' sin (0' A'),
Now, in the last term of equation (15), it will be sufficient to put
n_ N
n"~ N e "
and, introducing Y, Y', Y", it becomes
cosec -
It now remains to find the value of -77- From the second of equa-
tions (26) we find, to terms of the second order inclusive,
We have, also,
and hence
Therefore, the expression (35) becomes
But, according to equations (5),
NYY'-\-N"
and the foregoing expression reduces to
sin (/'-/) '
since Y' = R f sin (O r A f ). Hence the equation (15) becomes
1 \^sin(/-0Q ( .
-* (36)
180 THEORETICAL ASTRONOMY.
If we put
... _ _n_ sin (A' ^)
~~"ri" *sin(/l" /)'
JF-! _ . *" . ^ '
*
sn -
we have
= M=M 9 F. (37)
Let us now consider the equation (16), and let us designate by X,
X' y X" the abscissas of the earth, the axis of abscissas being directed
to that point of the ecliptic for which the longitude is O r , then
X = JR cos (0 00,
X'=K f ,
X"=JR"eos(0" 00.
It will be sufficient, in the last term of (16), to put
Jl JL
n" ~ N" '
and for 77- its value in terms of N" as already found. Then, since
this term reduces to
-l^rfV+yW 1 -1\ _ ^ta
B r n ^ * J\ pi E , 3 ] tan p cog y,_ , j _
and if we put
__n_ tan /3' cos (A Q r ) tan /? cos (A ; Q ; )
~~ "n 77 ' tan ft" cos (A' Q') tan p r cos (A" Q')' (38)
tan^'cos (X 0') tan/3oos(X' Q')
the equation (16) becomes
^F'. (39)
In the numerical application of these formulae, if the elements are
not approximately known, we first assume
n
when the intervals are nearly equal, and
OKBIT OF A HEAVENLY BODY. 181
n" " N" '
as given by (32), when the intervals are very unequal, and neglect
the factors F and F'. The values of t o and p fr which are thus ob-
tained, enable us to find an approximate value of r f , and with this a
more exact value of ,7 may be found, and also the value of F or F f .
Whenever equation (11) is not materially affected by errors of
observation, it will furnish the value of M with more accuracy than
the equations (37) and (39), since the neglected terms will not be so
great as in the case of these equations. In general, therefore, it is to
be preferred, and, in the case in which it fails, the very circumstance
that the geocentric path of the body is nearly in a great circle, makes
the values of F and F' differ but little from unity, since, in order
that the apparent path of the body may be nearly in a great circle,
r' must differ very little from R f .
66. When the value of M has been found, we may proceed to
determine, by means of other relations between p and p tr 9 the values
of the quantities themselves.
The co-ordinates of the first place of the earth referred to the third,
are
x, R" cos Q " R cos Q , , .^
y, = E"smO" KsmQ.
If we represent by g the chord of the earth's orbit between the places
corresponding to the first and third observations, and by G the longi-
tude of the first place of the earth as seen from the third, we shall
have
x, = g cos G, y, = ff sin G,
and, consequently,
12"cos(0" 0) jR = ?cos( O),
jR"sin(" 0) =gsm(G').
If ^ represents the angle at the earth between the sun and comet
at the first observation, and if we designate by w the inclination to
the ecliptic of a plane passing through the places of the earth, sun,
and comet or planet for the first observation, the longitude of the
ascending node of this plane on the ecliptic will be O, and we shall
have, in accordance with equations (81)^
cos * = cos fi cos (A 0),
sin 4 cos w = cos /? sin (A 0),
sin 4 sin w sin &
182 THEORETICAL ASTRONOMY.
from which
tan/5
tan w = -T
sin (A Q)'
tan(A- G )
tan 4 = .
costy
Since cos ft is always positive, cos ^ and cos (A 0) must have the
same sign; and, further, a// cannot exceed 180.
In the same manner, if w" and ij/' represent analogous quantities
for the time of the third observation, we obtain
tan/5"
tanw" = -
sin (A" Q")'
^ (43)
cos 4" = cos f cos (A" 0").
We also have
which may be transformed into
r a = G> sec R cos^ + .ft 2 sin**; (44)
and in a similar manner we find
r" 2 = (p" sec /3" #' cos V') 1 + #" sin 2 *". (45)
Let K designate the chord of the orbit of the body between the first
and third places, and we have
X > = (*" _ ,.). _|_ (f _ y). + (2 " _ f ).
But
= p cos A R cos 0,
2/=i0smA E sin O,
z = p tan /5,
and, since />" = Mp,
f = Mp sin X" jR" sin O",
from which we derive, introducing g and (r,
a/' # = JW/> cos A" /> cos A g cos $,
y" V = Mp sm *-" jO sin A g sin G,
z" z = Mp tan/5" p tan /?.
Let us now put
ORBIT OF A HEAVENLY BODY. 183
Mp cos A" p cos A = ph cos C cos H,
Mp sin A" /> sin A = ph cos C sin .5", (46)
Mp tan /3" p tan fi= ph sin C.
Then we have
#" x /oA cos C cos IT # cos 6r,
2/" y = ph cos C sin H g sin (2,
z" z = ph sin C.
Squaring these values, and adding, we get, by reduction,
x 2 = /> 2 /i 2 2g ph cos C cos ( G .H") + f ; (47)
and if we put
cos C cos ( G H} = cos ?>, (48^
we have
x 2 = (/>A y cos ^) 2 -f- <7 2 sin 2 y. (49)
If we multiply the first of equations (46) by cos^", and the
second by sin A", and add the products; then multiply the first by
sin X", and the second by cos K r , and subtract, we obtain
h cos C cos (H A") = M cos (A" A),
h cos C sin (H - n = sin (/" A), (50)
/i sin C =M tan /3" tan ft
by means of which we may determine A, f , and H.
Let us now put
g sin ^ = Ay
R sin 4* = B, h cos = b,
R" sin 4" = JB", - 6", (51)
# cos ?> 6jR cos ^ = c, g cos? b"R" cos 4" = c",
jO/i gr cos p = d,
and the equations (44), (45), and (49) become
The equations thus derived are independent of the form of the
orbit, and are applicable to the case of any heavenly body revolving
around the sun. They will serve to determine r and r" in all cases
in which the unknown quantity d can be determined. If p is known,
184 THEORETICAL ASTRONOMY.
d becomes known directly; but in the case of an unknown orbit,
these equations are applicable only when p or d may be determined
directly or indirectly from the data furnished by observation.
67. Since the equations (52) involve two radii-vectores r and r"
and the chord x joining their extremities, it is evident that an addi-
tional equation involving these and known quantities will enable us
to derive d, if not directly, at least by successive approximations.
There is, indeed, a remarkable relation existing between two radii-
vectores, the chord joining their extremities, and the time of describing
the part of the orbit included by these radii-vectores. In general,
the equation which expresses this relation involves also the semi-
transverse axis of the orbit; and hence, in the case of an unknown
orbit, it will not be sufficient, in connection with the equations (52),
for the determination of c?, unless some assumption is made in regard
to the value of the semi-transverse axis. For the special case of
parabolic motion, the semi-transverse axis is infinite, and the result-
ing equation involves only the time, the two radii-vectores, and the
chord of the part of the orbit included by these. It is, therefore,
adapted to the determination of the elements when the orbit is sup-
posed to be a parabola, and, though it is transcendental in form, it
may be easily solved by trial. To determine this expression, let us
resume the equations
lc(t-T}
1/2 jt
and, for the time t" }
= tan %v -j- I tan* ^v
1/2 gi
Subtracting the former from the latter, and reducing, we obtain
1/2 gf ~~ cos J-y" cos ^v \ ~q cos ^v" cos i> q /
and, since r = q sec 2 ^, this gives
On/ 1 t ' C ) Sill 77 \ t/ i/ / I/ ** I r / i * s tt ~\ / /7 I /" r* r\
^ = - = i: ^ 7=r^ 1 r -|- r'+ cos I (v vjvrr (53)
1/2 1/g \ /
But we have, also, from the triangle formed by the chord vc and the
radii-vectores r and r r/ ,
x 2 = r 2 -f- r" 2 2rr" cos (?/' v)
= (r + r") 2 4rr" cos' J W v).
PARABOLIC ORBIT. 185
Therefore,
21/W"
Let us now put
r + r" + x = m 3 , r + r" x = n,
m and n being positive quantities. Then we shall have
r + r"=J(m'+*),
2 cos J (v" v) Vrr" = mn ;
and, since m and n are always positive, it follows that the upper sign
must be used when v" v is less than 180, and the lower ^ign when
v" v is greater than 180. Combining the last equation with (53),
the result is
3k (f - = (m + m). (55)
y 2g
Now we have
sin |- (v" v) = sin v" cos ^v cos ^v" sin ^v.
Squaring this, and reducing, we get
sin 2 \ (v" v) = cos 2 ^v -\- cos 2 v" 2 cos Jv" cos |v cos J (t^ v),
or, introducing r and q,
9 1 s n \ Q ,
Bin 1 i (w " v) =1 -f
Therefore,
i " --7=
introducing this value into equation (55), we find
sin i (v" v) = --=% <> =1= n).
Replacing m and n by their values expressed in terms of r, r", and
x, this becomes
6k (if' -?> = (r + r" + x)t T (r + r" - x)t, (56)
the upper sign being used when v" v is less than 180. This
equation expresses the relation between the time of describing any
parabolic arc and the rectilinear distances of its extremities from each
other and from the sun, and enables us at once, when three of these
quantities are given, to find the fourth, independent of either the
186 THEORETICAL ASTRONOMY.
perihelion distance or the position of the perihelion with respect to
the arc described.
68. The transcendental form of the equation (56) indicates that,
when either of the quantities in the second member is to be found,
it must be solved by successive trials ; and, to facilitate these approxi-
mations, it may be transformed as follows :
Since the chord x can never exceed r + r", we may put
7Ippr = sin/, (57)
and, since x, r, and r" are positive, sin f r must aiways be positive.
The value of f must, therefore, be within the limits and 180.
From the last equation we obtain
cosV => + r") 2 -<
and substituting for Jt 2 its value given by
x 2 = (r + r") 2 4rr" cos 2 (v" v),
this becomes *
4rr" cos 2 W v)
(r + r ,,y
Therefore, we have
cos Y' = cos | (y" v) .,, (58)
and also
" r/ ""
Hence it appears that when v" v is less than 180, f belongs to
the first quadrant, and that when v" v is greater than 180, cosf
is negative, and f' belongs to the second quadrant.
Tf we introduce f' into the expressions for m 2 and n 2 , they becoma
^ = (r + ')(!- sin ,
which give
m *= ( r + r") (cos tf + sin J/),
W 2 = ( r 4. /') (-4- cos J/ ip sin yj ;
and, since f i g greater than 90 when v" v exceeds 180, the
equation (56) becomes
= (cosy + sin /)- (cos J/-siny)
(r
PARABOLIC ORBIT. 187
From this equation we get
= 6 cos 2 $ sin %/ -f- 2 sin* $/,
or
/> f
^TTT- = 6 sin !/ 4 sin 8 tf ;
and this, again, may be transformed into
6r'
(60)
Let us now put
sn
"
or
sin \*f' = l/ 2 sin #,
and we have
= 3 sin # 4 sin 3 x = sin 3#. (62)
is less than 180, f must be less than 90, and
hence, in this case, sin x cannot exceed the value |, or x must be
within the limits and 30. When v" v is greater than 180,
the angle f is within the limits 90 and 180, and corresponding to
these limits, the values of sin x are, respectively, \ and \ \/%. Hence,
in the case that v rr v exceeds 180, it follows that x must be within
the limits 30 and 45.
The equation
1/20
is satisfied by the values 3# and 180 3x; but when the first gives
x less than 15, there can be but one solution, the value 180 3x
being in this case excluded by the condition that 3x cannot exceed
135. When x is greater than 15, the required condition will be
satisfied by 3x or by 180 3#, and there will be two solutions,
corresponding respectively to the cases in which v ff v is less than
180, and in which v" v is greater than 180. Consequently,
wnen it is not known whether the heliocentric motion during the
interval t rf t is greater or less than 180, and we find 3x greater
than 45, the same data will be satisfied by these two different
solutions. In practice, however, it is readily known which of the
188 THEOEETICAL ASTRONOMY.
two solutions must be adopted, since, when the interval t" t is not
very large, the heliocentric motion cannot exceed 180, unless the
perihelion distance is very small; and the known circumstances will
generally show whether such an assumption is admissible.
We shall now put
o/
n = -, (63)
sin 3*= -L (64)
T/8
and we obtain
We have, also,
sin ^/ = |/2 sin x,
and hence
cos \r' = i/l 2 sin 2 x = I/cos 2x.
Therefore
sin = 2t sin x V cos
and, since JC = (r -|- r") sin f, we have
x = 2% (r + r") sin a; I/cos 2x.
If we put
3sin# / - ^- ,. x
^ = . o V cos 2x, (b5)
sinoa;
the preceding equation reduces to
From equation (64) it appears that rj must be within the limits
and Ji/8- We may, therefore, construct a table which, with ^ as
the argument, will give the corresponding value of //, since, with a
given value of 37, 3x may be derived from equation (64), and then
the value of // from (65). Table XI. gives the values of // corre-
sponding to values of r] from 0.0 to 0.9.
69. In determining an orbit wholly unknown, it will be necessary
to make some assumption in regard to the approximate distance of
the comet from the sun. In this case the interval t" t will gene-
rally be small, and, consequently, x will be small compared with r
and r". As a first assumption we may take r = 1, or r 4 r" = 2,
and fj. = 1, and then find x from the formula
FiRABOLIC ORBIT. 189
With this value of x we compute d, r, and r" by means of the
equations (52). Having thus found approximate values of r and r",
we compute y by means of (63), and with this value we enter Table
XI. and take out the corresponding value of p. A second value
for K is then found from (66), with which we recompute r and r" ', and
proceed as before, until the values of these quantities remain un-
changed. The final values will exactly satisfy the equation (56),
and will enable us to complete the determination of the orbit.
After three trials the value of r -f r" may be found very nearly
correct from the numbers already derived. Thus, let y be the true
value of log (r -j- r"), and let A?/ be the difference between any
assumed or approximate value of y and the true value, or
2/0 = y + Ay.
Then if we denote by y f the value which results by direct calculation
from the assumed value y Q , we shall have
2A>' y =/(y ) =
.Expanding this function, we have
2/o' 2/o =/(y) + 4 *y -4- Ay* + Ac.
But, since the equations (52) and (66) will be exactly satisfied when
the true value of y is used, it follows that
and hence, when AT/ is very small, so that we may neglect terms ot
the second order, we shall have
Vo' y, = 4 Ay = 4 (y y*).
Let us now denote three successive approximate values of log (r + r")
b 7 2A 2A/> 2/o"> and let
y ' yc = a> y" &' = '
then we shall have
a = A (y y),
a' = -4 (&'-?).
Eliminating A from these equations, we get
y ( a f o) = a'y - - ay ',
from which
' /'
(67;
190 THEORETICAL ASTRONOMY.
Unless the assumed values are considerably in error, the value of
y or of log (r + r") thus found will be sufficiently exact ; but should
it be still in error, we may, from the three values which approximate
nearest to the truth, derive y with still greater accuracy. In the
numerical application of this equation, a and a' may be expressed in
units of the last decimal place of the logarithms employed.
The solution of equation (56), to find t" t when JC is known, is
readily effected by means of Table VIII. Thus we have
= sin 3#.
l/2(r-f-*-") f
and, when f 1 is less than 90, if we put
. ---- . /~i
sin /
we get
T' = -i 1/2 N sin r ' (r + r") t, (68)
or
When f exceeds 90, we put
N' = sin 3s,
and we have
/ = 4 1/2 JF(r + /')*
in which log $ j/2 = 9.6733937. With the argument f we take
from Table VIII. the corresponding value of N or JV 7 , and by
means of these equations r' = k (t" t) is at once derived.
The inverse problem, in which r' is known and K is required, may
also be solved by means of the same table. Thus, we may for a first
approximation put
* = T ' 1/2.
and with this value of K compute d, r, and r" . The value of f 1 is
then found from
and the table gives the corresponding value of N QT JV 7 . A second
approximation to x will be given by the equation
S_
1/2 ' NVT-
PAEABOLIC CEBIT. 101
or by
3 /sin/
'v^'^'i/T+y'
in which log = = 0.3266063. Then we recompute d, r, and r",
v 2
and proceed as before until x remains unchanged. The approxima-
tions are facilitated by means of equation (67).
It will be observed that d is computed from
and it should be known whether the positive or negative sign must
be used. It is evident from the equation
d = ph g cos y>,
since /?, h, and g are positive quantities, that so long as , and the sign to be adopted must be determined
from the physical conditions of the problem.
If we suppose the chords g and x to be proportional to the linear
velocities of the earth and comet at the middle observation, we have,
the eccentricity of the earth's orbit being neglected,
= S r V
which shows that H is greater than g } and that d is positive, so long
as r f is less than 2. The comets are rarely visible at a distance from
the earth which much exceeds the distance of the earth from the sun,
and a comet whose radius-vector is 2 must be nearly in opposition in
order to satisfy this condition of visibility. Hence cases will rarely
occur in which d can be negative, and for those which do occur it
will generally be easy to determine which sign is to be used. How-
ever, if d is very small, it may be impossible to decide which of the
two solutions is correct without comparing the resulting elements
with other and more distant observations.
192 THEOKETICAL ASTKONOMY.
70. When the values of r and r" have been finally determined, as
just explained, the exact value of d may be computed, and then we
have
_ d + g cos
tan /?,
and also
r" cos b" cos (r O") = p" cos (A" O") R r ,
r" cos b" sin (r O") = p" sin (A" 0"), (72)
r"sin&" =ytan/9",
in which I and Z ;/ are the heliocentric longitudes and 6, b" the corre-
sponding heliocentric latitudes of the comet. From these equations
we find r, r", I, l ff , 6, and b" and the values of r and r" thus found,
should agree with the final values already obtained. When I" is less
than I, the motion of the comet is retrograde, or, rather, when the
motion is such that the heliocentric longitude is diminishing instead
of increasing.
From the equations (82) w we have
tan i sin (I ) = tan 6,
tan i sin (I" & ) = tan b",
which may be written
zt tan i(sin (I x) cos (x &) -f sin (re &) cos (I x)) = tan b,
tan i (sin (" a?) cos (a; & ) -f sin (re ) cos (r #)) = tan b".
Multiplying the first of these equations by sin (I" x) 9 and the second
by sin (I a?), and adding the products, we get
it tan i sin (x Q, ) sin (f f - - F) = tan b sin (I" x) tan b" sin (I a:) ;
and in a similar manner we find
it tan i cos (a; & ) sin (I" I) = tan 6" cos (I x) tan b cos (I" x).
Now, since x is entirely arbitrary, we may put it equal to I, and we
have
PAKABOLIC CEBIT. 193
tan i sin (I ft ) = db tan b,
. n ' tan 6" tan 6 cos (?' I) (74)
the lower sign being used when it is desired to introduce the distinc-
tion of retrograde motion.
The formulae will be better adapted to logarithmic calculation if
we put x = \(l"+ 0, whence l"x=l(l" l) and I x=\(l I")-,
and we obtain
taut sinQ(r-f Q ft) = : 5 - sin (^+6) -
2 cos 6 cos 6" cos J (/" I)
' ' ( jy \
tanicos(Kr + - O) = g CQ5i S C ' P 8 V, ~ |^.
These equations may also be derived directly from (73) by addition
and subtraction. Thus we have
db tan i (sin (I" ft ) -f sin (I ft )) = tan b" -f tan b,
tani(sin(r ft) sin (7 ft)) = tan&" tan6;
and, since
sin (r ft) -f sin (I ft) = 2 sin (f'+ ^ - 2ft ) cos i (r~ O,
sin(r ft) sin^ ft) =2cosi(^'+ ^ 2ft) sin J(^'
these become
< d.(, ( r +l )- a )- "_.
which may be readily transformed into (75). However, since b and
b" will be found by means of their tangents in the numerical appli-
cation of equations (71) and (72), if addition and subtraction loga-
rithms are used, the equations last derived will be more convenient
than in the form (75).
As soon as ft and i have been computed from the preceding equa-
tions, we have, for the determination of the arguments of the latitude
u and u",
COS I COS I
Now we have
u = v -f- o>,
in which CO = TC ft in the case of direct motion, and CD = ft K
13
194 THEORETICAL ASTRONOMY.
when the distinction of retrograde motion is adopted; and we shall
have
if.-ea-V' 4
and, consequently,
x 2 = r 2 + r" 2 - 2rr" cos (u" u), (78)
or
x 2 = (/' r cos O" it)) 2 + r 8 sin 2 (" u). (79)
The value of K derived from this equation should agree with that
already found from (66).
We have, further,
r = q sec 2 (u ), **' = ) ==: - T=, -= COS i (w" >) =
T/g 1/r 1/5
By addition and subtraction, we get, from these equations,
) + cos l(w o>)) = = 4-
frum which we easily derive
~ cos HK" +)-) cos K"- ) = 4=
1/3 t/r
But
1 1 1 / 4/7^ _ 4
T7 + ^~^\ X r
and if we put
4 f7r
since -yf will not differ much from 1, ^' will be a small angle; and
we shall have, since tan (45 + 0') cot (45 + 6') = 2 tan 20',
PARABOLIC ORBIT. 195
Therefore, the equations (80) become
1 . 1 e , f . . tan 2^
sin '
sin | (ti" u) Vrr 1 ' '
cos
cos w -
from which the values of q and w may be found. Then we shall
have, for the longitude of the perihelion
when the motion is direct, and
* = & *,
when i unrestricted exceeds 90 and the distinction of retrograde
motion is adopted.
It remains now to find T y the time of perihelion passage. We have
V = U - 0>, 1/'=u" - to.
With the resulting values of v and v" we may find, by means of
Table VI., the corresponding values of M (which must be distin-
guished from the symbol M already used to denote the ratio of the
curtate distances), and if these values are designated by M and M ",
we shall have
*
t-T=
m m
or
m m
f
in which m = -f , and log C Q 9.9601277. When v is negative, the
gi
corresponding value of M is negative. The agreement between the
two values of T will be a final proof of the accuracy of the numerical
calculation.
The value of T when the true anomaly is small, is most readily
and accurately found by means of Table VIII., from which we
derive the two values of ^V and compute the corresponding values
of T from the equation
2
in which logg, = 1.5883273. When v is greater than 90, we de-
196 THEORETICAL ASTRONOMY.
rive the values of N' from the table, and compute the corresponding
values of T from
71. The elements q and Tmay be derived directly from the values
of r, r", and x, as derived from the equations (52), without first
finding the position of the plane of the orbit and the position of the
orbit in its own plane. Thus, the equations (80), replacing u and u n
by their values v -\- co and v + &>", become
4dni (^ + tO sinj (v" - v) = *
V, Vr Vr
Adding together the squares of these, and reducing, we get
or
Q _ " '^~'_~~ " "
Combining this equation with (59), the result is
" r -f- r" >
and hence, since x = (r + r") sin 7-',
5 = sin 2 (i/' tO co
We have, further, from (78),
from which, putting
s i nv = r ^H^ t (84)
x
we derive
2i v rr /// > foK."\
cos v = sin J (v v). (85;
x
Therefore, the equation (83) becomes
PARABOLIC ORBIT. 197
= J (r + r") cos 2 tf cos'v, (86)
by means of which q is derived directly from r, r", and K, the value
of v being found by means of the formula (84), so that cosv ia
positive.
When f cannot be found with sufficient accuracy from the equa-
tion
we may use another form. Thus, we have
which give, by division,
tan(45-Hr') = \/r_ llllli (87)
~r~ -''--- X
In a similar manner, we derive
tan (45 + ") = \*_ S/Z ( 88 >
In order to find the time of perihelion passage, it is necessary first
to derive the values of v and v". The equations (59) and (85) give,
by multiplication,
tan (v" v) = tan -/ cos v, (89)
from which v" v may be computed. From (82) we get
If we put
tan /' = */_-., (9u)
this equation reduces to
tan $ (y" + v) = tan (/ 45) cot j (v" v), (91)
and the equations (81) give, also,
tan i (t/' + t>) = cot i (i/' v) sin 2^,
either of which may be used to find v" -f- v.
198 THEORETICAL ASTRONOMY.
From the equations
cos |v _ 1 cos \v" 1
Vq Vr Vq Vr"
by multiplying the first by sin \v" and the second by sin \v, add-
ing the products and reducing, we easily find
sin 4 (V' v) sin |t> cos \ (v" v) _ 1
Vq Vr
Hence we have
__ . , _
*
T/r
=COS^=-7=,
Vq Vr
which may be used to compute q, v, and v ff when v" v is known.
When \(v" v) and J(v" + v), and hence t>" and v, have been
determined, the time of perihelion passage must be found, as already
explained, by means of Table VI. or Table VIII.
It is evident, therefore, that in the determination of an orbit, as
soon as the numerical values of r, r", and Y. have been derived from
the equations (52), instead of completing the calculation of the ele-
ments of the orbit, we may find q and T, and then, by means of
these, the values of r f and v f may be computed directly. When this
has been effected, the values of n and n" may be found from (3), or
that of , from (34). Then we compute p by means of the first of
equations (70), and the corrected value of M from (33), or, in the
special cases already examined, from the equations (37) and (39). In
this way, by successive approximations, the determination of para-
bolic elements from given data may be carried to the limit of accuracy
which is consistent with the assumption of parabolic motion. In the
case, however, of the equations (37) and (39), the neglected terms
may be of the second order, and, consequently, for the final results
it will be necessary, in order to attain the greatest possible accuracy,
to derive
M=?"-
P
from (15) and (16). When the final value of M has been found, the
determination of the elements is completed by means of the formula;
already given.
PARABOLIC ORBIT. 199
72. EXAMPLE. To illustrate the application of the formulae for
the calculation of the parabolic elements of the orbit of a comet by
a numerical example, let us take the following observations of the
Fifth Comet of 1863, made at Ann Arbor:
Ann Arbor M. T. a 6
1864 Jan. 10 6* 57 m 20-.5 19* U m 4'.92 + 34 6' 27".4,
13 6 11 54 .7 19 25 2 .84 36 36 52 .8,
16 6 35 11 .6 19 41 4 .54 + 39 41 26 .9.
These places are referred to the apparent equinox of the date and
are already corrected for parallax and aberration by means of
approximate values of the geocentric distances of the comet. But
if approximate values of these distances are not already known, the
corrections for parallax and aberration may be neglected in the first
determination of the approximate elements of the unknown orbit of
a comet. If we convert the observed right ascensions and declina-
tions into the corresponding longitudes and latitudes by means of
equations (1), and reduce the times of observation to the meridian
of Washington, we get
Washington M. T. A (3
1864 Jan. 10 7 ft 24" 3' 297 53' 7".6 -f 55 46' 58".4,
13 6 38 37 302 57 51 .3 57 39 35 .9,
16 7 1 54 310 31 52 .3 + 59 38 18 .7.
Next, we reduce these places by applying the corrections for pre-
cession and nutation to the mean equinox of 1864.0, and reduce th<*
times of observation to decimals of a day, and we have
t = 10.30837, A = 297 52' 51".l, = + 55 46' 58".4,
1? = 13.27682, A' = 302 57 34 .4, /?' = 57 39 35 .9,
tf' = 16.29299, A" = 310 31 35 .0, "= + 59 38 18 .7.
For the same times we find, from the American Nautical Almanae,
O =290 6' 27".4, log E =9.992763,
0' =293 7 57 .1, logJ?' =9.992830,
O" = 296 12 15 .7, log #' = 9.992916,
which are referred to the mean equinox of 1864.0. It will gene-
rally be sufficient, in a first approximation, to use logarithms of five
decimals ; but, in order to exhibit the calculation in a more complete
form, we shall retain six places of decimals.
Since the intervals are very nearly equal, we may assume
200 THEORETICAL ASTRONOMY.
-
n"~r"~ N"'
Then we have
M== t"- t tan/5'sin(A O') tan/9sin(A' O')
H t tan ?' sin (A' ') tan jf sin (A" O')'
and
g sin ( - O) = R" sin (0" - Q),
g cos(G O) = R" cos(O" O) R;
h cos C cos (H A") = Jf _ cos (X" A),
A cos C sin (jff A") = sin (A" A),
AsinC = M tan/5" tan/3;
from which to find Jf, 6r, #, IT, C> an d ^ Thus we obtain
log M= 9.829827, JET=r 94 24' 1".8,
G = 22 58' 1".7, C = 40 28 21 .9,
log g = 9.019613, log h = 9.688532.
A" cos /?
Since r- = M -- -^ = 0.752. it appears that the comet, at the time
A cos p
of these observations, was rapidly approaching the earth. The
quadrants in which G O and H k" must be taken, are deter-
mined by the condition that g and h cos must always be positive.
The value of M should be checked by duplicate calculation, since an
error in this will not be exhibited until the values of X f and f} f are
computed from the resulting elements.
Next, from
cos 4. = cos /5 cos (A O), cos t" = cos /9" cos (A" 0";,
cos
/ jy i \ \ sec 20'
(I (t*" -f- ) ,) = - - - -j =,
cos I (t* w) K rr
we get
0' = 22' 47".4, w = 115 40' 6' ; .3, log q = 9.887378.
Hence we have
* = 01 -f = 60 23' 17". S,
204 THEORETICAL ASTRONOMY.
and
i = u > = 27 12' 6".l, v" = ti" = 37 38' 43 '.1.
Then we obtain
log m = 9.9601277 | log 5 = 0.129061,
and, corresponding to the values of v and v", Table VI. gives
log M = 1.267163, log M" = 1.424152.
Therefore, for the time of perihelion passage, we have
and
T=t = t 13.74364,
m
T=t' =f 19.72836.
m
The first value gives T= 1863 Dec. 27.56473, and the second gives
T= Dec. 27.56463. The agreement between these results is the final
proof of the calculation of the elements from the adopted value of
M= f -.
P
If we find T by means of Table VIII., we have
log N = 0.021616, log N" = 0.018210,
and the equation
2 2
T = t 3 Nr% sin v = if' 3^ N"i"* sin v",
in which log ^ = 1.5883273, gives for T the values Dec. 27.56473
and Dec. 27.56469.
Collecting together the several results obtained, we have the fol-
lowing elements:
T 1863 Dec. 27.56471 Washington mean time.
= 6023'17".8 _._ . ,
ic and Mean
log q = 9.887378.
Motion Direct.
73. The elements thus derived will, in all cases, exactly represent
the extreme places of the comet, since these only have been used in
finding the elements after p and p" have been found. If, by means
NUMERICAL EXAMPLES. 205
of these elements, we compute n and n n ', and correct the \alue of M,
the elements which will then be obtained will approximate nearer
the true values; and each successive correction will furnish more
accurate results. When the adopted value of M is exact, the result-
ing elements must by calculation reproduce this value, and since the
computed values of A, A", /9, and ft" will be the same as the observed
values, the computed values of X and ft' must be such that when
substituted in the equation for M, the same result will be obtained
as when the observed values of X' and ft' are used. But, according
to the equations (13) and (14), the value of M depends only on the
inclination to the ecliptic of a great circle passing through the places
of the sun and comet for the time ', and is independent of the angle
at the earth between the sun and comet. Hence, the spherical co-
ordinates of any point of the great circle joining these places of the
sun and comet wilj, in connection with those of the extreme places,
give the same value of M, and when the exact value of M has been
used in deriving the elements, the computed values of A' and ft' must
give the same value for w r as that which is obtained from observa-
tion. But if we represent by if/ the angle at the earth between the
sun and comet at the time ', the values of if/ derived by observation
and by computation from the elements will differ, unless the middle
place is exactly represented. In general, this difference will be small,
and since w' is constant, the equations
cos 4/ = cos p cos (A' O')>
sin 4-' cos w' = cos ft' sin (A' '), (93)
sin 4/ sin w' = sin ft,
give, by differentiation,
cos p dl' = cos w' sec p d$ r ,
dp = smufcoa(X O'HV-
From these we get
cosft'dA' tan(*' Q')
dp sin p
which expresses the ratio of the residual errors in longitude and
latitude, for the middle place, when the correct value of M has been
used.
Whenever these conditions are satisfied, the elements will be
correct on the hypothesis of parabolic motion, and the magnitude
of the final residuals in the middle place will depend on the deviation
of the actual orbit of the comet from the parabolic form. Further,
206 THEORETICAL ASTRONOMY.
when elements have been derived from a value of M which has not
been finally corrected, if we compute X r and $' by means of these
elements, and then
the comparison of this value of tan w f with that given by observa-
tion will show whether any further correction of M is necessary, and
if the difference is riot greater than what may be due to unavoidable
errors of calculation, we may regard M as exact.
To compare the elements obtained in the case of the example
given with the middle place, we find
v f = 32 31' 13".5, u' = 148 IV 19".8, log / == 9.922836.
Then from the equations
tan (f Q> ) = cos i tan u',
tan b f = tan i sin (l f & ),
we derive
I' = 109 46' 48".3, V = 28 24' 56".0.
By means of these and the values of O' and R f , we obtain
A' = 302 57' 41".l, p = 57 39' 37".0 ;
and, comparing these results with the observed values of X' and /?',
the residuals for the middle place are found to be
Comp. Obs.
cos p AA' = -f 3".6, A/9 = + I'M.
The ratio of these remaining errors, after making due allowance for
unavoidable errors of calculation, shows that the adopted value of
M. is not exact, since the error of the longitude should be less than
that of the latitude.
The value of w 1 given by observation is
log tan w' = 0.966314,
and that given by the computed values of A' and ft' is
log tan w' = 0.966247.
The difference being greater than what can be attributed to errors of
calculation, it appears that the value of M requires further cor-
NUMERICAL EXAMPLES. 207
rection. Since the difference is small, we may derive the correct
value of M by using the same assumed value of ,, and, instead of
Yb
the value of tan&/ derived from observation, a value differing as
much from this in a contrary direction as the computed value differs.
Thus, in the present example, the computed value of log tan w f is
0.000067 less than the observed value, and, in finding the new value
of Mj we must use
log tan w' = 0.966381
in computing /? and /?/' involved in the first of equations (14). If
the first of equations (10) is employed, we must use, instead of tan/9'
as derived from observation,
tan p = tan vf sin (A' 0'),
or
log tan f = 0.966381 + log sin (A' 0') = 0.198559,
the observed value of X' being retained. Thus we derive
log M= 9.829586,
and if the elements of the orbit are computed by means of this
value, they will represent the middle place in accordance with the
condition that the difference between the computed and the observed
value of tan w r shall be zero.
A system of elements computed with the same data from
log M= 9.822906 gives for the error of the middle place,
C.-O.
cos f A;/ = 1' 26".2, &p = 40".l.
If we interpolate by means of the residuals thus found for two values
of M, it appears that a system of elements computed from
log M= 9.829586
will almost exactly represent the middle place, so that the data are
completely satisfied by the hypothesis of parabolic motion.
The equations (34) and (32) give
log -^- = 0.006955, log -^ = 0.006831,
/ f\/ JLV
and from (10) we get
log M ' = 9.822906, log M" = 9.663729...
208 THEORETICAL ASTRONOMY.
Then by means of the equation (33) we derive, for the corrected
value of My
log M= 9.829582,
which differs only in the sixth decimal place from the result obtained
by varying tanw' and retaining the approximate values t = -r f = j'
74. When the approximate elements of the orbit of a comet are
known, they may be corrected by using observations which include
a longer interval of time. The most convenient method of effecting
this correction is by the variation of the geocentric distance for the
time of one of the extreme observations, and the formulae which
may be derived for this purpose are applicable, without modification,
to any case in which it is possible to determine the elements of the
orbit of a comet on the supposition of motion in a parabola. Since
there are only five elements to be determined in the case of parabolic
motion, if the distance of the comet from the earth corresponding to
the time of one complete observation is known, one additional com-
plete observation will enable us to find the elements of the orbit.
Therefore, if the elements are computed which result from two or
more assumed values of A differing but little from the correct value,
by comparison of intermediate observations with these different sys-
tems of elements, we may derive that value of the geocentric distance
of the comet for which the resulting elements will best represent the
observations.
In order that the formulae may be applicable to the case of any
fundamental plane, let us consider the equator as this plane, and,
supposing the data to be three complete observations, let A, A f , A"
be the right ascensions, and D, D', D" the declinations of the sun
for the times t, t', t". The co-ordinates of the first place of the earth
referred to the third are
x R" cos D" cos A" R cos D cos A,
y = R" cos D" sin A" EcosD sin A,
z=R"smD" EsmD.
If we represent by g the chord of the earth's orbit between the places
for the first and third observations, and by G and K, respectively,
the right ascension and declination of the first place of the earth as
seen from the third, we shall have
x = g cos K cos G,
y = g cos K sin G,
z = g sin K.
VAKIATION OF THE GEOCENTKIC DISTANCE. 209
and, consequently,
g cos K cos ( G A) = R" cos D" cos (A" A) R cos D,
g cos K sin ( G A) = jR" cos D" sin (A" A), (96)
g sin iT = .#" sin Z>" jR sin D,
from which #, JT, and G may be found.
If we designate by x n y n z, the co-ordinates of the first place of
the comet referred to the third place of the earth, we shall havf
x, = A cos 5 cos a -f g cos K cos G,
y, A cos d sin a -f- g cos K sin (r,
2, = A sin 5 -f- # sin K.
Let us now put
x, = h' cos C' cos IT,
^ == A' cos :' sin #',
z, = h' sin C',
and we get
h f cos % cos (.ff ' G) = A cos fl cos (a G) + cos JT,
A' cos C' sin GET G) = J cos sin (a (f). C97)
A.' sin C' = J sin '= cos C' cos H' cos <5" cos a"-\- cos C' sin JET' cos 5" sin a"-}- sin C' sin <5",
or
cos ?/ = cos ' cos ' = c,
Rsm*=B, R cos * = b, (103)
R" sin 4," = ", R" cos 4" = &",
and we shall have
= T/(J" c) 2 -f C 2 ,
&) 2 + 2 , (104)
'These equations, together with (56), will enable us to determine J"
by successive trials when J is given.
We may, therefore, assume an approximate value of A" by means
of the approximate elements known, and find r" from the last of
these equations, the value of T having been already found from the
assumed value of J. Then x is obtained from the equation
2r'
fj. being found by means of Table XI., and a second approximation
to the value of J" from
*. (105)
The approximate elements will give A" near enough to show whether
the upper or lower sign must be used. With the value of A" thus
found we recompute r" and x as before, and in a similar manner find
a still closer approximation to the correct value of A n '. A few trial?
will generally give the correct result.
When A" has thus been determined, the heliocentric places are
found by means of the formulae
r cos b cos (I A) = A cos d cos (a A) E cos D,
r cos b sin (/ A) = A cos d sin (a A),
r sin b = A sin <5 R sin D ;
VARIATION OF THE GEOCENTRIC DISTANCE. 211
r" cos V cos <7' A") = A" cos V cos (a" - A") K" cos ",
r" cos 6" sin (r A"} = A" cos d" sin (a" 4"), (107)
r" sin 6" = J" sin 5" " sin Z>",
in which 6, b" y I, I" are the heliocentric spherical co-ordinates re-
ferred to the equator as the fundamental plane. The values of r and
r" found from these equations must agree with those obtained from
(104).
The elements of the orbit may now be determined by means of the
equations (75), (77), and (81), in connection with Tables VI. and
VIII., as already explained. The elements thus derived will be re-
ferred to the equator, or to a plane passing through the centre of the
sun and parallel to the earth's equator, and they may be transformed
into those for the ecliptic as the fundamental plane by means of the
equations (109)^
75. With the resulting elements we compute the place of the comet
for the time t f and compare it with the corresponding observed place,
and if we denote the computed right ascension and declination by a/
and d Q ', respectively, we shall have
a' + <*'=', *+* = .',
in which a' and d' denote the differences between computation and
observation. Next we assume a second value of J, which we repre-
sent by J -f- d J, and compute the corresponding system of elements.
Then we have
a!' and d" denoting the differences between computation and obser-
vation for the second system of elements. We also compute a third
system of elements with the distance J J, and denote the differ-
ences between computation and observation by a and d; then we shall
have
and similarly for c?, d f , and d". If these three numbers are exactly
represented by the expression
m
in which J + x is the general value of the argument, since the values
}f a, a', and a" will be such that the third differences may be neg-
lected, this formula may be assumed to express exactly any value of
the function corresponding to a value of the argument not differing
212 THEORETICAL ASTRONOMY.
much from J, or within the limits x = dA and x = -f- <5J, the as-
sumed values J d J, J, and J + J being so taken that the correct
value of J shall be either within these limits or very nearly so.
To find the coefficients m, n, and o, we have
m n -f- o = a. m a', m -{- n-\- o a",
whence
? = a', n = (a" a), o = (a" -f a) a'.
Now, in order that the middle place may be exactly represented in
right ascension, we must have
which we find
or
In the same manner, the condition that the middle place shall be
exactly represented in declination, gives
In order that the orbit shall exactly represent the middle place, both
conditions must be satisfied simultaneously; but it will rarely happen
that this can be effected, and the correct value of x must be found
from those obtained by the separate conditions. The arithmetical
mean of the two values of x will not make the sum of the squares
of the residuals a minimum, and, therefore, give the most probable
value, unless the variation of cos 3 f AO/, for a given increment as-
signed to J, is the same as that of &d'. But if we denote the value
of x for which the error in a' is reduced to zero by x', and that for
which &d' = 0, by x", the most probable value of x will be
--
in which n = $(a ff a) and n f = \(d ff -- d). It should be observed
that, in order that the differences in right ascension and declination
shall have equal influence in determining the value of x 9 the values
of a, a', and a" must be multiplied by cos 8 f . The value of dA is
most conveniently expressed in units of the last decimal place of the
logarithms employed.
NUMERICAL EXAMPLE. 213
If the elements are already known so approximately that the first
assumed value of J differs so little from the true value that the
second differences of the residuals may be neglected, two assumptions
in regard to the value of A will suffice. Then we shall have o = 0,
and hence
m = a f , n = a" a'.
The condition that the middle place shall be exactly represented,
gives the two equations
"' ' 0,
Q.
The combination of these equations according to the method of least
squares will give the most probable value of x, namely, that for
which the sum of the squares of the residuals will be a minimum.
Having thus determined the most probable value of #, a final
system of elements computed with the geocentric distance A -f- #,
corresponding to the time t, will represent the extreme places exactly,
and will give the least residuals in the middle place consistent with
the supposition of parabolic motion. It is further evident that we
may use any number of intermediate places to correct the assumed
value of J, each of which will furnish two equations of condition
for the determination of x, and thus the elements may be found
which will represent a series of observations.
76. EXAMPLE. The formulae thus derived for the correction of
approximate parabolic elements by varying the geocentric distance,
are applicable to the case of any fundamental plane, provided that
a, d, A, D, &c. have the same signification with respect to this plane
that they have in reference to the equator. To illustrate their
numerical application, let us take the following normal places of
the Great Comet of 1858, which were derived by comparing an
ephemeris with several observations made during a few days before
and after the date of each normal, and finding the mean difference
between computation and observation :
Washington M. T.
1858 June 11.0
July 13.0
Aug. 14.0
a
141 18' 30".9
144 32 49 .7
152 14 12 .0
6
+ 24 46' 25".4,
27 48 .8,
-f 31 21 47 .9,
which are referred to the apparent equinox of the date. These
places are free from aberration.
214 THEOEETICAL ASTRONOMY.
We shall take the ecliptic for the fundamental plane, and con-
verting these right ascensions and declinations into longitudes and
latitudes, and reducing to the ecliptic and mean equinox of 1858.0,
the times of observation being expressed in days from the beginning
of the ye^tr, we get
t = 162.0, A = 135 51' 44".2, p = + 9 6' 57".8,
If = 194.0, *' = 137 39 41 .2, p = 12 55 9 .0,
*" = 226.0, A" = 142 51 31 .8, ^' = + 18 36 28 .7.
From the American Nautical Almanac we obtain, for the true places
of the sun,
Q = 80 24' 32".4, logJ? =0.006774,
O' =110 55 51 .2, log^R' =0.007101,
0" = 141 33 2 .0, log R" = 0.005405,
the longitudes being referred to the mean equinox 1858.0.
When the ecliptic is the fundamental plane, we have, neglecting
the sun's latitude, D = 0, and we must write ^ and ft in place of a
and <5, and O in place of A, in the equations which have been derived
for the equator as the fundamental plane. Therefore, we have
g cos (Q O) = R" cos (O" O) R,
<7 sin ( - 0) = R" sin (O" - O) ;
cos 4 = cos p cos (A O), cos 4/' = cos /3" cos (A" 0")
R cos 4 = b, 12" cos *" = &",
JB, 12"sin4/' = .B",
from which to find G, g, b, B, b", and B ff , all of which remain
unchanged in the successive trials with assumed values of J. Thus
we obtain
G = 201 T 57".4, log B = 9.925092, b = -f 0.568719,
log g = 0.013500, log B" = 9.510309, b" = -f 0.959342.
Then we assume, by means of approximate elements already
known,
log J = 0.397800,
and from
V cos :' cos (jff ' G) = J cos p cos (A G) + g,
V cos ' sin (H' G) = Acosp sin (A G\
h r sin C' =4 sin p,
we find U', C r , and A'. These give
H' = 153 46' 20".5, C' = + 7 24' 16".4, log A' = 0.487484.
NUMERICAL EXAMPLE. 215
Next, from
cos = 116 56' 33".7, o" = u" a> = 88 47' 55".l,
from which we get
T= 1858 Sept. 29.4274.
From these elements we find
log r' == 0.212844, v f = 107 7' 34".0, u' = 21 59' 12".3,
and from
tan (f &) cos i tan u',
tan b f = tan i sin (J! & ),
we get
I' = 154 56' 33".4, 6' = + 19 30' 22".l.
NUMERICAL EXAMPLE. 217
By means of these and the values of O' and R f , we obtain
A' = 137 39' 13".3, p = + 12 54' 45".3,
and comparing these results with observation, we have, for the error
of the middle place,
C. O.
/ = 27".2, A/5' = 23".7.
From the relative positions of the sun, earth, and cctnefc at the
time t" it is easily seen that, in order to diminish these residuals, the
geocentric distance must be increased, and therefore we assume, for
a second value of J,
log J = 0.398500,
from which we derive
H' = 153 44' 57".6, C' = -f 7 24' 26".l, log h' = 0.488026,
log C= 9.912587, logc = 0.472115, logr = 0.324207,
log A" = 0.311054, log r" = 0.054824, log x = 0.089922,
Then we find the heliocentric places
I = 159 40' 33".8, b = + 10 50' 8".6, logr = 0.324207,
I" = 144 17 12 .1, b" = + 35 8 37 .8, log r" = 0.054825,
and from these,
= 165 15' 41".l, i = 63 2' 49".2,
u = 12 10 30.8, w"=:40 13 26.0,
) cos sin ^
z, =(4 4>)sin/?,
the axis of x being directed to the vernal equinox. Let us now
designate by O the longitude of the sun as seen from the point of
reference in the ecliptic, and by R its distance from this point. Then
will the heliocentric co-ordinates of this point be
X= R cos 0,
t T/" T> Q^ s~\
The heliocentric co-ordinates of the centre of the earth are
X Q = R Q cos cos Q ,
Y Q = R cos T sin O ,
Z = R sin J .
But the heliocentric co-ordinates of the true place of observation
will be
X+x,, Y+y n Z + z,,
or
and, consequently, we shall have
R cos O (4, 4>) cos ^ cos A = /? cos r o cos Q p sin TT O cos 5 cos / ,
R sin O (4 ^ ) cos sin A = .R cos ^ sin O Po si n ^o cos ^o sm ^o>
( J, J ) sin /? = .Ro sin ^? sin ^ sin b .
If we suppose the axis of x to be directed to the point whose longi-
tude is O > these become
DETERMINATION OF AN OKBIT. 223
R cos (0 O ) (4 4>) cos 13 cos (A O ) =
E Q cos 2" p sin TT O cos 6 cos (7 O ),
.# sin (O Go) (4 4.) cos /? sin (A O ) = (2)
p sin TT O cos 6 sin (1 O )>
( A, J ) sin /? R Q sin 2" p Q sin TT O sin 6 ,
from which R and O may be determined. Let us now put
Z>; (3)
then, since ;r , S w and O O are small, these equations may be
reduced to
R = D cos (X O ) ^o PQ cos fy> cos (1 Q ) + R ,
R (O O ) = D sin (A Q ) 7r /? cos b sin (J Q 8 ),
Dtan/? 7r ^o sin6 + ^ ^o-
Hence we shall have, if TT O and 2" are expressed in seconds of arc,
(4)
000 . o
206264.8
, 206264.8 D sin (A Q ) TT O /QO cos b sin (^ )
W W ~T~ " p
from which we may derive the values of O and R which are to be
used throughout the calculation of the elements as the longitude and
distance of the sun, instead of the corresponding places referred to
the centre of the earth. The point of reference being in the plane
of the ecliptic, the latitude of the sun as seen from this point is zero,
which simplifies some of the equations of the problem, since, if the
observations had been reduced to the centre of the earth, the sun's
latitude would be retained.
We may remark that the body would not be seen, at the instant
of observation, from the point of reference in the direction actually
observed, but at a time different from t Q , to be determined by the
interval which is required for the light to pass over the distance
A, J . Consequently we ought to add to the time of observation
the quantity
( J, J ) 497'.78 = 497'.7S D sec /9, (5;
which is called the reduction of the time ; but unless the latitude of
the body should be very small, this correction will be insensible.
The value of A derived from equations (1) and the longitude O
224 THEORETICAL ASTRONOMY.
derived from (4) should be reduced by applying the correction for
nutation to the mean equinox of the date, and then both these and
the latitude /9 should be reduced by applying the correction for pre-
cession to the ecliptic and mean equinox of a fixed epoch, for which
the beginning of the year is usually chosen.
In this way each observed apparent longitude and latitude is to be
corrected for the aberration of the fixed stars, and the corresponding
places of the sun, referred to the point in which the line drawn from
the body through the place of observation on the earth's surface in-
tersects the plane of the ecliptic, are derived from the equations (4).
Then the places of the sun and of the planet or comet are reduced
to the ecliptic and mean equinox of a fixed date, and the results thus
obtained, together with the times of observation, furnish the data for
the determination of the elements of the orbit.
When the distance of the body corresponding to each of the
observations shall have been determined, the times of observation
may be corrected for the time of aberration. This correction is
necessary, since the adopted places of the body are the true places
for the instant when the light was emitted, corresponding respectively
to the times of observation diminished by the time of aberration,
but as seen from the places of the earth at the actual times of
observation, respectively.
When /? = 0, the equations (4) cannot be applied, and when the
latitude is so small that the reduction of the time and the correction
to be applied to the place of the sun are of considerable magnitude,
it will be advisable, if more suitable observations are not available,
to neglect the correction for parallax and derive the elements, using
the uncorrected places. The distances of the body from the earth
which may then be derived, will enable us to apply the correction for
parallax directly to the observed places of the body.
When the approximate distances of the body from the earth are
already known, and it is required to derive new elements of the
orbit from given observed places or from normal places derived from
many observations, the observations may be corrected directly for
parallax, and the times corrected for the time of aberration. We
shall then have the true places of the body as seen from the centre
of the earth, and if these places are adopted, it will be necessary, for
the most accurate solution possible, to retain the latitude of the sun
in the formulae which may be required. But since some of these
formulae acquire greater simplicity when the sun's latitude is not
introduced, if, in this case, we reduce the geocentric places to the
DETERMINATION OF AN ORBIT. 225
point in which a perpendicular let fall from the centre of the earth
to the plane of the ecliptic cuts that plane, the longitude of the sun
will remain unchanged, the latitude will be zero, and the distance R
will also be unchanged, since the greatest geocentric latitude of the
sun does not exceed 1". Then the longitude of the planet or comet
as seen from this point in the ecliptic will be the same as seen from
the centre of the earth, and if J, is the distance of the body from
this point of reference, and /9, its latitude as seen from this point, we
shall have
A, cos /?, = A cos /9,
A t sin ft, = A sin /5 J? sin S ,
from which we easily derive the correction /?, /9, or A/9, to be applied
to the geocentric latitude. Thus, we find
A/9 = -^> cos /9, (6)
~t
2 Q being expressed in seconds. This correction having been applied
to the geocentric latitude, the latitude of the sun becomes
2=0.
The correction to be applied to the time of observation (already
diminished by the time of aberration) due to the distance J, J
will be absolutely insensible, its maximum value not exceeding
O s .002. It should be remarked also that before applying the equa-
tion (6), the latitude JT should be reduced to the fixed ecliptic which
it is desired to adopt for the definition of the elements which deter-
mine the position of the plane of the orbit.
78. When these preliminary corrections have been applied to the
data, we are prepared to proceed with the calculation of the elements
of the orbit, the necessary formulae for which we shall now investi-
gate. For this purpose, let us resume the equations (6) 3 ; and, if we
multiply the first of these equations by tan /9 sin A" tan /9" sin A,
the second by tan/3" cos A tan /9 cos A", and the third by sin (X )"\
and add the products, we shall have
nR (tan /9" sin (A Q) tan /9 sin (A" Q))
p' (tan /? sin (A" A') tan jf sin (A" A) -f- tan $' sin (A' A))
R' (tan P" sin (A Q') tan ft sin (A" '))
+ n"R" (tan ft" sin (A Q") tan /5 sin (A" 0")).
It should be observed that when the correction for parallax is applied
15
226 THEORETICAL ASTRONOMY.
to the place of the sun, p f is the projection, on the plane of the
ecliptic, of the distance of the body from the point of reference to
which the observation has been reduced.
Let us now designate by jfiTthe longitude of the ascending node,
and by I the inclination to the ecliptic, of a great circle passing
through the first and third observed places of the body, and we have
tan p sin (A K} tan J,
tan 0" = sin (A" JE")tanJ..
Introducing these values of tan and tan ft" into the equation (7),
since
sin (A O) sin (A" K) sin (A" Q) sin (A JT) =
sin (A" A) sin (Q JT),
sin (A' A) sin (A" JT) + sin (A" A') sin (A JT) =
-f- sin (A" A) sin (A' K\
sin (A O') sin (A" K) sin (A" G') sin (A JT) =
sin (A" A) sin (G' JD,
sin (A 0") sin (A" K) sin (A" 0") sin (A K) =
sin (A" A) sin (G" K\
we obtain, by dividing through by sin (A" A) tan /,
= nR sin (G -ET) + p' (sin (A' K} tan p f cot I)
Let /9 denote the latitude of that point of the great circle passing
through the first and third places which corresponds to the longitude
^', then
tan ft = sin (A' K) tan J,
and the coefficient of p r in equation (9) becomes
sin (ft ff')
cos ft cos p r tan /
Therefore, if we put
we shall have
Bn -
"= cos /. tan/'
,,
This formula will give the value of p f t or of J', when the values of
n and n" have been determined, since a and K are derived from the
data furnished by observation.
DETERMINATION OF AN OEBIT. 227
To find K and 7, we obtain from equations (8) by a transformation
precisely similar to that by which the equations (75) 3 were derived,
We may also compute K and I from the equations which may be
derived from (74) 3 and (76) 3 by making the necessary changes in the
notation, and using only the upper sign, since I is to be taken always
less than 90.
Before proceeding further with the discussion of equation (11), let
us derive expressions for p and p" in terms of ,o', the signification of
p and p tf y when the corrections for parallax are applied to the places
of the sun, being as already noticed in the case of p f .
79. If we multiply the first of equations (6) 3 by sin 0" tan/3",
the second by cos 0" tan/3", and the third by sin (A" ")> and
add the products, we get
0=w/>(tan"sin(0" A) tan/9sin(0" *")) njRtan/9"8in(0" 0)
-V (tan p' sin (0" A') tan f sin (" A"))+# tan p sin (" Q'),
(13)
which may be written
0= np (tan /? sin (A" 0") tan /?" sin (X ")) rc^tan 0" sin (0" 0)
-f ft (tan /?" sin (A' Q ") tan ft sin (A" "))
-VCtan/S' tanft)sin(A" Q") + # tan/9' ; sin(O r; 00-
Introducing into this the values of tan ft, tan /3", and tan /3 in terms
of 1 and jfif, and reducing, the result is
Q = np sin (A" A) sin ( " K) nR sin ( " - ) sin (A" ")
+ R r sin (0" 00 sin W -K").
Therefore we obtain
_^/sin(^ AQ q sec^ sin(^ Q^) \
= ~ w \ sin (A" A) + sin (A" A) ' sin (Q" K} /
sn
/i sin (A" A)sin(O" JE")
But, by means of the equations (9) 3 , we derive
" 0) = (N ri) Rsm(O"- 0),
228 THEORETICAL ASTRONOMY.
and the preceding equation reduces to
sin (A" 0")
n\sin(A" A) ' sin(A" A)" s in(" _ y/ (u .
N\Ksm(Q" Q)8m(A" K)
"~rT/ sin (A" A)sin(O" K) '
To obtain an expression for p rf in terms of p r , if we multiply the
first of equations (6) 3 by sin tan/9, the second by cos tan/9,
and the third by sin (X ), and add the products, we shall have
0=V'(tan^sin(A' / ) tan /S" sin (A )) n"K"tansin(" )
*' ) tan/?' sin (A ))+ J R'tan/5sin(' ). (15)
Introducing the values of tan /9, tan /9 r , and tan /9 r/ in terms of K and
/, and reducing precisely as in the case of the formula already found
for p, we obtain
p' I sin (A' A) a sec j? sin (A ) \
n" \ sin (A" A) sin (A" A) sin ( K) /
T^et us now put, for brevity,
. Esm(Q-K) _R'sm(Q f K)
o = - j c i
a a
^_^sin(0 // g) sec/g f , _.Rfi"8in(0" 0)
a J "" sin (A" A)'
in(^ JQ , A sin (A .g)
-
and the equations (11), (14), and (16) become
(18)
If n and n" are known, these equations will, in most cases, be
sufficient to determine p, //, and //'.
DETEKMINATION OF AN OKBIT. 229
80. It will be apparent, from a consideration of the equations
which have been derived for p, p f , and p lf t that under certain circum-
stances they are inapplicable in the form in which they have been
given, and that in some cases they become indeterminate. When the
great circle passing through the first and third observed places of the
body passes also through the second plaoe, we have a = 0, and
equation (11) reduces to
n"R" sin (Q" K) + nR sin (0 K} = R' sin (0' K).
If the ratio of n" to n is known, this equation will determine the
quantities themselves, and from these the radius-vector r r for the
middle place may be found. But if the great circle which thus
passes through the three observed places passes also through the
second place of the sun, we shall have K== O', or K= 180 -j- O',
and hence
n"R" sin (O" 0') nR sin (' O) = 0,
or
n" Jg8in(0' 0)
n ~.R"sm(O" O')'
from which it appears that the solution of the problem is in this
case impossible.
If the first and third observed places coincide, we have ^ = X" and
/9 = /3", and each term of equation (7) reduces to zero, so that the
problem becomes absolutely indeterminate. Consequently, if the
data are nearly such as to render the solution impossible, according
to the conditions of these two cases of indetermination, the elements
which may be derived will be greatly affected by errors of observa-
tion. If, however, A is equal to A" and /9" differs from /9, it will be
possible to derive p f , and hence p and p"; but the formulae which
have been given require some modification in this particular case.
Thus, when J = A", we have K=X' = 1 9 7=90, and ft =90,
and hence a , as determined by equation (10), becomes - Still, in
this case it is not indeterminate, since, by recurring to the original
equation (9), the coefficient of p f , which is a sec ft', gives
a = sin p cot / cos /?' sin (/ K) t (19)
and when A = A", it becomes simply
= cos p sin (A' K).
/ N"\
\ n" /
230 THEORETICAL ASTRONOMY.
Whenever, therefore, the difference A" A is very small compared
with the motion in latitude, a should be computed by means of the
equation (19) or by means of the expression which is obtained
directly from the coefficient of p 1 in equation (7).
When A = X" = K, the values of Jf w Jtf/', M 2 , and M 2 " cannot
be found by means of the equations (17); but if we use the original
form of the expressions for p and p" in terms of p f , as given by
equations (13) and (15), without introducing the auxiliary angles,
we shall have
= ft_ tan p sin (A" Q") tan p' sin (A' Q")
n ' tan ft sin (A" 0") tan p' sin (A 0")
/ __N_\ _ jRtan"sin(0" Q) _
M n I tan ft sin (/I" 0") tan ft" sin (A - 0") 1
tan /? sin (A' Q) tan p sin (A Q)
tan /9 sin (A" 0) tan p" sin (A 0)
jR"tansin(0"-- 0)
tan /? sin (A" 0) tan p' sin (A )'
Hence
_ tan /?' sin (X* ") tan p' sin (A' Q")
1 ~~ tan sin (A" 0") tan ft" sin (A 0") '
_ tan ft sin (A' 0) tan /?' sin (A Q)
MI Z ' tan /S sin (A" - 0) - tan 0" sin (*)' , ,
_ J? tan ^ sin (0^0) _
8 ~ tan /5 sin (A" 0") tan ft" sin (A ")'
_ _ R" tan ft sin (Q" 0) __
a = = tan ft sin (A" 0) tan ft" sin (A ) '
are the expressions for M lt -Mi", M z , and M 2 " which must be used
when X = X' r or when A is very nearly equal to A"; and then p and p rr
will be obtained from equations (18). These expressions will also be
used when A" X = 180, this being an analogous case.
When the great circle passing through the first and third observed
places of the body also passes through the first or the third place of
the sun, the last two of the equations (18) become indeterminate, and
other formula must be derived. If we multiply the second of equa-
tions (7) 3 by tan/3" and the fourth by sin (A" O r ), and add the
products, then multiply the second of these equations by tan /? and
the fourth by sin (A '), and add, and finally reduce by means
of the relation
NE sin (' - 0) = N"R" sin (0" - '),
we get
o
DETERMINATION OF AN ORBIT. 231
tan p' sin (A' ') tan ft' sin (A" ')
n tan ft" sin (A ') tan ft sin (A" ')
.#" tan/S" sin (" 0')
tan ft" sin (A ') tan ft sin (A"
tan jt sin (A ') tan ft sin (A' ')
tan ft" sin (A ') tan ft sin (A" ')
R tan /5 sin (')
(21)
tan/S" sin (A 0') tan/? sin (A" 0')
These equations are convenient for determining p and p' r from // ;
but they become indeterminate when the great circle passing through
the extreme places of the body also passes through the second place
of the sun. Therefore they will generally be inapplicable for the
cases in which the equations (18) fail.
If we eliminate p" from the first and second of the equations (6) 3
we get
Q = np sin (A" A) nR sin (A" Q) p' sin (A" A')
+ R' sin (A" 0') ri'R" sin (A" 0"),
from which we derive
p > sin(A"-AQ
-;Tsin(A"-A)
nR sin (A" Q ) R sin (A" Q') + ri'R' sin (I" Q ")
n sin (A" A)
Eliminating p between the same equations, the result is
P' sin (A'- A)
p - n " sin (A" -A)
nR sin (A Q) R' sin (A Q') -f- n"R" sin (A Q")
n" sin (A" A)
These formulae will enable us to determine p and p" from p 1 in the
special cases in which the equations (18) and (21) are inapplicable;
but, since they do not involve the third of equations (6) 3 , they are
not so well adapted to a complete solution of the problem as the
formulae previously given whenever these may be applied.
If we eliminate successively p" and p between the first and fourth
of the equations (7) 3 , we get
tan P" cos (X Q ') tan p cos (X 1 Q ')
_
p ~
_
P ~"
n tan /3" cos (A Q') tan p cos (A" Q')
tan/3" nR cos (0' Q ) R + n"R" cos(Q" Q')
n tan /5" cos (A Q ') tan p cos (A" O') '
tan j3' cos (A Q') tan /9 cos (A' Q ')
n"' tan ft" cos (A Q ') tan ft cos (A" Q')
tan/9 nR cos (Q' Q) JT + ri'R" cos (0" 0')
n" ' tan 0" cos (A ') tan/? cos (A" 0') '
232 THEORETICAL ASTRONOMY.
which may also be used to determine p and p ff when the equations
(18) and (21) cannot be applied. When the motion in latitude is
greater than in longitude, these equations are to be preferred instead
of (22) and (23.)
81. It would appear at first, without examining the quantities in-
volved in the formula for p f , that the equations (26) 3 will enable us
sec /?' = (n + n") - n - c, (28)
in which, if we introduce the values of and n + n rr as given by
fi
(26) and (27), only terms of the fourth order with respect to the
234 THEORETICAL ASTRONOMY.
times will be neglected, and consequently the resulting value of />'
will be affected with only an error of the second order when a is of
the third order. Further, if the intervals between the observations
are not very unequal, r 2 r" 2 will be a quantity of an order superior
to r 2 , and when these intervals are equal, we have, to terms of the
fourth order,
?L = !L'
n r'
The equation (27) gives
2/ 8 (n + n" 1) = TT".
Hence, if we put
P- n -
n' (29)
we may adopt, for a first approximation to the value of p',
and p f will be affected with an error of the first order when the in-
tervals are unequal ; but of the second order only when the intervals
are equal. It is evident, therefore, that, in the selection of the
observations for the determination of an unknown orbit, the in-
tervals should be as nearly equal as possible, since the nearer they
approach to equality the nearer the truth will be the first assumed
values of P and , thus facilitating the successive approximations ;
and when a is a very small quantity, the equality of the intervals
is of the greatest importance.
From the equations (29) we get
1 1/ _^ ( Tf-\ 1 * ~T ;\ t-t \\ /QIA
and introducing P and Q into (28), there results
This equation involves both p f and r' as unknown quantities, but
by means of another equation between these quantities p f may be
eliminated, thus giving a single equation from which r f may be
found, after which p f may also be determined.
DETERMINATION OF AN ORBIT. 235
82. Let ty represent the angle at the earth between the sun and
planet or comet at the second observation, and we shall have, from
the equations (93) 3 ,
tan/9 7
tan w r =
sin (A' Q')
-, (33^
cos *' = cos p cos (A' 0'),
by means of which we may determine ij/, which cannot exceed 180.
Since cos ft' is always positive, cos i// and cos (A' O ') must have the
same sign.
We also have
/2 = j. + jgr. _ 2 A'R cos 4,',
which may be put in the form
r' 2 = 0>' sec /3' R cos 4? -f R 2 sin 2 4',
from which we get
P ' sec ,3' = R' cos 4' VV* jR"sin4/. (34)
Substituting for /) r sec ^ r its value given by equation (32), we have
For brevity, let us put
4, c = k m (35)
' J c oS = 4>
and we shall have
k ^ = R cos V 1/r' 2 jR' 2 sm 2 V. (36)
When the values of P and Q have been found, this equation will
give the value of r' in terms of quantities derived directly from the
data furnished by observation. We shall now represent by z' the
angle at the planet between the sun and earth at the time of the
second observation, and we shall have
sm z'
236 THEORETICAL ASTRONOMY.
Substituting this value of r f , in the preceding equation, there results
(& - E' cos 40 sin z' + K sin *' cos z' = j^jj^-,> C38 )
and if we put
TJ Q sin C = K sin V,
, o cos Z = lc Q R' cos *', (39)
the condition being imposed that ra shall always be positive, we
have, finally,
sin (X q= C) = m sinV. (40)
In order that m may be positive, the quadrant in which is taken
must be such that % shall have the same sign as 1 , since sin i// is
always positive.
From equation (37) it appears that sin z' must always be positive,
or 2' < 180; and further, in the plane triangle formed by joining
the actual places of the earth, sun, and planet or comet corresponding
to the middle observation, we have
y / sin (X + 4/) K sin (z r + *Q
sin 4/ sin z'
Therefore,
and, since p r is always positive, it follows that sin (z' + ty) must be
positive, or that z 1 cannot exceed 180 ty.
When the planet or comet at the time of the middle observation is
both in the node and in opposition or conjunction with the sun, we
shall have /?' = 0, '\J/ = 180 when the body is in opposition, and
vJ/ = when it is in conjunction. Consequently, it becomes impos-
sible to determine r r by means of the angle z r ; but in this case the
equation (36) gives
*.-=-#+.',
when the body is in opposition, the lower sign being excluded by the
condition that the value of the first member of the equation must be
positive, and for \/ = 0,
the upper sign being used when the sun is between the earth and the
DETERMINATION OF AN ORBIT. 237
planet, and the lower sign when the planet is between the earth and
the sun. It is hardly necessary to remark that the case of an obser-
vation at the superior conjunction when /9' = 0, is physically impos-
sible. The value of r' may be found from these equations by trial ;
and then we shall have
when the body is in opposition, and
/ T}f fiJ
when it is in inferior conjunction with the sun.
For the case in which the great circle passing through the extreme
observed places of the body passes also through the middle place,
which gives a = 0, let us divide equation (32) through by c, and we
have
*" c 1 p' sec /?'
~+7~ ~~T~'
The equations (17) give
and if we put
c c _ r
~1 ; 7T ^>n-
we shall have
since c = GO when a = 0. Hence we derive
(42)
But when the great circle passing through the three observed places
passes also through the second place of the sun, both c and C be-
come indeterminate, and thus the solution of the problem, with the
given data, becomes impossible.
83. The equation (40) must give four roots corresponding to each
sign, respectively; but it may be shown that of these eight roots at
least four will, in every case, be imaginary. Thus, the equation may
be written
m sin 4 z' sin z' cos C + cos z' sin C,
238 THEORETICAL ASTRONOMY.
and, by squaring and reducing, this becomes
ra 2 sin 8 z' 2m cos C sin 5 / -f- sm2 *' sm2 C = 0.
When is within the limits 90 and + 90, cos will be positive,
and, m being always positive, it appears from the algebraic signs of
the terms of the equation, according to the theory of equations, that
in this case there cannot be more than four real roots, of which three
will be positive and one negative. When exceeds the limits 90
and -f- 90, cos will be negative, and hence, in this case also, there
cannot be more than four real roots, of which one will be positive
and three negative. Further, since sin 2 is real and positive, there
must be at least two real roots one positive and the other negative
whether cos be negative or positive.
"We may also remark that, in finding the roots of the equation (40),
it will only be necessary to solve the equation
sin (z = ra sin* /, (43)
since the lower sign in (40) follows directly from this by substituting
180 z r in place of 2'; and hence the roots derived from this will
comprise all the real roots belonging to the general form of the
equation.
The observed places of the heavenly body only give the direction
in space of right lines passing through the places of the earth and
the corresponding places of the body, and any three points, one in
each of these lines, which are situated in a plane passing through the
centre of the sun, and which are at such distances as to fulfil the
condition that the areal velocity shall be constant, according to the
relation expressed by the equation (30) 1? must satisfy the analytical
conditions of the problem. It is evident that the three places of the
earth may satisfy these conditions ; and hence there may be one root
of equation (43) which will correspond to the orbit of the earth, or
give
,' = 0.
Further, it follows from the equation (37) that this root must be
and such would be strictly the case if, instead of the assumed values
of P and , their exact values for the orbit of the earth were adopted,
and if the observations were referred directly to the centre of the
earth, in the correction for parallax, neglecting also the perturbation*
in the motion of the earth.
DETERMINATION OF AN ORBIT. 239
In the case of the earth,
sm(Q" Q)'
sm(O" O')
inCO" O)'
and the complete values of P and Q become
_ ^'sinCQ'-G)
~ f "" ''
, 3 / ER' sin(0'- Q) + RR' B in(0"- Q')
~ ' \ '" ;
and since the approximate values
P=C Q = rr"
differ but little from these, as will appear from the equations (27) 3 ,
there will be one root of equation (43) which gives z r nearly equal
to 180 ij/. This root, however, cannot satisfy the physical con-
ditions of the problem, which will require that the rays of light in
coming from the planet or comet to the earth shall proceed from
points which are at a considerable distance from the eye of the
observer. Further, the negative values of sin z 1 are excluded by the
nature of the problem, since r f must be positive, or z' < 180 ; and
of the three positive roots which may result from equation (43), that
being excluded which gives z' very nearly equal to 180 ij/, there
will remain two, of which one will be excluded if it gives z' greater
than 180 ^ r , and the remaining one will be that which belongs
to the orbit of the planet or comet. It may happen, however, that
neither of these two roots is greater than 180 ^'j in which case
both will satisfy the physical conditions of the problem, and hence
the observations will be satisfied by two wholly different systems of
Clements. It will then be necessary to compare the elements com-
puted from each of the two values of z' with other observations in
order to decide which actually belongs to the body observed.
In the other case, in which cos is negative, the negative roots
being excluded by the condition that r' is positive, the positive root
must in most cases belong to the orbit of the earth, and the three
observations do not then belong to the same body. However, in the
case of the orbit of a comet, when the eccentricity is large, and the
intervals between the observations are of considerable magnitude, if
240 THEORETICAL ASTRONOMY.
the approximate values of P and Q are computed directly, by means
of approximate elements already known, from the equations
p _ r/ sin (u r u)
-/^sirTK^')'
^ _ sin (u f - u) + *V' sin (u"- u'}
- 2r l
it may occur that cos is negative, and the positive root will actually
belong to the orbit of the comet. The condition that one value of
z f shall be very nearly equal to 180 $', requires that the adopted
values of P and Q shall differ but little from those derived directly
from the places of the earth ; and in the case of orbits of small
eccentricity this condition will always be fulfilled, unless the intervals
between the observations and the distance of the planet from the sun
are both very great. But if the eccentricity is large, the difference
may be such that no root will correspond to the orbit of the earth.
84. We may find an expression for the limiting values of m c and
, within which equation (43) has four real roots, and beyond which
there are only two, one positive and one negative. This change in
the number of real roots will take place when there are two equal
roots, and, consequently, if we proceed under the supposition that
equation (43) has two equal roots, and find the values of m and
which will accord with this supposition, we may determine the limits
required.
"Differentiating equation (43) with respect to z f , we get
cos (d C) = 4m sin V cos z' ;
and, in the case of equal roots, the value of z f as derived from this
must also satisfy the original equation
sin (z' C) = m sin V.
To find the values of m and which will fulfil this condition, if we
eliminate m between these equations, we have
sin z' cos (z* C) = 4 cos z' sin (z' C),
from which we easily find
sin (2z' = 1 sin C. (45)
This gives the value of in terms of z' for which equation (43) haa
DETERMINATION OF AN ORBIT. 241
equal roots, and at which it ceases to have four real roots. To find
the corresponding expression for m , we have
_ sin (Y C) cos (Y C)
; sin 4 z' ~ 4 sin V cos /
in which we must use the value of given by the preceding equation.
Now, since sin (2z f ) must be within the limits 1 and -f- 1, the
limiting values of sin will be + f and |, or must be within the
limits + 36 52'.2 and 36 52'.2, or 143 7'.8 and 216 52'.2. If
is not contained within these limits, the equation cannot have equal
roots, whatever may be the value of ra , and hence there can only be
two real roots, of which one will be positive and one negative. If
for a given value of we compute z' from equation (45), and call
this ZQJ or
sin O ' C) = | sin C,
we may find the limits of the values of m , within which equation
(43) has four real roots. The equation for Z Q ' will be satisfied by
the values
2* '-C, 180-(2z '-C);
and hence there will be two values of m , which we will denote by
m l and m 2 , for which, with a given value of , equation (43) will
have equal roots. Thus we shall have
m= sin(y-C)
sin \ f '
and, putting in this equation 180 (2z/ ) instead of 2z ' , or
90 (V C) in place of z ',
cos<
It follows, therefore, that for any given value of f, if m is not
within the limits assigned by the values of m x and m 2 , equation (43)
will only have two real roots, one positive and one negative, of
which the latter is excluded by the nature of the problem, and the
former may belong to the orbit of the earth. But if P and Q differ
so much from their values in the case of the orbit of the earth that
z 1 is not very nearly equal to 180 tj/, the positive root, when
exceeds the limits -f 36 52'.2 and 36 52'.2, may actually satisfy
the conditions of the problem, and belong to the orbit of the body
observed.
16
242 THEORETICAL ASTRONOMY.
When C is within the limits 143 7'.8 and 216 52'.2, there will
be four real roots, one positive and three negative, if m is within the
limits m l and m 2 ; but, if m surpasses these limits, there will be only
two real roots.
Table XII. contains for values of from 36 52'.2 to + 36 52'.2
the values of m l and m 2 , and also the values of the four real roots
corresponding respectively to m 1 and m 2 .
In every case in which equation (43) has three positive roots and
one negative root, the value of m must be within the limits indicated
by m 1 and m 2 , and the values of z f will be within the limits indicated
by the quantities corresponding to m l and m 2 for each root, which
we designate respectively by z/, z/, z 3 ', and z/. The table will show,
from the given values of m and 180 ij/, whether the problem
admits of two distinct solutions, since, excluding the value of z',
which is nearly equal to 180 ij/, and corresponds to the orbit of
the earth, and also that which exceeds 180, it will appear at once
whether one or both of the remaining two values of z 1 will satisfy
the condition that z' shall be less than 180 <$,'. The table will
also indicate an approximate value of z', by means of which the
equation (43) may be solved by a few trials.
For the root of the equation (43) which corresponds to the orbit
of the earth, we have p f = 0, and hence from (36) we derive
Substituting this value for k in the general equation (32), we have
-
and, since p r must be positive, the algebraic sign of the numerical
value of 1 Q will indicate whether r f is greater or less than R f . It is
easily seen, from the formulae for l w b, d, &c., that in the actual
application of these formulae, the intervals between the observations
not being very large, 1 will be positive when /9' /? and sin (O' K)
have contrary signs, and negative when /9 r /9 has the same sign as
sin (O' jfif). Hence, when O' K is less than 180, r r must be
less than R r if ft' /9 is positive, but greater than R f if /?' /9 is
negative. When O' K exceeds 180, r r will be greater than R'
if /9' /? is positive, and less than &' if /?' /9 is negative. We
may, therefore, by means of a celestial globe, determine by inspection
whether the distance of a comet from the sun is greater or less than
DETERMINATION OF AN ORBIT. 243
that of the earth from the sun. Thus, if we pass a great circ'e
through the two extreme observed places of the comet, r f must be
greater than R' when the place of the comet for the middle observa-
tion is on the same side of this great circle as the point of the
ecliptic which corresponds to the place of the sun. But when the
middle place and the point of the ecliptic corresponding to the place
of the sun are on opposite sides of the great circle passing through
the first and third places of the comet, r' must be less than R'.
85. From the values of // and r' derived from the assumed values
T "
P = and Q = rr", we may evidently derive more approximate
values of these quantities, and thus, by a repetition of the calcula-
tion, make a still closer approximation to the true value of p r . To
derive other expressions for P and Q which are exact, provided that
r f and p r are accurately known, let us denote by s ff the ratio of the
sector of the orbit included by r and r' to the triangle included by
the same radii-vectores and the chord joining the first and second
places ; by s' the same ratio with respect to r and r", and by s this
ratio with respect to r' and r". These ratios s, s f , s" must neces-
sarily be greater than 1, since every part of the orbit is concave
toward the sun. According to the equation (30)^ we have for the
areas of the sectors, neglecting the mass of the body,
and therefore we obtain
s "[r/]=:TV> /[W'J^rV^ s[tV']=Ty$. (46)
Then, since
we shall have
and, consequently,
[r/]
-
.. T 8 f , _^
= =- (47)
T-T" / oV
p sn
or
_ . / rr"sin(tt" M) \ _ 1
= \2v / r/ 7 cosJ(w" u)l p sin* %("
Therefore, the equation (59) becomes
1 _ E- sin "-
C ] ' ( J
Let Jt r be the chord of the orbit between the first and third places,
and we shall have
x' 2 = (r + r") 2 4rr" cos 2 J (n" w).
Now, since the chord #' can never exceed r + r /r , we may put
x'=(r-fr")sin r ', (61)
and from this, in combination with the preceding equation, we derive
7 cos J (M" M) = (r + /') cos /. (62)
DETERMINATION OF AN ORBIT. 249
Substituting this value, and \rr"~\ = -, Vp, in equation (60), it
reduces to
E"-E- S m(E"-E) r" 1 1 _
sin 3 1 (E" E) "(r+ r") 8 cos 3 p * 7* "*" 7
The elements a and e are thus eliminated, but the resulting equation
involves still the unknown quantities E" E and s f . It is neces-
sary, therefore, to derive an additional equation involving the same
unknown quantities in order that E" E maybe eliminated, and
that thus the ratio s r , which is the quantity sought, may be found.
From the equations
r a ae cos E, r" = a ae cos E",
we get
r " + r = 2a 2ae cos J (E" -f E) cos | (E" E).
Substituting in this the value of ecos%(E"-}- E) from (58), we have
r " 4. r = 2 a s i n * j (E" E) + 21/77' cos j (w' ; t*) cos J (^" ^),
and substituting for sin l(E tr E) its value from (57), there results
But, since
t*"~ u) (l-2sin
r sin' j (u ;/ M) _ (K r ]) 2 = 2r y2 / _ 1
^> ~ 2prr ff cos 2 ^ (t*" u)~ s' 2 \ 21/^y 7 cos
we have
from which we derive
which is the additional equation required, involving E" j&and '
as unknown quantities.
Let us now put
(65)
, sin'V
J= "W F "
250 THEORETICAL ASTRONOMY.
and the equations (63) and (64) become
.-
y f s' 3 '"'
(66)
When the value of y f is known, the first of these equations will
enable us to determine s', and hence the value of #', or sin 2 (i" E} t
from the last equation.
The calculation of f ma 7 be facilitated by the introduction of an
additional auxiliary quantity. Thus, let
(67)
and from (62) we find
cos / = cos (u" u) ., = 2 cos | (u" u} cos'/ tan /,
or
cos / = sin 2/ cos (u" u). (68)
We have, also,
*' 2 = (r + r'J 4rr" cos 2 J (t*" ),
which gives
x ' 2 = ( r r") 2 + 4rr" sin a J (*" u}.
Multiplying this equation by cos 2 J(it /r u) and the preceding one
by sin 2 \ (u" u), and adding, we get
x' 2 = (r + r") 2 sin 2 (w" u) + (r / r ) a cos 2 ^ (w v it). (69)
From (67) we get
and, therefore,
cos 2/=L-=i
r -f- r"
so that equation (69) may be written
_f 2 =: sin 2 / = sin 2 (w /; w) + cos 2 2/ cos 2 j (u" u).
We may, therefore, put
sin / cos G 1 = sin ^ (w" w),
sin / sin G' = cos (w" u) cos 2/, (70)
cos / = cos I (" w) sin 2/,
DETERMINATION OF AN ORBIT. 251
from which p' may be derived by means of its tangent, so that sin f f
shall be positive. The auxiliary angle G' will be of subsequent use
in determining the elements of the orbit from the final hypothesis for
P and Q.
88. We shall now consider the auxiliary quantity y' introduced
into the first of equations (66). For brevity, let us put
and we shall have
,_ sm 9 m
2g sin 2g
This gives, by differentiation,
dy' n 4 sin 2 q dq
4- = o cot q do -. g-i
y ^9 sm *>9
or
di/
-f- = 2>y cot g 4y 2 cosec g.
dg
The last of equations (65) gives a/ sin 2 #, and hence
dg
- = 2 cosec g.
Therefore we have
dy' _ Qy f cos g 8y fa = 3 (1 - 2aQ y' 4y"
dx' ~ sin 2 g 2x' (1 #')
It is evident that we may expand y f into a series arranged in refer-
ence to the ascending powers of x 1 ', so that we shall have
y' =. a, -f- fix' -f- ?%'* ~\- ^' 3 + ^'* -f- C' 5 -f~ &C.
Differentiating, we get
and substituting for -g* the value already obtained, there results
afc-r + (4r 2/9) ^' 2 + (6* 4r) ^' 3 + (8e 65) ^ + (IOC
= (3a 4a 2 ) + (3^ 6a 8a/9) a;' + (3r 6^ 4/5 2
+ (35 6r 8/?r 8a^) x' 3 -f (3e 65 4f 8/?<5 8ae) ^
4. (3C 6-- 8r<5 8/?e 8aC) x f& + &c.
Since the coefficients of like powers of x' must be equal, we nave
3 tt _ 4 a 2 = o, 3{3 6a 8a = 2A
3r 6,5 4;5 2 8<* r = 2 (2 r ), &c. ;
252 THEORETICAL ASTRONOMY.
and hence we derive
= i, P = T 9 o, r=T-5, d = T
cr _ 6228 f _ 265896 , _ 191390
e
336875*
_ 265896
2TS3e573
Therefore we have
If we multiply through by V, and put
*" + dii-fts*' 5
<=., (72;
we obtain
Combining this with the second of equations (66), the result is
VY+= +/+?'.
If we put
V = ____, (74)
we shall have
But from the first of equations (66) we get
7 -*(/-i;;
and therefore we have
^^- (75)
H- ^
As soon as 3/ is known, this equation will give the corresponding
value of s'.
Since ' is a quantity of the fourth order in reference to the differ-
ence | (E" E\ we may evidently, for a first approximation to the
value of y f , take
and with this find s f from (75), and the corresponding value of x*
from the last of equations (66). With this value of x f we find the
corresponding value of r , and recompute ?/, s', and x f ; and, if tL
DETERMINATION OF AN ORBIT. 253
value of ' derived from the last value of x f differs from that already
used, the operation must be repeated.
It will be observed that the series (72) for ' converges with great
rapidity, and that for E" j=94 the term containing # /6 amounts
to only one unit of the seventh decimal place in the value of c'. Table
XIV. gives the values of ' corresponding to values of x f from 0.0
to 0.3, or from E" E=0 to E" E = 132 50'.6. Should a
case occur in which E" E exceeds this limit, the expression
* ~ E"E sin (E" E)
may then be computed accurately by means of the logarithmic tables
ordinarily in use. An approximate value of x' may be easily found
with which y' may be computed from this equation, and then ' from
(73). With the value of ' thus found, if may be computed from
(74), and thus a more approximate value of x' is immediately
obtained.
The equation (75) is of the third degree, and has, therefore, three
roots. Since s' is always positive, and cannot be less than 1, it
follows from this equation that rf is always a positive quantity. The
equation may be written thus :
8* a" ,Y 1,/=0,
and there being only one variation of sign, there can be only one
positive root, which is the one to be adopted, the negative roots being
excluded by the nature of the problem. Table XIII. gives the
values of logs' 2 corresponding to values of if from j/=0 to J/=0.6.
When r/ exceeds the value 0.6, the value of s f must be found directly
from the equation (75).
89. We are now enabled to determine whether the orbit is an
ellipse, parabola, or hyperbola. In the ellipse x = sin* \(E" E)
is positive. In the parabola the eccentric anomaly is zero, and hence
x = 0. In the hyperbola the angle which we call the eccentric
anomaly, in the case of elliptic motion, becomes imaginary, and
hence, since sin \ (E" E) will be imaginary, x' must be negative.
It follows, therefore, that if the value of x' derived from the equa-
tion
is positive, the orbit is an ellipse ; if equal to zero, the orbit is a
parabola ; and if negative, it is a hyperbola.
254 THEORETICAL, ASTRONOMY.
For the case of parabolic motion we have x r 0, and the second
of equations (66) gives
" = j' (76)
If we eliminate s r by means of both equations, since, in this case,
y' = f, we get
m'l =/* -f- |/l
Substituting in this the values of m and I given by (65), we obtain
q f
= 3 sin y cos r r + 4 sin 3 1/,
which gives
6 '
= 6 sin i/ cos 2 tf + 2 sin" i/,
or
_f T , = (sin J/ + cos i/) 8 + (sin i/ - cos tf?.
This may evidently be written
= sn + - sn
the upper sign being used when f is less than 90, and the lower
sign when it exceeds 90. Multiplying through by (r -j- r /; )^, and
replacing (r -f- r") sin ^ by X, we obtain
6r ' = ( r + r " + x)f qz (r + r" - x)f,
which is identical with the equation (5G) 3 for the special case of
parabolic motion.
Since x' is negative in the case of hyperbolic motion, the value of
' determined by the series (72) will be different from that in the
case of elliptic motion. Table XIV. gives the value of ' corre-
sponding to both forms; but when x' exceeds the limits of this table,
it will be necessary, in the case of the hyperbola also, to compute the
value of ' directly, using additional terms of the series, or we may
modify the expression for y' in terms of E ff and E so as to be
applicable.
If we compare equations (44)! and (56) 1? we get
tan E= l/^I tan F;
DETERMINATION OF AN ORBIT. 255
and hence, from (58) w
We have, also, by comparing (65)! with (41 ) w since a is negative m
the hyperbola,
2<7 '
which gives
Now, since
c<
in which e is the base of Naperian logarithms, we have
E 1/^1 = loge (cos E -f 1/^T sin E\
which reduces to
or
E= I/HI log.
from which to obtain p. If we compute s and s" also, we shall have
/ sr'r" sin (u" u')\ 2 I s"rr f sin (u r u)\*
and the mean of the two values of p obtained from this expression
should agree with that found from (83), thus checking the calcula-
tion and showing the degree of accuracy to which the approximation
to P and Q has been carried.
The last of equations (65) gives
sm|(" Jg?) = T/^, (84)
from which E" E may be computed. Then, from equation (57),
since e = sin may be found.
Since
o , r r" - o f
cos 2/ = ; n, sm 2y = ; j
r + r ' r + r"
we have
g , 1> o^ _ 2 ^ o
r T F V / r/ / sin2/
p p _ 2p cot 2/
" '
and from equations (70),
. , sin i (it" w) tan 6r f -of cos /
cot 2/ == - - / - , sm 2/ = - T7-77 - r.
cos / cos ^ (u u)
Therefore the formulae (87) reduce to
* sin , - " t* = - == tan G'
(88)
cos (w (?/' + w)) = - x _ sec J (w v w;,
cos/F rr"
from which also 6 and co may be derived. Then
sin ) 8
~~
COS 2 ~ p
or it may be computed directly from the equation
4s'W cos 2 ^ (u" u) sm*$ (E" JB
which results from the substitution, in the last term of the preceding
equation, of the expressions for a cos , V f = U to, v" = u" ct> '
and if we compute r, r' 9 r" from these by means of the polar equa-
tion of the conic section, the results should agree with the values of
the same quantities previously obtained. According to the equation
(45)j, we have
tan \E = tan (45 ?) tan Jv,
tan \E' = tan (45 J?) tan jv', (90)
tan E" == tan (45 j?) tan W',
from which to find E, E' y and E" . The difference E" E should
agree with that derived from equation (84) within the limits of
accuracy afforded by the logarithmic tables. Then, to find the mean
anomalies, we have
M =E eswE,
M'=E'esmE', (91;
M" = E"esmE";
and, if M Q denotes the mean anomaly corresponding to any epoch T t
we have, also,
? n the application of which the values of t, t> ', and t" must be those
which have been corrected for the time of aberration. The agree-
262 THEORETICAL ASTRONOMY.
ment of the three values of M will be a final test of the accuracy of
the entire calculation. If the final values of P and Q are exact,
this proof will be complete within the limits of accuracy admitted
by the logarithmic tables.
When the eccentricity is such that the equations (91) cannot be
solved with the requisite degree of accuracy, we must proceed accord-
ing to the methods already given for finding the time from the peri-
helion in the case of orbits differing but little from the parabola.
For this purpose, Tables IX. and X. will be employed. As soon as
v, v f , and v" have been determined, we may find the auxiliary angle
V for each observation by means of Table IX. ; and, with V as the
argument, the quantities M 9 M f , M" (which are not the mean anoma-
lies) must be obtained from Table VI. Then, the perihelion distance
having been computed from
P
we shall have
in which log (7 = 9.96012771, for the determination of the time of
perihelion passage. The times t, t', t u must be those which have
been corrected for the time of aberration, and the agreement of the
three values of T is a final proof of the numerical calculation.
If Table X. is used, as soon as the true anomalies have been found,
the corresponding values of log B and log C must be derived from
the table. Then w is computed from
and similarly for w f and w" ; and, with these as arguments, we derive
M, M r , M" from Table VI. Finally, we have
MBq* _
(93)
for the time of perihelion passage, the value of O being the same as
in (92).
When the orbit is a parabola, e = I and p 2g, and the elements
Q and '
is in each case the declination. From these we derive the longitude
and latitude of the zenith for each observation, namely,
4,= 6033'.9, 4'= 35035'.2, ^' = 342 59'.2,
6 = + 22 25.0, & '=4-50 50.9, & "= + 53 41.6.
Then, by means of equations (4), we obtain
= 18".92, A' = 36".94, A0" = 25".76,
A log E Q = 0.0001084, A log Rj = 0.0002201,
A log # " = 0.0002796.
For the reduction of time, we have the values + OM5, -j~ 0*.28, and
-f- 8 .34, which are so small that they may be neglected.
2G6 THEOEETICAL ASTRONOMY.
Finally, the longitudes of both the sun and planet are reduced to
the mean equinox of 1863.0 by applying the corrections
50".95, -51". 52, 52".14;
and the latitudes of the planet are reduced to the ecliptic of the same
date by applying the corrections O r/ .15, O r/ .14, and O r/ .14,
respectively.
Collecting together and applying the several corrections thus ob-
tained for the places of the sun and of the planet, reducing the un-
corrected times of observation to the meridian of Washington, and
expressing them in days from the beginning of the year, we have the
following data :
t Q = 257.68079, A = 17 46' 28".17, = + 3 8' 43".51,
tj = 264.42570, A' = 16 40 25 .19, f = 2 52 27 .62,
C' = 271.38625, A" = 15 1544.03, P" = + 2 3242.98,
O =172 0'32".23, log =0.0021056,
O' =178 35 48 .74, logjR' =0.0011656,
0"=185 25 36 .90, log R" = 0.0002378.
The numerical values of the several corrections to be applied to
the data furnished by observation and by the solar tables should be
checked by duplicate calculation, since an error in any of these re-
ductions will not be indicated until after the entire calculation of the
elements has been effected.
By means of the equations
m(0" 0Q _ EE f sm(Q' Q)
" RR" sin (O" O) ' RR" sin (" 0)'
tan/3' tan (A' -Q')
cosrf '
we obtain
log N= 9.7087449, log N" = 9.6950091,
4/ = 16142'13".16,
log (R' sin V) = 9.4980010, log (R cos 4/) = 9.9786355..
The quadrant in which -\}/ must be taken is determined by the con-
ditions that i// must be less than 180, and that cos^' and cos (A' O')
must have the same sign. Then from
NUMEKICAL EXAMPLE.
tan Jsin Q (/' + A) - K) = |^gf , sec J (/' - i),
tanlcosQ (A" + A) - K) = " , cosec | -
267
JST)
G" JjQ sec/3' jfcft" sm(G" 0)
J~ sin (A" A)' a sin (/>/' A)~~
we compute K, I, $,, a , 6, c, c?, /, and h. The angle J must be less
than 90, and the value of ft must be determined with the greatest
possible accuracy, since on this the accuracy of the resulting elements
principally depends. Thus we obtain
K= 4 47' 29".48, log tan 1= 9.3884640,
A 2 52' 59"f if , log a = 6.8013583 n ,
log 6 = 2.5456342 n , log c = 2.2328550 W ,
log d = 1.2437914, log/= 1.3587437 n , log h = 3.924769L
The formulae
, _ sin (A ; /I) ^ sin (/I Q)
" J "
gve
sin (A" A)
sin (/" K)
log M, = 9.8946712,
log Jf a = 1.9404111,
hsm(l
log Mf = 9.6690383,
log M z " = 0.7306625 n .
The quantities thus far obtained remain unchanged in the suc-
cessive approximations to the values of P and Q.
For the first hypothesis, from
i? sin Z,=R sin 4/,
T)Q cos C = & -R' cos 4*',
268 THEORETICAL ASTRONOMY.
we obtain
log r = 9.0782249, log T" = 9.0645575,
log P = 9.9863326, log Q = 8.1427824,
log c == 2.2298567 n , log k = 0.0704470,
log 1 Q = 0.0716091, log 7y = 0.3326925,
C = 8 24' 49".74, Iogm == 1.2449136.
The quadrant in which must be situated is determined by the con-
dition that ^ shall have the same sign as b
The value of z 1 must now be found by trial from the equation
sin (si C) = m sin 4 sf.
Table XII. shows that of the four roots of this equation one exceeds
180, and is therefore excluded by the condition that sin/ must be
positive, and that two of these roots give z f greater than 180 ij/,
and are excluded by the condition that z r must be less than 180 -J/.
The remaining root is that which belongs to the orbit of the planet,
and it is shown to be approximately 10 40' ; but the correct value
is found from the last equation by a few trials to be
z' = 9 1'22".96.
The root which corresponds to the orbit of the earth is 18 20' 41",
and differs very little from 180 ty.
Next, from
we derive
log r' = 0.3025672, log P ' = 0.0123991,
'' log n = 9.7061229, log n" = 9.6924555,
log p = 0.0254823, log f = 0.0028859.
The values of the curtate distances having thus been found, the
heliocentric places for the three observations are now computed from
NUMERICAL EXAMPLE. 269
r cos b cos (I O) =p cos (A Q) R,
r cos b sin (I Q) =^sin(A Q),
r sin & = f> tan /? ;
/ cos 6' cos (V Q =y cos (A' ') #,
/ cos V sin (r .0') = P f sin (A' '),
r' sin b' />' tan p ;
r" cos 6" cos (r Q") = P " cos (A" 0") J2",
r" cos 6" sin (f 0") = /e," sin (A" 0"),
which give
J = 5 14' 39".53, log tan b =8.4615572, logr =0.3040994,
V = 7 45 11 .28, log tan b' =8.4107555, logr' =0.3025673,
I" = 10 21 34 .57, log tan b" = 8.3497911, log r" = 0.3011010.
The agreement of the value of logr' thus obtained with that already
found, is a proof of part of the calculation. Then, from
(\ nn . 7^ r^\ tan 6"-}- tan 6
nG(/ + - Q) = . .
cos i cos ^ cos i
we get
ft = 207 2' 38".16, t = 4 27' 23".84,
u = 158 8' 25".78, u f = 160 39' 18".13, u" = 163 16' 4".42.
The equation
tan b' = tan i sin (T ft )
gives log tan b r = 8.4107514, which differs 0.0000041 from the value
already found directly from //. This difference, however, amounts
to only O r/ .05 in the value of .the heliocentric latitude, and is due to
errors of calculation. If we compute n and n" from the equations
r'r" sin (u" u') rr r sin (u' u)
n = . ; , n" =
rr" sin (u" U*)' ~ rr" sin (" u) '
the results should agree with the values of these quantities previously
computed directly from P and Q. Using the values of u, u', and
u" just found, we obtain
log n = 9.7061158, log n" = 9.6924683,
270 THEORETICAL ASTRONOMY.
which differ in the last decimal places from the values used in finding
p and p ff . According to the equations
d log n = 21.055 cot (u" ') du',
d log n" = -f 21.055 cot (u r u) du',
the differences of logn and logn" being expressed in units of the
seventh decimal place, the correction to u' necessary to make the two
values of logn agree is 0".15; but for the agreement of the two
values of log??/', u' must be diminished by 0".26, so that it appears
that this proof is not complete, although near enough for the first
approximation. It should be observed, however, that a great circle
passing through the extreme observed places of the planet passes
very nearly through the third place of the sun, and hence the values
of p and p" as determined by means of the last two of equations (18)
are somewhat uncertain. In this case it would be advisable to com-
pute p and p", as soon as p f has been found, by means of the equa-
tions (22) and (23). Thus, from these equations we obtain
log p = 0.025491 8, log p" = 0.0028874,
and hence
I = 514'40".05, log tan b =8.4615619, log r = 0.3041042,
J"=10 2134.19, log tan b" =8.3497919, log/' =0.3011017,
& == 207 2' 32".97, i = 4 27' 25".13,
u = 158 8' 31".47, u' = 160 39' 23".31, u" = 163 16' 9".22.
The value of log tan b f derived from X f and these values of Q, and i,
is 8.4107555, agreeing exactly with that derived from p f directly.
The values of n and n" given by these last results for u, u' and u" y
are
log n = 9.7061144, log n" = 9.6924640 ;
and this proof will be complete if we apply the correction du f = 0".18
to the value of u', so that we have
u" u' = 2 36' 46".09, u' u = 2 30' 51".66.
The results which have thus been obtained enable us to proceed to
a second approximation to the correct values of P and , and we
may also correct the times of observation for the time of aberration
by means of the formulae
t = t Q Cp sec /?, t = t Q ' Cp' sec p, t" = t " Cp" sec p',
wherein log C= 7.760523, expressed in parts of a day. Thus we get
t = 257.67467, if = 264.41976, *" = 271.38044,
NUMERICAL EXAMPLE. 271
and hence
log r = 9.0782331, log r' = 9.3724848, log T" = 9.0645692.
Then, to find the ratios denoted by s and s", we have
sin Y cos Q = sin | (u" t/)>
sin Y sin G = cos J (u" u') cos 2/,
cos 7- = cos ^ (u" u') sin 2/ ;
tan/' =
sin f' cos G" = sin ^ (V w),
sin r" sin G" = cos J (V t*) cos 2/',
cos /' = cos (M' w) sin 2/' ;
r 2 . sin 2
") 3 cos 3 /
from which we obtain
y = 44 57 f 6".00, /' = 44 56 f 57".50,
r= 1 18 35 .90, r "= 1 15 40 .09,
logm = 6.3482114, logm' ; = 6.3163548,
log.; = 6.1163135, log/' = 6.0834230.
From these, by means of the equations
m m
V
using Tables XIII. and XIV., we compute s and s". First, in the
case of s, we assume
y = _ = 0.0002675,
and, with this as the argument, Table XIII. gives log s 2 = 0.0002581 .
Hence we obtain x f = 0.000092, and, with this as the argument,
Table XIV. gives c = 0.00000001 ; and, therefore, it appears that a
repetition of the calculation is unnecessary. Thus we obtain
logs =0.0001290, logs" =0.0001 200.
When the intervals are small, it is not necessary to use the formulae
272 THEORETICAL ASTRONOMY.
in the complete form here given, since these ratios may then be found
by a simpler process, as will appear in the sequel. Then, from
TT" r"
ss" rr" cos J (it" w') cos J (it" tt) cos ^ (M' )'
we find
log P = 9.9863451, log Q = 8.1431341,
with which the second approximation may be completed. We now
compute c , 7c Q , 1 , z' ', &c. precisely as in the first approximation ; but
we shall prefer, for the reason already stated, the values of p and p"
computed by means of the equations (22) and (23) instead of those
obtained from the last two of the formulae (18). The results thus
derived are as follows :
log c = 2.2298499 n , log Jc = 0.0714280,
log 1 = 0.0719540, log i? = 0.3332233,
C = 8 24' 12".48, log m = 1.2447277,
z f = 9 0' 30".84,
log / 0.3032587, log p' = 0.0137621,
log n == 9.7061153, log n"= 9.6924604,
log/> = 0.0269143, log/' = 0.0041748,
I = 5M5'57".26, log tan b =8.4622524, logr =0.3048368,
I' = 7 46 2.76, log tan V = 8.4114276, log/ = 0.3032587,
l rf = 10 22 .91, log tan b" = 8.3504332, log r" = 0.3017481,
a = 207 0' 0".72, * = 4 28 r 35' r .20,
u = 158 12' 19".54, u' = 160 42' 45".82, u" = 163 19' 7".14.
The agreement of the two values of logr' is complete, and the value
of log tan b' computed from
tan b' = tan i sin (l r Q> ),
is log tan b f = 8. 41 1427 9, agreeing with the result derived directly
from p r . The values of n and n" obtained from the equations (54)
are
log n = 9.7061156, log n" = 9.6924603,
which agree with the values already used in computing p and p", and
the proof of the calculation is complete. We have, therefore,
tt "_ u > = 2 36' 21".32, u' u = 2 30' 26".28, u" u = 5& 47".60.
From these values of ii n u r and u f u, we obtain
log s = 0.0001284, logs" = 0.0001193,
NUMERICAL EXAMPLE. 273
and, recomputing P and Q, we get
log P == 9.9863452, log Q = 8.1431359,
which differ so little from the preceding values of these quantities
that another approximation is unnecessary. We may, therefore, from
the results already derived, complete the determination of the elements
of the orbit.
The equations
sin / cos G' = sin -^ (u" u\
sin / sin G r = cos J (u" u) cos 2/,
cos / = cos (u" Uj sin 2/,
m , = v _sm
" 3 J ""
(r-j-r") 3 cosV
give
/ = 44 53' 53".25, / = 2 33' 52".97, log tan G' = 8.9011435
log m' = 6.9332999, log/ = 6.7001345.
From these, by means of the formulae
'_ m ' '
""!+/+* ~*' a '^'
and Tables XIII. and XIV., we obtain
logs' 2< =: 0.0009908, loga/ = 6.549411U
Then from
we get
logjs = 0.3691818.
The values of logp given by
s/r" sin (i*" w') \ 2 / s'Vr' sin (w f u) \"
are 0.3691824 and 0.3691814, the mean of which agrees with the
result obtained from u" u, and the differences between the separate
results are so small that the approximation to P and Q is sufficient,
The equations
P
cos = 190 15' 39".57, log e = log sin ? = 9.2751434,
?> == 10 51 39 .62, TT = , -f- = 37 15' 40".29.
This value of ^ gives log cos
, v" = u" cw,
according to which we have
v = 327 56' 39".97, v' = 330 27' 6".25, i" = 333 3' 27".57.
If we compute r, r f , and r" from these values by means of the polar
equation of the ellipse, we get
log r = 0.3048367, log r' = 0.3032586, log r" = 0.3017481,
and the agreement of these results with those derived directly from
p, p f , and p rr is a further proof of the calculation.
The equations
tan { E = tan (45 ?) tan v,
tan ^E' = tan (45 1$0 tan ^,
tan JJE" = tan (45 ?) tan W
give
E = 333 17' 28' .18, E' = 335 24' 38".00, E" = 337 36' 19".78.
NUMERICAL EXAMPLE. 275
The \alue of J (E" E) thus obtained differs only 0".003 from that
computed directly from x' '.
Finally, for the mean anomalies we have
M = E e sin E, M' = E' e sin E' y M" = E" e sin E'\
from which we get
M = 338 8' 36".71, M' = 339 54' 10".61, M" = 341 43' 6".97 ;
and if Jf denotes the mean anomaly for the date T=1863 Sept. 21.5
Washington mean time, from the formulae
M=M n(t r>
we obtain the three values 339 55' 25".97, 339 55' 25".96, and
339 55' 25".96, the mean of which gives
M 9 = 339 55' 25".96.
The agreement of the three results for Jf is a final proof of the
accuracy of the entire calculation of the elements.
Collecting together the separate results obtained, we have the fol-
lowing elements :
Epoch = 1863 Sept. 21.5 Washington mean time.
M = 339 55' 25".96
*= 37 15 40 .29)
ft - 207 72 V Ech P tlc and Mean
06 * J " ' v v I *i / . . -or\
t= 4 28 35.20J Equinox 1863.0.
(99)
in which log JA = 8.8596330. We have, also, to the same degree of
approximation,
!< '=71' to^ = -5> (100)
For the values
log r = 9.0782331, log r f = 9.3724848, log r" ** 9.0645692,
log/ = 0.3032587,
these formulae give
log s = 0.0001277, log s' = 0.0004953, log " = 0.0001199,
which differ but little from the correct values 0.0001284, 0.0004954,
and 0.0001193 previously obtained.
Since
sec 8 / = 1 + 6 sin 2 tf + &c.,
the second of equations (65) gives
sin ' &c -
Substituting this value in the first of equations (66), we get
If we neglect terms of the fourth order with respect to the time, it
will be sufficient in this equation to put y r = f, according to (71), and
hence we have
4
3
and, since s f 1 is of the second order with respect to r', we have,
to terms of the fourth order,
280 THEORETICAL ASTRONOMY.
Therefore,
which, when the intervals are small, may be used to find s f from r
and r". In the same manner, we obtain
l^r,- (102)
For logarithmic calculation, when addition and subtraction loga-
rithms are not used, it is more convenient to introduce the auxiliary
angles , r , and ", by means of which these formulae become
K tW (103)
in which log |^ = 9.7627230. For the first approximation these
equations will be sufficient, even when the intervals are considerable,
to determine the values of s and s" required in correcting P and Q.
The values of r, r', r", and r" above given, in connection with
log r = 0.3048368, log r" = 0.3017481,
give
log 5 = 0.0001284, log s f = 0.0004951, log s" = 0.0001193.
These results for logs and logs" are correct, and that for logs' differs
only 3 in the seventh decimal place from the correct value.
ORBIT FEOM FOUK OBSERVATIONS.
CHAPTER V.
DETERMINATION OF THE ORBIT OF A HEAVENLY BODY FROM FOUR OBSERVATIONS,
OF WHICH THE SECOND AND THIRD MUST BE COMPLETE.
95. THE formulae given in the preceding chapter are not sufficient
to determine the elements of the orbit of a heavenly body when its
apparent path is in the plane of the ecliptic. In this case, however,
the position of the plane of the orbit being known, only four ele-
ments remain to be determined, and four observed longitudes will
furnish the necessary equations. There is no instance of an orbit
whose inclination is zero ; but, although no such case may occur, it may
happen that the inclination is very small, and that the elements
derived from three observations will on this account be uncertain,
and especially so, if the observations are not very exact. The diffi-
culty thus encountered may be remedied by using for the data in the
determination of the elements one or more additional observations,
and neglecting those latitudes which are regarded as most uncertain.
The formulae, however, are most convenient, and lead most expe-
ditiously to a knowledge of the elements of an orbit wholly unknown,
when they are made to depend on four observations, the second and
third of which must be complete ; but of the extreme observations
only the longitudes are absolutely required.
The preliminary reductions to be applied to the data are derived
precisely as explained in the preceding chapter, preparatory to a de-
termination of the elements of the orbit from three observations.
Let t, t', t", t'" be the times of observation, r, r', r", r'" the radii-
vectores of the body, u, u f , u n ', u'" the corresponding arguments of
the latitude, R, R', R", R" r the distances of the earth from the sun,
and O, O', O", O'" the longitudes of the sun corresponding to
these times. Let us also put
= r'r'"sm(u'" u'\
[rV"] = rV" sin (u" r u"\
and
282 THEORETICAL ASTRONOMY.
Then, according to the equations (5) 3 , we shall have
nx x' -f- ri'x" = 0,
ny _' cos A' R' cos O')
+ n"G
= n sin A J2 sin O ) (f? sin A r R sin O'
+ w"
= n ' G/ cos / - R cos O') (p" cos A" #' cos O") (3)
-f n'" (p'" cos X'" R" cos 0'"),
= w' (p r sin A' ^ sin O') (p" sin X" R" sin 0")
4- n'" (/>"' sin A'" R" sin 0'").
If we multiply the first of these equations by sin ^, and the second
by cos ^, and add the products, we get
= nR sin (A Q) (p* sin (X X) + R f sin (A 0'))
+ n" (P" sin (A" A) -f- #' sin (A ")) ; (4)
and in a similar manner, from the third and fourth equations, we
find
= n' (p f sin (X" X ) R' sin (X" ')) (5j)
- (p" sin (/" A") K' sin (/" ")) - n'"R" sin (A'" 0'").
Whenever the values of n, n' 9 n", and n //r are known, or may be
determined in functions of the time so as to satisfy the conditions of
motion in a conic section, these equations become distinct or inde-
Dendent of each other ; and, since only two unknown quantities p 1
OEBIT FROM FOUR OBSERVATIONS. 283
and p n are involved in them, they will enable us to determine these
curtate distances.
Let us now put
cos p sin (/ /I) = A, cos ft" sin (A" X)=B,
cos ft" sin (/'" A") = C, cos p sin (*'" X ) = D,
and the preceding equations give
Ap' sec ^ Bn" P " sec /5" = wJ5 sin (A ) R r sin (A Q')
+ n"J2"Biny 0"),
!>>' sec /?' Q/' sec /9"= n'tf sin (/" ') R" sin (A'" Q") (7)
+ n'"R'" sin (/" Q'").
Tf we assume for n and n" their values in the case of the orbit of
the earth, which is equivalent to neglecting terms of the second order
in the equations (26) 3 , the second member of the first of these equa-
tions reduces rigorously to zero ; and in the same manner it can be
shown that when similar terms of the second order in the corre-
sponding expressions for n f and n" are neglected, the second member
of the last equation reduces to zero. Hence the second member of
each of these equations will generally differ from zero by a quantity
which is of at least the second order with respect to the intervals of
time between the observations. The coefficients of p' and p" are of
the first order, and it is easily seen that if we eliminate p" from
these equations, the resulting equation for // is such that an error of
the second order in the values of n and n" may produce an error of
the order zero in the result for p f , so that it will not be even an
approximation to the correct value ; and the same is true in the case
of p rr . It is necessary, therefore, to retain terms of the second order
in the first assumed values for n, n f , n", and ii' 1 ' \ and, since the
terms of the second order involve r f and r f/ , we thus introduce two
additional unknown quantities. Hence two additional equations in-
volving r f , r", p'j p" and quantities derived from observation, must
be obtained, so that by elimination the values of the quantities sought
may be found.
From equation (34) 4 we have
P' sec p = R' cos 4/ V r' 2 R r * sin 2 4/, (8)
which is one of the equations required; and similarly we find, for
the other eauation,
P " sec ft" = R" cos V ' db y r'" 2 R" 2 sin 2 4". (9)
284 THEORETICAL ASTRONOMY.
Introducing these values into the equations (7), and putting
x' = l/V 2 -.fl' 2 sin 2 47,
x''=V / r"* # 2 sin 2 4/',
we get
Ax' Bri'x" = nR sin (A O) R' sin (A 0')
-f n"R" sin (A O") AR' cos V + n"BR" cos *",
ZtoV Cx" = n'R' sin (X" 0') jR" sin (A"' 0")
-j- n'"R"' sin (A'" 0'") n'DR' cos 4' + CR" cos 4,".
Let us now put
B -h' D V>
A ht C =h >
or
, _ cos /3" sin (A" A) , _ cos ff sin (r r AQ
" cos f sin (A' A) ' ~ cos jt' sin (A'" A")'
and we have
x' = Kn"x" -f ndT of + n'V,
a;" = A" n y + n '"d" a" -f nV.
These equations will serve to determine x r and x", and hence r f and
r r/ , as soon as the values of n f n f , n ff , and n" 1 are known.
96. In order to include terms of the second order in the values of
n and n", we have, from the equations (26) 3 ,
and, putting
these give
ORBIT FROM FOUR OBSERVATIONS. 285
Lot us now put
T '" = k cr o, v = (*"' o
and, making the necessary changes in the notation in equations (26) s ,
we obtain
,r-"(r ' + r) , T"' (T"*+ T"'T - 1) dr" \
~* fn. -- - i^j --- ar*"f
r ** at i ,.
I . r ( r ' + ^'") , .r(r' + rr'"-r'"0 dr"
fs ^^ ~& 7r ~ "df
From these we get, including terms of the second order,
and hence, if we put
P" = ~, " = (' + '"-!)/', (17)
we shall have, since r ' = r + r' /r ,
= n.
When the intervals are equal, we have
P' P"
= 777 -pm
and these expressions may be used, in the case of an unknown orbit,
for the first approximation to the values of these quantities.
The equations (13) and (17) give
and, introducing these values, the equations (12) become
286 THEORETICAL ASTEOKOMY.
* = W-pr ( 1 + % } (*V + P'd' + c') -
1 ff'\
x " = TTpr ( 1 + ;* ) (A ' V + p " d " + c " }
Let us now put
P'cf+c'
An
c" -f
C 1 I TV/ J
(21;
1 + P" 1 +
and we shall have
(22)
+e ")-a".
We have, further, from equations (10),
r''=(*/'+P' 2 sinV)',
r f; =(a/'+JB"sinV / ) f ,
If we substitute these values of r' 3 and r" z in equations (22), the two
resulting equations will contain only two unknown quantities x' and
x". when P', P", ', and Q" are known, and hence they will be
sufficient to solve the problem. But if we effect the elimination of
either of the unknown quantities directly, the resulting equation
becomes of a high order. It is necessary, therefore, in the numerical
application, to solve the equations (22) by successive trials, which
may be readily effected.
If z' represents the angle at the planet between the sun and the
earth at the time of the second observation, and z" the same angle at
the time of the third observation, we shall have
, _ E' sin 4/
~'
(24)
,,_'
sin z"
Substituting these values of r f and r" in equations (10), we get
*'=r>z> (25)
x" = r" cos z ",
and hence
ORBIT FKOM FOUR OBSERVATIONS. 287
, R' sin 4,'
tan z== -- -, - ,
(26)
by means of which we may find z' and z" as ,soon as x f and x" shall
have been determined ; and then r' and r" are obtained from (24) or
(25). The last equations show that when x r is negative, z r must be
greater than 90, and hence that in this case r' is less than R f .
In the numerical application of equations (22), for a first approxi-
mation to the values of x r and x", since Q' and Q" are quantities of
the second order with respect to r or r r// , we may generally put
and we have
*'
*"
or, by elimination,
,
1 -/'/"
1 /'/"
With the approximate values of x' and x" derived from these equa-
tions, we compute first r r and r" from the equations (26) and (24),
and then new values of x' and x" from (22), the operation being
repeated until the true values are obtained. To facilitate these ap-
proximations, the equations (22) give
~/'(;+f ). >;
*/ = . " "^
Let an approximate value of x f be designated by # ', and let the
value of x" derived from this by means of the first of equations (27)
be designated by x ". With the value of x " for x" we derive a
new value of x f from the second of these equations, which we denote
by a?/. Then, recomputing x" and x', we obtain a third approximate
value of the latter quantity, which may be designated by a?,'; and,
if we put
a?/ x ' = a Q , #,' xj ',
288 THEORETICAL ASTRONOMY.
we shall have, according to the equation (67) 3 , the necessary changes
being made in the notation,
z' = < ?^- = x; PL-. (28)
< o o o
The value of x' thus obtained will give, by means of the first of
equations (27), a new value of x", and the substitution of this in the
last of these equations will show whether the correct result has been
found. If a repetition of the calculation be found necessary, the
three values of x' which approximate nearest to the true value will,
by means of (28), give the correct result. In the same manner, if
we assume for x" the value derived by putting Q f = and Q" = 0,
and compute x f , three successive approximate results for x" will
enable us to interpolate the correct value.
When the elements of the orbit are already approximately known,
the first assumed value of x' should be derived from
instead of by putting Q f and Q" equal to zero.
97. It should be observed that when A' = A or A'" = A", the equa-
tions (22) are inapplicable, but that the original equations (7) give,
in this case, either p" or p' directly in terms of n and n ff or of n'
and n'" and the data furnished by observation. If we divide the
first of equations (22) by h f , we have
The equations (21) give
f 1 /' P 17 ~r~ 17
h' ~~ 1 -I- P" A' ~ 1 + P'
and from (11) we get
of _ R' cos V B sin (I Q')
A' ~ A' B
1 = tf'cos*" + ^" sin ^- Q "), (29;
d' J?sin(A O)
A'~
Then, if we put
ORBIT FROM FOUR OBSERVATIONS.
its value may be found from the results for and ^ derived by
means of these equations, and we shall have
When A' = I, we have h' == oo, and this formula becomes
=( 1 + -J ) (*" + C ') - d + P'),
the value of j-, being given by the first of equations (29) This
equation and the second of equations (22) are sufficient to determine
x f and x" in the special case under consideration.
The second of equations (22) may be treated in precisely the same
manner, so that when X ffr = X", it becomes
o
=(
and this must be solved in connection with the first of these equations
in order to find x f and x ff .
98. As soon as the numerical values of x r and x n have been
derived, those of r f and r" may be found by means of the equations
(26) and (24). Then, according to (41) 4 , we have
X*& MO
The heliocentric places are then found from p f and p" by means of
the equations (71) 3 , and the values of r' and r" thus obtained should
agree with those already derived. From these places we compute
the position of the plane of the orbit, and thence the arguments of
the latitude for the times t f and t".
The values of r', r", u', u", n, n", n f , and n"' enable us to deter-
mine r, r" r , u t and u" f . Thus, we have
and, from the equations (1) and (3) 3 ,
19
MJK) THEORETICAL ASTRONOMY.
[rr"] =
Therefore,
n
r sin (u f u) = r" sin (u" w')
n
r sin (u" u) = - r' sin (it" it'),
(32)
i>" sin ("' ") = 4? r' sin (" '),
71
r'" sin (t*'" *') = ^7 r" sin (u" ')
From the first and second of these equations, by addition and sub-
traction, we get
r sin ((u f u) -J- 3 (w" w')) = sm 2 V w ')
/- re 'V" (33)
r cos ((i/ u) -f- 2 (w" i*')) = cos J (w" it'),
71
from which we may find r, u r it, and u = u r (u f it).
In a similar manner, from the third and fourth of equations (32),
we obtain
r m sin ((u m t*") + J (it" i/)) = V sin J (i*" - w'),
,/ w (34)
r'" cos ((i* w M ") + A (w" u')) = - m^- cos J (w" w'),
71
from which to find r'" and it'".
When the approximate values of r, r f , r", r f ", and w, it', it", u fn
have been found, by means of the preceding equations, from the
assumed values of P', P", Q f , and Q", the second approximation to
the elements may be commenced. But, in the case of an unknown
orbit, it will be expedient to derive, first, approximate values of
and r", using
p' _!_, p" = _L,
and then recompute P f and P" by means of the equations (14) and
ORBIT FROM FOUR OBSERVATIONS. 291
(18), before finding u f and u". The terms of the second order will
thus be completely taken into account in the first approximation.
99. If the times of observation have not been corrected for the
time of aberration, as in the case of an orbit wholly unknown, this
correction may be applied before the second approximation to the
elements is effected, or at least before the final approximation is com-
menced. For this purpose, the distances of the body from the earth
for the four observations must be determined ; and, since the curtate
distances p f and p ff are already given, there remain only p and p' tf to
be found. If we eliminate p f from the first two of equations (3), the
result is
nR sin (A' Q) R sin (A' Q') -f n" R" sin (X 0").
and, by eliminating p" from the last two of these equations, we also
obtain
, , n' sin (A" -A')
' : = / V'sm(A'"-/')
n 1 R' sin (A" - Q') R" sin (A" Q ") -f ri" R" f sin (A" "')
n'" sin (A'" /I")
by means of which p and p" f may be found. The combination of
the first and second of equations (3) gives
COS (X
nR cos a Q) R' cos (A 0') + n" R" cos (A 0")
/o = ^ cos (A' A) 2_- cos (A" A) (37)
n n
+
and from the third and fourth we get
/>'' cos( r'_A'o-$'
n' R 1 cos (A'" 0') R' cos (A'" 0") -f n'" R'" cos (A'" Q '")
.'" = ^7 cos (A'" - A") ^ cos (A'" A') (38)
Further, instead of these, any of the various formulae which have
been given for finding the ratio of two curtate distances, may be
employed; but, if the latitudes /?, /9', &c. are very small, the values
of p and p"' which depend on the differences of the observed longi-
tudes of the body must be preferred.
292 THEORETICAL ASTRONOMY.
T1ie values of p f and p rrf may also be derived by computing the
heliocentric places of the body for the times t and t' n by means of
the equations (82) 1? and then finding the geocentric places, or those
which belong to the points to which the observations have been
reduced, by means of (90) b writing p in place of Jcos/9. This
process affords a verification of the numerical calculation, namely,
the values of X and X' fr thus found should agree with those furnished
by observation, and the agreement of the computed latitudes ft and
ft'" with those observed, in case the latter are given, will show how
nearly the position of the plane of the orbit as derived from the
second and third observations represents the extreme latitudes. If
it were not desirable to compute X and X" in order to check the
calculation, even when ft and ft flf are given by observation, we mighi
derive p and p" r from the equations
p = r sin u sin i cot y5,
p'" = r'" sin u" f sin i cot p",
when the latitudes are not very small.
In the final approximation to the elements, and especially when
the position of the plane of the orbit cannot be obtained with the
required precision from the second and third observations, it will be
advantageous, provided that the data furnish the extreme latitudes
ft and ft" f , to compute p and p f " as soon as p f and p ff have been
found, and then find I, l f ", 6, and b'" directly from these by means
of the formulae (71) 3 . The values of & and i may thus be obtained
from the extreme places, or, the heliocentric places for the times t r
and t'" being also computed directly from p' and p", from those
which are best suited to this purpose. But, since the data will be
more than sufficient for the solution of the problem, when the extreme
latitudes are used, if we compute the heliocentric latitudes b f and b'"
from the equations
tan b' = tan i sin (I' & ),
tan b" = tan i sin (I" &),
they will not agree exactly with the results obtained directly from p 1
and /?", unless the four observations are completely satisfied by the
elements obtained. The values of r' and r n ', however, computed
directly from p r and p" by means of (71) 3 , must agree with those
derived from x r and x n .
The corrections to be applied to the times of observation on account
ORBIT FROM FOUR OBSERVATIONS. 293
of aberration may now be found. Thus, if t w t f , t Q ", and t Q '" are
the uncorrected times of observation, the corrected values will be
t = t Cp sec/5,
wherein log C== 7.760523, and from these we derive the corrected
values of r, r', r", r'", and r '.
100. To find the values of P', P", Q', and Q", which will be
exact when r, r', r", r f// , and w, u' 9 u", u'" are accurately known, we
have, according to the equations (47) 4 and (51) 4 , since Q f = |,
_ _ _
~" * ss" ' rr" cos J (t*" w') cos i (t*" M) cos j (' )'
In a similar manner, if we designate by s fff the ratio of the sector
formed by the radii-vectores r rr and r'" to the triangle formed by
the same radii-vectores and the chord joining their extremities, we
find
_ i rr'" r^
V : ~ 2 ^777 ' r y/, C o S l (>'" M ") COS i (U'" t*') COS J (u" 1*')'
The formulae for finding the value of s' rr are obtained from those for
s by writing / /r/ , f ff , G" f , &c. in place of /, /-, (r, &c., and using
r", r'", w //; u" instead of r', r", and it" M', respectively.
By means of the results obtained from the first approximation to
the values of P', P", Q', and ", we may, from equations (41) and
(42), derive new and more nearly accurate values of these quantities,
and, by repeating the calculation, the approximations to the exact
values may be carried to any extent which may be desirable. When
three approximate values of P' and ', and of P" and ", have
been derived, the next approximation will be facilitated by the use
of the formulae (82) 4 , as already explained.
When the values of P', P", Q f , and Q" have been derived with
sufficient accuracy, we proceed from these to find the elements of the
orbit. After &, i, r, r f , r" , r'", u, u', u" ', and u f/f have been found,
the remaining elements may be derived from any two radii-vectores
294 THEORETICAL ASTRONOMY.
and the corresponding arguments of the latitude. It will be most
accurate, however, to derive the elements from r, r /r/ , u, and u" f .
If the values of P', P", Q', and Q" have been obtained with great
accuracy, the results derived from any two places will agree with
those obtained from the extreme places.
In the first place, from
cos G = sin (u" f u), (43)
sin YQ sin O = cos (V" w) cos 2/ ,
cos r = cos | (u'" u) sin 2/ ,
we find p and 6r . Then we have
r '")S
"to (44 )
from which, by means of Tables XIII. and XIV., to find s and a? .
We have, further,
and the agreement of the value of p thus found with the separate
results for the same quantity obtained from the combination of any
two of the four places, will show the extent to which the approxima-
tion to P ; , P") Q f , and Q" has been carried. The elements are now
to be computed from the extreme places precisely as explained in the
preceding chapter, using r'" in the place of r" in the formula there
given and introducing the necessary modifications in the notation,
which have been already suggested and which will be indicated at
once.
101. EXAMPLE. For the purpose of illustrating the application
of the formulae for the calculation of an orbit from four observations,
let us take the following normal places of Eurynome derived by
comparing a series of observations with an ephemeris computed from
approximate elements.
Greenwich M. T. a {,
1863 Sept. 20.0 14 30' 35".6 + 9 23' 49".7,
Dec. 9.0 9 54 17 .0 2 53 41 .8,
1864 Feb. 2.0 28 41 34 .1 962 .8,
April 30.0 74 29 58 .9 -f- 19 35 41 .5.
NUMERICAL EXAMPLE. 295
These normals give the geocentric places of the planet referred to the
mean equinox and equator of 1864.0, and free from aberration. For
the mean obliquity of the ecliptic of 1864.0, the American Nautical
Almanac gives
e = 23 27' 24".49,
and, by means of this, converting the observed right ascensions and
declinations, as given by the normal places, into longitudes and lati-
tudes, we get
Greenwich M. T.
1863 Sept. 20.0
Dec. 9.0
1864 Feb. 2.0
April 30.0
A
16 59' 9".42
10 14 17 .57
29 53 21 .99
75 23 46 .90
ft
4- 2 56' 44".58,
1 15 48 .82,
2 29 57 .38,
3 4 44 .49.
These places are referred to the ecliptic and mean equinox of 1864.0,
and, for the same dates, the geocentric latitudes of the sun referred
also to the ecliptic of 1864.0 are
+ 0".60, -f-0".53, -f-0".36, + 0".19.
For the reduction of the geocentric latitudes of the planet to the
point in which a perpendicular let fall from the centre of the earth
to the plane of the ecliptic cuts that plane, the equation (6) 4 gives the
corrections 0".57, 0".38, 0".18, and 0".07 to be applied tc
these latitudes respectively, the logarithms of the approximate dis-
tances of the planet from the earth being
0.02618, 0.13355, 0.29033, 0.44990.
Thus we obtain
t = 0.0, A = 16 59' 9".42, ft = + 2 56' 44".01,
if = 80.0, X =10 14 17 .57, f = 1 15 49 .20,
" = 135.0, A" = 29 53 21 .99, f = 2 29 57 .56,
f" = 223.0, r = 75 23 46 .90, jf n = 3 4 44 .56 ;
and, for the same times, the true places of the sun referred to the
mean equinox of 1864.0 are
=177 0'58".6, logE =0.0015899.
0' =256 58 35.9, log# =9.9932638,
0" =312 57 49 .8, log-R" =9.9937748,
0'"= 40 21 26.8, log #" = 0.0035149,
296
THEORETICAL ASTRONOMY.
From the equations
tan/5'
tan w f = .
tan to" =
. ,., -j^>
sm(A' O)
tan"
. , ' ,
sm(A" O )
tan (A' Q')
tan * = -- - - 7=^1
cosw/
tan(A" O")
tan V = - - - \
cosw"
we obtain
4/ = 113 15' 20".10,
4,"= 76 5617.75,
cos V) = 9.5896777 n ,
log (R sin 4,') =9.9564624,
log (#' cos 4") = 9.3478848,
log (R" sin *") = 9.9823904.
The quadrant in which ty must be taken, is indicated by the condi-
tion that cosij/ and cos (A' O') must have the same sign. The
same condition exists in the case of i//' Then, the formulaB
A = cos ff sin (^ A),
C = cos /9" sin (A'" A"),
^-A'
A - h,
B = cos p' sin (A" jl),
Z> = cos f sin (A'" A'),
,,
O)
, ft
" Q'")
give the following results :
log A = 9.0699254 n ,
log B = 9.3484939,
log h' = 0.2785685 n ,
log a! = 0.8834880 n ,
log d = 0.9012910 n ,
log d' = 0.4650841,
log C = 9.8528803,
log D = 9.9577271,
log h" = 0.1048468,
log a" = 9.9752915 n ,
log c" = 9.7267348 w ,
log d" = 9.9096469 n .
We are now prepared to make the first hypothesis in regard to the
values of P', ', P f/ , and Q". If the elements were entirely un-
known, it would be necessary, in the first instance, to assume for these
quantities the values given by the expressions
NUMERICAL EXAMPLE. 297
then approximate values of r' and r" are readily obtained by means
of the equations (27), (26), and (24) or (25). The first assumed
value of x' to be used in the second member of the first of equations
(27), is obtained from the expression which results from (22) by
putting Q f = and Q" = 0, namely,
x' = *
after which the values of x f and x" will be obtained by trial from
(27). It should be remarked, further, that in the first determination
of an orbit entirely unknown, the intervals of time between the ob-
servations will generally be small, and hence the value of x f derived
from the assumption of Q f = and Q" will be sufficiently ap-
proximate to facilitate the solution of equations (27).
As soon as the approximate values of r f and r" have thus been
found, those of P' and P" must be recomputed from the expressions
With the results thus derived for P' and P", and with the values of
Q f and Q" already obtained, the first approximation to the elements
must be completed.
When the elements are already approximately known, the first
assumed values of P', P", Q', and Q f/ should be computed by means
of these elements. Thus, from
_rV'smQ/' t/) r/sinQ/ 1>)
~ rr" sin (" v) ' ~ rr" sin (v" v) '
,rV"sin(y" f/Q w _ rV ; sin(^ i;Q
~
r r sin (v v) TT sin (v v )
we find n, n' 3 n rf , and n"'. The approximate elements of Eurynom*
givo
v = 322 55' 9".3, logr =0.308327,
it =353 19 26 .3, log/ =0.294225,
v"= 14 45 8.5, log/' =0.296088,
v'"= 47 2332.8, log /" = 0.317278,
298 THEORETICAL ASTRONOMY.
and hence we obtain
log n = 9.653052, log n" = 9.806836,
log n 1 = 9.825408, logn'" = 9.633171.
Then, from
P" = ,
we get
log P' = 9.846216, log Q f = 9.840771,
log P"= 9.807763, log Qf' = 9.882480.
The values of these quantities may also be computed by means of the
equations (41) and (42).
Next, from
, _ P'd' + c' , _ h f
C ~~ = 1 + P" * ~ 1 -f- P''
,_Pd"+c" h"
c o = = x + P n > J - fqrp>
we find
log c ' = 0.541344 n , log/ = 0.047658.,
log C Q " = 9.807665 n , log/" = 9.889385.
Then we have
'
*= *"+"" <
, ,,
tan / = --, tan z" =
,_R' sin V _ a/ ,,_jrsinV_ a"
sin/ cos/' sin/ 7 cos/"
from which to find r' and r rr . In the first place, from
x' = v 1 jR"sinV,
we obtain the approximate value
log x f = 0.242737.
Then the first of the preceding equations gives
log *" = 0.237687.
NUMERICAL EXAMPLE. 299
From this we get
z" = 29 3' 11". 7, log r" = 0.296092 ;
and then the equation for x' gives
logo;' = 0.242768.
Hence we have
z' = 27 20' 59".6, log r' = 0.294249 ;
and, repeating the operation, using these results for x r and r', we get
log x" = 0.237678, log of = 0.242757.
The correct value of log a?' may now be found by means of equation
(28). Thus, in units of the sixth decimal place, we have
= 242768 242737 = + 31, a Q ' = 242757 242768 = 11,
and for the correction to be applied to the last value of log x', in
units of the sixth decimal place,
Therefore, the corrected value is
log af= 0.242760,
and from this we derive
log a" = 0.237681.
These results satisfy the equations for x' and x", and give
z' = 27 21' 1".2, log/ = 0.294242,
z" = 29 3 12 .9, log r" = 0.296087.
To find the curtate distances for the first and second observations,
the formulae are
oo o ,, = -
sin z' sm z"
which give
log p' = 0.133474, log p" = 0.289918.
Then, by means of the equations
300 THEORETICAL ASTRONOMY.
r' cos V cos (f 00 = p' cos (X K 9
r' cos V sin (? ') = p' sin (A' QO,
/ sin V = p' tan p,
r" cos ft" cos (r Q'O = P " cos (A" 0") J2",
r" cos b" sin (/" Q") = p" sin (A" "),
r" sin ft" = //'tan/S",
we find the following heliocentric places :
r = 37 35' 26".4, log tan ft' = 8.182861 n , log / = 0.294243,
r = 58 5815.3, logtanft" = 8.634209 n , log r" = 0.296087.
The agreement of these values of log r' and log r" with those obtained
directly from x' and x" is a partial proof of the numerical calcula-
tion.
From the equations
tan i sin ( (I" -f /') ) = J (tan ft" + tan ftO sec j (r 7),
tan i cos Q (J 1 -f I') & ) = J (tan ft" tan ftO cosec i (/" 0,
tan (^ &0 tan(r ft)
tan w = - - - r-^ , tan u =
cos i cos t
we obtain
& = 206 42' 24".0, i = 4 36' 47".2,
u r =190 55 6 .6 u" = 212 20 53 .5.
Then, from
we get
log n" = 9.806832, logw =9.653048,
log n' = 9.825408, log n'" = 9.633171,
and the equations
r sin ((u r - u) + i (u" - w')) = /+ J' V ' sin J (i*" - 0,
r cos ((i*' - u) + i (w" - tO) = ^= cos i (u" - tif ),
r"' sin ((*"' - w'O + J (u" - wO) - , sin j (t" - wO,
r"' cos ((u'" - i/O -f i (u" - 1*0) = cos i (u" -
NUMERICAL EXAMPLE. 301
give
logr = 0.308379, u = 160 30 ; 57".6,
log r"' = 0.317273, t*'"=244 5932.5.
Next, by means of the formulae
tan (I & ) = cos i tan u, tan 6 = tan i sin (/ ft ),
tan (Y" ) = cos i tan w'", tan 6'" = tan i sin ('" ft ),
^ cos (A o ) = f cos 6 cos (7 0) + -K,
jo sin (A Q ) = r cos 6 sin (7 O),
/o tan /? = r sin & ;
p ' C os (A"' O'") = r'" cos 6'" cos (f" "0 + 12"',
p' s in (A'" O'") = r'" cos 6"' sin (f" 0'"),
//"tan/5"' = r"'sin&'",
we obtain
J = 7 16' 51".8, r = 91 37' 40".0,
b = + 1 32 14 .4, 6"' = 4 10 47 .4,
A = 16 59 9 .0, /" 75 23 46 .9,
/? = + 2 56 40 .1, /?'" = 3 4 43 .4,
log P = 0.025707, log p'" = 0.449258.
The value of X" f thus obtained agrees exactly with that given by
observation, but / differs O r/ .4 from the observed value. This differ-
ence does not exceed what may be attributed to the unavoidable
errors of calculation with logarithms of six decimal places. The
differences between the computed and the observed values of /9 and
ft" show that the position of the plane of the orbit, as determined
by means of the second and third places, will not completely satisfy
the extreme places.
The four curtate distances which are thus obtained enable us, in
the case of an orbit entirely unknown, to complete the correction for
aberration according to the equations (40).
The calculation of the quantities which are independent of P',
P ff j Q f j and Q /f , and which are therefore the same in the successive
hypotheses, should be performed as accurately as possible. The
s*
value of -^-> required in finding x" from x', may be computed
directly from
jf .
the values of 77 and 77 being found by means of the equations (29);
302 THEORETICAL ASTRONOMY.
and a similar method may be adopted in the case of j, r > Further,
in the computation of x f and x", it may in some cases be advisable
to employ one or both of the equations (22) for the final trial. Thus,
in the present case, x" is found from the first of equations (27) by
means of the difference of two larger numbers, and an error in the
last decimal place of the logarithm of either of these numbers affects
in a greater degree the result obtained. But as soon as r" is known
Q"
so nearly that the logarithm of the factor 1 -f -^ remains unchanged,
the second of equations (22) gives the value of x" by means of the
sum of two smaller numbers. In general, when two or more for-
mulae for finding the same quantity are given, of those which are
otherwise equally accurate and convenient for logarithmic calculation,
that in which the number sought is obtained from the sum of smaller
numbers should be preferred instead of that in which it is obtained
by taking the difference of larger numbers.
The values of r, r f , r /f , r'", and u, u f y u ff , u fff , which result from
the first hypothesis, suffice to correct the assumed values of P f , P",
', and Q". Thus, from
"r 77 " '/ , /r 777 "
sin Y cos G = sin (u" u'), sin /' cos G" = sin A (u r u),
sin f sin G = cos J (u" u') cos 2/, sin /' sin G" cos (u r u) cos 2/',
cos f = cos | (u" u') sin 2/, cos f" = cos \ (V u) sin 2,%",
sin f cos G'" = sin I (u" r w"),
sin f sin G"' = cos (u" f u"} cos 2/"
cos /" = cos J (u"' u") sin 2/" ;
T 2 COS 6 / __ T" 2 COS 6 /'
m =
COS?'
m
r' /s cos 3 /'
t+j + ? *+/'+*
in connection with Tables XIII. and XIV. we find s, s", and '".
The results are
NUMEKICAL EXAMPLE.
303
log T = 9.9759441,
/ = 45 3'39".l,
r=-10 42 55 .9,
log m = 8.186217,
log; = 7.948097,
log s = 0.0085248,
log r"= 0.1386714,
7"=4432' 1".4,
/'IS 13 45 .0,
log m"= 8.516727,
log/'= 8.260013,
log "== 0.0174621,
log T"'= 0.1800641,
/"== 45 41' 55".2,
y"'=16 22 48 .5.
log m'"= 8.590596,
log/"= 8.325365,
log s'"= 0.020406?.
Then, by means of the formulae
a -*
V - 2 ,,
y, = , nl' _
2 ss "f r y // cog ^ (yftr u n^ cog ^ (jjin ^ CQg ^ ^/ % /y
we obtain
log P' = 9.8462100,
log P" = 9.8077615,
log ' = 9.8407536,
log " = 9.8824728,
with which the next approximation may be completed.
We now recompute c ', c/', /', /", x f , x n ', &c. precisely as already-
illustrated; and the results are
log c ' = 0.5413485 n ,
log/ = 0.0476614 n ,
log x' = 0.2427528,
z f = 27 21' 2".71,
log/ =0.2942369,
log />' = 0.1334635,
log n = 9.6530445,
log n' = 9.8254092,
log c " = 9.8076649 n ,
log/" = 9.8893851,
log x" = 0.2376752,
z" = 29 3' 14".09,
logr" =0.2960826,
log p" = 0.2899124,
log n" = 9.8068345,
log ri" = 9.6331707.
Then we obtain
/' = 37 35' 27".88,
I" 58 58 16 .48,
log tan V = 8.1828572 n ,
log tan b" = 8.6342073 n ,
log/ = 0.2942369,
log /' = 0.2960827.
These results for log r f and log r" agree with those obtained directly
from z f and z", thus checking the calculation of ty and ty f and of
the heliocentric places.
Next, we derive
206 42' 25".89,
55 6.27,
t = 4 36' 47".20,
2052.96,
304 THEORETICAL ASTRONOMY.
and from u ff u', r', r", n, n", n f , and n" f , we obtain
logr =0.3083734, u = 160 30' 55".45,
log/" =0.3172674, w'"=244 5931.98.
For the purpose of proving the accuracy of the numerical results,
we compute also, as in the first approximation,
/= 716'51".54, /'" = 91 37' 41".20,
b = + 1 32 14 .07, b'"= 4 10 47 .36,
A= 16 59 9.38, A'" = 75 2346.99,
= + 2 56 39 .54, p"= 3 4 43 .33,
log P = 0.0256960, log p" r = 0.4492539.
The values of ^ and \ tn thus found differ, respectively, only 0".04
and O r/ .09 from those given by the normal places, and hence the
accuracy of the entire calculation, both of the quantities which are
independent of P', P" ', ', and ", and of those which depend on
the successive hypotheses, is completely proved. This condition,
however, must always be satisfied whatever may be the assumed
values of P', P", Q f , and ".
From TJ r'j u, it', &c., we derive
log a = 0.0085254, log s" = 0.0174637, log s m = 0.0204076,
and hence the corrected values of P', P", ', and Q" become
logP' = 9.8462110, log q = 9.8407524,
log P" = 9.8077622, log " = 9.8824726.
These values differ so little from those for the second approximation,
the intervals of time between the observations being very large, that
a further repetition of the calculation is unnecessary, since the results
which would thus be obtained can differ but slightly from those
which have been derived. We shall, therefore, complete the deter-
mination of the elements of the orbit, using the extreme places.
Thus, from
sin YQ cos G = sin J (u tn u),
sin Y O sin G = cos J (u" f u) cos 2/ ,
cos YO cos (u"' u) sin 2/ ,
5 i i
~r Jo
? - _
"
m _ m a
~~ 7 ~
NUMERICAL EXAMPLE. 305
we get
log r = 0.5838863, log tan G Q = 8.0521953 W>
n = 42 14' 30".17, log m = 9.7179026,
log s 2 = 0.2917731, log x = 8.9608397.
The formula
s rr m sm(u m t*)\
gves
log p = 0.371 2401;
and if we compute the same quantity by means of
grV'sm(X' iQ \ 2 / g'Vr'sinK u) V I s'VV'sin (u'"-
u"} \
I*
the separate results are, respectively, 0.3712397, 0.3712418, and
0.3712414. The differences between these results are very small, and
arise both from the unavoidable errors of calculation and from the
deviation of the adopted values of P', P x/ , Q', and Q" from the
limit of accuracy attainable with logarithms of seven decimal places.
A variation of only O r/ .2 in the values of u' u and u fff u ff wil?
produce an entire accordance of the particular results.
From the equations
smi(V" u) /T,,
C 8 p = sin K"' -"
P
cos
) tan (45 ?),
t&n%E' = tani(X oO tan (45 ?>),
tan IE" = tan J O" - o) tan (45 ' p),
tan E'" = tan (u" r o) tan (45 ?>)
from which the results are
E = 329 11' 46".01, .E" 12 5' 33".63,
' = 354 29 11 .84, '" = 39 34 34 .65.
The value of J (E" f E) thus derived differs only 0".03 from that
obtained directly from x .
For the mean anomalies, we have
which give
, M" = E" esmE",
= E r e sin E', M'" = E'" e sin E'",
M = 334 55' 39".32, M" = 9 44' 52".82,
M' = 355 33 42 .97, JJf'" = 32 26 44 .74.
Finally, if M Q denotes the mean anomaly for the epoch T= 1864
Jan. 1.0 mean time at Greenwich, from
M Q = M fi (t T) =M t p(fT)
= M" fJL (f T} = M'" p. (f" T),
we obtain the four values
Jf = l29'39".40
39 .49
39 .40
39 .40,
the agreement of which completely proves the entire calculation of
the elements from the data. Collecting together the several results,
we have the following elements :
NUMERICAL EXAMPLE. 307
Epoch = 1864 Jan. 1.0 Greenwich mean time.
M = 1 29' 39".42
=
?= 11 15 52 .22
log a = 0.3881359
log ; = 2.9678027
A* = 928".54447.
102. The elements thus derived completely represent the four ob-
served longitudes and the latitudes for the second and third places,
which are the actual data of the problem ; but for the extreme lati-
tudes the residuals are, computation minus observation,
A = 4".47, A,?'" = + 1".23.
These remaining errors arise chiefly from the circumstance that the
position of the plane of the orbit cannot be determined from the
second and third places with the same degree of precision as from
the extreme places. It would be advisable, therefore, in the final
approximation, as soon as p f , p", n, n' f , n f , and n rrr are obtained, to
compute from these and the data furnished directly by observation
the curtate distances for the extreme places. The corresponding
heliocentric places may then be found, and hence the position of the
plane of the orbit as determined by the first and fourth observations.
Thus, by means of the equations (37) and (38), we obtain
log P = 0.0256953, log p'" = 0.4492542.
With these values of p and //", the following heliocentric places are
obtained :
I = 716'51".54, log tan b =8.4289064, logr =0.3083732,
l f " = 91 37 40 .96, log tan b'" = 8.8638549 n , log r"' = 0.3172678.
Then from
tan i sin (J (f" + 1) ft) = J (tan V" -f tan 6) sec J (f" Z),
tan i cos Q (f" -f & ) = i (tan b"' tan 6) cosec J (f I),
we get
^ = 206 42' 45".23, i = 4 36' 49".76.
For the arguments of the latitude the results are
u = 160 30' 35".99, u'" = 244 59' 12".53.
398 THEORETICAL ASTRONOMY.
The equations
tan V = tan i sin (I 1 Q ),
tan V = tan i sin (I" ),
give
log tan &' = 8.1827129^ log tan b" = 8,6342104 n ,
and the comparison of these results with those derived directly from
p' and p" exhibits a difference of -f 1".04 in &', and of 0".06 in
6". Hence, the position of the plane of the orbit as determined from
the extreme places very nearly satisfies the intermediate latitudes.
If we compute the remaining elements by means of these values
of r, r f ", and u, u" r , the separate results are :
log tan G = 8.0522282 n , log m = 9.7179026,
log sj = 0.2917731, log x = 8.9608397,
log j9 = 0.3712405, I (E" E) = 17 35' 42".12,
log (a cos ?) = 0.3796884, log cos
9 i, u, and u".
Then, if the value of a computed from the last result for u" u
differs from the last assumed value, a further repetition of the calcu-
CIKCULAB CEBIT. 313
latiori becomes necessary. But when three successive approximate
values of a have been found, the correct value may be readily inter-
polated according to the process already illustrated for similar cases.
As soon as the value of a has been obtained which completely
satisfies equation (5), this result and the corresponding values of &,
i, and the argument of the latitude for a fixed epoch, complete the
system of circular elements which will exactly satisfy the two observed
places. If we denote by u the argument of the latitude for the epoch
T, we shall have, for any instant t,
u being the mean or actual daily motion computed from
Jc
The value of u thus found, and r = a, substituted in the formulse for
computing the places of a heavenly body, will furnish the approxi-
mate ephemeris required.
The corrections for parallax and aberration are neglected in the
first determination of circular elements ; but as soon as these approxi-
mate elements have been derived, the geocentric distances may be
computed to a degree of accuracy sufficient for applying these cor-
rections directly to the observed places, preparatory to the determi-
nation of elliptic elements. The assumption of r f = a will also be
sufficient to take into account the term of the second order in the first
assumed value of P, according to the first of equations (98) 4 .
104. When approximate elements of the orbit of a heavenly body
have been determined, and it is desired to correct them so as to satisfy
as nearly as possible a series of observations including a much longer
interval of time than in the case of the observations used in finding
these approximate elements, a variety of -methods may be applied.
For a very long series of observations, the approximate elements
being such that the squares of the corrections which must be applied
to them may be neglected, the most complete method is to form the
equations for the variations of any two spherical co-ordinates which
fix the place of the body in terms of the variations of the six ele-
ments of the orbit; and the differences between the computed places
for different dates and the corresponding observed places thus furnish
equations of condition, the solution of which gives the corrections to
be applied to the elements But when the observations do not in-
314 THEORETICAL ASTRONOMY.
elude a very long interval of time, instead of forming the equations
for the variations of the geocentric places in terms of the variations
of the elements of the orbit, it will be more convenient to form the
equations for these variations in terms of quantities, less in number,
from which the elements themselves are readily obtained. If no as-
sumption is made in regard to the form of the orbit, the quantities
which present the least difficulties in the numerical calculation are
the geocentric distances of the body for the dates of the extreme
observations, or at least for the dates of those which are best adapted
to the determination of the elements. As soon as these distances are
accurately known, the two corresponding complete observations are
sufficient to determine all the elements of the orbit.
The approximate elements enable us to assume, for the dates t and
t n , the values of A and J"; and the elements computed from these
by means of the data furnished by observation, will exactly represent
the two observed places employed. Further, the elements may be
supposed to be already known to such a degree of approximation that
the squares and products of the corrections to be applied to the
assumed values of A and A" may be neglected, so that we shall have,
for any date,
.. da, . da
COS S Aa = COS d - A J -j- COS d -=--- rr A A ,
d d fR\
d* d3
** = dA^ + ^F AJ '
If, therefore, we compare the elements computed from A and A" with
any number of additional or intermediate observed places, each ob-
served spherical co-ordinate will furnish an equation of condition for
the correction of the assumed distances. But in order that the equa-
tions (6) may be applied, the numerical values of the partial differen-
tial coefficients of a and d with respect to A and d" must be found.
Ordinarily, the best method of effecting the determination of these is
to compute three systems of elements, the first from A and A", the
second from A + D and J", and the third from A and A" -f D", D
and D" being small increments assigned to A and A" respectively.
] f now, for any date t' y we compute a/ and d f from each system of
elements thus obtained, we may find the values of the differential
coefficients sought. Thus, let the spherical co-ordinates for the time
t f computed from the first system be denoted by a' and 3 f ; those
computed from the second system of elements, by a/ -f a sec 3 f and
6' + d; and those from the third system, by a'+ a!' sec d' and d'-\- d".
Then we shall have
VARIATION OF TWO GEOCENTRIC DISTANCES. 315
.. da! a dd d
,,a__ -..
dA"~ D'" dA"~ D"'
and the equations (6) give
COS d r Aa' = ~ A A -f -^77 A J",
d d" (8)
In the same manner, computing the places for various dates, for
which observed places are given, by means of each of the three systems
of elements, the equations for the correction of A and A" , as deter-
mined by each of the additional observations employed, may be
formed.
105. For the purpose of illustrating the application of this method,
let us suppose that three observed places are given, referred to the
ecliptic as the fundamental plane, and that the corrections for parallax,
aberration, precession, and nutation have all been duly applied. By
means of the approximate elements already known, we compute the
values of A and A" for the extreme places, and from these the helio-
centric places are obtained by means of the equations (71) 3 and (72) 3 ,
writing A cos /9 and A" cos ft" in place of p and p". The values of
&, i, u, and u" will be obtained by means of the formulae (76) 3 and
(77) 3 ; and from r, r" and u" u the remaining elements of the
orbit are determined as already illustrated. The first system of ele-
ments is thus obtained. Then we assign an increment to J, which
we denote by D, and with the geocentric distances A -f- D and A"
we compute in precisely the same manner a second system of ele-
ments. Next, we assign to A" an increment D", and from A and
A" + D" a third system of elements is derived. Let the geocentric
longitude and latitude for the date of the middle observation com-
puted from the first system of elements be designated, respectively,
by V and /9/ ; from, the second system of elements, by V and /? 2 ' ;
and from the third system, by ^/ and /9 3 '. Then from
It ( -\ I i l\ of fjff O f O f \** s
we compute a, a", d, and d" ', and by means of these and the values
of D and D" we form the equations
316 THEORETICAL ASTRONOMY.
= CM
for the determination of the corrections to be applied to the f.rst
assumed values of A and A n ', by means of the differences between
observation and computation. The observed longitude and latitude
being denoted by X and /?', respectively, we shall have
COS f A/ = (/ A/) COS /?',
ff = ff B' (11)
for finding the values of the second members of the equations (10),
and then by elimination we obtain the values of the corrections A J
and AJ" to be applied to the assumed values of the distances.
Finally, we compute a fourth system of elements corresponding to
the geocentric distances A -f A A and A" -f A A" either directly from
these values, or by interpolation from the three systems of elements
already obtained ; and, if the first assumption is not considerably in
error, these elements will exactly represent the middle place. It
should be observed, however, that if the second system of elements
represents the middle place better than the first system, A/ and /? 2 r
should be used instead of ^/ and /9/ in the equations (11), and, in
this case, the final system of elements must be computed with the
distances A -f- D + A A and A" + A A". Similarly, if the middles
place is best represented by the third system of elements, the cor-
rections will be obtained for the distances used in the third hy-
pothesis.
If the computation of the middle place by means of the final ele-
ments still exhibits residuals, on account of the neglected terms of
the second order, a repetition of the calculation of the corrections
A/f and AJ", using these residuals for the values of the second
members of the equations (10), will furnish the values of the dis-
tances for the extreme places with all the precision desired. The
increments D and D" to be assigned successively to the first assumed
values of A and A" may, without difficulty, be so taken that the
true elements shall differ but little from one of the three systems
computed ; and in all the formulae it will be convenient to use, in-
stead of the geocentric distances themselves, the logarithms of these
distances, and to express the variations of these quantities in units
of the last decimal place of the logarithms.
These formulae will generally be applied for the correction of
VARIATION OF T\VO GEOCENTRIC DISTANCES. 317
approximate elements by means of several observed places, which
may be either single observations or normal places, each derived from
several observations, and the two places selected for the computation
of the elements from A and A" should not only be the most accurate
possible, but they should also be such that the resulting elements are
not too much affected by small errors in these geocentric places.
They should moreover be as distant from each other as possible, the
other considerations not being overlooked. When the three systems
of elements have been computed, each of the remaining observed
places will furnish two equations of condition, according to equations
(10), for the determination of the corrections to be applied to the
assumed values of the geocentric distances ; and, since the number
of equations will thus exceed the number of unknown quantities,
the entire group must be combined according to the method of least
squares. Thus, we multiply each equation by the coefficient of AJ
in that equation, taken with its proper algebraic sign, and the sum
of all the equations thus formed gives one of the final equations
required. Then we multiply each equation by the coefficient of AJ"
in that equation, taken also with its proper algebraic sign, and the
sum of all these gives the second equation required. From these
two final equations, by elimination, the most probable values of AJ
and AZ/ ;/ will be obtained ; and a system of elements computed with
the distances thus corrected will exactly represent the two funda-
mental places selected, while the sum of the squares of the residuals
for the other places will be a minimum. The observations are thus
supposed to be equally good; but if certain observed places are
entitled to greater influence than the others, the relative precision
of these places must be taken into account in the combination of the
equations of condition, the process for which will be fully explained
in the next chapter.
When a number of observed places are to be used for the correction
of the approximate elements of the orbit of a planet or comet, it w* ! ll
be most convenient to adopt the equator as the fundamental plane.
In this case the heliocentric places will be computed from the assumed
values of A and J", and the corresponding geocentric right ascensions
and declinations by means of the formulae (106) 3 and (107) 3 ; and the
position of the plane of the orbit as determined from these by means
of the equations (76) 3 will be referred to the equator as the funda-
mental plane. The formation of the equations of condition for the
corrections AJ and A A" to be applied to the assumed values of the
distances will then be effected precisely as in the case of I and /?, the
318 THEORETICAL ASTRONOMY.
necessary changes being made in the notation. In a similar manner,
the calculation may be effected for any other fundamental plane which
may be adopted.
It should be observed, further, that when the ecliptic is taken as
the fundamental plane, the geocentric latitudes should be corrected
by means of the equation (6) 4 , in order that the latitudes of the sun
phall vanish, otherwise, for strict accuracy, the heliocentric places
must be determined from A and A" in accordance with the equations
(39):-
106. The partial differential coefficients of the two spherical co-
ordinates with respect to A and A" may be computed directly by
means of differential formula; but, except for special cases, the
numerical calculation is less expeditious than in the case of the indi-
rect method, while the liability of error is much greater. If we
adopt the plane of the orbit as determined by the approximate values
of A and A" as the fundamental plane, and introduce / as one of the
elements of the orbit, as in the equations (72) 2 , the variation of the
geocentric longitude 6 measured in this plane, neglecting terms of the
second order, depends on only four elements; and in this case the
differential formulae may be applied with facility. Thus, if we ex-
press r and v in terms of the elements ^, M Q) and /*, we shall have
and
or
In like
dr
dA
dv
dA
_dr
dtp
dv
dtp
dr
dM
dr
h d,u
dv
dp.
dp.
' dA
'dA
->
dv
dA
' dA'
dp.
dx
dM Q
d
'dA
dM
dA
1 dp.
__ _
d? ' dA dM ' dA " dfi ' dA'
dA ~ d
TT> an( i - n -- are
dA dA dA dA
known, the equations necessary for finding the differential coefficients
of the elements , = o.
dA aA
To find -7-7- and - 7 . , from the equations
dA dA
A cos ^ cos = x -f- X,
A cos >? sin = y -{- F",
in which ^ is the geocentric latitude in reference to the plane of the
orbit computed from A and A" as the fundamental plane, and X, Y
the geocentric co-ordinates of the sun referred to the same plane, we
get
dx = cos 7] cos 6 dA,
dy = cos T? sin 6 dA,
or, substituting for dx and dy their values given by (73) 2 ,
cos t] cos 6 dA = cos u dr r sin u d (v -f- /),
cos r) sin O d A = sin udr -\- r cos it c? (v -f~ /).
Eliminating, successively, c? (v + #) and c?r, we get
dr , n x
- = cos ^ cos (6 u),
d
I . , .
= - cos ^ sm (6 u).
7 -
dA r
Therefore, we shall have
dy , dv dv dv
dip dA ' dM Q dA dp. dA ,_
d X _ _dif_ d?_ d^_ dM^,dv^_ ^_
dA H " ' dA + ^jf o jj ' ' ^ ' ^j
_dr^_ ^__
' ~
^ dA dif dA dfji dA
and if we compute the numerical values of the differential coefficients
of r, r", v, and v tr with respect to the elements Sm(t U } '
dr" d, and , respectively, we compute the values of 0, 37, and f for the
dates of the several places to be employed. Then the residuals for
each of the observed places are found from the formulae
cos f) A0 = sin Y A<5 -f- cos f cos 8 Aa,
Aiy = cos Y A# sin f cos d Aa,
the values of Aa and A for each place being found by subtracting
from the observed right ascension and declination, respectively, the
right ascension and declination computed by means of the elements
derived from A and A". The values of 9 f], and f being required
only for finding cos y A#, A/7, and the differential coefficients of d and
ry, with respect to the elements of the orbit, need not be determined
with great accuracy.
Next, we compute -^ and - k , . from equations (12), and from
,+ n ^ .1 i dr dr" dv dv" dr , / i i
(16), the values of , , , -= , -r^, &c,, by means of which,
d, A!/ O , and
may be obt
we compute
may be obtained. If, from the values of -- - and -p
VARIATION OF TWO GEOCENTRIC DISTANCES. 323
and apply these corrections to the values of v and v" found from J
and J", we obtain the true anomalies corresponding to the distances
A -}- A J and J" + A J". The corrections to be applied to the values
of r and r" derived from A and A" are given by
If A J and AJ" are expressed in seconds of arc, the corresponding
values of Ar and Ar" must be divided by 206264.8. The corrected
results thus obtained should agree with the values of r and r" com-
puted directly from the corrected values of v, v", p, and e by means
of the polar equation of the conic section. Finally, we have
dz = sin rj dd,
and similarly for dz ff and the last of equations (73) 2 gives
r sin u Ai' r cos u sin i' A & ' == sin 7 A J,
r" sin u" Ai' r" cos u" sin i' A ft ' sin ^ A J",
from which to find AI V and A & ', u and u" being the arguments of
the latitude in reference to the equator. We have also, according to
(72)*
A a/ = A/ COS i' A & ',
4^ = A/ + 2 Sin 1 t* A &',
from which to find the corrections to be applied to a) 1 and n f . The
elements which refer to the equator may then be converted into those
for the ecliptic by means of the formula which may be derived from
(109)! by interchanging & and &' and 180 i' and i.
The final residuals of the longitudes may be obtained by substi-
tuting the adopted values of A J and A A" in the several equations of
condition, or, which affords a complete proof of the accuracy of the
entire calculation, by direct calculation from the corrected elements ;
and the determination of the remaining errors in the values of rj will
show how nearly the position of the plane of the orbit corresponding
to the corrected distances satisfies the intermediate latitudes.
Instead of and i, the equations may be formed by
means of which the corrections to be applied to the assumed values
of the two geocentric distances, or to those of & and i, will be
obtained.
110. The formulae which have thus far been given for the correc-
tion of an approximate orbit by varying the geocentric distances,
depend on two of these distances when no assumption is made in
regard to the form of the orbit, and these formulae apply with equal
facility whether three or more than three observed places are used.
But when a series of places can be made available, the problem may
be successfully treated in a manner such that it will only be necessary
to vary one geocentric distance. Thus, let x, y, z be the rectangular
heliocentric co-ordinates, and r the radius-vector of the body at the
time t, and let X, F, Z be the geocentric co-ordinates of the sun at
the same instant. Let the geocentric co-ordinates of the body be
designated by x 09 y Q , Z Q , and let the plane of the equator be taken as
the fundamental plane, the positive axis of x being directed to the
vernal equinox. Further, let p denote the projection of the radius-
vector of the body on the plane of the equator, or the curtate dis-
tance with respect to the equator; then we-shall have
x = p cos a, y = p sin a, z p tan d. (32)
If we represent the right ascension of the sun by A, and its declina-
tion by _D, we also have
VARIATION OF ONE GEOCENTRIC DISTANCE. 329
Z=RsmD. (33)
The fundamental equations for the undisturbed motion of the planet
or comet, neglecting its mass in comparison with that of the sun, are
but since
and, neglecting also the mass of the earth,
these become
Substituting for a? , y , and 2; their values in terms of a and d, and
putting
we get
+ ^co Sa + , = 0,
+ sina + , = 0, ' (36)
Differentiating the equations (32) with respect to t, we find
dx n dp . da
-d?= c
f
31)0 THEORETICAL ASTRONOMY.
Differentiating again with respect to t, and substituting in the equa
tions (36) the values thus found, the results are
If we multiply the first of these equations by sin a, and the second
by cos a, and add the products, we obtain
d' 2 a
j gBin.-?COB.-/-
dt 2 rff
57
Now, from (35) we get
sin a ^ cos a = & 2 1 -= -- 3 I -K cos D sin (a A),
and the preceding equation becomes
The value of -vr thus found is independent of the differential co-
dp
efficients of d with respect to t. To find another value of -J-, using
dt
all three of equations (38), we multiply the first of these equations
by sin A tan 8, the second by cos A tan , and the third by
sin (a A). Then, adding the products, since sin A = TJ cos A,
the result is
from which we get
% - cot (a - A) % + **( 2 ' + cot % ) + i cot*
a/> _ , rr> and -77, the equations
at at at
X= Rcos Q,
Y= jRsin O cose,
Z = R sin O sin e,
give, by differentiation,
dX dR ^dO
^^eosQ.. _ jRsm0 _,
^= sin Q cose^ + R cos cose^ (43)
at at at
dZ dR . n .dQ
-,. = sm O sm e ,, + R cos O sin e -=-.
at at at
332 THEORETICAL ASTRONOMY.
Now, according to equation (52) 1? we have
dQ _ T(l-e 2 )(l + m )
~dT z ~ir~
m Q denoting the mass of the earth, and e Q the eccentricity of its orbit.
The polar equation of t^ie conic section gives
dr _ r 2 e sin v dv
~dt~ p "ST
Let -T denote the longitude of the sun's perigee, and this equation
gives
.sin(0-r, (45)
If we neglect the square of the eccentricity of the earth's orbit, we
have simply
_
-7-
(4<5)
The values of -^- and -7- having been found by means of these
dX dY
formulae, the equations (43) give the required results for - , -, and
dZ
-j7, and hence, by means of (42), we obtain the velocities of the
uit
comet or planet in directions parallel to the co-ordinate axes.
112. The values of x, y, and 2 may be derived by means of the
equations
x = A cos <5 cos a X,
y A cos d sin a Y,
z = A sin d Z,
and from these, in connection with the corresponding velocities, the
elements of the orbit may be found. The equations (32), give im-
mediately the values of the inclination, the semi-parameter, and the
right ascension of the ascending node on the equator. Then, the
position of the plane of the orbit being known, we may compute r
and u directly from the geocentric right ascension and declination by
means of the equations (28) and (30). But if we use the values of
the heliocentric co-ordinates directly, multiplying the first of equa-
tions (93) x by cos &, and the second by sin &, and adding the pro-
ducts, we have
VAEIATION OF ONE GEOCENTRIC DISTANCE. 333
r sin u = z cosec i,
r cos u = x cos & + y sin & >
from which r and w may be found, the argument of the latitude u
being referred to the plane of xy as the fundamental plane. The
equation
r 2 = x* -f if + s a
gives
dr_x dx.y dy , * dz
cos 4 ,
F 2
e cos (u w) = - - r sin 2 4/ 1,
and, since
these are easily transformed into
2ae sin (w a>} = (2a r) sin 24 ,
2ae cos (it w) = (2a r) cos 24> r.
If we multiply the first of these equations by cos u and the second
by sin w, and add the products; then multiply the first by sin it and
the second by cos it, and add, we obtain
2ae sin w = (2a r) sin (2^ + w) r sin it, / ^N
2ae cos w = (2a r) cos (2^ + u) r cos u,
These equations give the values of a) and e.
113. We have thus derived all the formulae necessary for finding
the elements of the orbit of a heavenly body from one geocentric
distance, provided that the first and second differential coefficients of
a and d with respect to the time are accurately known. It remains,
VARIATION OF ONE GEOCENTRIC DISTANCE. 335
therefore, to devise the means by which these differential coefficients
may be determined with accuracy from the data furnished by obser-
vation. The approximate elements derived from three or from a
small number of observations will enable us to correct the entire
series of observations for parallax and aberration, and to form the
normal places which shall represent the series of observed places.
We may now assume that the deviation of the spherical co-ordinates
computed by means of the approximate elements from those which
would be obtained if the true elements were used, may be exactly
represented by the formula
*0 = A + Bh+ Ch\ (53)
h denoting the interval between the time at which the deviation is
expressed by A and the time for which this difference is A#. The
differences between the normal places and those computed with the
approximate elements to be corrected, will then suffice to form equa-
tions of condition by means of which the values of the coefficients
A, B, and C may be determined. The epoch for which h = may
be chosen arbitrarily, but it will generally be advantageous to fix it
at or near the date of the middle observed place. If three observed
places are given, the difference between the observed and the com-
puted value of each right ascension will give an equation of condition,
according to (53), and the three equations thus formed will furnish
the numerical values of A, J3, and C. These having been deter-
mined, the equation (53) will give the correction to be applied to the
computed right ascension for any date within the limits of the
extreme observations of the series. When more than three normal
places are determined, the resulting equations of condition may be
reduced by the method of least squares to three final equations, from
which, by elimination, the most probable values of A, j5, and C will
bo derived. In like manner, the corrections to be applied to the
computed latitudes may be determined. These corrections being
applied, the ephemeris thus obtained may be assumed to represent
the apparent path of the body with great precision, and may be em-
ployed as an auxiliary in determining the values of the differential
coefficients of a and d with respect to t.
Let f(a) denote the right ascension of the body at the middle
epoch or that for which h = Q, and let /(a dz no)) denote the value of
a for any other date separated by the interval na) 9 in which at is the
interval between the successive dates of the ephemeris. Then, if we
put n successively equal to 1, 2, 3, &c., we shall have
336 THEORETICAL ASTRONOMY.
Function. I. Diff. II. Diff. III. Diff. IV. Diff. V. Diff.
/(a 3a>) ff
/(a -f 3a>)
The series of functions and differences may be extended in the same
manner in either direction. If we expand /(a + nco) into a series.
the result is
/(a-ftt0^:a-|-~nw + 2^ w2ty2 + ^ wVJ 4- A ^^* cy * + &c -
or, putting for brevity A = -j-a) ) B = \ -^ - a) 2 , &c.,
/(a + nco) = a -f An -f 5^ 2 + Oi 3 -f Dn 4 + &c.
If we now put n successively equal to 4, 3, 2, 1, 0, +1,
&c., we obtain the values of /(a 4co),f(a 3w), ...... /(a -f 4a>)
in terms of A, B, C, &c. Then, taking the successive orders of
differences and symbolizing them as indicated above, we obtain a
series of equations by means of which A, B, C, &c. will be deter-
mined in terms of the successive orders of differences. Finally, re-
placing Ay B, C, &G. by the quantities which they represent, and
putting
ir (<*-*) + if '(<*+jo =/'(),
if'" ( - *-) + if '" ( + i) =/"' (a), &c. (
we obtain
= A- (/'(a) - I/'" (a) + 3 > 5 /'(oO - Ti^f-(a) + &c.),
= (/"() - */"() + *? 5 /"" () - *<>.),
= (/"() -1/-W
VARIATION OF ONE GEOCENTRIC DISTANCE. 337
by means of which the successive differential coefficients of a with
respect to t may be determined. The derivation of these coefficients
in the case of d is entirely analogous to the process here indicated for
a. Since the successive differences will be expressed in seconds of
arc, the resulting values of the differential coefficients of a and d with
respect to t will also be expressed in seconds, and must be divided by
^06264.8 in order to express them abstractly.
We may adopt directly the values of -JT> -^p -yr. and -^ determined
by means of the corrected ephemeris, or, if the observed places do
not include a very long interval, we may determine only the values
of -jp, -j, &G. by means of the ephemeris, and then find -=- and -^-
directly from the normal places or observations. Thus, let a, a/, a"
be three observed right ascensions corresponding to the times t, t', t",
and we shall have
which give
These equations, being solved numerically, will give the values of -7-
d*a
and , and we may thus by triple combinations of the observed
places, using always the same middle place, form equations of con-
dition for the determination of the most probable values of these
differential coefficients by the solution of the equations according to
the method of least squares.
7 /72/^
In a similar manner the values of -^ and -jr may be derived.
114. In applying these formulae to the calculation of an orbit,
after the normal places have been derived, an ephemeris should be
computed at intervals of four or eight days, arranging it so that one
of the dates shall correspond to that of the middle observation or
normal place. This ephemeris should be computed with the utmost
338 THEORETICAL ASTRONOMY.
care, since it is to be employed as an auxiliary in determining quan-
tities on which depends the accuracy of the final results. The com-
parison of the ephemeris with the observed places will furnish, by
means of equations of the form
A + Eh + Ch 2 = Aa',
A _f- B'h + C'K = A*,
k being the interval between the middle date t f and that of the place
used, the values of A, J5, (7, A', &c.; and the corrections to be
applied to the ephemeris will be determined by
A -f Bna> -f Oi 2 " 8 = Aa,
A'
The unit of h may be ten days, or any other convenient interval,
observing, however, that no) in the last equations must be expressed
in parts of the same unit. With the ephemeris thus corrected, we
da d?a dd _ d?8
compute the values of -j-, -^-, ^7, and j- as already explained, inese
differential coefficients should be determined with great care, since it
is on their accuracy that the subsequent calculation principally de-
. . dX dY , dZ .
pends. We compute, also, the velocities -jr t -rr, and rr by means
7x-v 7 -p dt at at
of the formulaB (43), ^- and being computed from (46). The
quantities thus far derived remain unchanged in the two hypotheses
with regard to J.
Then we assume an approximate value of J, and compute
p J cos d ;
and by means of the equation (40) or (39) we compute the value ol
- It will be observed that if we put the equation (40) in the form
dp P C
_ = _, + _
p
the coefficient -^ remains the same in the two hypotheses. The three
equations (38) may be so combined that the resulting value of
_J2 ^*^
will not contain -T~> This transformation is easily effected, and may
(J tL
be advantageous in special cases for which the value of -^ is very
uncertain.
The heliocentric spherical co-ordinates will be obtained from the
RELATION BETWEEN TWO PLACES IN THE OEBIT. 339
assumed value of J by means of the equations (106) 31 and the rec-
tangular co-ordinates from
x = r cos b cos I,
y = r cos b sin I,
z ==r sin b.
The velocities -57, ~, and -r- will be given by (42), and from these
and the co-ordinates x, y, z the elements of the orbit will be com-
puted by means of the equations (32) 1? (47), (49), &c. With the
elements thus derived we compute the geocentric places for the dates
of the normals, and find the differences between computation and
observation. Then a second system of elements is computed from
J 4- $d, and compared with the observed places. Let the difference
between computation and observation for either of the two spherical
co-ordinates be denoted by n for the first system of elements, and by
n f for the second system. The final correction to be applied to J, in
order that the observed place may be exactly represented, will be
determined by
('_) + n-0. (56)
Each observed right ascension and each observed declination will
thus furnish an equation of condition for the determination of A J,
observing that the residuals in right ascension should in each case be
multiplied by cos d. Finally, the elements which correspond to the
geocentric distance J -f- A A will be determined either directly or by
interpolation, and these must represent the entire series of observed
places.
115. The equations (52) s enable us to find two radii-vectores when
the ratio of the corresponding curtate distances is known, provided
that an additional equation involving r, r", K, and known quantities
is given. For the special case of parabolic motion, this additional
equation involves only the interval of time, the two radii-vectores,
and the chord joining their extremities. The corresponding equation
for the general conic section involves also the semi-transverse axis
of the orbit, and hence, if the ratio M of the curtate distances is
known, this equation will, in connection with the equations (52) 3 ,
enable us to find the values of r and r" corresponding to a given
value of a. To derive this expression, let us resume the equations
THEORETICAL ASTRONOMY.
4 = E" - E - 2e sin (E" - E} cos J (" +
a?
r -f r" = 2a 2ae cos (" ) cos J (" -f- JS).
For the chord x we have
x 1 = (r + r") 2 4rr" cos* J (u" w),
which, by means of (58) 4 , gives
cos 4
and, substituting for r + ^ r/ its value given by the last of equations
(57), we get
x 2 = 4a 2 sin 2 (" JE) (1 e 2 cos 2 &E' + E)). (58)
J^et us now introduce an auxiliary angle ^, such that
cos h = e cos CE" H- J5;),
the condition being imposed that h shall be less than 180, and put
then the equations (57) and (58) become
= 2g 2 sin g cos h,
r + r" = 2a (1 cos g
x = 2a sin ^ sin h.
" = 2a (1 cos g cos A),
Further, let us put
h ff = $,
and the last two of equations (59) give
Introducing and e into the first of equations (59), it becomes
s = sin e) (<5 sin 5). (61)
a?
The formulae (60) enable us to determine e and d from r -f r", x,
and a, and then the time r' = k (t ff t) may be determined from
(61). Since, according to (58) 4 ,
Vrr" cos J (u" u) = a (cos ^ cos 7i) = 2 sin ^e Bin = r^-T - ~ V rr",
8)
which may be used to determine . In this case, and H will be referred to the
equator as the fundamental plane. The angles ^ and i//' will be
obtained from the equations (102) 3 , or from equations of the form
350 THEORETICAL ASTRONOMY.
of (26), and finally the auxiliary quantities A, B, B", &c. will be
obtained from (51) 3 , writing d and d /f in place of ft and ft", respect-
ively.
As soon as these auxiliary quantities have been determined, by
means of (52) 3 the value of % must be found which will exactly
satisfy equation (65). To effect this, we first compute e from
sin ^e =
and, if it be required, we also find 3 from
using approximate values of r + r" and x. Then we find Q from
(66), and Ar ' from (76) or from (78), the logarithms of the auxiliary
quantities B Q and N being found by means of Table XV. with the
argument e. The value of r/ having been found from (77), the
equations (73) and (74), in connection with Table XI., enable us to
obtain a closer approximation to the correct value of x. With this
we compute new values of r and r ff , and repeat the determination
of x. A few trials will generally give the correct result, and these
trials may be facilitated by the use of the formula (67) 3 . It will be
observed, also, that Q and Ar ' are very slightly changed by a small
change in the values of r -j- r fl and x, so that a repetition of the
calculation of these quantities pnly becomes necessary for the final
trial in finding the value of x which completely satisfies the equa-
tions (52) 3 and (65). When the value of a is such that the values
of Q and ^V exceed the limits of Table XV., the equation (61) may
be employed, and, in the case of hyperbolic motion, when Q and '
exceed the limits of Table XVI., we may employ the complete ex-
pression for the time r' in terms of m and n as given by (79).
The values of r, r /r , and x having thus been found, the equations
will determine the curtate distances p and />". When the equator is
the fundamental plane, we have
p = A cos 9, p" = J" cos d".
From p, p", and the corresponding geocentric spherical co-ordinates,
the radii- vectores and the heliocentric spherical co-ordinates I, I", 6,
and b" will be obtained, and thence &, i, u, u ff , and the remaining
VARIATION OF THE SEMI-TRANSVERSE AXIS. 351
elements of the orbit, as already illustrated. In the case of elliptic
motion, if we compute the auxiliary quantities e and d by means of
the equations (60), we shall have
e cos J (E" + J) = cos J (e + d),
from which e and \(E fr -\~ E} may be found, and hence, since
i(J" J) = i(e ), we derive .# and J0". The values of q and
i? may then be found directly from these and quantities already
obtained. Thus, the last of equations (43)! gives
cos ^v cos \E cos ^v" cos ^E"
V~q ' 1/r Vq 1/7'
Multiplying the first of these expressions by sin Ju", and the second
oy sin J0, adding the products, and reducing, we obtain
smi(v" v)sin%v _ cos j (v" v) cos %E cos \E"
l/q Vr V7' '
Therefore, we shall have
_1_ g . n _ cosjJg cos E"
V~q Sm 3V " ~ 1/r tan J (M" *) T/7' sin J (t*" uj
1 cos IE
-= cos Iv = -,
VQ V
from which q and v may be found as soon as cos \E and cos \E" are
known. In the case of parabolic motion the eccentric anomaly is
equal to zero, and these equations become identical with (92) 3 . The
angular distance of the perihelion from the ascending node will be
obtained from
(a = u V.
Since r = a ae co$E, and q = a(l e), we have
and hence
1-1
a
(89)
352 THEORETICAL ASTRONOMY.
When the eccentricity is nearly equal to unity, the value of q given
by approximate elements will be sufficient to compute cos^ and
cos^E" by means of these equations, and the results thus derived
will be substituted in the equations (88), from which a new value of
q results. If this should differ considerably from that used in com-
puting cos \E and cos \E", a repetition of the calculation will give
the correct result.
In the case of hyperbolic motion, although E and E f> are imagi-
nary, we may compute the numerical values of cos^ and cos^E''
from the equations (89), regarding a as negative, and the results will
be used for the corresponding quantities in (88) in the computation
of q and v for the hyperbolic orbit.
Next, we compute a second system of elements from M and/ -f Sf,
and a third system from M -f- dM and/, df and dM denoting the
arbitrary increments assigned to / and M respectively. The com-
parison of these three systems of elements with additional observed
places of the comet, will enable us to form the equations of condition
for the determination of the most probable values of the corrections
&M and A/ to be applied to M and /respectively. The formation of
these equations is effected in precisely the same manner as in the case
of the variation of the geocentric distances or of & and i, and it does
not require any further illustration. The final elements will be ob-
tained from M-\- &M, and/-f- A/, either directly or by interpolation.
We may remark, further, that it will be convenient to use log M as
the quantity to be corrected, and to express the variations of log M
in units of the last decimal place of the logarithms.
When the orbit differs very little from the parabolic form, it will
be most expeditious to make two hypotheses in regard to M, putting
in each case = 0, and only compute elliptic or hyperbolic elements
in the third hypothesis, for which we use J/and f=df. The first
and second systems of elements will thus be parabolic.
120. Instead of M and - we may use A and - as the quantities to
a a
be corrected. In this case we assume an approximate value of J by
means of elements already known, and by means of (96) 3 , (98) 3 , (102),,
and (103) 3 , we compute the auxiliary quantities C, B, B", &c., re-
quired in the solution of the equations (104) 3 . We assume, also, an
approximate value of J r/ and compute the corresponding value of r",
the value of r having been already found from the assumed value of
J. Then, by trial, we find the value of x which, in connection witn
EQUATIONS OF CONDITION. 353
the assumed value of -, will satisfy the equations (104) 3 and (65) or
(61). The corresponding value of A" is given by
J" = e i/x 2 C\
When A" has thus been determined, the heliocentric places will be
obtained by means of the equations (106) 3 and (107) 3 , and, finally,
the corresponding elements of the orbit will be computed. If the
ecliptic is taken as the fundamental plane, we put D = Q, A = O,
and write ^ and /9 in place of a and d respectively.
If we now compute a second system of elements from A + d J and
f = -, and a third system from J and /+
by ; then we shall have
1 , 1
y = i*~7
and the equations for the residuals are transformed into the form
A f f\ I f \ f(\-\ \
A0 = (m nj ) -f- n j z*. (91)
If we now assign to 2, successively, the values 1, 2, 3, &c., the re-
siduals thus obtained will indicate the value of z which best satisfies
the series of observations, and hence how many revolutions of the
comet have taken place during the interval denoted by T.
122. In the determination of the orbit of a comet from three ob-
served places, a hypothesis in regard to the semi-transverse axis may
w r ith facility be introduced simultaneously with the computation of
the parabolic elements. The numerical calculation as far as the form-
ation of the equations (52) s will be precisely the same for both the
parabolic and the elliptic or hyperbolic elements. Then in the one
case we find the values of r, r f/ , and x which will satisfy equation
(56) 3 , and in the other case we find those which will satisfy the equa-
tion (65), as already explained. From the results thus obtained, the
two systems of elements will be computed. Let /= -> then in the
case of the system of parabolic elements Ave have/=0, and the com-
parison of the middle place with these and also with the elliptic or
hyperbolic elements will give the value of
in which 6 l denotes the geocentric spherical co-ordinate computed
from the parabolic elements, and 2 that computed from the other
system of elements. Further, let A# denote the difference between
computation and observation for the middle place, and the correction
to be applied to /, in order that the computed and the observed
values of 6 may agree, will be given by
Hence, the two observed spherical co-ordinates for the middle place
will give two equations of condition from which A/ may be found,
356 THEORETICAL ASTRONOMY.
and the corresponding elements will be those which best represent
the observations, assuming the adopted value of M to be correct.
123. The first determination of the approximate elements of the
orbit of a comet is most readily effected by adopting the ecliptic as
the fundamental plane. In the subsequent correction of these ele-
ments, by varying - and M or J, it will often be convenient to use
a
the equator as the fundamental plane, and the first assumption in
regard to M will be made by means of the values of the distances
given by the approximate elements already known. But if it be
desired to compute M directly from three observed places in reference
to the equator, without converting the right ascensions and declina-
tions into longitudes and latitudes, the requisite formula may be
derived by a process entirely analogous to that employed when the
curtate distances refer to the ecliptic. The case may occur in which
only the right ascension for the middle place is given, so that the
corresponding longitude cannot be found. It will then be necessary
to adopt the equator as the fundamental plane in determining a
system of parabolic elements by means of two complete observations
and this incomplete middle place. If we substitute the expressions
for the heliocentric co-ordinates in reference to the equator in the
equations (4) 3 and (5) s , we shall have
n (p cos a R cos D cos A} (p r cos a' R f cos U cos J/)
-f n" (P" sin a" R" cos D" cos A"\
= n (p sin a R cos D sin A) (p f sin a' E' cos D' sin A') (92)
-f n" (p" sin a" R" cos D" sin A"),
Q = n(pttmd R sin D} (p tan S' R' sin Z>')
-f- n" (p" tan a" R" sin IT),
in which p y //, p tf denote the curtate distances with respect to the
equator, A, A f , A" the right ascensions of the sun, and D, Z>', I/'
its declinations. These equations correspond to (6) 3 , and may be
treated in a similar manner.
From the first and second of equations (92) we get
= n (p sin (a' a) RcosD sin (ofA)) -f R cos D' sin (a' 4')
n" (P" sin (a" a') -f R" cosZ>" sin (a' A")),
and hence
=? = ^ >4 na-a
p n" Sin (a" a')
?i ft cos D sin (a' A) R r cosD' sin (a'- A'}-\-ri'R" cos D" sin (a' A")
pri'sm(a" a')
VARIATION OF TWO RADII- VECTORES. 357
This formula, being independent of the decimation d', may be used
to compute M when only the right ascension for the middle place ia
given. For the first assumption in the case of an unknown orbit,
we take
1f sin (a' a)
M=
if t sin(a' / a')'
and, by means of the results obtained from this hypothesis, the com-
plete expression (93) may be computed. By a process identical with-
that employed in deriving the equation (36) 3 , we derive, from (93),
the expression
p" = p--j$~$- ) (94)
j TT f^\tt\i^- \ \R r cos D' sin (a' A!}
-g^F^ ^ ; \r* #/ sin (a" a')
and, putting
,_. n sin (a' a)
n" ' sin (a" a') '
F I ' rr * (?' T ^ cos D' sin (a' A') ^/1__1_\
^ g n ' r" ^ T sin (a' a) p \r' 3 R' 3 f'
we have
(95)
The calculation of the auxiliary quantities in the equations (52) 3
will be effected by means of the formulae (96) 3 , (86), (87), (102) 3 , and
(51) 3 . The heliocentric places for the times t and t" will be given
by (106) 3 and (107) 3 , and from these the elements of the orbit will
be found according to the process already illustrated.
124. The methods already given for the correction of the approxi-
mate elements of the orbit of a heavenly body by means of additional
observations or normal places, are those which will generally be
applied. There are, however, modifications of these which may be
advantageous in rare and special cases, and which will readily suggest
themselves. Thus, if it be desired to correct approximate elements
by varying two radii-vectores r and r", we may assume an approxi-
mate value of each of these, and the three equations (88)j will con-
tain only the three unknown quantities d, b, and /. By elimination,
these unknown quantities may be found, and in like manner the
S58 THEORETICAL ASTRONOMY.
values of J", b"j and I". It will be most convenient to compute
the angles ^ an< ^ Vj anc ^ then find z and z" from
R sin * R" sin V
sm z = - , sm z = - ^ ,
or, putting # 2 = r 2 J? 2 sin 2 ^, and x" 2 = r" 2 -R" 2 sin 2 ^", from
sin 4 " sin V
tan z = - , tan z = - .
x x"
The curtate distances will be given by the equations (3), and the
heliocentric spherical co-ordinates by means of (4), writing r in place
of a. From these u" u may be found, and by means of the values
of r, r n ', and u" u the determination of the elements of the orbit
may be completed. Then, assigning to r an increment dr, we com-
pute a second system of elements, and from r and r" -f- dr ff a third
system. The comparison of these three systems of elements with an
additional or intermediate observed place will furnish the equations
for the determination of the corrections Ar and Ar /r to be applied to
r and r", respectively. The comparison of the middle place may be
made with the observed geocentric spherical co-ordinates directly, or
with the radius-vector and argument of the latitude computed directly
from the observed co-ordinates; and in the same manner any number
of additional observed places may be employed in forming the equa-
tions of condition for the determination of Ar and Ar r/ .
Instead of r and r", we may take the projections of these radii-
vectores on the plane of the ecliptic as the quantities to be corrected.
Let these projected distances of the body from the sun be denoted
by r and r ", respectively ; then, by means of the equations (88)^
we obtain
(96)
from which I may be found ; and in a similar manner we may find
I". If we put
we have
tan(;-,>) = * sin(A - 0) . (97,
X Q
Let 8 denote the angle at the sun between the earth and the place
of the planet or comet projected on the plane of the ecliptic ; then
we shall have
VARIATION OF TWO RADII-VECTORES. 359
=180 + O I,
==: R8in(Z- L 0) (98)
and
p tan ft
tan 6 = , (99)
r o
by means of which the heliocentric latitudes b and b" may be found.
The calculation of the elements and the correction of r and r " are
then effected as in the case of the variation of r and r" .
In the case of parabolic motion, the eccentricity being known, we
may take q and T as the quantities to be corrected. If we assume
approximate values of these elements, r, r', r ff , and v, v f , v" will be
given immediately. Then from r, r r , r' 1 and the observed spherical
co-ordinates of the body we may compute the values of u" u' and
u' u. In the same manner, by means of the observed places, wo
compute the angles u" u' and u f u corresponding to q-\-dq and T y
and to q and T-+- d T } 3q and dT denoting the arbitrary increments
assigned to q and T, respectively. The comparison of the helio-
centric motion, during the intervals t" t 1 and t' i, thus obtained,
in the case of each of the three systems of elements, from the ob-
served geocentric places with the corresponding results given by
w" u' = v" v',
enables us to form the equations by which we may find the cor-
rections A^ and &T to be applied to the assumed values of q and T,
respectively, in order that the values of u" u f and u' u computed
by means of the observed places shall agree with those given by the
true anomalies computed directly from q and T.
360 THEORETICAL ASTRONOMY.
CHAPTER VII.
METHOD OP LEAST SQUARES, THEORY OF THE COMBINATION OP OBSERVATIONS, AND
DETERMINATION OF THE MOST PROBABLE SYSTEM OF ELEMENTS FROM A SERIES
OF OBSERVATIONS.
125. WHEN the elements of the orbit of a heavenly body are known
to such a degree of approximation that the squares and products of
the corrections which should be applied to them may be neglected,
by computing the partial differential coefficients of these elements
with respect to each of the observed spherical co-ordinates, we may
form, by means of the differences between computation and observa-
tion, the equations for the determination of these corrections. Three
complete observations will furnish the six equations required for the
determination of the corrections to be applied to the six elements of
the orbit ; but, if more than three complete places are given, the
number of equations will exceed the number of unknown quantities,
and the problem will be more than determinate. If the observed
places were absolutely exact, the combination of the equations of
condition in any manner whatever would furnish the values of these
corrections, such that each of these equations would be completely
satisfied. The conditions, however, which present themselves in the
actual correction of the elements of the orbit of a heavenly body by
means of given observed places, are entirely different. When the
observations have been corrected for all known instrumental errors,
and when all other known corrections have been duly applied, there
still remain those accidental errors which arise from various causes,
such as the abnormal condition of the atmosphere, the imperfections
of vision, and the imperfections in the performance of the instrument
employed. These accidental and irregular errors of observation cannot
be eliminated from the observed data, and the equations of condition
for the determination of the corrections to be applied to the elements
of an approximate orbit cannot be completely satisfied by any system
of values assigned to the unknown quantities unless the number of
equations is the same as the number of these unknown quantities.
It becomes an important problem, therefore, to determine the par-
ticular combination of these equations of condition, by means of which
METHOD OF LEAST SQUARES. 361
the resulting values of the unknown quantities will be those which,
while they do not completely satisfy the several equations, will afford
the highest degree of probability in favor of their accuracy. It will
be of interest also to determine, as far as it may be possible, the
degree of accuracy which may be attributed to the separate results.
But, in order to simplify the more general problem, in which the
quantities sought are determined indirectly by observation, it will be
expedient to consider first the simpler case, in which a single quantity
is obtained directly by observation.
126. If the accidental errors of observation could be obviated, the
different determinations of a magnitude directly by observation would
be identical ; but since this is impossible when an extreme limit of
precision is sought, we adopt a mean or average value to be derived
from the separate results obtained. The adopted value may or may
not agree with any individual result, since it is only necessary that
the residuals obtained by comparing the adopted value with the
observed values shall be such as to make this adopted value the most
probable value. It is evident, from the very nature of the case, that
we approach here the confines of the unknown, and, before we pro-
ceed further, something additional must be assumed.
However irregular and uncertain the law of the accidental errors
of observation may be, we may at least assume that small errors are
more probable than large errors, and that errors surpassing a certain
limit will not occur. We may also assume that in the case of a large
number of observations, errors in excess will occur as frequently as
errors in defect, so that, in general, positive and negative residuals
of equal absolute value are equally probable. It appears, therefore,
that the relative frequency of the occurrence of an accidental error A
in the observed value will depend on the magnitude of this error,
and may be expressed by
(n' x)
Q = (n x) 7 - ^rjr, -- ^ + (ri x) 7-? & \\, , , 4- &c.,
(n x) d (n x) (ri x) d (ri x) '
and the comparison of this with (6) shows that
d log
(ri x)
(n x)d(n x) (ri x)d(ri x) "
k being a constant quantity. Hence we derive
d loge 1 O ^ 1 2 S ' ^ 1 2 ^J 4 *'
which converges rapidly when T is small. To find the value of T
which corresponds to the value 0.44311 assigned to the integral, we
compute the value of the series (17) for the values 0.45, 0.47, and
0.49 assigned to T, successively, and from the results thus obtained
it is easily seen that when the sum of the terms of the series is
0.44311, we have
T=Jir = 0.47694,
or
0.47694
r = -j , (18)
which determines the relation between the probable error and the
measure of precision.
The probability that the error of an observation, without regard to
sign, does not exceed nr, is expressed by
, > I <-/ wi/j ( J.t7 )
VTT^O
and this integral, therefore, indicates the ratio of the number of obser-
vations affected with an error which does not exceed nr to the whole
number of observations. Hence, if we assign different values to n,
the integral (19) computed for the several assumed values of
nhr = 0.47694n
will give the relative number of errors of a given magnitude. Thus,
if we put n = \) we obtain
0.2385
-4: JV" eft = 0.264.
from which it appears that in a series of 1000 observations there
ought to be 264 observations in which the error does not exceed \r.
It has been found, in this manner, that in the case of an extended
series of observations the number of errors of a given magnitude
assigned by theory agrees very closely with that actually given by
the series of observations ; and hence we conclude that the error com-
mitted in extending the limits of the summation in the expression (1)
to oo and -f oo, instead of the finite limits which it is presumed
that the actual errors cannot exceed, is very slight, so that the form
368 THEORETICAL ASTRONOMY.
of the function (p (J) which has been derived may be regarded as that
which best satisfies all the conditions of the problem.
129. The relative accuracy of different series of observations may
also be indicated by means of what are called the mean error and the
mean of the errors for each series, the former being the error whose
square is equal to the mean of the squares of all the errors of the
series, and the latter the mean of these errors without reference to
their algebraic sign.
Let denote the mean error ; then, since the number of observa-
tions having the error J is m
V x m
and we have
In the case of a single observation, if P denotes the probability of
the error zero, and P f the probability of the error d, we have
Hence it appears that if h denotes the measure of precision of the
arithmetical mean of m observations, the relation between h and A,
the measure of precision of an observation, is given by
h * = mh*; (27)
and if r is the probable error of the arithmetical mean, and its
mean error, we have, according to the equations (18) and (20),
(28)
These expressions determine the probable and the mean error of the
arithmetical mean of a number of observations when these errors in
the case of a single observation are known.
131. The expressions for the relation between the mean and pro-
bable errors have been derived for the case of a very large number
of observations, a number so great that the error of the arithmetical
mean becomes equal to zero. In the case of a limited number of
observed values of #, the residuals given by comparing the arith-
metical mean with the several observations will not, in general, give
the true errors of the observations ; but the greater the number of
observations, the nearer will these residuals approach the absolute
errors. If J, J r , A n ', &c. are the actual errors of the observations,
and v, v f , v ff , &c. those which result from the most probable value of
x, we shall have, denoting the arithmetical mean by x w and the true
value by X Q -f 3,
A=v S, A' = v' 3, A" = v " d, &c.;
METHOD OF LEAST SQUABES. 371
hence
me 2 = [ A J] = [ W ] + m<* 2 . (29)
This equation will enable us to determine the mean error of an ob-
servation when 3 is given ; but, since this is necessarily unknown,
some assumption in regard to its value must be made. If we assume
it to be equal to the mean error of the arithmetical mean, the re-
maining error will be wholly insensible, and hence the equation (29)
becomes
we 2 = [iw] + me 2 = [vv~\ -f- 3 .
Therefore, we shall have
<
and, according to (21),
r = 0.6745
These equations give the values of the mean and probable errors of
a single .observation in terms of the actual residuals found by com-
paring the arithmetical mean with the several observed values.
The probable and the mean error of the arithmetical mean will be
given by
' ^ (32;
r = 0.6745^ |^j_ .
When the number of observations is very large, the probable error
of an observation and also that of the arithmetical mean may be de-
termined by means of the mean of the errors. If we suppose the
number of positive errors to be the same as the number of negative
errors, the mean of the errors without reference to the algebraic sign
gives
and hence we have, according to (23),
r = 0.8453 &i (33)
m
For the mean error of an observation we have
e = l/ii = 1.2533 4 (34)
372 THEORETICAL ASTRONOMY.
If the number of observations is very great, the results given oy
these equations will agree with those given by (30) and (31); but for
any limited series of observed values, the results obtained by means
of the mean error will afford the greatest accuracy.
132. The relative accuracy of two or more observed values of a
quantity may be expressed by means of what are called their weights.
If the observations are made under precisely similar circumstances,
so that there is no reason for preferring one to the other, they are said
to have the same weight. The weight must therefore depend on the
measure of precision of the observations, and hence on their probable
errors. The unit of the weight is entirely arbitrary, since only the
relative weights are required, and if we denote the weight by p, the
value of p indicates the number of observations of equal accuracy
which must be combined in order that their arithmetical mean may
have the same degree of precision as the observation whose weight is
p. Hence, if the weight of a single observation is 1, the arithmetical
mean of m such observations will have the weight m. Let the pro-
bable error of an observation of the weight unity be denoted by r,
and the probable error of that whose weight is p f by r' then, ac-
cording to the first of equations (28), we shall have
or
For the case of an observation whose weight is p rr and whose pro-
bable error is r", we have
from which it appears that the weights of two observations are to each
other inversely as the squares of their probable or mean errors, and f
according to (18), directly as the squares of their measures of precision.
Let us now consider two values of #, which may be designated by
x r and a?", the mean errors of these values being, respectively, e' and
e": then, if we put
X=x'x"
and suppose that both a;' and x tr have been derived from a large num-
ber m of observations (and the same number in each case), so that the
residuals v 9 v/, v n ', &c. in the case of x f and the residuals v,, v/, v/' t
&c. in the case of x n may be regarded as the actual errors of obser-
METHOD OF LEAST SQUARES.
vatiou, the errors of the value of X t as determined from the several
observations, will be
V V,, V' V,', V" V,", &G.
Let the mean error of X be denoted by E; then we have
mE* =S(v v,y = [w] zh 2 [>,] + [v,v t ] ;
and since the number of observed values is supposed to be so great
that the frequency of negative products w, is the same as that of the
similar positive products, so that [vvj] = 0, this equation gives
or
E 2 = e" -I- e" 2 .
Combining X with a third value x" r whose mean error is e'", the
mean error of x f x" x 1 " will be found in the same manner to be
equal to e /2 + e //2 -j- s ///2 ; and hence we have, for the algebraic sum
of any number of separate values,
E = * + e' 2 + e" 2 + Ac., (35)
and, according to the last of equations (21),
R = ir 2 -hr' 2 + r" 2 + &c., (36)
R being the probable error of the algebraic sum. If the probable
errors of the several values are the same, we have
and the probable error of the sum of m values will be given by
E = rl/m.
Hence tne probable error of the arithmetical mean of m observed
values will be
R r
*= =77
m Vm
which agrees with the first of equations (28).
Let P denote the weight of the sum X, p f the weight of a?', and p /f
that of x" ; then we shall have
p _
"
374 THEORETICAL ASTRONOMY.
from which we get
Since the unit of weight is arbitrary, we may take
and hence we have, for the weight of the algebraic sum of any
number of values,
= ^ == t /1 + r" 2 -f-r'" 2 -{-&c.'
or, whatever may be the unit of weight adopted,
. I * I __ I
p' "" p" ' p'" '
In the case of a series of observed values of a quantity, if we
designate by r f the probable error of a residual found by comparing
the arithmetical mean with an observed value, by r the probable
error of the observation, by X Q the arithmetical mean, and by n any
observed value, the probable error of
n=x +v,
according to (36), will be
.+**=; + ",
r being the probable error of the arithmetical mean. Hence we derive
m
m 1
and if we adopt the value
/ = 0.8453 ^3 ;
m'
the expression for the probable error of an observation becomes
r = 0.8453 M (40)
l/m(m 1)
in which [v] denotes the sum of the residuals regarded as positive,
and m the number of observations.
133. Let n, n f , n", &c. denote the observed values of x, and let p,
p', p ff } &c. be their respective weights ; then, according to the defi-
METHOD OF LEAST SQUARES. 375
nition of the weight, the value n may be regarded as the arithmetical
mean of p observations whose weight is unity, and the same is true
in the case of n f , n f/ , &c. We thus resolve the given values into
p + p f + p" + observations of the weight unity, and the arith-
metical mean of all these gives, for the most probable value of x,
_pn + p'ri + p"n" + Ac. _ [pn\ ,
' ' &c. ' '
The unit of weight being entirely arbitrary, it is evident that the
relation given by this equation is correct as well when the quantities
P) p'y P"> & c - are fractional as when they are whole numbers. The
weight of X Q as determined by (41) is expressed by the sum
and the probable error of x is given by
r ' (42)
when r, denotes the probable error of an observation whose weight
is unity. The value of r, must be found by means of the observa-
tions themselves. Thus, there will be p residuals expressed by
n x Q , p' residuals expressed by n' o? , and similarly in the case of
n", n" r , &c. Hence, according to equation (31), we shall have
r, = 0.6745
in which m denotes the number of values to be combined, or the
number of quantities n, n f , n", &c. For the mean error of # , we
have the equations
(44;
If different determinations of the quantity x are given, for which
the probable errors are r, r', r", &c., the reciprocals of the squares
of these probable errors may be taken as the weights of the respective
values n, n', n rf , &c., and we shall have
!L L ?L- I ^ L
.,2 ~r ^/ 2 ~r v //2 T *
376 THEORETICAL ASTRONOMY.
with the probable error
= ' (46)
The mean errors may be used in these equations instead of the pro-
bable errors.
134. The results thus obtained for the case of the direct observa-
tion of the quantity sought, are applicable to the determination of
the conditions for finding the most probable values of several un-
known quantities when only a certain function of these quantities is
directly observed. In the actual application of the formulae it will
always be possible to reduce the problem to the case in which the
quantity observed is a linear function of the quantities sought. Thus,
let V be the quantity observed, and , y, , &c. the unknown quan-
tities to be determined, so that we have
Let , y w f , &c. be approximate values of these quantities supposed
to be already known by means of previous calculation, and let x, y,
z, &c. denote, respectively, the corrections which must be applied to
these approximate values in order to obtain their true values. Then,
if we suppose that the previous approximation is so close that the
squares and products of the several corrections may be neglected, we
have
_ T , dV , dV . dV
v - v =d! x +^y+~dt z +-->
and thus the equation is reduced to a linear form. Hence, in general,
if we denote by n the difference between the computed and the ob-
served value of the function, and similarly in the case of each obser-
vation employed, the equations to be solved are of the following
form :
ax -f- by "h cz ~f~ du -\- ew -f- / -j~ n 0,
a'x + b'y -f c'z + d'u + e'w +ft + ri = 0, (47)
a " x 4. y'y 4- c " z 4. d"u -f e"w +ft + n" = 0,
&c. &c.
which may be extended so as to include any number of unknown
quantities. If the number of equations is the same as the number
of unknown quantities, the resulting values of these will exactly
satisfy the several equations; but if the number of equations exceeds
the number of unknown quantities, there will not be any system of
METHOD OF LEAST SQUARES. 377
values for these which will reduce the second members absolutely to
zero, and we can only determine the values for which the errors for
the several equations, which may be denoted by v, v', v", &c., will be
those which we may regard as belonging to the most probable values
of the unknown quantities.
Let J, J r , J", &c. be the actual errors of the observed quantities;
then the probability that these occur in the case of the observations
used in forming the equations of condition, will be expressed by
and the most probable values of the unknown quantities will be those
which make P a maximum. The form of the function
shall be a minimum. Hence it appears that when the observations are
equally precise, the most probable values of the unknown quantities
are those which render the sum of the squares of the residuals a
minimum, and that, in general, if each error is multiplied by its
measure of precision, the sum of the squares of the products thus
formed must be a minimum.
If we denote the actual residuals by v, v f , v", &c., and regard the
observations as having the same measure of precision, the condition
that the sum of their squares shall be a minimum gives
dM =0 ^M = 0> ^W^o.&c.,
dx * dy dz
378 THEOEETICAL ASTRONOMY.
or
dv . ,dv' ,,dv" ,
V ~T- + v -J- + *> -T- + 0,
eta ' eta n da? ~
dv . ,dv r dv"
I p **/ . ff UU . -
V ^ + V ^ + V dF +---- = >
&c. &c.
If we differentiate the equations
ax -{-by + cz -f- du -j- ew -(-/ + n = v,
a'z + 6'y -f- c'z -f d'w + e'w -\-ft + ^' =v', (49)
a "a; 4. 5" y _j_ c " z 4_ rf" M + e "w; +f"t + n" = i;' r ,
&c. &c.
with respect to x, y, z, &c., successively, we obtain
dv _ dv' f dv" o
(50)
dv , dv f dv lf
dy dy dy
&c. &c. &c.
Introducing these values into the equations (48), and substituting for
0, v r , v ff , &c. their values given by (49), we get
[aa] x -j- [a&] y -f [ac] z -j- [ad] w + [ae] w + [a/] < + [a^] = 0,
[a6] x + [65] y + [6c] 2 + [ W] w + \be] w + [6/] t + [6/1] = 0,
[ac] a; + [6c] y + [cc] z -f [cd] u + [ce] ti; + [c/] * + [c/i] = 0,
\_ad~] x + [6d] y + [cd] 3 + [dd] u + [de] w + [d/] * + [dw] = 0,
[ae] a; + {be] y + [ce] 2 + \de\ u + [ee] w -f [e/] < + [en] = 0,
[/]* -h P/]y + ['/]* + [d/] + [>/]* + [//] * + [>] - 0,
in which
[aa] = aa -f- a'a' -j- a" a" -f- . . . .
[ac] =ac-f-aV+a'V' + ....
[66] = 66 -f W -f V'b" + ____
&c. &c.
The equations of condition are thus reduced to the same number as
the number of the unknown quantities, and the solution of these
will give the values for which the sum of the squares of the residuals
will be a minimum. These final equations are called normal equations.
When the observations are not equally precise, in accordance with
the condition that AV + h /2 v' 2 -\- h tt2 v m -\- &c. shall be a minimum,
METHOD OF LEAST SQUARES. 379
each equation of condition must be multiplied by the measure of
precision of the observation; or, since the weight is proportional to
the square of the measure of precision, each equation of condition
must be multiplied by the square root of the weight of the observa-
tion, and the several equations of condition, being thus reduced to
the same unit of weight, must be combined as indicated by the equa-
tions (51).
135. It will be observed that the formation of the first normal
equation is eifected by multiplying each equation of condition by
the coefficient of x in that equation and then taking the sum of all
the equations thus formed. The second normal equation is obtained
in the same manner by multiplying by the coefficient of y; and thus
by multiplying by the coefficient of each of the unknown quantities
the several normal equations are formed. These equations will gene-
rally give, by elimination, a system of determinate values of the
unknown quantities #, y, z, &c. But if one of the normal equations
may be derived from one of the others by multiplying it by a con-
stant, or if one of the equations may be derived by a combination of
two or more of the remaining equations, the number of distinct rela-
tions will be less than the number of unknown quantities, and the
problem will thus become indeterminate. In this case an unknown
quantity may be expressed in the form of a linear function of one or
more of the other unknown quantities. Thus, if the number of
independent equations is one less than the number of unknown
quantities, the final expressions for all of these quantities except one,
will be of the form
X = a + Pt, y = a' -f fit, S = a" -f fi't, &C. (53^)
The coefficients a, /9, a/, /9', &c. depend on the known terms and co-
efficients in the normal equations, and if by any means t can be de-
termined independently, the values of x, y, z, &c. become determinate.
It is evident, further, that when two of the normal equations may be
rendered nearly identical by the introduction of a constant factor, the
problem becomes so nearly indeterminate that in the numerical appli-
cation the resulting values of the unknown quantities will be very
uncertain, so that it will be necessary to express them as in the equa-
tions (53).
The indeterrnination in the case of the normal equations results
necessarily from a similarity in the original equations of condition,
and when the problem becomes nearly indeterminate, the identity of
380 THEORETICAL ASTRONOMY.
the equations will be closer in the normal equations than in the equa-
tions of condition from which they are derived. It should be observed,
also, that when we express x } y, z, &c. in terms of t, as in (53), the
normal equation in t, which is the one formed by multiplying by the
coefficient of t in each of the equations of condition, is not required.
136. The elimination in the solution of the equations (51) is most
conveniently effected by the method of substitution. Thus, the first
of these equations gives
lab'] [ae] [ad] [ae] [a/] [an] .
it/ - 7/ ~ "p ^r Z ' ~p - ^ Zv ~" "* r- -. IV ~^~* 1^ ^^ .
[aa] [aa] [aa] \_aa\ \_aa] [aa]
and if we substitute this for x in each of the remaining normal equa-
tions, and put
[66] - |g| [at] = [M.1], [fc] - [gj [ao] = [Jc.1],
(54)
r i r i
[] - gj [ae] = [ce.ll, [ffl - jgj- [/] = [c/1] ;
[dd] - [afl = [!] - 0.
These equations are symmetrical, and of the same form as the normal
equations, the coefficients being distinguished by writing the numeral
1 within the brackets.
The unknown quantity x is thus eliminated, and by a similar pro-
cess y may be eliminated from the equations (59), the resulting equa-
tions being rendered symmetrical in form by the introduction of the
numeral 2 within the brackets. Thus, we put
. = [cc . 2 ], .
= [ce.2], [/.!] - [i/.l] =
. .
and the equations become
(.63)
[cc.2] z + [crf.2] t* + [ce.2] ti; -f [c/.2] < 4- [cn.2] = 0,
1 [cd.2] z 4 [dd.2] M 4 [rfe-2] w 4- [rff.2] < 4 [d-2] = 0,
[ce.2] z + [rfc.2] u 4 [ec.2] to 4 [e/.2] * 4 [cn-2] = 0,
[c/.2] z 4 Hf-2] i* 4 [e/2] w 4- [//2] + [>.2] = 0.
To eliminate z from these equations, we put
382 THEORETICAL ASTRONOMY.
[ee.2] - l [e( ,2] = [ ee . 3 ],
.2] = [dn.3], [en.2] -
W21
[/-2]-g|-[e.2] = [/n.
and we have
[eta.3] M + [de.3] w -f [d/.3] < + [d.3] = 0,
[efe.3] ti + [ee.3] w + [e/.3] + [en.3] = 0, (68)
Again we put, in a similar manner,
.3] - [dn.3] = [6n.4], (69)
and the equations are
+ [671.4] = 0,
. . .
Finally, to eliminate w, we put
= [/ re . 5 ] ) (71)
and the resulting equation is
0, (72)
which gives
[/n.5]
The value of t thus found enables us to derive that of w by means
of the first of equations (70). The value of w being found, that of
u will be obtained from the first of equations (68). In like manner,
the remaining unknown quantities will be determined by means of
the equations (64), (59), and (51). The determination of the unknown
quantities is thus reduced to the solution of the following system of
equations :
METHOD OF LEAST SQUARES. 383
&&, + M M + M W +m t +M =0
[aa] r [aa] [aa] ' [aa] r [aa]
3+ "' f W+ <+ =0>
[cc.2] [cc.2] [cc.2] '
the coefficients of which will have been found in the process of de-
termining the several auxiliary quantities. It will be observed,
further, that both in the normal equations and in those which result
after each successive elimination, the coefficients which appear in a
horizontal line, with the exception of the coefficient involving the
absolute terms of the equations of condition, are found also in the
corresponding vertical line. The form of the notation [66.1], [&c.l],
&c. may be symbolized thus :
= [fir. (ft + 1)1 (75)
in which oc, /9, y, denote any three letters, and fj, any numeral.
The equations (74) are derived for the case of six unknown quan-
tities, which is the number usually to be determined in the correction
of the elements of the orbit of a heavenly body; but there will be
no difficulty in extending the process indicated to the case of a greater
number of unknown quantities, except that the number of auxiliaries
symbolized generally by (75) increases very rapidly when the number
of unknown quantities is increased.
137. In the numerical application of the formulae, when so many
quantities are to be computed, it becomes important to be able to
check the accuracy of the calculation in its successive stages. First,
then, to prove the calculation of the coefficients in the normal equa-
tions, we put
a+&+c+d-fe-f/=s,
a! -|- j' _j_ c ' + d' + e r +/' = s f , &c.
If we multiply each of the sums thus formed by the corresponding
absolute term n, and take the sum of all the products, we have
384 THEORETICAL ASTRONOMY.
[an] + [5n] + [en] + [an] + M + Lfii] = [m]. (76)
In a similar manner, multiplying by each of the coefficients in the
original equations of condition, we find
[oa] + [aft] + M + [ad] + M + [of] = M,
[a6] + [ftft] + [ftc] + [ftd] + [be] + [ft/] - [ft*],
[ac] -f M + [cc] + [cd] + [ce] + [cf] = [,],
[ad] + [ftd] + [cd] + [dd] -f [de] + [df ] = [da],
[ae] + [fte] + M + [dc] + [ee\ + [e/] - [ea],
[/] + P/1 + [cf] + Hf] + [/] + [//] -
Hence it appears that if we compute the sums , s r , s r/ , s' /r , &c., and
form [as], [6s], [cs], &c. simultaneously with the calculation of the
coefficients in the normal equations, the equation (76) must be satis-
fied when the absolute terms of the normal equations are correct;
and the equations (77) must be satisfied when the coefficients of the
unknown quantities in the normal equations are correct.
The accuracy of the calculation of the auxiliary quantities sym-
bolized by the equation (75) may be proved in a similar manner.
Thus, we have
which, by means of the first and second of equations (77), becomes
= [W] _ [a4] + M _ gi [ac]
or
[ba.l] = [ftft.1] -f [6c.l] + [ftd.l] + [fte.l] + [ft/.l] ; (78)
and similarly we derive the expressions for [cs.l], [cfe.l], &c. It is
obvious, therefore, that the calculation of the coefficients in the equa-
tions (59), (64), (68), and (70) will be checked as in the case of the
coefficients in the normal equations, the auxiliaries depending on 8
being determined as if s, s f , s ff , &c. were the coefficients of an addi-
tional unknown quantity in the several equations of condition. Hence
we must have, finally,
[*.5] - 1>5]. (79)
If we multiply each of the equations (49) by its v, and take the
sum of the several products, we get
[av] x -f [bv] y + [cv] * + [dv] u -j- [ev] w -f- [/v] t + \vn\ = [w],
METHOD OF LEAST SQUARES. 385
But, according to the equations (48) and (50), we have, for the most
probable values of the unknown quantities,
[av] = 0, |>] = 0, [eo] = 0, &c. ;
and hence
[vn~] = [w]. (80)
If we multiply each of the equations (49) by its n, and take the sum
of all the products thus formed, substituting [vv] for [wi], there re-
sults
[an] x -f- [bri] y -f- [en] + [dn] u -f- [eri] w -f- \Jri\ t ~j- [nn] = [vv].
Substituting in this the value of x given by the first normal equa-
tion, it becomes
[bn.Y] y + [cn.l] z + \_dn.Y] u + [en.l] w + [/n.l] < + [nn.l] = [w],
in which
[n.l] = [*m]-^[.3] t + [nn.3] = [w],
[e.4]tc + [>.43<-{-[nn.4] = M, (82)
|>.5]< + [nn.5] = M,
[nn.6] = [vv].
The expressions for the auxiliaries [ww.2], [nn.3], &c. are
[.2] = [.!] - M [in.1], [nn.8] = [n.2] - M [m.2],
[nn.4] = [nn.8] - [dn.3], [n.5] = [nn.4] - [on.4],
.5]. (83)
The process here indicated may be readily extended to the case of a
greater number of unknown quantities, and we have, in general, when
u denotes the number of unknown quantities,
[vv] = [nn.fi"]. (84)
25
386 THEORETICAL ASTRONOMY.
This equation affords a complete verification of the entire numerical
calculation involved in the determination of the unknown quantities
from the original equations of condition. Thus, after the elimination
has been completed, we substitute the resulting values of x, y y 2, &c.
in the equations of condition, and derive the corresponding values
of the residuals v, v f , v", &c. Then, taking the sum of the squares
of these, the equation (84) must be satisfied within the limits of the
unavoidable errors of calculation with the logarithmic tables em-
ployed. If this condition is satisfied, it may be inferred that the
entire calculation of the values of the unknown quantities from the
given equations of condition is correct.
138. If the values of x, y, z, &c. thus found were the absolutely
exact values, the residuals v, v', v", &c. would be the actual errors
of observation. But since the results obtained only furnish the most
probable values of the unknown quantities, the final residuals may
differ slightly from the accidental errors of observation. Further,
it is evident that the degree of precision with which the several
unknown quantities may be determined by means of the data of the
problem may be very different, so that it is desirable to be able to
determine the relative weights of the different results.
It will be observed that the expressions for either of the unknown
quantities resulting from the elimination of the others is a linear
"unction of n, n f , n", &c., so that we have
x + an + a!n' + a"n" + a" V" + ....== 0, (85)
in wLich the coefficients a, a/, a/', &c. are functions of the several
coefficients of the unknown quantities in the equations of condition.
If we now suppose the equations of condition to be reduced to the
same unit of weight, the mean error of the several absolute terms of
the equations will be the same, and will be the mean error of an
observation whose weight is unity. Thus, if e denotes the mean
error of an observation of the weight unity, the mean error of an
will be ae, that of a'n' will be o/e', and similarly for the other terms
of (85, ; and, according to the equation (35), the mean error of x
will bt)
z x = ]/a 2 -f- a' 2 -f a" 2 -f- &C. = l/[aa]. (86)
Hence the weight of x will be expressed by
'
METHOD OF LEAST SQUARES. 387
Let x, denote the true value of x, namely, that which would be
obtained if the true values of v, v f , v", &c. were retained in the
second members of the equations of condition instead of putting
them equal to zero ; then it is evident that the expression for x, must
be that which would result by substituting n v in place of n in the
formulae for the most probable value as determined from the actual
data. Hence we have
Xf + n ( n v -) + a>'-t/) + . . . . = 0,
and comparing this with the expression (85), we obtain
Substituting in this the values of v, v f , v", &c. given by the equations
(49), there results
x, = x+ [aa] X, -f [06] y, + [ac] z, + [ad] u, -f [oe] w, -f [a/] t, + [aw],
and since, according to (85), x -j- [an] = 0, in order to satisfy this
expression for x n we must evidently have
[aa] = l, [06] =0, [ac]=0, [ad] = 0, [ae] = 0, [a/] = 0. (88)
Since the values of the unknown quantities as determined by the
normal equations must be the same by whatever mode the elimination
may have been performed, let us suppose the method of indeterminate
multipliers to be applied for the determination of a?, and let these
multipliers be designated by 9, q f , q", &c. ; then, the values of these
factors are determined by the condition that the coefficient of x in
the final equation shall be unity, and that the coefficients of the other
unknown quantities shall be zero. Hence we shall have
[aa] q + [oft] q f + [ac] g" + [ad] q'" + .... = 1,
lab] q + [56] ^ + [be] q" + [bd] q'" +.... = 0, (89)
[ac] q + [be] q' + M 5" + M 5"' -f . . . . = 0,
&c. &c.
and also, retaining the residuals v, v f , v", &c. in the formation of the
normal equations,
Therefore, since
Xf + [an] = [av],
and since the first member of this equation must be identical with
the first member of (90), we have
388 THEORETICAL ASTRONOMY.
which gives, by expanding the several sums,
aq + bq' + cq" + dq m +.... = ,
a'q -f Vq' + e'q" + d'q'" + .... = ', (91)
a"? + &'Y + *'Y' + <*'Y" 4- = a",
&c. &c.
Multiplying each of these equations by its cc, and adding the pro-
ducts, the result is
[a] q + [06] q f + [c] 3 " + [ad] 9 '" + ....== [aa],
which, by means of the equations (88), reduces to
9 4- (92)
Hence it appears that the eliminating factor q is the reciprocal of the
weight of x, and, since the coefficients of q, q r , q ff , &c. in the equa-
tions (89) are the same as those of x, y, z, &c. in the normal equa-
tions, that if we put [an] = 1, \bn\ = 0, [en] = 0, &c., in the
normal equations, the resulting value of x will be the reciprocal of
the weight of the most probable value of this quantity.
The equation (90) shows that if, in the general elimination, by
whatever method it may have been effected, we write [av], [bv], &c.
instead of zero in the second members of the normal equations re-
spectively, the coefficient of [at?] is the reciprocal of the weight of x.
It is obvious that it will not be necessary to know the numerical
values of [av\ 9 [bv], &c., since only the coefficient q is required. The
most probable value of x is found from (90) by the condition of a
minimum of the squares of the residuals, namely, that
[av] = 0, [bv] = 0, [co] = 0, &c.
The process here indicated for the determination of the weight of
the final value of x is general, and applies to the case of any other
unknown quantity provided that the necessary changes are made in
the notation. Thus, the reciprocal of the weight of y is determined
by writing, in the normal equations, 1 in place of [bri], and putting
[an], [en], &c. equal to zero, and completing the elimination. It
is also the coefficient of [bv] in the value of y when the elimination
is effected with the symbols [av], [bv], &c. retained in the second
members of the normal equations.
139. It may be easily shown that when the elimination is effected
by the method of successive substitution, as already explained, the
METHOD OF LEAST SQUARES. 3] = I>-1] = l>-2] = |>.3] = |>.4] = [/n.5] = 1,
and hence, according to (72), for the reciprocal of the weight of t,
which gives
(93)
The weight of t is therefore equal to its coefficient in the final equa-
tion which results from the elimination of the other unknown quan-
tities by successive substitution. Hence, by repeating the elimination,
successively changing the order of the quantities, so that each of the
unknown quantities may have the last place, the weights will be
determined independently, and the agreement of the several sets of
values for the unknown quantities will be a proof of the accuracy of
the calculation. It is not necessary, however, to make so many
repetitions of the elimination, since, in each case, the weights of two
of the unknown quantities will be given by means of the auxiliaries
used in the elimination. Thus, the reciprocal of the weight of w is
obtained by putting [eri\ = 1, and the other absolute terms of the
normal equations equal to zero, and finding the corresponding value
of w. This operation gives
Hence the equation (73) becomes
t = =-
and substituting this value of t in the last of equations (70), we get
f.41
'
or
<*.4], (94)
390 THEOEETICAL ASTKONOMY.
which gives the weight of w in terms of the auxiliary quantities
required in the determination of its most probable value.
If the order of elimination is now completely reversed, so that x
is made the last in the elimination, the weights of x and y will be
determined by the equations
p.==[oa.5],
^=M [M . 4] .
* [aa.4] L
A third elimination, in which z and u are the unknown quantities
first determined, will give the weights of these determinations. It
appears, therefore, that when only four unknown quantities are to be
found, a single repetition of the elimination, the order of the quan-
tities being completely reversed, will furnish at once the weights of
the several results, and check the accuracy of the calculation. When
there are only two unknown quantities, the elimination gives directly
the values of these quantities and also of their weights.
140. In the case of three or more unknown quantities, the weights
of all the results may be determined without repeating the elimina-
tion when certain additional auxiliary quantities have been found.
The weights of the two which are first determined are given in terms
of the auxiliaries required in the elimination, that of the quantity
which is next found will require the value of an additional auxiliary
quantity, the succeeding one will require two additional auxiliaries,
and so on. The equations (74) show that when the substitution is
effected analytically the final value of x will have the denominator
D = [ad] [66.1] [cc.2] [eW.3] [ee.4] [jgf.5],
and this denominator, being the determinant formed from all the
coefficients in the normal equations, must evidently have the same
value whatever may be the order in which the unknown quantities
are eliminated. Let us now suppose that each of the unknown
quantities is, in succession, made the last in the elimination, and let
the auxiliaries in each elimination be distinguished from those when
t is last eliminated by annexing the letter which is the coeffi 3ient rf
the quantity first determined ; then we shall have
D = [ad] [66.1] [cc.2] [dd.3] \_eeA~] [ff.5]
= M, [W.H [<*.2]. [
by means of which the weights of the six unknown quantities may
be determined. The process here indicated may be readily extended
to the case of a greater number of unknown quantities. The equa-
tion for p w is identical with (94), the expression for p u introduces the
new auxiliary quantity [,^.4]^, and that for p s introduces two new
auxiliaries.
The expressions for the new auxiliaries \JfA~\ d ^ [jf/".4] c , [ee.3] c , &c.
are easily formed by observing that all the auxiliaries as far as those
which are designated by the numeral 4 are not affected by putting e
or /last, that, as far as those which contain the numeral 3, it makes
no difference whether d, e, or / is placed last, that those distinguished
by the numerals 1 and 2 are not affected by making c, c/, e, or /the
last, and that those designated by the numeral 1 are unchanged
unless a is made the last. Thus, we obtain
= [ ^-p]^ 3] ' (97)
J92 THEORETICAL ASTRONOMY.
and, also,
Egl
= [ee.2] -
In like manner we may derive the expressions for the new auxiliaries
introduced into the equations for p y and p x . It will be expedient,
however, in the actual application of the formulae, to eliminate first
in the order a?, y, z, u, w, t, and the weights of the results for u, w,
and t will be obtained by means of the first three of equations (96),
the single additional auxiliary required being found by means of
(97). Then the elimination should be performed in the order t, w, u,
Zj y, x, and we shall have
(99)
by means of which the weights of #, y, and 2 will be determined.
The agreement of the two sets of values of the unknown quantities
will prove the accuracy of the numerical calculation in the process
of elimination.
141. The weights of the most probable values of the unknown
quantities may also be computed separately when certain auxiliary
factors have been found, and these factors are those which are intro-
duced when the equations (74) are solved by the method of inde-
terminate multipliers instead of by successive substitution. Thus,
in order to find x, let the first of these equations be multiplied by 1,
the second by A', the third by A n ', the fourth by A'", and so on,
and let the sum of all these products be taken ; then the equations
of condition for the determination of the several eliminating factors
will be
(ioo)
_ , A> , - An , - , jiv
- [aa] + t bb.l \ A f [cc.2] * *" ^ 1 '
_ Ca/1 , [V-l] ., , [#2] ,,
U "" [aa] + IWj + [cc.2] ^ ^ [dd.3] [ee.4]
METHOD OF LEAST SQUARES. 393
To determine y from the last five of equations (74), let the eliminating
factors be denoted by B ', "', B* 9 and . v , and we shall have
[6e.l] R ,,
"
_ . [eeJ.2]
- + ^ ? >
[C6.2] , [fo.3] , ,
f * '
, , [6/4] Iv
fj6 fjE
In a similar manner, we obtain the following equations for the de-
termination of the eliminating factors necessary for finding the values
of the remaining unknown quantities :
[ce.2] [rfe.3] ~,,,
"
[6/4] 1T
"
(102)
[.4J
The expressions for the values of the unknown quantities will there-
fore become
_
-
[an] [6ro.l] ,, [ot.2] , [dro.3] . w [en.4] .,
" 1 " ""^
[66.1] ' [cc.2]
r 7 en r AT rf\~\ (1^3)
\_dn.6} \enA\ ^. v L/^AI ^ r
LW1
394 THEOKETICAL ASTRONOMY.
The tiit of these equations will give the reciprocal of the weight of
x, when we put [an] = 1, and the other absolute terms of the
normal equations equal to zero; the second will give the reciprocal
of the weight of y by putting [bn] = - 1, and the other absolute
terms of the normal equations equal to zero ; and, continuing the
process, finally the last equation will give the reciprocal of the weight
of t when we put fn = 1, and [an], [bn], [en], &c. equal to zero.
It remains, therefore, to determine the particular values of [6n.l],
[en. 2], &c., and the expressions for the weights will be complete.
If we multiply the first of equations (100) by [an], it becomes
\bn.l-] = [an-] A' + [bn]. 104)
Multiplying the second of equations (100) by [an], and the first of
(101) by [bn], adding the products, and introducing the value of
[6n.l] just found, we get
[en-] - [cn.l] + j^l [bn.l] + [an] A" + [bn] B" = 0,
which reduces to
[an~] A" + [bn-] B" -f [en] = [cn.2]. (105)
Multiplying the third of equations (100) by [an], the second of (101)
by [bn], and the first of (102) by [en], adding the products, and re-
ducing by means of (104) and (105), we obtain
= Idn] - [] = [en.4],
[an] A* + [bn] B v + [en] <7 V + \dn\ D v -f [en] E* + [>] = [/7i.5]. ^
The equations (104), (105), (106), and (107), enable us to find the
particular values of [6n.l], [cn.2], &c. required in the expressions for
the reciprocals of the weights. Thus, for the weight of x, we have
Ian] = 1, [bn] = [m] = \dn\ = [en] = [fn] = ;
METHOD OF LEAST SQUARES. 395
and these equations give
[^.1] = _ A' t [en.2] = A", [dn.3] = - A'",
[6W.4] = A iv , L/w.5] = A\
For the case of the weight of y, we have
[bn] = 1, [cm] = [CTI] = [drc] = [en] = [/w] = 0,
and the same equations give
p ?l .l] = 1, [cn.2] = J3", [dn.3] = J3'",
[en.4] = B iv , L/k5] = v .
We have, also, for the weight of z,
|_cn.2] = 1, [cfri.3]= C"", [en.4] = C iv , |>.5] = C\
for the weight of w,
[dn.3] = 1, [m.4] = D ly , [./h.5] = 7) T ;
for the weight of w,
[i.4] = -l, [/n.5] = -^;
and finally, for the weight of ,
[/.5]= -1.
Introducing these particular values into the equations (103), the cor-
responding values of the unknown quantities are the reciprocals of
the weights of their most probable values, respectively; and hence
we derive
1 AA ' A " A ' A '" A> " ^ 1Vj[iV AVA *
]7 ~ [oa] ^ [56.1] "*" [cc.2] n " [dd.3] T [ee.4] T [jfif.5] '
1 _J __ jy^ y B'^^^ B*B*
y ~ [SO] + [^2j "" [dd.3] + [e6.4] "
1 1 C'" C'" C IV C IV C V C V
[SSI H ' LPT
1 _ 1 E V E V
~Wl
1 1
The equations (103) and (108) will serve to determine separately
the value of each unknown quantity and also that of its weight, the
396 THEORETICAL ASTRONOMY.
auxiliary factors A' ', A", B", &c. having been found from the equa-
tions (100), (101), and (102). If we reverse the operation and re-
compose the equations (74) by means of the expressions for the un-
known quantities given by (103), the conditions which immediately
follow furnish another series of equations for the determination of the
auxiliary factors. The ea nations thus derived will give first the values
of A', B", v ; and so
on. They are equally as convenient as those already given, provided
that the values of all the unknown quantities are required as well as
their respective weights.
142. The formulae already given for the relations between the data
of the problem and the weights of the most probable values of the
unknown quantities, are those which are of the greatest practical
value. It will be apparent from what has been derived that there
must be a variety of methods which may be applied, but that all of
these methods involve essentially the same numerical operations.
The peculiar symmetry of the normal equations affords also a variety
of expressions applicable to the different phases under which the
problem presents itself.
According to the general theory of elimination, the expression for
any unknown quantity, as determined from the normal equations,
may be put in the form
*= -|[]-f [i]-^M-&c, (109)
in which D is the determinant formed from all the coefficients of tho
unknown quantities in the normal equations, and in which A, A', A",
&c. are the partial determinants required in the elimination. Thus,
A is the determinant formed from the coefficients of all the unknown
quantities except x, in all the equations except the first; A" is the
determinant formed from the coefficients of y, z, &c. in all the equa-
tions except the second ; and the values of A n r , A f ", &c. are formed
in a similar manner. Now, since the value of x which results when
we put [an] 1, and the other absolute terms of the normal
equations equal to zero, is the reciprocal of the weight of the most
probable value of this unknown quantity as given by (109), we have
In like manner, the expression for the most probable value of y will be
METHOD OF LEAST SQUARES, 397
y= -M- [^]-M-&c., (ill)
B, B', B", &c. being the partial determinants formed when the co-
efficients of y are omitted; and for its weight we have
The formulae for the most probable value of z and for its weight are
entirely analogous to those for x and y y so that the process here indi-
cated may be extended to the case of any number of unknown quan-
tities. It appears, therefore, that the weight of the most probable
value of any unknown quantity is found by dividing the complete
determinant of all the coefficients by the partial determinant formed
when we omit the normal equation corresponding particularly to this
unknown quantity, and when we omit also the coefficients of this
quantity in the remaining normal equations.
The peculiar arrangement of the coefficients in the normal equa-
tions abbreviates somewhat the expressions for the several determi-
nants. Thus, in the case of three unknown quantities, we have
A = [66] [cc] [6c] 2 , B f = [ad] [cc] [ac] 2 , C" = [aa] [66] [a6] 2 ,
D = [aa] [66] [cc] + 2[a6] [be] [ac] [ad] [6c] 2 [66] [ac] 2 [cc] [a6] 2 ,
which are all the quantities required for finding simply the weights
of the most probable values of a?, y, and z. The expression for the
weight of z is
D
When there are but two unknown quantities, we have
A = [66], B' = [ad], D = [aa] [66] [a6] 2 ,
and hence
_ [aa] [66] [a6] 2 _ [oa] [66] [a6] 2
Px ~ [66] P *~ [aa]
When the number of unknown quantities is increased, the expressions
for the determinants necessarily become much more complicated, and
hence the convenience of other auxiliary quantities is manifest.
143. The case has been already alluded to in which the determina-
tion of the values of the unknown quantities is rendered uncertain
by the similarity of the signs and coefficients in the normal equations,
398 THEORETICAL ASTRONOMY.
and in which the problem becomes nearly indeterminate. Sometimes
it will be possible to overcome the difficulty thus encountered by a
suitable change of the elements to be determined ; but, generally, for
a complete and satisfactory solution, additional data will be required.
It often happens, however, that several of the unknown quantities
may be accurately determined from the given equations when the
values of the others are known, but that the certainty of the deter-
mination of the same quantities is very greatly impaired when all
the unknown quantities are derived simultaneously from the same
equations. Let us suppose that one of the unknown quantities is,
from the very nature of the problem, not susceptible of an accurate
determination from the data employed. The equations will then
present themselves in a form approaching that in which the number
of independent relations is one less than the number of unknown
quantities, so that it will be necessary to determine the other unknown
quantities in terms of that whose value is necessarily uncertain. In
this case the elimination should be so arranged that the quantity
which is regarded as uncertain is that whose value would be first
determined. Then, if its coefficient in the final equation, corre-
sponding to (72), is very small, a circumstance which indicates at
once the existence of the uncertainty when it is not otherwise sus-
pected, the process of elimination should not be completed, and the
auxiliary quantities should be determined only as far as those re-
quired in the formation of the equation which corresponds to the first
of (70). Thus, let t be the uncertain quantity, and we have
[6/4] len.4]
W = - - - t - p - -pr-,
[ee.4] [ee.4]
which must be substituted for w in the first of equations (68). AVe
thus obtain w, u, z, y, and x as functions of t. If the solution is
effected by means of the equations (103), let x w y w Z Q , &c. denote the
values of these unknown quantities when we put = 0; and then
we shall have
_ [an] _ [bnA.~\ [c?i.2] , [cfoi.3] ., [en A] , lv
[cm] [55.1] [ce.2] [cW.3] [ee.4]
[fokl] [e*.2] [d-3] , [en.4] v riij ,
2/0 ~ ~ \m\~teS\* ~\ddX\" [66.4]*'
l>*.2] [.4]
[ (115)
in which (e x ), (e y ), &c. denote the mean errors of x w y w &c. These
formulae show, also, that when one of the variables is neglected, the
equations assign too great a degree of precision to the results thus
obtained.
When there are two or more unknown quantities which cannot be
determined from the data with sufficient certainty, the problem must
be treated in a manner entirely analogous to that here indicated; but,
since cases of this kind will rarely, if ever, occur, it is not necessary
to pursue the subject further.
144. The weights which are obtained for the most probable values
of the unknown quantities enable us to find the mean and probable
errors of these values. Let s denote the mean error of an observa-
tion whose weight is unity; then the mean error of x will be
(116)
and, in like manner, the expressions "for the mean errors of y, z, u,
&c. will be
e =" '. = -?- . = -=,&c. (117)
1/P, Vp. Vp,
It remains, therefore, to determine the value of e by means of the
final residuals obtained by comparing the observed values of the
function with those given by the most probable values of the va-
400 THEORETICAL ASTRONOMY.
riables. If these residuals were the actual fortuitous errors of obser-
vation, the mean error of an observation would be
M
m being the number of equations of condition. This value is evi-
dently an approximation to the correct result; but since by supposing
the residuals v, v f , v", &c. to be the actual errors of the several ob-
served values of the function, we assign too high a degree of pre-
cision to the several results, the true value of e must necessarily be
greater than that given by this equation. Let the true values of the
unknown quantities be x -f- A#, y -f- Ay, z -f- AZ, &c., the substitution
of which in the several equations of condition would give the
residuals J, J r , J", &c. ; then we shall have
&A?/ -f- cAz -f- d&u ---- -f v = A,
' '
&c. &c.
If we multiply each of these equations by its J, and take the sura
of all the products, we get
[o J] *x -f [6 J] Ay -f [cJ] A^ + [dJ] AW + .... -f [vJ] = [J J].
But if we multiply each of the same equations by its v, take the sum
of the products, and reduce by means of (48) and (50), we obtain
_ = [>*];
and hence we derive
[J J] = [VV] 4 [O J] AX -f [6 J] Ay + [C J] A3 + [d J] AW + .... (119)
If we form the normal equations from (118), it will be observed that
they are of the same form as the normal equations formed from the
original equations of condition, provided that we write A in place
of n; and hence, according to (85), we have
A* = aJ + a'J' + a"J'' + .....
We have, also,
[o J] = aA + a' A' 4 a" A" 4 ,
and the product of these equations gives
[aJ] AX aa J 2 4- aVJ' 2 4 a"a" J" 2 + /. . .
4-aa'JJ'-f aa"JJ"4-....
The mean value of the terms containing JJ', Jd",.&c. is zero, and
COMBINATION OF OBSERVATIONS. 40J
for the mean values of J 2 , J' 2 , J" 2 , &c. we must, in each case, write
e 2 . Hence the mean value of the product [a J] A# will be
and this, by means of the first of equations (88), is further reduced to
[a J] &x 2 .
In a similar manner, we obtain the value e 2 for the mean value of
each of the products [ftzfJA?/, [cJJA2, &c. Now, the terms added to
[vv] in the second member of the equation (119) are necessarily very
small, and, although their exact value cannot be determined, we may
without sensible error adopt the mean values of the several terms as
here determined, so that the equation becomes
[ J J] = [tw] + pe*, (120)
u being the number of unknown quantities. Therefore, since
[ J J] = me 2 , we shall have
m fj. * m /j.
by means of which the mean error of an observation whose weight
is unity may be determined. When p = 1, this equation becomes
identical with (30).
For the determination of the probable errors of the final values of
the unknown quantities, if r denotes the probable error of an obser-
vation of the weight unity, we have the following equations:
r = 0.67449
r
= i/?
145. The formulae which result from the theory of errors according
to which the method of least squares is derived, enable us to combine
the data furnished by observation so as to overcome, in the greatest
degree possible, the effect of those accidental errors which no refine-
ment of theory can successfully eliminate. The problem of the cor-
rection of the approximate elements of the orbit of a heavenly body
by means of a series of observed places, requires the application of
nearly all the distinct results which have been derived. The first
approximate elements of the orbit of the body will be determined
om three or four observed places according to the methods which
26
102 THEORETICAL ASTRONOMY.
have been already explained. In the case of a planet, if the inclina-
tion is not very small, the method of three geocentric places may be
employed, but it will, in general, afford greater accuracy and require
but little additional labor to base the first determination on four
observed places, according to the process already illustrated. In the
case of a comet, the first assumption made is that the orbit is a
parabola, and the elements derived in accordance with this hypothesis
may be successively corrected, until it is apparent whether it is ne-
cessary to make any further assumption in regard to the value of the
eccentricity. In all cases, the approximate elements derived from a
few places should be further corrected by means of more extended
data before any attempt is made to obtain a more complete determi-
nation of the elements. The various methods by which this pre-
liminary correction may be effected have been already sufficiently de-
veloped.
The fundamental places adopted as the basis of the correction may
be single observed places separated by considerable intervals of time;
but it will be preferable to use places which may be regarded as the
average of a number of observations made on the same day or during
a few days before and after the date of the average or normal place.
The ephemeris computed from the approximate elements known may
be assumed to represent the actual path so closely that, for an interval
of a few days, the difference between computation and observation
may be regarded as being constant, or at least as varying proportion-
ally to the time. Let n, n', n", &c. be the differences between com-
putation and observation, in the case of either spherical co-ordinate,
for the dates t, t f , t n ', &c., respectively; then, if the interval between
the extreme observations to be combined in the formation of the
normal place is not too great, and if we regard the observations as
equally precise, the normal difference n Q between computation and
observation will be found by taking the arithmetical mean of the
several values of n, and this being applied with the proper sign to
the computed spherical co-ordinate for the date , which is the mean
of t, I', t", &c.j will give the corresponding normal place. But when
different weights p, p', p ff , &c. are assigned to the observations, the
value of n must be found from
_ np -f n'p' + n"p" + . . . .
>
and the weight of this value will be equal to the sum
P+p'
COMBINATION OF OBSERVATIONS. 403
The date of the normal place will be determined by
_pt+ P -e + prr + ....
* P+P'+P" + .... '
]f the error of the ephemeris can be considered as nearly constant,
it is not necessary to determine with great precision, since any date
not differing much from the average of all may be adopted with suf-
ficient accuracy. It should be observed further that, in order to
obtain the greatest accuracy practicable, the spherical co-ordinates of
the body for the date t should be computed directly from the elements,
so that the resulting normal place may be as free as possible from the
effect of neglected differences in the interpolation of the ephemeris.
When the differences between the computed and the observed
places to be combined for the formation of a normal place cannot be
considered as varying proportionally to the time, we may derive the
error of the ephemeris from an equation of the form of (53) 6 , namely,
the coefficients J., jB, and C being found from equations of condition
formed by means of the several known values of A# in the case of
each of the spherical co-ordinates.
146. In this way we obtain normal places at convenient intervals
throughout the entire period during which the body was observed.
From three or more of these normal places, a new system of elements
should be computed by means of some one of the methods which
have already been given; and these fundamental places being judi-
ciously selected, the resulting elements will furnish a pretty close
approximation to the truth, so that the residuals which are found by
comparing them with all the directly observed places may be regarded
as indicating very nearly the actual errors of those places. We may
then proceed to investigate the character of the observations more
fully. But since the observations will have been made at many dif-
ferent places, by different observers, with instruments of different
sizes, and under a variety of dissimilar attendant circumstances, it
may be easily understood that the investigation will involve much
that is vague and uncertain. In the theory of errors which has been
developed in this chapter, it has been assumed that all constant
errors have been duly eliminated, and that the only errors which
remain are those accidental errors which must ever continue in a
greater or less degree undetermined. The greater the number and
404 THEORETICAL ASTRONOMY.
perfection of the observations employed, the more nearly will these
errors be determined, and the more nearly will the law of their dis-
tribution conform to that which has been assumed as the basis of
the method of least squares.
When all known errors have been eliminated, there may yet remain
constant errors, and also other errors whose law of distribution is
peculiar, such as may arise from the idiosyncrasies of the different
observers, from the systematic errors of the adopted star-places' in
the case of differential observations, and from a variety of other
sources; and since the observations themselves furnish the only means
of arriving at a knowledge of these errors, it becomes important to
discuss them in such a manner that all errors which may be regarded,
in a sense more or less extended, as regular may be eliminated.
When this has been accomplished, the residuals which still remain
will enable us to form an estimate of the degree of accuracy which
may be attributed to the different series of observations, in order that
they may not only be combined in the most advantageous manner,
but that also no refinements of calculation may be introduced which
are not warranted by the quality of the material to be employed.
The necessity of a preliminary calculation in which a high degree
of accuracy is already obtained, is indicated by the fact that, however
conscientious the observer may be, his judgment is unconsciously
warped by an inherent desire to produce results harmonizing well
among themselves, so that a limited series of places may agree to
such an extent that the probable error of an observation as derived
from the relative discordances would assign a weight vastly in excess
of its true value. The combination, however, of a large number of
independent data, by exhibiting at least an approximation to the
absolute errors of the observations, will indicate nearly what the
measure of precision should be. As soon, therefore, as provisional
elements which nearly represent the entire series of observations have
been found, an attempt should be made to eliminate all errors which
may be accurately or approximately determined. The places of the
comparison-stars used in the observations should be determined with
care from the data available, and should be reduced, by means of the
proper systematic corrections, to some standard system. The reduc-
tion of the mean places of the stars to apparent places should also be
made by means of uniform constants of reduction. The observations
will thus be uniformly reduced. Then the perturbations arising from
the action of the planets should be computed by means of formula
which will be investigated in the next chapter, and the observed
COMBINATION OF OBSERVATIONS. 405
places should be freed from these perturbations so as to give the
places for a system of osculating elements for a given date.
147. The next step in the process will be to compare the pro-
visional elements with the entire series of observed places thus cor-
rected; and in the calculation of the ephemeris it will be advan-
tageous to correct the places of the sun given by the tables whenever
observations are available for that purpose. Then, selecting one or
more epochs as the origin, if we compute the coefficients A, B, C in
the equation
A0 = A + Br + Cr\ (125)
in the case of each of the spherical co-ordinates, by means of equa-
tions of condition formed from all the observations, the standard
ephemeris may be corrected so that it may be regarded as representing
the actual path of the body during the period included by the obser-
vations. When the number of observations is sonsiderable, it will be
more convenient to divide the observations into groups, and use the
differences between computation and observation for provisional
normal places in the formation of the equations of condition for the
determination of A, B, and C. It thus appears that the corrected
ephemeris which is so essential to a determination of the constant
errors peculiar to each series of observations, is obtained without first
having determined the most probable system of elements. The cor-
rections computed by means of the equation (125) being applied to
the several residuals of each series, we obtain what may be regarded
as the actual errors of these observations. The arithmetical or pro-
bable mean of the corrected residuals for the series of observations
made by each observer may be regarded as the average error of obser-
vation for that series. The mean of the average errors of the several
series may be regarded as the actual constant error pertaining to all
the observations, and the comparison of this final mean with the
means found for the different series, respectively, furnishes the pro-
bable value of the constant errors due to the peculiarities of the
observers; and the constant correction thus found for each observer
should be applied to the corresponding residuals already obtained.
In this investigation, if the number of comparisons or the number
of wires taken is known, relative weights proportional to the number
of comparisons may be adopted for the combination of the residuals
for each series. In this manner, observations which, on account of
the peculiarities of the observers, are in a certain sense heterogeneous,
may be rendered homogeneous, being reduced to a standard which
406 THEORETICAL ASTRONOMY.
approaches the absolute in proportion as the number and perfection
of the distinct series combined are increased. Whatever constant
error remains will be very small, and, besides, will affect all places
alike.
The residuals which now remain must be regarded as consisting
of the actual errors of observation and of the error of the adopted
place of the comparison-star. Hence they will not give the probable
error of observation, and will not serve directly for assigning the
measures of precision of the series of observations by each observer.
Let us, therefore, denote by e, the mean error of the place of the
comparison-star, by e, the mean error of a single comparison; then
will 4=- be the mean error of m comparisons, and the mean error of
V m
the resulting place of the body will, according to equation (35), be
given by
. = + '.'. (126)
The value of e , in the case of each series, will be found by means of
the residuals finally corrected for the constant errors, and the value
of e s is supposed to be determined in the formation of the catalogue
of star-places adopted. Hence the actual mean error of an observa-
tion consisting of a single comparison will be
e, = l/m(> 2 -e, 2 ). (127)
The value of e, for each observer having been found in accordance
with this equation, the mean error of an observation consisting of m
comparisons will be
Sf
Vm
The mean error of an observation whose weight is unity being de-
noted by e, the weight of an observation based on m comparisons will
be
, = (128)
The value of e may be arbitrarily assigned, and we may adopt for it
10" or any other number of seconds for which the resulting values
of p will be convenient numbers.
When all the observations are differential observations, and the stars
of comparison are included in the fundamental list, if we do not take
into account the number of comparisons on which each observed
COMBINATION OF OBSERVATIONS. 407
place depends, it will not be necessary to consider e t) and we may
then derive e, directly from the residuals corrected for constant errors.
Further, in the case of meridian observations, the error which corre-
sponds to e s will be extremely small, and hence it is only when these
are combined with equatorial observations, or when equatorial obser-
vations based on different numbers of comparisons are combined, that
the separation of the errors into the two component parts becomes
necessary for a proper determination of the relative weights.
According to the complete method here indicated, after having
eliminated as far as possible all constant errors, including the correc-
tions assigned by equation (125) to be applied to the provisional
ephemeris, we find the value of e, given by the equation
ne,* = [row] [m] e, 2 , (129)
in which n denotes the number of observations; m, m', m", &c. the
number of comparisons for the respective observations; and v, v', v ff ,
&c. the corresponding residuals. Then, by means of equation (128),
assuming a convenient number for e, we compute the weight of each
observation. Thus, for example, let the residuals and corresponding
values of m be as follows :
Ad m Ad m
+ 2".0 5, 1".0 7,
1 .8 5, + 1 .5 5,
.4 10, +4 .1 8,
5 .5 5, .0 5.
Let the mean error of the place of a comparison-star be
then we have n ~ 8, and, according to (129),
8s, 2 =341.78 200.0,
which gives
*,= :4".2.
Let us now adopt as the unit of weight that for which the mean erroi is
then we obtain by means of equation (128), for the weights of the
observations,
2.5, 2.5, 5.1, 2.5, 3.6, 2.5, 4.1, 2.5,
respectively.
408 THEORETICAL ASTRONOMY.
In this manner the weights of the observations in the series made
by each observer must be determined, using throughout the same
value of e. Then the differences between the places computed from
the provisional elements to be corrected and the observed places cor-
rected for the constant error of the observer, must be combined ac-
cording to the equations (123) and (125), the adopted values of p, p',
p"j &c. being those found from (128). Thus will be obtained the
final residuals for the formation of the equations of condition from
which to derive the most probable value of the corrections to be
applied to the elements. The relative weights of these normals will
be indicated by the sums formed by adding together the weights of
the observations combined in the formation of each normal, and the
unit of weight will depend on the adopted value of e. If it be de-
sired to adopt a different unit of weight in the case of the solution
of the equations of condition, such, for example, that the weight of
an equation of average precision shall be unity, we may simply divide
the weights of the normals by any number p which will satisfy the
condition imposed. The mean error of an observation whose weight
is unity will then be given by
the value of e being that used in the determination of the weights p,
p', &o.
148. The observations of comets are liable to be affected by other
errors in addition to those which are common to these and to planet-
ary observations. Different observers will fix upon different points
as the proper point to be observed, and all of these may differ from
the actual position of the centre of gravity of the comet; and fur-
ther, on account of changes in the physical appearance of the comet,
the same observer may on different nights select different points.
These circumstances concur to vitiate the normal places, inasmuch as
the resulting errors, although in a certain sense fortuitous, are yet
such that the law of their distribution is evidently different from
that which is adopted as the basis of the method of least squares.
The impossibility of assigning the actual limits and the law of dis-
tribution of many errors of this class, renders it necessary to adopt
empirical methods, the success of which will depend on the discrimi-
nation of the computer.
If e denotes the mean jerror of an observation based on m com-
COMBINATION OF OBSERVATIONS. 409
parisons, and e c the mean error to be feared on account of the pecu-
liarities of the physical appearance of the comet,
will express the mean error of the residuals; and if n of these
residuals are combined in the formation of a normal place, the mean
error of the normal will be given by
e n ' = M -{-./. (130)
The value of e c 2 may be determined approximately from the data
furnished by the observations. Thus, if the mean error of a single
comparison, for the different observers, has been determined by means
of the differences between single comparisons and the arithmetical
mean of a considerable number of comparisons, and if the mean error
of the place of a comparison-star has also been determined, the
equation (126) will give the corresponding value of 2 ; then the
actual differences between computation and observation obtained by
eliminating the error of the ephemeris and such constant errors as
may be determined, will furnish an approximate value of e c by means
of the formula
in which n denotes the number of observations combined.
Sometimes, also, in the case of comets, in order to detect the opera~
tion of any abnormal force or circumstance producing different effects
in different parts of the orbit, it may be expedient to divide the
observations into two distinct groups, the first including the observa-
tions made before the time of perihelion passage, and the other
including those subsequent to that epoch.
149. The circumstances of the problem will often suggest appro-
priate modifications of the complete process of determining the rela-
tive weights of the observations to be combined, or indeed a relaxa-
tion from the requirements of the more rigorous method. Thus, if
on account of the number or quality of the data it is not considered
necessary to compute the relative weights with the greatest precision
attainable, it will suffice, when the discussion of the observations has
been carried to an extent sufficient to make an approximate estimate
of the relative weights, to assume, without considering the number
of comparisons, a weight 1 for the observations at one observatory, a
410 THEORETICAL ASTRONOMY.
weight | for another class of observations, J for a third class, and so
on. It should be observed, also, that when there are but few obser-
vations to be combined, the application of the formulae for the mean
or probable errors may be in a degree fallacious, the resulting values
of these errors being little more than rude approximations ; still the
mean or probable errors as thus determined furnish the most reliable
means of estimating the relative weights of the observations made
by different observers, since otherwise the scale of weights would
depend on the arbitrary discretion of the computer. Further, in a
complete investigation, even when the very greatest care has been
taken in the theoretical discussion, on account of independent known
circumstances connected with some particular observation, it may be
expedient to change arbitrarily the weight assigned by theory to
certain of the normal places. It may also be advisable to reject
entirely those observations whose weight is less than a certain limit
which may be regarded as the standard of excellence below which
the observations should be rejected; and it will be proper to reject
observations which do not afford the data requisite for a homogeneous
combination with the others according to the principles already
explained. But in all cases the rejection of apparently doubtful
observations should not be carried to any considerable extent unless
a very large number of good observations are available. The mere
apparent discrepancy between any residual and the others of a series,
is not in itself sufficient to warrant its rejection unless facts are
known which would independently assign to it a low degree of pre-
cision.
A doubtful observation will have the greatest influence in vitiating
the resulting normal place when but a small number of observed
places are combined ; and hence, since we cannot assume that the law
of the distribution of errors, according to which the method of least
squares is derived, will be complied with in the case of only a few
observations, it will not in general be safe to reject an observation pro-
vided that it surpasses a limit which is fixed by the adopted theory
of errors. If the number of observations is so large that the dis-
tribution of the errors may be assumed to conform to the theory
adopted, it will be possible to assign a limit such that a residual
which surpasses it may be rejected. Thus, in a series of m observa-
tions, according to the expression (19), the number of errors greater
than nr will be
COMBINATION OF OBSERVATIONS.
4J1
and when n has a value such that the value of this expression is less
than 0.5, the error nr will have a greater probability against it than
for it, and hence it may be rejected. The expression for finding the
limiting value of n therefore becomes
2m
(131)
By means of this equation we derive for given values of m the cor-
responding values of nhr = 0.47694n, and hence the values of n.
For convenient application, it will be preferable to use e instead of r,
and if we put n f = 0.67449w, the limiting error will be n'e, and the
values of n' corresponding to given values of m will be as exhibited
in the following table.
TABLE.
m
'
m
n'
m
n'
m
' |
i
6
1.732
20
2.241
55
2.608
90
2.773
8
1.863
25
2.326
60
2.638
95
2.791
10
1.960
30
2.394
65
2.665
100
2.807
12
2.037
35
2.450
70
2.690
200
3.020
14
2.100
40
2.498
75
2.713
300
3.143
16
2.154
45
2.539
80
2.734
400
3.224
18
2.200
50
2.576
85
2.754
500
3.289
According to this method, we first find the mean error of an obser-
vation by means of all the residuals. Then, with the value of m as
the argument, we take from the table the corresponding value of n',
and if one of the residuals exceeds the value n'e it must be rejected,
Again, finding a new value of e from the remaining m 1 residuals,
and repeating the operation, it will be seen whether another observa-
tion should be rejected ; and the process may be continued until a
limit is reached which does not require the further rejection of ob-
servations. Thus, for example, in the case of 50 observations in
which the residuals 11". 5 and +7". 8 occur, let the sum of the
squares of the residuals be
[w] = 320.4.
Then, according to equation (30), we shall have
* = 2".56.
412 THEORETICAL ASTRONOMY.
Corresponding to the value m = 50, the table gives n f -- 2.576, and
the limiting value of the error becomes
and hence the residuals ll /7 .5 and + 7".8 are rejected. Recom-
puting the mean error of an observation, we have
320-4 -193.0S = 1".65.
In the formation of a normal place, when the mean error of an
observation has been inferred from only a small number of observa-
tions, according to what has been stated, it will not be safe to rely
upon the equation (131) for the necessity of the rejection of a doubt-
ful observation. But if any abnormal influence is suspected, or if
any antecedent discussion of observations by the same observer, made
under similar circumstances, seems to indicate that an error of a given
magnitude is highly improbable, the application of this formula will
serve to confirm or remove the doubt already created. Much will
therefore depend on the discrimination of the computer, and on his
knowledge of the various sources of error which may conspire con-
tinuously or discontinuously in the production of large apparent
errors. It is the business of the observer to indicate the circum-
stances peculiar to the phenomenon observed, the instruments em-
ployed, and the methods of observation; and the discussion of the
data thus furnished by different observers, as far as possible in ac-
cordance with the strict requirements of the adopted theory of errors,
will furnish results which must be regarded as the best which can be
derived from the evidence contributed by all the observations.
150. When the final normal places have been derived, the differ-
ences between these and the corresponding places computed from the
provisional elements to be corrected, taken in the sense computation
minus observation, give the values of n, n f , n" } &c. which are the
absolute terms of the equations of condition. By means of these
elements we compute also the values of the differential coefficients of
each of the spherical co-ordinates with respect to each of the elements
to be corrected. These differential coefficients give the values of the
coefficients a, 6, c, a', &', &c. in the equations of condition. The
mode of calculating these coefficients, for different systems of co-or-
dinates, and the mode of forming the equations of condition, have
been fully developed in the second chapter. It is of great import-
CORRECTION OF THE ELEMENTS. 413
ance that the numerical values of these coefficients should be care-
fully checked by direct calculation, assigning variations to the ele-
ments, or by means of differences when this test can be successfully
applied. In assigning increments to the elements in order to check
the formation of the equations, they should not be so large that the
neglected terms of the second order become sensible, nor so small that
they do not afford the required certainty by means of the agreement
of the corresponding variations of the spherical co-ordinates as
obtained by substitution and by direct calculation.
As soon as the equations of condition have been thus formed, we
multiply each of them by the square root of its weight as given by
the adopted relative weights of the normal places; and these equa-
tions will thus be reduced to the same weight. In general, tho
numerical values of the coefficients will be such that it will be con-
venient, although not essential, to adopt as the unit of weight that
which is the average of the weights of the normals, so that the
numbers by which most of the equations will be multiplied will not
differ much from unity. The reduction of the equations to a uniform
measure of precision having been effected, it remains to combine them
according to the method of least squares in order to derive the most
probable values of the unknown quantities, together with the relative
weights of these values. It should be observed, however, that the
numerical calculation in the combination and solution of these equa-
tions, and especially the required agreement of some of the checks of
the calculation, will be facilitated by having the numerical values of
the several coefficients not very unequal. If, therefore, the coefficient
a of any unknown quantity x is in each of the equations numerically
much greater or much less than in the case of the other unknown
quantities, we may adopt as the corresponding unknown quantity to
be determined, not x but vz, i/ being any entire or fractional number
such that the new coefficients -, T , &c. shall be made to agree in
magnitude with the other coefficients. The unknown quantity whose
value will then be derived by the solution of the equations will be
vx, and the corresponding weight will be that of vx. To find the
weight of x from that of vx, we have the equation
?. = >*-. (132)
In the same manner, the coefficient of any other unknown quantity
may be changed, and the coefficients of all the unknown quantities
may thus be made to agree in magnitude within moderate limits, th*
414 THEORETICAL ASTRONOMY.
advantage of which, in the numerical solution of the equations, will
be apparent by a consideration of the mode of proving the calcula-
tion of the coefficients in the normal equations. It will be expedient,
also, to take for v some integral power of 10, or, when a fractional
value is required, the corresponding decimal. It may be remarked,
further, that the introduction of v is generally required only when
the coefficient of one of the unknown quantities is very large, as
frequently happens in the case of the variation of the mean daily
motion //.
When the coefficients of some of the unknown quantities are
extremely small in all the equations of condition to be combined, an
approximate solution, and often one which is sufficiently accurate for
the purposes required, may be obtained by first neglecting these
quantities entirely, and afterwards determining them separately. In
general, however, this can only be done when it is certainly known
that the influence of the neglected terms is not of sensible magnitude,
or when at least approximate values of these terms are already given.
When we adopt the approximate plane of the orbit as the funda-
mental plane, the equations for the longitude involve only four ele-
ments, and the coefficients of the variations of these elements in the
equations for the latitudes are always very small. Hence, for an
approximate solution, we may first solve the equations involving four
unknown quantities as furnished by the longitudes, and then, substi-
tuting the resulting values in the equations for the latitudes, they
will contain but two unknown quantities, namely, those which give
the corrections to be applied to & and i.
151. When the number of equations of condition is large, the
computation of the numerical values of the coefficients in the normal
equations will entail considerable labor; and hence it is desirable to
arrange the calculation in a convenient form, applying also the checks
which have been indicated. The most convenient arrangement will
be to write the logarithms of the absolute terms n, n f , n", &c. in a
horizontal line, directly under these the logarithms of the coefficients
a, a', a", &c., then the logarithms of 6, &', 6", &c., and so on. Then
writing, in a corresponding form, the values of logn, logn r , &c. on a
slip of paper, by bringing this successively over each line, the sums
[nri], [an], [6n], &c. will be readily formed. Again, writing on
another slip of paper the logarithms of a, a', a", &c., and placing
this slip successively over the lines containing the coefficients, we
derive the values [aa], [a&], [ac], &c. The multiplication by 6, c, d t
CORRECTION OF THE ELEMENTS. 415
&c. successively is effected in a similar manner; and thus will be
derived [66], [6c], [bcl], &c., and finally \_ff] in the case of six un-
known quantities. In forming these sums, in the cases of sums of
positive and negative quantities, it is convenient as well as conducive
to accuracy to write the positive values in one vertical column and
the negative values in a separate column, and take the difference of
the sums of the numbers in the respective columns. The proof of
the calculation of the coefficients of the normal equations is effected
by introducing s, s', s", &c., the algebraic sums of all the coefficients
in the respective equations of condition, and treating these as the
coefficients of an additional unknown quantity, thus forming directly
the sums [sn], [as], [6s], [cs], &c. Then, according to the equations
(76) and (77), the values thus found should agree with those obtained
by taking the corresponding sums of the coefficients in the normal
equations.
The normal equations being thus derived, the next step in the
process is the determination of the values of the auxiliary quantities
necessary for the formation of the equations (74). An examination
of the equations (54), (55), &c., by means of which these auxiliaries
are determined, will indicate at once a convenient and systematic
arrangement of the numerical calculation. Thus, we first write in a
horizontal line the values of [aa], [a6], [ac], . . . [as], [an], and di-
rectly under them the corresponding logarithms. Next, we write
under these, commencing with [a6], the values of [66], [6c], [bd],
. . [6s], [6n] ; then, adding the logarithm of the factor = - to the
\_aa\
logarithms of [a6], [ac], &c. successively, we write the value of
p=r [a6] under [66], that of p =r [ac] under [6c], and so on. Sub-
[oa] L J [aa] L J
tracting the numbers in this line from those in the line above, the
differences give the values of [66.1], [6c.l], . . . [6s.l], [6n.l], to be
written in the next line, and the logarithms of these we write directly
under them. Then we write in a horizontal line the values of [cc],
[cd], . . [cs], [en], placing [cc] under [6c.l], and, having added the
logarithm of ^ - to the logarithms of [ac], [ad], &c. in succession,
we derive, according to the equations (55) and (58), the values of
[cc.l], [coM], . . [cs.l], [cn.l], which are to be placed under the cor-
responding quantities [cc], [cd], &c. Next, we subtract from these,
respectively, the products t
[6c - 1] r6on [6e - 1] rwn [ic - 1] rfcn [6 ^r6n
f6UJ [6 [663] Lftrf - 1J> ' ' [663] L * S ' 1J ' f 66.11 L '
416 THEORETICAL ASTRONOMY.
and thus derive the values of [cc.2], [cd2], . . [es.2], [cn.2], which
are to be written in the next horizontal line and under them their
logarithms. Then we introduce, in a similar manner, the coefficients
[dd], [de] y . . [dn], writing \_dd~] under [ccZ.2] ; and from each of these
in succession we subtract the products
thus finding the values of [ we g e ^ [ ee '^]j [^/-l]) [ 6S '1]> an( i [0W-.1]; then subtracting
from these the products
j ' ' [66.1]
we obtain the values of [ee.2], [e/.2], [es.2], and [en.2]. Again,
subtracting
we have the values of [ee.3], [^f.3], [es.3], [cn.3]; and finally, sub-
tracting from these the products
j ' ' E3Z3] ' >
we derive the results for [66.4], [e/.4], [es.4], and [en.4]; under which
the corresponding logarithms are to be written.
If there are six unknown quantities to be determined, we must
further write in a horizontal line the values of [jj/*], [/s], and [/n],
CORRECTION OF THE ELEMENTS. 41*
placing [jgf] under [e/.4], and by means of five successive subtrac-
tions entirely analogous to what precedes, and as indicated by the
remaining equations for the auxiliaries, we obtain the values of [j^.5],
[/a.5], and [/n.5].
The values of [fo.l], [cs.l], [cs.2], &c. serve to check the calcula-
tion of the successive auxiliary coefficients. Thus we must have
[56.1] + [6c.l] -f- [fcd.1] + [6e.l] + [6/.1] = [6s.l]
[6c.l] -f [cc.l] + [cdl] -f [ce.l] -f [c/.l] = [.l], &c.,
[cc.2] + [ed.2] -f [ce.2] -f [c/.2] = [cs.2],
[cd.2] + [eta.2] + [efe.2] + [d/.2] = [cfo.2], Ac.
Hence it appears that when the numerical calculation is arranged as
above suggested, the auxiliary containing s must, in each line, be
equal to the sum of all the terms to the left of it in the same line
and of those terms containing the same distinguishing numeral found
in a vertical column over the last quantity at the left of this line.
There will yet remain only the auxiliaries which are derived from
\_sn~] and [nri] to be determined. These additional auxiliaries will
be found by means of the formulae
. , - .
.3] = [m.2] - [es . 2] , [m . 4] = [m.3] - [efe.3], (133)
0.5] = [w.4] - [.4], [w.8] = [w.5] -
and the equations (81) and (83). The arrangement of the numerical
process should be similar to that already explained.
The values of [sw.l], [sw.2], &c. check the accuracy of the results
for [6n.l], [cn.l], [cn.2], [e?n.3], &c. by means of the equations
[bn.l] + [cn.l] -f \dn.V\ + [67i.l] + [>.l] = [i.l],
[c7i.2] + [dn.Z] -f- [e^.2] -f [/7i.2] == [sw.2],
[dw.3] -}- [e.3] + LM] = [m.3], (134;
It appears further, that, in the case of six unknown quantities, sinct
[/s.5] = [jgr.5], we have [n.6] = 0.
Having thus determined the numerical values of the auxiliaries
required, we are prepared to form at once the equations (74), by means
of which the values of the unknown quantities will be determined
27
418 THEORETICAL ASTRONOMY.
by successive substitution, first finding t from the last of these equa-
tions, then substituting this result in the equation next to the last
and thus deriving the value of w, and so on until all the unknown
quantities have been determined. It will be observed that the loga-
rithms of the coefficients of the unknown quantities in these equa-
tions will have been already found in the computation of the aux-
iliaries.
If we add together the several equations of (74), first clearing them
of fractions, we get
= [ad] x + ([06] + [56.1]) y + ([oc] + [6c.l] -f [ee.2]) z
+ (M + [W-l] +
+ (M + [&e-i] +
-f ([/] + P/1] +
+ [aw] + [6n.l] + [m.2] -+ [d.3] + [m.4] + [>.5] ;
and this equation must be satisfied by the values of #, y, z, &c. found
from (74).
152. EXAMPLE. The arrangement of the calculation in the case
of any other number of unknown quantities is precisely similar; and
to illustrate the entire process let us take the following equations,
each of which is already multiplied by the square root of its weight:
0.707z + 2.052y 2.3723 0.221w + 6".58 = 0,
0.471z + 1.347y 1.715s 0.085u + 1 .63 = 0,
0.260* + 0.770y 0.356z -f 0.483w 4 .40 = 0,
0.092* + 0.343?/ -f 0.2352 -f 0.469w 10 .21 = 0,
0.414* -f 1.204y 1.5063 0.205w + 3 .99 = 0,
0.040* -f 0.150# + 0.1042 + 0.206w 4 .34 = 0.
First, we derive
[nn] = 204.313,
[ol = + 4.815, [oa]^ + 0.971,
[6n] = + 12.961, [06] = + 2.821, [66] = + 8.208,
[en] - 25.697, [oc]= 3.175, [6c] = 9.168, [cc] = + 11.028,
[dn]= -10.218, [od] = 0.104, [W] = 0.261, [cd] = + 0.938, [dd] = + 0.594,
[w] =-18.139, [as] = + 0.513, [6s] = + 1.610, [c] = 0.377, [as] =+1.177.
The values of [sn], [as], [6s], [cs], and [cfe], found by taking the
sums of the normal coefficients, agree exactly with the values com-
puted directly, thus proving the calculation of these coefficients.
The normal equations are, therefore,
NUMERICAL EXAMPLE. 419
0.971* -f 2.821y 3.1752 0.104w -f 4.815 = 0,
2.821ar -f- 8.208y 9.168s O.'Zdlu -f 12.961 = 0,
3.175* 9.168y + 11.0282 -f- 0.938it 25.697 = 0,
- 0.104* 0.251y + 0.9382 -f- 0.594u 10.218 = 0.
It will be observed that the coefficients in these equations are nu-
merically greater than in the equations of condition; and this will
generally be the case. Hence, if we use logarithms of five decimals
in forming the normal equations, it will be expedient to use tables
of six or seven decimals in the solution of these equations.
Arranging the process of elimination in the most convenient form,
the successive results are as follows :
= + 0.0123, [fcc.l] = + 0.0562, [6*1] = + 0.0511, [bs.l] = + 0.1196, [fenl] = 1.0278,
[cc.l] = -f 0.6463, [ccZ.l] = + 0.5979, [cs.l] = + 1.3004, [cn.l] = 9.9528,
[cc.2] = + 0.3895, [cd.2] = + 0.3644, [cs.2] = + 0.7539, [cn.2] = - 5.2567,
[dd.l] = + 0.5829, [ds.l] = + 1.2319, [dn.l] = 9.7023,
[dd.2] = + 0.3706, [ds.2] = + 0.7350, [dre.2] = 5.4323,
[cW.3] = + 0.0297, [ds.3] = -f- 0.0297 [dn.3] = 0.5143,
[nw.l] = 180.436, [sn.l] = 20.6828,
[nn.2]= 94.552, [sn.2] = 10.6889,
[nn.3]= 23.608, [i.3]= 0.5143,
[wi.4]= 14.698, [sn.4] = 0.
The several checks agree completely, and only the value of [rw.4]
remains to be proved. The equations (74) therefore give
* -f 2.9052y 3.26982 0.1071w + 4.9588 == 0,
y + 4.56912 + 4.1545w 83.5610 = 0,
z + 0.9356w 13.4960 = 0,
u 17.3165 = 0,
and from these we get
u = -f 17".316, z = 2".705, y = + 23".977, * = 81".608.
Then the equation (135) becomes
r= + 0.9710* + 2.8333y 2.72932 + 0.3412w 1.9838,
which is satisfied by the preceding values of the unknown quantities.
If we substitute these values of x, y, z, and u in the equations of
condition already reduced to the same weight by multiplication by
the square roots of their weights, we obtain the residuals
i Q" gj -^f g^ i 2" 17 2" 01 0".40 0".72
The sum of the squares of these gives
[w] = {nnA'] = 11.672,
and the difference between this result and the value 14.698 already
420 THEORETICAL ASTRONOMY.
found is due to the decimals neglected in the computation of the
numerical values of the several auxiliaries. The sum of all the
equations of condition gives generally
to* + l^y + M* + ld]u + ....+ [n] - 0], (136)
which may be used to check the substitution of the numerical values
in the determination of v, v' 9 &c. Thus, we have, for the values
here given,
1.984s + 5.866y 5.610z + 0.647w 6.75 = [>] = l."63.
It remains yet to determine the relative weights of the resulting
values of the unknown quantities. For this purpose we may apply
any of the various methods already given. The weights of u and z
may be found directly from the auxiliaries whose values have been
computed. Thus, we have
p. = [*-3] 0.00056, p 9 = |^|| [66.2] 0.0072.
We may also compute the weights by means of the equations (96).
Thus, to find the weight of y, we have
[cW.21 = [cM.l] - - [crf.l] = + 0.02977,
and hence
The equations (103) and (108) are convenient for the determination
of the values and weights of the unknown quantities separately.
CORRECTION OF THE ELEMENTS. 421
Thus, by means of the values of the auxiliaries obtained in the first
elimination, we find from the equations (100), (101), and (102),
A' =. 2.9052, A" = -f 16.5442, A'" = 3.3012,
#' = 4.5691, JB"' = + 0.1202, C"" = 0.9356,
and then the equations (103) and (108) give
e = 81".609, y = + 23".977, z = 2".705, u =-- + 17".316 f
Pm = 0.00057, p v = 0.0074, A = 0.0312, Pv = 0.0297,
agreeing with the results obtained by means of the other methods.
The weights are so small that it may be inferred at once that the
values of x, y, z, and u are very uncertain, although they are those
which best satisfy the given equations. It will be observed that if
we multiply the first normal equation by 2.9, the resulting equation
will differ very little from the second normal equation, and hence we
have nearly the case presented in which the number of independent
relations is one less than the number of unknown quantities.
The uncertainty of the solution will be further indicated by deter-
mining the probable errors of the results, although on account of the
small number of equations the probable or mean errors obtained may
be little more than rude approximations. Thus, adopting the value
of [vv] obtained by direct substitution, we have
and hence
r= 1".629,
which is the probable error of the absolute term of an equation of
condition whose weight is unity. Then the equations
r r r
r = . , r = . , r, = , > &c.,
* Vpl ' VP, ' Vp.
give
r x = 68".25, r y = 18".94, r z = 9".22, r u = 9".45.
It thus appears that the probable error of z exceeds the value obtained
for the quantity itself, and that although the sum of the squares of
the residuals is reduced from 204.31 to 11.67, the results are still
quite uncertain.
153. The certainty of the solution will be greatest when the coef-
ficients in the equations of condition and also in the normal equation**
422 THEORETICAL ASTRONOMY.
differ very considerably both in magnitude and in sign. In the cor-
rection of the elements of the orbit of a planet when the observa-
tions extend only over a short interval of time, the coefficients will
generally change value so slowly that the equations for the direct
determination of the corrections to be applied to the elements will
not afford a satisfactory solution. In such cases it will be expedient
to form the equations for the determination of a less number of
quantities from which the corrected elements may be subsequently
derived. Thus we may determine the corrections to be applied to
two assumed geocentric distances or to any other quantities which
afford the required convenience in the solution of the problem,
various formulae for which have been given in the preceding chapter.
The quantities selected for correction should be known functions of
the elements, and such that the equations to be solved, in order to
combine all the observed places, shall not be subject to any uncer-
tainty in the solution. But when the observations extend over a long
period, the most complete determination of the corrections to be
applied to the provisional elements will be obtained by forming the
equations for these variations directly, and combining them as already
explained. A complete proof of the accuracy of the entire calcula-
tion will be obtained by computing the normal places directly from
the elements as finally corrected, and comparing the residuals thus
derived with those given by the substitution of the adopted values
of the unknown quantities in the original equations of condition.
If the elements to be corrected differ so much from the true values
that the squares and products of the corrections are of sensible mag-
nitude, so that the assumption of a linear form for the equations does
not afford the required accuracy, it will be necessary to solve the
equations first provisionally, and, having applied the resulting cor-
rections to the elements, we compute the places of the body directly
from the corrected elements, and the differences between these and
the observed places furnish new values of n, n f , n", &c., to be used
in a repetition of the solution. The corrections which result from
the second solution will be small, and, being applied to the elements
as corrected by the first solution, will furnish satisfactory results. In
this new solution it will not in general be necessary to recompute the
coefficients of the unknown quantities in the equations of condition,
since the variations of the elements will not be large enough to affect
sensibly the values of their differential coefficients with respect to
the observed spherical co-ordinates. Cases may occur, however, in
vhich it may become necessary to recompute the coefficients of one
CORRECTION OF THE ELEMENTS. 423
or more of the unknown quantities, but only when these coefficients
are very considerably changed by a small variation in the adopted
values of the elements employed in the calculation. In such cases
the residuals obtained by substitution in the equations of condition
will not agree with those obtained by direct calculation unless the
corrections applied to the corresponding elements are very small. It
may also be remarked that often, and especially in a repetition of the
solution so as to include terms of the second order, it will be suffi-
ciently accurate to relax a little the rigorous requirements of a com-
plete solution, and use, instead of the actual coefficients, equivalent
numbers which are more convenient in the numerical operations re-
quired. Although the greatest confidence should be placed in the
accuracy of the results obtained as far as possible in strict accordance
with the requirements of the theory, yet the uncertainty of the deter-
mination of the relative weights in the combination of a series of
observations, as well as the effect of uneliminated constant errors,
may at least warrant a little latitude in the numerical application,
provided that the weights of the results are not thereby much affected.
A constant error may in fact be regarded as an unknown quantity to
be determined, and since the effect of the omission of one of the
unknown quantities is to diminish the probable errors of the resulting
values of the others, it is evident that, on account of the existence of
constant errors not determined, the values of the variables obtained
by the method of least squares from different corresponding series of
observations may differ beyond the limits which the probable errors
of the different determinations have assigned. Further, it should be
observed that, on account of the unavoidable uncertainty in the esti-
mation of the weights of the observations in the preliminary combi-
nation, the probable error of an observed place whose weight is
unity as determined by the final residuals given by the equations of
condition, may not agree exactly with that indicated by the prior
discussion of the observations.
154. In the case of very eccentric orbits in which the corrections
to be applied to certain elements are not indicated with certainty by
the observations, it will often become necessary to make that whose
weight is very small the last in the elimination, and determine the
other corrections as functions of this one; and whenever the coeffi-
cients of two of the unknown quantities are nearly equal or have
nearly the same ratio to each other in all the different equations of
condition, this method is indispensable unless the difficulty is reme
424 THEORETICAL ASTRONOMY.
died by other means, such as the introduction of different elements or
different combinations of the same elements. The equations (113)
furnish the values of the unknown quantities when we neglect that
which is to be determined independently; and then the equations
(114) give the required expressions for the complete values of these
quantities. Thus, when a comet has been observed only during a
brief period, the ellipticity of the orbit, however, being plainly indi-
cated by the observations, the determination of the correction to be
applied to the mean daily motion as given by the provisional ele-
ments, in connection with the corrections of the other elements, will
necessarily be quite uncertain, and this uncertainty may very greatly
affect all the results. Hence the elimination will be so arranged that
A// shall be the last, and the other corrections will be determined as
functions of this quantity. The substitution of the results thus
derived in the equations of condition will give for each residual an
expression of the following form :
Therefore we shall have
[>] = [ Vo ] + 2 [ V ] AM -f
which may be applied more conveniently in the equivalent form
M = [ Vo ] - d [v] + M AM + [ '. (138)
The most probable value of A/Z will be that which renders [vv] a
minimum, or
(139)
and the corresponding value of the sum of the squares of the
residuals is
l- (140)
The correction given by equation (139) having been applied to /*,
the result may be regarded as the most probable value of that ele-
ment, and the corresponding values of the corrections of the other
elements as determined by the equations (114) having been also duly
applied, we obtain the most probable system of elements. These,
however, may still be expressed in the form
-f QAM, &c.
CORRECTION OF THE ELEMENTS. 425
the coefficients A QJ B w C , &c. being those given by the equations
(114), and thus the elements may be derived which correspond to any
assumed value of p differing from its most probable value. The
unknown quantity A/* will also be retained in the values of the
residuals. Hence, if we assign small increments to //, it nrjy easily
be seen how much this element may differ from its mo^fi probable
value without giving results for the residuals which are i/v,ompatible
with the evidence furnished by the observations.
If the dimensions of the orbit are expressed by mearx.* of the ele-
ments q and e, it may occur that the latter will not be determined
with certainty by the observations, and hence it should be treated as
suggested in the case of //; and we proceed in a similar manner when
the correction to be applied to a given value of tL; semi-transverse
axis a is one of the unknown quantities to be dete mined.
426 THEORETICAL ASTRONOMY,
CHAPTER VIII.
INVESTIGATION OP VARIOUS FORMULAE FOR THE DETERMINATION OF THE SPECIAL
PERTURBATIONS OF A HEAVENLY BODY.
155. WE have thus far considered the circumstances of the undis-
turbed motion of the heavenly bodies in their orbits; but a complete
determination of the elements of the orbit of any body revolving
around the sun, requires that we should determine the alterations in
its motion due to the action of the other bodies of the system. For
this purpose, we shall resume the general equations (18) 1? namely,
m) ,
which determine the motion of a heavenly body relative to the sun
when subject to the action of the other bodies of the system. We
have, further,
m' /I xaf + yy + t \ , m" / 1 xx"+ yf+ zz" \
'-lm\ r' 3 J^l m \' r "* P
which is called the perturbing function, of which the partial differen-
tial coefficients, with respect to the co-ordinates, are
dQ_ m' Ix'-x ^\ m " lx"-x x"
fa-l+m\ ff r'*l + l + m\ p" r">
dQ _ m' [z' z J_\ , m" jz" z 4' \ , ^
S"~rfm\ P 3 r^/^l+ml ?'* r"* / "
and in which m', m r/ , &c. denote the ratios of the masses of the
several disturbing planets to the mass of the sun, and m the ratio of
the mass of the disturbed planet to that of the sun. These partial
differential coefficients, when multiplied by & 2 (l-f-w), express thn
PERTURBATIONS. 427
sum of the components of the disturbing force resolved in directions
parallel to the three rectangular axes respectively.
When we neglect the consideration of the perturbations, the general
equations of motion become
dt*
/7/2 ' ^ ' J ,, 3 ' \*^y
the complete integration of which furnishes as arbitrary constants of
integration the six elements which determine the orbitual motion of a
heavenly body. But if we regard these elements as representing the
actual orbit of the body for a given instant of time t, and conceive
of the effect of the disturbing forces due to the action of the other
bodies of the system, it is evident that, on account of the change
arising from the force thus introduced, the body at another instant
different from the first will be moving in an orbit for which the
elements are in some degree different from those which satisfy the
original equations. Although the action of the disturbing force is
continuous, we may yet regard the elements as unchanged during the
element of time dt, and as varying only after each interval dt. Let
us now designate by t the epoch to which the elements of the orbit
belong, and let these elements be designated by M w TT O , & , i w e , and
a ; then will the equations (3) be exactly satisfied by means of the
expressions for the co-ordinates in terms of these rigorously-constant
elements. These elements will express the motion of the body sub-
ject to the action of the disturbing forces only during the infinitesimal
interval dt, and at the time t Q + dt it will commence to describe a
new orbit of which the elements will differ from these constant ele-
ments by increments which are called the perturbations.
According to the principle of the variation of parameters, or of
the constants of integration, the differential equations (1) will be
satisfied by integrals of the same form as those obtained when the
second members are put equal to zero, provided only that the arbitrary
constants of the latter integration are no longer regarded as pure
constants but as subject to variation. Consequently, if we denote the
variable elements by M, TT, &, i, e, and a, they will be connected
with the constant elements, or those which determine the orbit at the
instant , by the equations
428 THEORETICAL ASTRONOMY.
di
(4)
in which 7-, -r- . &c. denote the differential coefficients of the ele-
at at
ments depending on the disturbing forces. When these differential
coefficients are known, we may determine, by simple quadrature, the
perturbations dM, dn, &c. to be added to the constant elements in
order to obtain those corresponding to any instant for which the
place of the body is required. These differential coefficients, however,
are functions of the partial differential coefficients of Q with respect
to the elements, and before the integration can be performed it
becomes necessary to find the expressions for these partial differential
coefficients. For this purpose we expand the function Q into a con-
verging series and then differentiate each term of this series relatively
to the elements. This function is usually developed into a converg-
ing series arranged in reference to the ascending powers of the eccen-
tricities and inclinations, and so as to include an indefinite number
of revolutions; and the final integration will then give what are
called the absolute or general perturbations. When the eccentricities
and inclinations are very great, as in the case of the comets, this
development and analytical integration, or quadrature, becomes no
longer possible, and even when it is possible it may, on account of
the magnitude of the eccentricity or inclination, become so difficult
that we are obliged to determine, instead of the absolute perturbations,
what are called the special perturbations, by methods of approxima-
tion known as mechanical quadratures, according to which we deter-
mine the variations of the elements from one epoch t Q to another
epoch t. This method is applicable to any case, and may be advan-
tageously employed even when the determination of the absolute
perturbations is possible, and especially when a series of observations
extending through a period of many years is available and it is
desired to determine, for any instant , a system of elements, usually
called osculating elements, on which the complete theory of the motion
may be based.
Instead of computing the variations of the elements of the orbit
directly, we may find the perturbations of any known functions of
these elements; and the most direct and simple method is to deter-
mine the variations, due to the action of the disturbing forces, of
any system of three co-ordinates by means of which the position of
PERTURBATIONS. 429
the body or the elements themselves may be found. We shall, there-
fore, derive various formulae for this purpose before investigating th<
formulae for the direct variation of the elements.
156. Let X Q , y , z be the rectangular co-ordinates of the body at
the time t computed by means of the osculating elements M w TT O , Q ,
&c., corresponding to the epoch t . Let x, y, z be the actual co-ordi-
nates of the disturbed body at the time t; and we shall have
dx, dy, and dz being the perturbations of the rectangular co-ordinates
from the epoch t to the time t. If we substitute these values of x,
y, and z in the equations (1), and then subtract from each the corre-
sponding one of equations (3), we get
Let us now put r = r -f- dr; then to terms of the order ^r 2 , which is
equivalent to considering only the first power of the disturbing force,
we have
and hence
We have also from
r'-z' + ^
neglecting terms of the second order,
(7)
430 THEORETICAL ASTRONOMY.
The integration of the equations (6) will give the perturbations dx,
dy, and dz to be applied to the rectangular co-ordinates x w y w z com-
puted by means of the osculating elements, in order to find the actual
co-ordinates of the body for the date to which the integration belongs.
But since the second members contain the quantities dx 9 %, dz which
are sought, the integration must be effected indirectly by successive
approximations; and from the manner in which these are involved
in the second members of the equations, it will appear that this inte-
gration is possible.
If we consider only a single disturbing planet, according to the
equations (2), we shall have
*' \
7')'
and these forces we will designate by X, Y, and Z respectively ; then,
if in these expressions we neglect the terms of the order of the
square of the disturbing force, writing .r , y , Z Q in place of x, y, z,
the equations (6) become
df r,
(9)
which are the equations for computing the perturbations of the rec-
tangular co-ordinates with reference only to the first power of the
masses or disturbing forces. We have, further,
, = (gf _ .,.) + ft - y ) -I- (/ _ z y, (10)
in which, when terms of the second order are neglected, we use the
values x Q , y , z for x, y, and z respectively.
1 57. From the values of 8x, dy, and dz computed with regard to
the first power of the masses we may, by a repetition of part of the
calculation, take into account the squares and products and even the
higher powers of the disturbing forces. The equations (5) may be
written thus:
VAEIATION OF CO-OKDINATES. 431
d?
in which nothing is neglected. In the application of these formulae,
as soon as dx, dy, and $2 have been found for a few successive inter-
vals, we may readily derive approximate values of these quantities
for the date next following, and with these find
x = x -\-dx, y = 2/ + dy, Z = Z Q -\- dz,
and hence the complete values of the forces X, Y, and Z, by means
of the equations (8). To find an expression for the factor
-5f
which will be convenient in the numerical calculation, we have
and therefore
. = x 2 g * (y
2 ~T~
Let us now put
?=
and
/}=i
then we shall have
and the values of/ may be tabulated with the argument q. The
equations (11) therefore become
(14)
432 THEOKETICAL ASTEONOMY.
The coefficients of 8x, dy, and dz in equation (12) may be found at
once, with sufficient accuracy, by means of the approximate values
of these quantities; and having found the value of / corresponding
to the resulting value of g, the numerical values of , 2 > , f > and
d*8z
-ITT, which include the squares and products of the masses, will be
obtained. The integration of these will give more exact values of
dx, dy, and dz, and then, recomputing q and the other quantities which
require correction, a still closer approximation to the exact values of
the perturbations will result.
Table XVII. gives the values of log/ for positive or negative
values of q at intervals of 0.0001 from q = to q = 0.03. Unless
the perturbations are very large, q will be found within the limits of
this table; and in those cases in which it exceeds the limits of the
table, the value of
may be computed directly, using the value of r in terms of r Q and
dx, dy, dz.
In the application of the preceding formulae, the positions of the
disturbed and disturbing bodies may be referred to any system of
rectangular co-ordinates. It will be advisable, however, to adopt
either the plane of the equator or that of the ecliptic as the funda-
mental plane, the positive axis of x being directed to the vernal
equinox. By choosing the plane of the elliptic orbit at the time t
as the plane of xy, the co-ordinate z will be of the order of the per-
turbations, and the calculation of this part of the action of the dis-
turbing force will be very much abbreviated; but unless the inclina-
tion is very large there will be no actual advantage in this selection,
since the computation of the values of the components of the dis-
turbing forces will require more labor than when either the equator
or the ecliptic is taken as the fundamental plane. The perturbations
computed for one fundamental plane may be converted into those
referred to another plane or to a different position of the axes in the
same plane by means of the formulae which give the transformation
of the co-ordinates directly.
158. We shall now investigate the formulae for the integration of
the linear differential equations of the second order which express the
variation of the co-ordinates, and generally the formulae for finding
the integrals of expressions of the form J f(x)dx and JJ J(x)dx*
MECHANICAL QUADRATURE. 433
when the values of f(x) are computed for successive values of x in-
creasing in arithmetical progression. First, therefore, we shall find
the integral of f(x) dx within given limits.
Within the limits for which x is continuous, we have
f(x) = a + {3x + r x* + W+ex*+....; (15)
and if we consider only three terms of this series, the resulting equa-
tion
f(x) ^a + px + rx*
is that of the common parabola of which the abscissa is x and the
ordinate /(#), and the integral of f(x) dx is the area included by the
abscissa, two ordinates, and the included arc of this curve. Gene-
rally, therefore, we may consider the more complete expression for
f(x) as the equation of a parabolic curve whose degree is one less
than the number of terms taken. Hence, if we take n terms of the
series as the value of/(#), we shall derive the equation for a parabola
whose degree is n 1, and which has n points in common with the
curve represented by the exact value of f(x).
If we multiply equation (15) by dx and integrate between the
limits and x', we get
dx =
If now the values of f(x) for different values of x from to x f are
known, each of these, by means of equation (15), will furnish an
equation for the determination of a, /?, 7-, &c. ; and the number of
terms which may be taken will be equal to the number of different
known values of /(a?). As soon as a, /?, 7-, &c. have thus been found,
the equation (16) will give the integral required.
If the values of f(x) are computed for values of x at equal inter-
vals and we integrate between the limits x = 0, and x = n&x, &x
being the constant interval between the successive values of x y and
n the number of intervals from the beginning of the integration, we
obtain
x
Cf(x) dx =
Let us now suppose a quadratic parabola to pass through the points
of the curve represented by f(x), corresponding to x = 0, x = &x,
28
434 THEORETICAL ASTRONOMY.
and x = 2&x; then will the area included by the arc of this parabola,
the extreme ordinates, and the axis of abscissas be
'Ibx
J/(aO'dn,
I f(x) dx = (o I /(a -j- TWO) dn.
If we expand the function /(a + no)), we have
na>
f .
., (20)
436 THEOKET1CAL ASTRONOMY.
and hence
//(a + n) dn = C + n/(a) + in- ^ + JnV
C being the constant of integration. The equations (54) 6 give
* ^ =/() - ir w + w() - T
-' =/"() - A/*() + */"() - */"() + .
=/" (a) - \r () + T !*/' u ()-...,
(22)
=/ " (a) - * /VI(a) + ' / "" (a) ~
=/' () -if'" ()+.-.,
iii which the functional symbols in the second members denote the
diiferent orders of finite differences of the function. Hence we obtain
+ no*) cfo = + nf(a)
+ in 1 (/(a) - I/" (a) + A/* (a) - T ^/ vU (a) + . . .)
+ Jn(/"(a) - A/ 1 ' (a) + ^/"(a) - v hf^(a) + ...)
Ef we take the integral between the limits n r and +n', the terms
containing the even powers of n disappear. Further, sinc the values
of the function are supposed to be known for a series of values of n
at intervals of a unit, it will evidently be convenient to determine
the integral between the required limits by means of the sum of a
series of integrals whose limits are successively increased by a unit,
such that the difference between the superior and the inferior limit
of each integral shall be a unit. Hence we take the first integral
between the limits J and +J, and the equation (23) gives, after
reduction,
MECHANICAL QUADRATURE. 43?
-f *
f/(a + n0 dn -/(a) + /"() ~ s4Is/ iv () +
J -i (24)
-4 g f!fl!^/ Tiil W + &c.
It is evident that by writing, in succession, a -\- a), a + 2o>, ....
a -f- ia> in place of a, we simply add 1 to each limit successively, so
that we have
% < + 1 + *
J/(a + na) cfo ==(/(( + + ( i) ) d (n t)
But since
1 *+*
-* -i *
if we give to i successively the values 0, 1, 2, 3, &c. in the preceding
equation, and add the results, we get
i -f i n = i n = i
J/(a + n>) dn = ^f(a + n0 -f- ^ ? ^/' (a + na>)
t n = i
rCa + ) -I- ^W 5 ^/ vl ( + n) - Ac.
n=0 n=0
Let us now consider the functions /(a), /(a + not), &c. as being
themselves the finite differences of other functions symbolized by '/,
the first of which is entirely arbitrary, so that we may put, in accord-
ance with the adopted notation,
/O) = '/( -f M - '/( - J0,
/(a + ) = '/(a + j,) - '/(a + J),
/(a + tia,) = '/(a + ( n + J '/(a + ( - J) >).
Therefore we shall have
n = t
Y/O + no) = '/(a + (i + i) ,) - '/(a - i),
n =
and also
n = i
) =/ (a + (t + J) ) -/ (a - ^),
433 THEORETICAL ASTRONOMY.
Further, since the quantity f f(a \a>) is entirely arbitrary, we may
assign to it a value such that the sum of all the terms of the equation
which have the argument a \a> shall be zero, namely,
(26)
Substituting these values in (25), it reduces to
I f(x) dx to I /(a -|~ nw) efoi
a-fc. -i (27)
= {'/( + (* + i + &/( + (*' + J
In the calculation of the perturbations of a heavenly body, the
dates for which the values of the function are computed may be so
arranged that for n = J, corresponding to the inferior limit, the
integral shall be equal to zero, the epoch of f(a \co) being that of
the osculating elements. It will be observed that the equation (26)
expresses this condition, the constant of integration being included
in '/(a \to). If, instead of being equal to zero, the integral has a
given value when n = J, it is evidently only necessary to add this
value to f f(a Jo>) as given by (26).
160. The interval co and the arguments of the function may always
be so taken that the equation (27) will furnish the required integral,
either directly or by interpolation ; but it will often be convenient to
integrate for other limits directly, thus avoiding a subsequent inter-
polation. The derivation of the required formulae of integration
may be effected in a manner entirely analogous to that already indi-
cated. Thus, let it be required to find the expression for the integra 1
taken between the limits \ and i.
The general formula (23) gives
J,
" (a)-
and since, according to the notation adopted,
/ () = 1 CT ( - i") +/'( + i-))
= f( + i-) -iT(), (28)
/"()=ro+i) -*/"(),
/'() =/'( + i0 - if" (), &e.,
MECHANICAL QUADRATURE. 439
this becomes
i
//(a+n-O cfo=l/(a)+ J/ (a+ j0-&r (a)-,!,/'' (+
(29 >
Therefore we obtain
< + *
/(a-j-nw) dn=^f(a+ic
Now we have
\ f( a + nuj ) dn= I /(a -|- nai] dn ( /(a -f- na>) dn ;
-i -i
and if we substitute the values already found for the terms in the
second member, and also
/" ( + fa) = f( a + (, + !))_ f( a + (i-l) a,),
/( + i.) =/'" (a + (t + i) -/"' (< + (t - i) ),
we get
a + i -TT-> and ;-> and another integration
becomes necessary in order to obtain the values of dx, dy, and dz.
We will therefore proceed to derive formulae for the determination
of the double integral directly.
440 THEORETICAL ASTRONOMY.
For the double integral j J f(x) dx 2 we have, since dx* = a) 2 dn 2 ,
The value of the function designated by /(a) being so taken that
when n = \,
Cf(a + n0 dn 0,
the equation (23) gives
Therefore, the general equation is
o
J/(a -f nJ o>i ; = ^ J(CL -\- nw) 2? ^ / ( a ~t~ ^ w )
A/ >^ *i
,17 Q n= \_, (35)
-f-&c.
n=0 =0
We may evidently consider '/(a Ja>), '/(a + Jw), &c. ae the differ-
ences of other functions, the first of which is arbitrary; 40 that we
have
7 (a) = f(a + -J0 + i'/O - i) - i"/(a + ) - if ( - ),
'/(a + *) = J'/(a + i) + i'/(a + ^) - J"/(a + 2) - ^/" (a),
-f i'/( + (^-i)) = i7(a + (
- r/(a + (?l _l )w) .
Therefore
n =
Substituting these values in equation (35), and observing that
700 + 7( - ) = 2"/( - ) + '/( - -^)>
/(a) + /(a - a,) = 2/(a) - / (a - Jo,),
/' W +/" () 2/' (a) -/"' (a - i0, Ac.,
and that, since "f(a to) is arbitrary, we may put
"/(a - ,) =
() + 2/" (a )) - &o.,
(36)
442 THEORETICAL ASTRONOMY.
the integral becomes
* + + *><> r / + *
JJ /(*) dx* = > 2 JJ /(a + no,) dn*
l)>) (37)
which is the expression for the double integral between the limits
\ and i + J.
The value of "/(a to) given by equation (36) is in accordance
with the supposition that for n = \ the double integral is equal to
zero, and this condition is fulfilled in the calculation of the pertur-
bations when the argument a \a> corresponds to the date for which
the osculating elements are given. If, for n = J, neither the single
nor the double integral is to be taken equal to zero, it is only neces-
sary to add the given value of the single integral for this argument
to the value of '/(a Jai) given by equation (26), and to add the
given value of the double integral for the same argument to the value
of "/(a ai) given by (36).
162. In a similar manner we may find the expressions for the
double integral between other limits. Thus, let it be required to
find the double integral between the limits J and i.
Between the limits and we have
jjf(a
which gives
dn + J/(a) + &.
+ ssW + 1*4*3* + Ac.
GO (38 )
and this again, by means of (28), gives
**
MECHANICAL QUADRATURE. 443
Therefore, since
CC rr rr
JJ /O -f no) dn* =JJ f(a + no) drc, 2 JJ /( a + n0 d,
-* -t
and
'/( + (* + i) = 7( + (* + 1; ) 7( + '),
A* -K* + *>)== /(a + (t + l /(<* + *),
/'" (a + (i + i) a,) =/" (a + (i + 1) w) -f" (a -f ,), &c.
we shall have
a -f i w ^
(^) da? = *0/(a + nai) (fo 8
*w -i (39)
which gives the required integral between the limits J and i.
163. It will be observed that the coefficients of the several terms
of the formulae of integration converge rapidly, and hence, by a
proper selection of the interval at which the values of the function
are computed, it will not be necessary to consider the terms which
depend on the fourth and higher orders of differences, and rarely
those which depend on the second and third differences. The value
assigned to the interval a) must be such that we may interpolate with
certainty, by means of the values computed directly, all values of the
function intermediate to the extreme limits of the integration; and
hence, if the fourth and higher orders of differences are sensible, it
will be necessary to extend the direct computation of the values of
the function beyond the limits which would otherwise be required,
in order to obtain correct values of the differences for the beginning
and end of the integration. It will be expedient, therefore, to take
(o so small that the fourth and higher differences may be neglected,
but not smaller than is necessary to satisfy this condition, since other-
wise an unnecessary amount of labor would be expended in the
direct computation of the values of the function. It is better, how-
ever, to have the interval at smaller than what would appear to be
strictly required, in order that there may be no uncertainty with
respect to the accuracy of the integration. On account of the rapidity
with which the higher orders of differences decrease as we diminish
to, a limit for the magnitude of the adopted interval will speedily be
obtained. The magnitude of the interval will therefore be suggested
by tne rapidity of the change of value of the function. In the coin-
444 THEORETICAL ASTRONOMY.
ptitation of the perturbations of the group of small planets between
Mars and Jupiter we may adopt uniformly an interval of forty days;
but in the determination of the perturbations of comets it will evi-
dently be necessary to adopt different intervals in different parts of
the orbit. When the comet is in the neighborhood of its perihelion,
and also when it is near a disturbing planet, the interval must neces-
sarily be much smaller than when it is in more remote parts of its
orbit or farther from the disturbing body.
It will be observed, further, that since the double integral contains
the factor o> 2 , if we multiply the computed values of the function by
o> 2 , this factor will be included in all the differences and sums, and
hence it will not appear as a factor in the formulae of integration.
If, however, the values of the function are already multiplied by w 2 ,
and only the single integral is sought, the result obtained by the
formula of integration, neglecting the factor o> 2 , will be to times the
actual integral required, and it must be divided by a) in order to
obtain the final result.
164. In the computation of the perturbations of one of the asteroid
planets for a period of two or three years it will rarely be necessary
to take into account the effect of the terms of the second order with
respect to the disturbing force. In this case the numerical values of
the expressions for the forces will be computed by using the values
of the co-ordinates computed from the osculating elements for the
beginning of the integration, instead of the actual disturbed values
of these co-ordinates as required by the formulae (8). The values of
the second differential coefficients of dx, %, and dz with respect to
the time, will be determined by means of the equations (9). If the
interval CD is such that the higher orders of differences may be neg-
lected, the values of the forces must be computed for the successive
dates separated by the interval o>, and commencing with the date
t Q \co corresponding to the argument a co, t Q being the date to
which the osculating elements belong. Then, since the last terms
.. . . . ,
of the formulae for ^~, jp and 7^- involve ox, oy, and 02, which
are the quantities sought, the subsequent determination of the differ-
ential coefficients must be performed by successive trials. Since the
integral must in each case be equal to zero for the date t Q , it will be
admissible to assume first, for the dates t \co and t Q -f- \co corre-
sponding to the arguments a a> and a, that dx 0, dy = 0, and
dz = 0, and hence that the three differential coefficients, for each
VARIATION OF CO-ORDINATEb. 445
date, are respectively equal to X w Y Q , and Z . We may now by inte-
gration derive the actual or the very approximate values of the
variations of the co-ordinates for these two dates. Thus, in the case
of each co-ordinate, we compute the value of f f(a \co) by means
of the equation (26), using only the first term, and the value of
"f(a to) from (36), using in this case also only the first term. The
value of the next function symbolized by "f will be given by
Then the formula (39), putting first * = 1 and then i = 0, and
neglecting second differences, will give the values of the variations
of the co-ordinates for the dates a co and a. These operations will
be performed in the case of each of the three co-ordinates; and, by
means of the results, the corrected values of the differential coeffi-
cients will be obtained from the equations (9), the value of 3r being
computed by means of (7). With the corrected values thus derived
a new table of integration will be commenced ; and the values of
r f(a \co) and "f(a to) will also be recomputed. Then we obtain,
also, by adding '/(a ;) to /(a), the value of f f(a -f- Jw), and, by
adding this to "f(a), the value of "f(a -f to).
An approximate value of f(a + to) may now be readily estimated,
and two terms of the equation (39), putting i= 1, will give an ap-
proximate value of the integral. This having been obtained for
each of the co-ordinates, the corresponding complete values of the
differential coefficients may be computed, and these having been
introduced into the table of integration, the process may, in a similar
manner, be carried one step farther, so as to determine first approxi-
mate values of 3x, 8y, and dz for the date represented by the argu-
ment a + 2, and then the corresponding values of the differential
coefficients. We may thus by successive partial integrations deter-
mine the values of the unknown quantities near enough for the cal-
culation of the series of differential coefficients, even when the inte-
grals are involved directly in the values of the differential coefficients.
If it be found that the assumed value of the function is, in any case,
much in error, a repetition of the calculation may become necessary ;
but when a few values have been found, the course of the function
will indicate at once an approximation sufficiently close, since what-
ever error remains affects the approximate integral by only one-
twelfth part of the amount of this error. Further, it is evident
that, in cases of this kind, when the determination of the values of
the differential coefficients requires a preliminary approximate inte-
446 THEORETICAL ASTRONOMY.
gratiou, it is necessary, in order to avoid the effect of the errors in
the values of the higher orders of differences, that the interval (o
should be smaller than when the successive values of the function to
be integrated are already known. In the case of the small planets
an interval of 40 days will afford the required facility in the approxi-
mations; but in the case of the comets it may often be necessary to
adopt an interval of only a few days. The necessity of a change in
the adopted value of co will be indicated, in the numerical applica-
tion of the formula?, by the manner in which the successive assump-
tions in regard to the value of the function are found to agree with
the corrected results.
The values of the differential coefficients, and hence those of the
integrals, are conveniently expressed by adopting for unity the unit
of the seventh decimal place of their values in terms of the unit of
space.
165. Whenever it is considered necessary to commence to take into
account the perturbations due to the second and higher powers of the
disturbing force, the complete equations (14) must be employed. In
this case the forces Jf, Y 9 and Z should not be computed at once for
the entire period during which the perturbations are to be determined.
The values computed by means of the osculating elements will be
employed only so long as simply the first power of the disturbing
force is considered, and by means of the approximate values of dx,
dy, and dz which would be employed in computing, for the next place,
the last terms of the equations (9), we must compute also the cor-
rected values of JT, F, and Z. These will be given by the second
members of (8), using the values of x, y, and z obtained from
We compute also q from (12), and then from Table XVII. find the
corresponding value of /. The corrected values of , 2 , ,, 2 > and
Ej- will be given by the equations (14), and these being introduced,
in the continuation of the table of integration, we obtain new values
of fix, %, and dz for the date under consideration. If these differ
much from those previously assumed, a repetition of the calculation
will be necessary in order to secure extreme accuracy. In this repe-
tition, however, it will not be necessary to recompute the coefficients
of dx, dy, and dz in the formula for q, their values being given with
sufficient accuracy by means of the previous assumption ; and gene-
VARIATION OF CO-ORDINATES. 447
rally a repetition of the calculation of X, Y, and Z will not be
required.
Next, the values of dx, %, and dz may be determined approxi-
mately, as already explained, for the following date, and by mean*
of these the corresponding values of the forces Jf, F, and Z will be
found, and also/ and the remaining terms of (14), after which the
integration will be completed and a new trial made, if it be con-
sidered necessary. In the final integration, all the terms of the for-
mulae of integration which sensibly aifect the result may be taken
into account. By thus performing the complete calculation of each
successive place separately, the determination of the perturbations in
the values of the co-ordinates may be effected in reference to all
powers of the masses, provided that we regard the masses and co-or-
dinates of the disturbing bodies as being accurately known ; and it is
apparent that this complete solution of the problem requires very
little more labor than the determination of the perturbations when
only the first power of the disturbing force is considered. But
although the places of the disturbing bodies as given by the tables
of their motion may be regarded as accurately known, there are yet
the errors of the adopted osculating elements of the disturbed body
to detract from the absolute accuracy of the computed perturbations;
and hence the probable errors of these elements should be constantly
kept in view, to the end that no useless extension of the calculation
may be undertaken. When the osculating elements have been cor-
rected by means of a very extended series of observations, it will be
expedient to determine the perturbations with all possible rigor.
When there are several disturbing planets, the forces for all of
these may be computed simultaneously and united in a single sum,
so that in the equations (14) we shall have ZX, SY, and 2Z instead
of X, Y, and Z respectively; and the integration of the expressions
fJ^dx d^ftii d^ds
for fif-, ~j and ^- will then give the perturbations due to the
Cit Cut ut
action of all the disturbing bodies considered. However, when the
interval to for the different disturbing planets may be taken differently,
it may be considered expedient to compute the perturbations sepa-
rately, and especially if the adopted values of the masses of some of
the disturbing bodies are regarded as uncertain, and it is desired to
separate their action in order to determine the probable corrections
to be applied to the values of m, m f , &c., or to determine the effect
of any subsequent change in these values without repeating the cal-
culation of the perturbations.
448
THEORETICAL ASTRONOMY.
166. EXAMPLE. To illustrate the numerical application of the
formulae for the computation of the perturbations of the rectangular
co-ordinates, let it be required to compute the perturbations of
Eurynome @ arising from the action of Jupiter from 1864 Jan. 1.0
Berlin mean time to 1865 Jan. 15.0 Berlin mean time, assuming the
osculating elements to be the following :
Epoch = 1864 Jan. 1.0 Berlin mean time.
M 9 = 1 29' 5".65
7r = 44 17 12 .17) T
r\ of\a oo K a I Ecliptic and Mean
&6 = ZUO oa O .oy *
i,= 4 36 52.11.
Po = 11 15 51 .02
log a = 0.3881319
Equinox 1860.0
From these elements we derive the following values :
Berlin Mean Time. x y z
Ios;r
1863 Dec.
12.0
+ 1.53616
+ 1.23012
0.03312
0.294084,
1864 Jan.
21.0
1.15097
1.59918
0.07369
0.294837,
March
1.0
0.69518
1.87033
0.10978
0.300674,
April
10.0
+ 0.19817
2.03141
0.13936
0.310864,
May
20.0
0.31012
2.08092
0.16134
0.324298,
June
29.0
0.80326
2.02602
0.17523
0.339745,
Aug.
8.0
1.26055
1.87959
0.18122
0.356101,
Sept.
17.0
1.66729
1.65711
0.17990
0.372469,
Oct.
27.0
2.01414
1.37473
0.17209
0.388214,
Dec.
6.0
2.29597
1.04766
0.15870
0.402894,
1865 Jan.
15.0
2.51077
+ 68978
0.14066
0.416240.
The adopted interval is a) = 40 days, and the co-ordinates are re-
ferred to the ecliptic and mean equinox of 1860.0. The first date,
it will be observed, corresponds to t Q \to y and the integration is to
commence at 1864 Jan. 1.0.
The places of Jupiter derived from the tables give the following
values of the co-ordinates of that planet, with which we write also
the distances of Eurynome from Jupiter computed by means of the
formula
,. = (of - x? +
Aug.
8.0
57.30
35.92
1.66,
a + 6>
Sept.
17.0
59.09
32.47
2.08,
a + 7"
Oct.
27.0
61.55
28.60
2.43,
a-{-8a>
Dec.
6.0
64.85
24.34
2.69,
a + 9w
1865 Jan.
15.0
+ 69.09
+ 19.78
-f 2.83,
which are expressed in units of the seventh decimal place.
"We now, for a first approximation, regard the perturbations as
450 THEORETICAL ASTRONOMY.
being equal to zero for the dates Dec. 12.0 and Jan. 21.0, and, in the
case of the variation of x, we compute first
'/(a - K> = - A/' (a j0 = - A (53.71 - 53.00) = - 0.03,
and the approximate table of integration becomes
/( ~ = + 53.00 m _ >} __ 03 "/( - *) =
/() = + 53.71 K "/(a) -=
Then the formula (39), putting first i = 1, and then i = 0, gives
Dec. 12.0 to= + 2.24 + -^p = + 6.66,
Jan. 21.0 dx= + 2.21 + -^ = + 6.69.
In a similar manner, we find
Dec. 12.0 Sy = + 5.85 dz = 0.16,
Jan. 21.0 fy = + 5.82 & = 0.14.
By means of these results we compute the complete values of the
second members of equations (40), dr being found from
and thus we obtain
(PJoj cPtJy .cfrte
Pate " "^ "^ W ^
Dec. 12.0 -|- 53.86 + 47.76 1.45 -f 8.85,
Jan. 21.0 -f 54.23 + 47.25 0.96 + 8.63.
We now commence anew the table of integration, namely,
/ 7 7 / '/ 7 / '/ 7
+53.86 _ 002 + 2.26, +47.76 , 02 + 1.97, -1.45 _ Q2 -0.04,
+54.23 5421 + 2.24, +47.25 ^ ' + 1.99, -0.96 _ Q98 -0.06,
+56.45, +49.26, -1.04.
the formation of which is made evident by what precedes.
We may next assume for approximate values of the differential
coefficients, for the date March 1.0, +54.6, +46.7, and 0.5,
respectively; and these give, for this date,
NUMERICAL EXAMPLE. 451
dx = -f 56.45 + ~ = + 61.00,
1Z
fy = + 49.26 + ^. = + 53.15,
fe = l.04~4^ = 1.08.
By means of these approximate values we obtain the following
results :
1 864 March 1.0 " 2 = + 55.01, " 2 r= + 53.86, ^~ = - 1.00,
3r = + 71.03.
Introducing these -into the table of integration, we find, for the corre-
sponding values of the integrals,
tx = + 61.03, fy = + 53.75, da = 1.12.
These results differ so little from those already derived from the
assumed values of the function that a repetition of the calculation is
unnecessary. This repetition, however, gives
55.04,
Assuming, again, approximate values of the differential coefficients
for April 10.0, and computing the corresponding values of dx, dy,
and dz, we derive, for this date,
= + 48.06, = + 63.19,
Introducing these into the table of integration, and thus deriving
approximate values of dx, dy, and dz for May 20, we carry the pro-
cess one step further. In this manner, by successive approximations,
we obtain the following results :
Date.
1863 Dec. 12.0
+ 53.86
+ 47.76
1.45,
1864 Jan. 21.0
54/23
47.25
0.96,
March 1.0
55.04
53.91
1.00,
April 10.0
48.06
63.19
1.54,
May 20.0
32.85
65.40
2.07,
June 29.0
16.74
54.48
1.75,
Aug. 8.0
8.62
31.39
0.36,
Sept. 17.0
+ 14.20
+ 2.09
+ 1.86,
452
THEORETICAL ASTRONOMY.
Date. *
1864 Oct. 27.0 + 34.84
Dec. 6.0 68.79
1865 Jan. 15.0 + 112.64
26.32
47.87
58.39
w2 ^
+ 4.44,
6.86,
+ 8.68.
The complete integration may now be effected, and we may use both
equation (37) and equation (39), the former giving the integral for
the dates Jan. 1.0, Feb. 10.0, March 21.0, &c., and the latter the
integrals for the dates in the foregoing table of values of the function.
The final results for the perturbations of the rectangular co-ordinates,
expressed in units of the seventh decimal place, are thus found to be
the following :
Berlin Mean Time.
6x
6y
9*
1863 Dec. 12.0
+ 6.7
+ 5.9
-0.2,
1864 Jan. 1.0
0.0
0.0
0.0,
21.0
+ 6.8
5.9
0.1,
Feb. 10.0
27.1
23.5
0.5,
March 1.0
61.0
53.7
1.1,
21.0
108.9
97.4
2.0,
April 10.0
169.7
155.7
3.1,
30.0
242.7
229.9
4.7,
May 20.0
325.7
320.3
6.7,
June 9.0
417.1
427.2
9.3,
29.0
514.6
549.1
12.3,
July 19.0
616.1
684.9
15.7,
Aug. 8.0
720.8
831.4
19.5,
28.0
827.4
986.0
23.4,
Sept. 17.0
936.8
1144.6
27.0,
Oct. 7.0
1049.4
1303.8
30.2,
27.0
1168.2
1460.0
32.6,
Nov. 16.0
1295.4
1609.4
33.9,
Dec. 6.0
1435.6
1749.6
33.8,
26.0
1592.8
1877.6
32.0,
1865 Jan. 15.0
+ 1^72.6
+ 1992.3
28.2.
During the interval included by these perturbations, the terms of
the second order of the disturbing forces will have no sensible effect;
but to illustrate the application of the rigorous formulae, let us com-
mence at the date 1864 Sept. 17.0 to consider the perturbations of
the second order.
In the first place, the components of the disturbing force must be
computed by means of the equations
NUMERICAL EXAMPLE. 453
*Z= rfm'tf ( Z -^ 4, V
\ ^s 575 j
The approximate values of Sx, dy, and dz for Sept. 17.0 given imme-
diately by the table of integration extended to this date, will suffice
to furnish the required values of the disturbed co-ordinates by means
of
and to find p = p Q + dp y we have
or
in which ^ is the modulus of the system of logarithms. Thus we
obtain, for Sept. 17.0,
d log ^ = + 0.0000084,
* Y= + 32.48, >*Z = + 2.08,
which require no further correction.
Next, we compute the values of
which also will not require any further correction, and thus we form,
according to (12), the equation
q = 0.29996&C + 0.29815fy 0.03237&.
The approximate values of dx, dy, and dz being substituted in thia
equation, we obtain
q= + 0.0000061,
corresponding to which Table XVII. gives
log/= 0.477115.
Hence we derive
7.2 /M 2 P
(f qx -3x*) = - 44.87, S3* (fqy - 9y) = - 30.40,
454 THEORETICAL ASTRONOMY.
and the equations (14) give
= + 14.22, = + 2.08,
These values being introduced into the table of integration, the
resulting values of the integrals are changed so little that a repetition
of the calculation is not required.
We now derive approximate values of dx, %, and dz for Oct. 27.0,
and in a similar manner we obtain the corrected values of the differ-
ential coefficients for this date ; and thus by computing the forces for
each place in succession from approximate values of the perturbations,
and repeating the calculation whenever it may appear necessary, we
may determine the perturbations rigorously for all powers of the
masses. The results in the case under consideration are the follow-
ing:
J^/CttC/*
dP
dp
dt z
1864 Sept. 17.0
+ 14.22
+ 2.08
+ 1.87,
Oct. 27.0
34.84
26.31
4.44,
Dec. 6.0
68.77
47.86
6.86,
1865 Jan. 15.0 + 112.60 58.39 + 8.68.
Introducing these results into the table of integration, the integrals
for Jan. 15.0 are found to be
fa = + 1772.6, dy = -f 1992.3, 8z = 28.2,
agreeing exactly with those obtained when terms of the order of the
square of the disturbing forces are neglected.
If the perturbations of the rectangular co-ordinates referred to the
equator are required, we have, whatever may be the magnitude of the
perturbations,
dx, = dx,
dy f = COS e dy sin e dz, (41 )
dz f = sin e dy -f- cos e dz,
x n y,, z, being the co-ordinates in reference to the equator as the fun
damental plane. Thus we obtain, for 1865 Jan. 15.0,
dx, = + 1772.6, dy, = + 1838.9, 8z, == + 767.2.
These values, expressed in seconds of arc of a circle whose radius is
the unit of space, are
dx= + 36".562, dy, = + 37".930, dz, = + 15".825.
VARIATION OF CO-ORDINATES. 455
The approximate geocentric place of the planet for the same date is
a = 183 28', S = 5 39', log A = 0.3229,
and hence, neglecting terms of the second order, we derive, by means
of the equations (3) 2 , for the perturbations of the geocentric right
ascension and declination,
Aa = 17".03, A* = + 5".67.
167. The values of dx, dy, and dz, computed by means of the co-
ordinates referred to the ecliptic and mean equinox of the date t, must
be added to the co-ordinates given by the undisturbed elements and
referred to the same mean equinox. The co-ordinates referred to the
ecliptic and mean equinox of t may be readily transformed into those
referred to the ecliptic and mean equinox of another date t'. Thus,
let # denote the longitude of the descending node of the ecliptic of t f
on that of , measured from the mean equinox of t, and let TJ be the
mutual inclination of these planes; then, if we denote by a?', y f , z r
the co-ordinates referred to the ecliptic of t as the fundamental plane,
the positive axis of x 9 however, being directed to the point whose
longitude is 6, we shall have
x' = x cos -f- y sin 0,
(42)
Let us now denote by a?", y", z" the co-ordinates when the ecliptic
of t is the plane of xy, the axis of x remaining the same as in the
system of x', y f , z'. Then we shall have
y" = y f cos i) z sin 17, (43)
2" = y' sin f) -f- d cos ^.
Finally, transforming these so that the axis of z remains unchanged
while the positive axis of x is directed to the mean equinox of t, and
denoting the new co-ordinates by a?,, y,, z n we get
. x, = x" cos (0 + p) f sin (0 + p\
y, = x" sin (0 + p) + y" cos (0 + p\ (44)
,=",
in which p denotes the precession during the interval t' t. Elimi-
nating x", y", and z" from these equations by means of (43) and (42),
observing that, since y is very small, we may put cos 7 = 1, we get
456 DHEOKETICAL ASTRONOMY.
x, x cosp y sin p -f- - z sin (0 -f~ P)>
8
y, xsmp-\-y cosp - z cos (0 + P)> (45)
s
z, =z - x sin 6 -f - y cos 0,
9 S
in which s = 206264.8, rj being supposed to be expressed in seconds
of arc. If we neglect terms of the order j? 3 , these equations become
A^2 (V\ y*
x, = x ^x - y + - ( s i n ^ + P cos 0) 2 >
"3 S S
& = y i|rY+f*- 7(008* .p sin*)*, (46)
o S o
2, = 2 - x sin + - y cos 0.
s s
These formulae give the co-ordinates referred to the ecliptic and mean
equinox of one epoch when those referred to the ecliptic and mean
equinox of another date are known. For the values of p, y, and 0,
we have
p = (50".21129 + 0".0002442966r) (f t\
TJ=( (T.48892 0".000006143r) (*'-*),
= 351 36' 10" + 39".79 (t 1750) 5".21 (if 0,
in which r = \(t f t) 1750, t and t' being expressed in years from
the beginning of the era. If we add the nutation to the value of p,
the co-ordinates will be derived for the true equinox of t'.
The equations (45) and (46) serve also to convert the values of dx y
%, and dz belonging to the co-ordinates referred to the ecliptic and
mean equinox of t into those to be applied to the co-ordinates re-
ferred to the ecliptic and mean equinox of t' . For this purpose it
is only necessary to write dx, <%, and dz in place of x, y y and z re-
spectively, and similarly for x,, y n z,.
In the computation of the perturbations of a heavenly body during
a period of several years, it will be convenient to adopt a fixed equi-
nox and ecliptic throughout the calculation ; but when the perturba-
tions are to be applied to the co-ordinates, in the calculation of an
ephemeris of the body taking into account the perturbations, it will
be convenient to compute the co-ordinates directly for the ecliptic
and mean equinox of the beginning of the year for which the
ephemeris is required, and the values of dx, dy, and dz must be
reduced, by means of the equations (45), as already explained, from
the ecliptic and mean equinox to which they belong, to the ecliptic
and mean equinox adopted in the case of the co-ordinates required.
VARIATION OF CO-ORDINATES. 457
In a similar manner we may derive formulae for the transformation
of the co-ordinates or of their variations referred to the mean equinox
and equator of one date into those referred to the mean equinox
and equator of another date; but a transformation of this kind will
rarely be required, and, whenever required, it may be effected by first
converting the co-ordinates referred to the equator into those referred
to the ecliptic, reducing these to the equinox of t 1 by means of (45)
or (46), and finally converting them into the values referred to the
equator of t'. Since, in the computation of an ephemeris for the
comparison of observations, the co-ordinates are generally required
in reference to the equator as the fundamental plane, it would appear
preferable to adopt this plane as the plane of xy in the computation
of the perturbations, and in some cases this method is most advan-
tageous. But, generally, since the elements of the orbit of the dis-
turbed planet as well as the elements of the orbits of the disturbing
bodies are referred to the ecliptic, the calculation of the perturbations
will be most conveniently performed by adopting the ecliptic as the
fundamental plane. The consideration of the change of the position
of the fundamental plane from one epoch to another is thus also ren-
dered more simple. Whenever an ephemeris giving the geocentric
right ascension and declination is required, the heliocentric co-ordi-
nates of the body referred to the mean equinox and equator of the
beginning of the year will be computed by means of the osculating
elements corrected for precession to that epoch, and the perturbations
of the co-ordinates referred to the ecliptic and mean equinox of any
other date will be first corrected according to the equations (46), and
then converted into those to be applied to the co-ordinates referred to
the mean equinox and equator. If the perturbations are not of con-
siderable magnitude and the interval t' t is also not very large, the
correction of dx, dy y and dz on account of the change of the position
of the ecliptic and of the equinox will be insignificant; and the
conversion of the values of these quantities referred to the ecliptic
into the corresponding values for the equator, is effected with great
facility.
In the determination of the perturbations of comets, ephemerides
being required only during the time of describing a small portion of
their orbits, it will sometimes be convenient to adopt the plane of the
undisturbed orbit as the fundamental plane. In this case the posi-
tive axis of x should be directed to the ascending node of this plane
on the ecliptic, and the subsequent change to the ecliptic and equinox,
whenever it may be required, will be readily effected.
458 THEORETICAL ASTRONOMY.
168. The perturbations of a heavenly body may thus be deter-
mined rigorously for a long period of time, provided that the oscu-
lating elements may be regarded as accurately known. The peculiar
object, however, of such calculations is to facilitate the correction of
the assumed elements of the orbit by means of additional observa-
tions according to the methods which have already been explained;
and when the osculating elements have, by successive corrections,
been determined with great precision, a repetition of the calculation
of the perturbations may become necessary, since changes of the ele-
ments which do not sensibly affect the residuals for the given differ-
ential equations in the determination of the most probable corrections,
may have a much greater influence on the accuracy of the resulting
values of the perturbations.
When the calculation of the perturbations is carried forward for a
long period, using constantly the same osculating elements, and
those which are supposed to require no correction, the secular per-
turbations of the co-ordinates arising from the secular variation of
the elements, and the perturbations of long period, will constantly
affect the magnitude of the resulting values, so that fix, Sy, and $2
will not again become simultaneously equal to zero. Hence it
appears that even when the adopted elements do not differ much
from their mean values, the numerical amount of the perturbations
may be very greatly increased by the secular perturbations and by
the large perturbations of long period. But when the perturbations
are large, the calculation of the complete values of , , 2 > , 2 > and
j- (which is effected indirectly) cannot be performed with facility,
requiring often several repetitions in order to obtain the required
accuracy, since any error in the value of the second differential coeffi-
cient produces, by the double integration, an error increasing propor-
tionally to the time in the values of the integral. Errors, therefore,
in the values of the second differential coefficients which for a mode-
rate period would have no sensible effect, may in the course of a long
period produce large errors in the values of the perturbations, and it
is evident that, both for convenience in the numerical calculation and
for avoiding the accumulation of error, it will be necessary from time
to time to apply the perturbations to the elements in order that the
integrals may, in the case of each of the co-ordinates, be again equal
to zero. The calculation will then be continued until another change
of the elements is required.
CHANGE OF THE OSCULATING ELEMENTS. *59
The transformation from a system of osculating elements for one
epoch to that for another epoch is very easily effected by means of
the values of the perturbations of the co-ordinates in connection
with the corresponding values of the variations of the velocities
-T-, -J-, and -rr- The latter will be obtained from the values of the
at at at
second differential coefficients by means of a single integration ac-
cording to the equations (27) and (32). Thus, in the case of the
example given, we obtain for the date 1865 Jan. 15.0, by means of
(32), in units of the seventh decimal place,
= + 385.9, 40^ = + 214.6, 40^ = + 9.7.
at at
The velocities in the case of the disturbed orbit will be given by the
formulae
dx_ _dx d8x dy dy d$y dz _ dz ddz ( .
"dt~~~3r~T"dT ~dt~~~dt~~~dt' ~dt '"" "dt '" W ^ '
To obtain the expressions for the components of the velocity
resoHed parallel to the co-ordinates, we have, according to the equa-
tions (6) 2 ,
dx . . f . , dr . f . , dv
-j- = sma sm ( A -f u) -f- r sin a cos ( A -f- u) -j-,
at at at
-~ sin b sin (5 -J- u) r- -f- r sin b cos (L -f- u) =-
at at at
dz . rsi . \dr * ff , \ dv
-jr = sm c sm ( C + w) -TT + r sm e cos ( C -f- u) -j--
These equations are applicable in the case of any Fundamental plane,
if the auxiliaries sin a, sin 6, sin c, A, B, and C are determined in
reference to that plane. To transform them still further, we have
rfr
dt
=
dt
in which to denotes the angular distance of the J erihelion from the
ascending node. Substituting these values, we ot tain, by reduction,
460 THEORETICAL ASTRONOMY.
dx
/- ((e cos -f- cos u) cosB (e sin w -f- sin u) sin 5) sin 6,
efe kv\ -\-iMf, N /~ / -f- sin w) = Fsin U,
(48)
(e cosa> -f- cos if) = Fcos i7,
KJ
and we have
= Fsin a cos (J. -f U"),
-|- = Fsin b cos (5 + CO, (49)
dt
^=Fsinccos(a4- CO-
These equations determine the components of the velocity of a hea-
venly body resolved in directions parallel to the co-ordinate axes,
and for any fundamental plane to which the auxiliaries A, B, &c.
belong. When the ecliptic is the fundamental plane, we have
sin c = sin i, (7=0.
The sum of the squares of the equations (48) gives
(*-)'
P
and hence it appears that F is the linear velocity of the body.
The determination of the osculating elements corresponding to any
date for which the perturbations of the co-ordinates and of the veloci-
ties have been found, is therefore effected in the following manner :
First, by means of the osculating elements to which the perturba-
tions belong, we compute accurate values of r , a? , y Q , z , and by
means of the equations (48) and (49) we compute the values of -~,
-jp and -7' Then we apply to these the values of the perturba-
tions, and thus find x, y, z, -J-, ~jt, and -^-- These having been
CHANGE OF THE OSCULATING ELEMENTS. 461
found, the equations (32)! will furnish the values of ft, i, and p;
and the remaining elements may be determined as explained in Art*
112. Thus, from
Vr sin * = kVp (1
dx . dy , dz
we obtain Vr and and from
r sin u = ( x sin ft + y cos ft) sec t,
r cos it = a; cos ft
we derive r and u; and hence Ffrom the value of Vr. When i is
not very small, we may use, instead of the preceding expression for
r sin u,
r sin u = z cosec i.
Next, we compute a from
2a r = ;
and from
2ae sin o> = (2a r) sin (2^ -}- w) r sin w,
2ae cos a> = (2a r) cos (2^ + M ) r cos w >
we find to and e. The mean daily motion and the mean anomaly or
the mean longitude for the epoch will then be determined by means
of the usual formulae.
In the case of a very eccentric orbit, after r and u have been found,
-r- will be given by equations (48) 6 , and the values of e and v will
be given by the equations (49) 6 . Then the perihelion distance will
be found from
P
and the time of perihelion passage will be found from v and e by
means of Table IX. or Table X.
In the numerical values of the velocities -rr -77, &c., more decimals
at at
must be retained than in the values of the co-ordinates, and enough
must be retained to secure the required accuracy of the solution. If
it be considered necessary, the different parts of the calculation may
be checked by means of various formulse which have already been
given. Thus, the values of ft and i must satisfy the equation
462 THEORETICAL ASTRONOMY.
z cos i y sin i cos & -f x sin i sin & = 0.
We have, also,
r'^ + ^-f z\
z = r sin it sin i,
which must be satisfied by the resulting values of F, r, and u; and
the values of a and e must satisfy the equation
p = a (1 e 2 ) a cos* ?>.
169. When the plane of the undisturbed orbit is adopted as the
fundamental plane, we obtain at once the perturbations
d (r cos u), 8 (r sin u), fa,
and from these the perturbations of the polar co-ordinates are easily
derived. There are, however, advantages which may be secured by
employing formulae which give the perturbations of the polar co-or-
dinates directly, retaining the plane of the orbit for the date t as the
fundamental plane.
Let w denote the angle which the projection of the disturbed
radius-vector on the plane of xy makes with the axis of x, and /9 the
latitude of the body with respect to the plane of xy; then we shall
have
x = r cos ft cos w,
y = r cos ft sin w, (50)
z = r sin ft.
Let us now denote by X, Y y and Zj respectively, the forces which are
expressed by the second members of the equations (1), and the first
two of these equations give
C being the constant of integration. The equations (50) give
dx d(rcosft} Q . dw
- = cos w j. - r cos ft sm w -r
dt dt at
dy . d(rcoaft) . Q dw
~- = smw j. - -f r cos ft cosw -=-,
dt at at
and hence
dy dx . dw
- -
VARIATION OF POLAR CO-ORDINATES. 463
Therefore we have
r'cos 2 ^T= (Yx Xy)dt+ C.
ut %/
If we denote by S the component of the disturbing force in a direc-
tion perpendicular to the disturbed radius-vector and parallel with
the plane of xy, we shall have
X = S sin iv, Y= S cos w,
and
Therefore
r 2 cos 2 /5^-= Cs 9 r f'g"=l-^, (61)
we have
'"=rw- (63)
and hence
Finally, we have, from the last of equations (1),
db *'(l+m)
3? = *- -? -- *> (64)
by means of which the value of z may be found, since, in the case of
the undisturbed motion, we have z = 0.
The values of/' corresponding to different values of q' may be
tabulated with the argument 2 under the sign of integration, this integral,
omitting the factor at in the formulae of integration, will become
(*)J S r cos /9 dtj as required. The last term of the equation will be
multiplied by a).
In the case of dr, each term of the equation for - must contain
(Jut
the factor co 2 . If the second of equations (65) is employed, the first
and third terms of the second member will be multiplied by co 2 -, but
since the value of S Q is supposed to be already multiplied by o> 2 , the
second term will only be multiplied by a).
The perturbations may be conveniently determined either in units
of the seventh decimal place, or expressed in seconds of arc of a
circle whose radius is unity. If they are to be expressed in seconds,
the factor s = 206264.8 must be introduced so as to preserve the
homogeneity of the several terms, and finally dr and dz must be con-
verted into their values in terms of the unit of space.
172. It remains yet to derive convenient formula for the deter-
mination of the forces S , R, and Z. For this purpose, it first becomes
necessary to determine the position of the orbit of the disturbing
planet in reference to the fundamental plane adopted, namely, the
plane defined by the osculating elements of the disturbed orbit at the
instant 1 Q . Let V and & ' denote the inclination and the longitude of
the ascending node of the disturbing body with respect to the ecliptic,
and let /denote the inclination of the orbit of the disturbing body
with respect to the fundamental plane. Further, let N denote the
longitude of its ascending node on the same plane measured from the
ascending node of this plane on the ecliptic or from the point whose
longitude is & , and let N f be the angular distance between the as-
cending node of the orbit of the disturbing body on the ecliptic and
the ascending node on the fundamental plane adopted. Then, from
the spherical triangle formed by the intersection of the plane of the
THEORETICAL ASTRONOMY.
ecliptic, the fundamental plane, and the plane of the orbit of the dis-
turbing body with the celestial vault, we have
sin ^Isin J (.y + N') = sin (&' - ) sin J (i' + i ),
from which to find N 9 N f , and /.
Let ft' denote the heliocentric latitude of the disturbing planet
with respect to the fundamental plane, w r its longitude in this plane
measured from the axis of x, as in the case of w, and u Q f the argu-
ment of the latitude with respect to this plane. Then, according to
the equations (82) w we have
tan (V N) = tan u ' cos I,
If u r denotes the argument of the latitude of the disturbing planet
with respect to the ecliptic, we have
u ' = u' N'. (68)
This formula will give the value of u f , and then w f and ft' will be
found from (67). We have, also,
cos u Q r = cos ft cos (w' N\
which will serve to indicate the quadrant in which w r J\Tmust be
taken.
The relations here derived are evidently applicable to the case in
which the elements of the orbits of the disturbed and disturbing
planets are referred to the equator, the signification of the quantities
involved being properly considered.
The co-ordinates of the disturbing planet in reference to the plane
of the disturbed orbit at the instant t as the fundamental plane will
be given by
x' = r' cos ft cos w'j
i/ = r' cospsinw', (691
To find the force R, we have
R = X-+ Y y - + Z-.
r ' r ' r
VARIATION OF POLAK CO-ORDINATES. 469
and
Substituting in these the values of x' t y f , z' given by (69), and the
corresponding values of x, y y z given by (50), and putting
* = ^~' (70)
we get
. R = m'k* I h r' cos /3' cos /? cos (w w) + /i r' sin sin /5' -^ ). (71)
The equation
S rcosj3= Yx- Xy
gives
S Q = m 'k* h r' cos f sin (w r w), (72)
from which to find S . Finally, we have
Z=m f k* lhr f sin p - 8 ), (73)
from which to find Z.
When we determine the perturbations only with respect to the
first power of the disturbing force, the expressions for R, S Q , and Z
become
jB = w'* i ( hr' cospcosW wJ^},
\ Po I (74)
S = m'k* h r' cos jf sin (w f wj,
Z =
To compute the distance p, we have
, = (-,/ _ .) + (y' -
which gives
^j _ r ' _j_ r 2 _ 2 r / cos ^9 cos ft cos (/ w) 2r r' sin sin ^, (75)
and, if we neglect terms of the second order, we have
P * = r' 2 + r 2 2r / cos p cos (w f w ). (76)
If we put
cos r = cos cos ' cos (w' w) + sin sin f, (77)
we have
2rr'cosr
+ (r r' cos r) 8 ;
470 THEOEETICAL ASTKONOMY.
and hence we may readily find p from
^ '
the exact value of the angle n, however, not being required.
Introducing f into the expression for R, it becomes
(79)
by means of which R may be conveniently determined.
173. When we neglect the terms depending on the squares and
higher powers of the masses in the computation of the perturbations,
the forces jR, S Q , and Z will be computed by means of the equations
(74), p Q being found from (76) or from (78), when we put
cos f = cos p cos (w f w ).
But when the terms of the order of the square of the disturbing
force are to be taken into account, the complete equations must be
used. Thus, we find p from (78), S from (72), Z from (73), and R
from (71) or (79). The values of dw, dr, and z, computed to the
point at which it becomes necessary to consider the terms of the
second order, will enable us at once to estimate the values of the
perturbations for two or three intervals in advance to a degree of
approximation sufficient for the calculation of the forces; and the
values of R, 8 Q , and Z thus found will not require any further cor-
rection.
When the places of the disturbing planet are to be derived from
an ephemeris giving the heliocentric longitudes and latitudes, the
values of & ' and i r will be obtained from two places separated by a
considerable interval, and then the values of u f will be determined
by means of the first of equations (82) L or by means of (85) x . When
the inclination V is very small, it will be sufficient to take
u' = l'R' + 8 tan 2 Jt* sin 2 (l r - ft'),
in which s = 206264.8. But when the tables give directly the lon-
gitude in the orbit, u f + &', by subtracting &' from each of these
longitudes we obtain the required values of u r .
It should be observed, also, that the exact determination of the
values of the forces requires that the actual disturbed values of r',
10', and /?' should be used. The disturbed radius-vector r' will be
VARIATION OF POLAR CO-ORDINATES. 471
given immediately by the tables of the motion of the disturbing
body, but the determination of the actual values of w' and ft' re-
quires that we should use the actual values of N', N y and I in the
solution of the equations (68) and (67). Hence the disturbed values
of & ' and i f should be used in the determination of these quantities
for each date by means of (66). It will, however, generally be tho
case that for a moderate period the variation of &' and i f may be
neglected; and whenever the variation of either of these has a sensi-
ble effect, we may compute new values of N 9 N r , and / from time to
time, by means of which the true values may be readily interpolated
for each date. We may also determine the variations of N, N f , and
/ arising from the variation of &' and i 1 , by means of differential
formulae. Thus the relations will be similar to those given by the
equations (71) 2 , so that we have
sin-ZV'
. ,_., - ;
sin(' Q.) smJ
31 = sin N' sin i' $&' -f cos N' 8i r ,
fh>m which to find dN', 3N, and 81.
When the perturbations are computed only in reference to the first
power of the mass, the change of Q f and i f may be entirely neg-
lected; but when the perturbations are to be computed for a long
period of time, and the terms depending on the squares and products
of the disturbing forces are to be included, it will be advisable to
take into account the values of dN, 3N f , and dl, and, using also the
value of u' in the actual orbit of the disturbing body, compute the
actual values of w' and ft'.
In the case of several disturbing bodies, the forces will be deter-
mined for each of these, and then, instead of R, S , and Z, in the
formulae for the differential coefficients, 2R, 28 W and 2 Z will be used.
174. By means of the values of dw, fir, and z, the heliocentric or
(he geocentric place of the disturbed planet may be readily found.
Thus, let the positive axis of x be directed to the ascending node of
the osculating orbit at the instant t on the plane of the ecliptic;
then, in the undisturbed orbit, we shall have
u denoting the argument of the latitude. Let # y,, z, be the co-or-
472 THEORETICAL ASTRONOMY.
dinates of the body referred to a system of rectangular co-ordinates
in which the ecliptic is the plane of xy, and in which the positive
axis of x is directed to the vernal equinox. Then we shall have
x, = x cos & y cosi sin & + z sin i Q sin ,
y, = x sin & + y cosi Q cos & * sin i Q cos & ,
z,=y sin i + z cos t ,
or, introducing the values of x and y given by (50),
Xf=r cos /9 cos w cos & r cos ft sin w cos i sin ^ -f- 2 sin i sin ^ ,
y, =r cos /? cos w sin Q -f- r cos /5 sin w cos i cos & z sin i cos & , (81)
z t =r cos /? sin w sin i -f- 2 cos i .
Introducing also the auxiliary constants for the ecliptic according to
the equations (94^ and (96) w we obtain
x t = r cos /? sin a sin ( J. -(- w) -f- 2 cos a,
y, =r cos /3 sin 6 sin (^ -f w) -f z cos 6, (82)
2, = r cos /? sin i sin w + 2 cos i ,
by means of which the heliocentric co-ordinates in reference to the
ocliptic may be determined.
If the place of the disturbed body is required in reference to the
equator, denoting the heliocentric co-ordinates by x fn y,, y z,,, and the
obliquity of the ecliptic by e, we have
x n = x,
y,,=y f cose z, sine,
z n = y f sin e -f- z, cos e.
Substituting for x n y n z, their values given by (81), and introducing
the auxiliary constants for the equator, according to the equations
'99) x and (101) w we get
x n = r cos /? sin a sin (A -f- w} -\- z cos a,
y lt = r cos /3 sin b sin (B -f- w) -f- z cos b, (83)
z t , = r cos ft sin c sin ( (7 -j- w) -\- z cos c.
The combination of the values derived from these equations with the
corresponding values of the co-ordinates of the sun, will give the
required geocentric places of the disturbed body. These equations
are applicable to the case of any fundamental plane, provided that
the auxiliary constants a, A, b, jB, &c. are determined with respect
to that plane. In the numerical application of the formulae, the
value of w will be found from
w = it, -f- dw t
VAEIATION OF POLAR CO-ORDINATES. 473
U Q being the argument of the latitude for the fundamental osculating
elements, and care must be taken that the proper algebraic sign is
assigned to cos a, cos b, and cos c.
If the values of TT O , & , and i Q used in the calculation of the per-
turbations are referred to the ecliptic and mean equinox of the date
t ', and the rectangular co-ordinates of the disturbed body are required
in reference to the ecliptic and mean equinox of the date t ", the
value of w must be found from
the value of W Q referred to the ecliptic of t f being reduced to that of
", by means of the first of equations (115)^ Then & and i Q should
be reduced from the ecliptic and mean equinox of t ' to the ecliptic
and mean equinox of t Q " by means of the second and third of the
equations (115)^ and, using the values thus found in the calculation
of the auxiliary constants for the ecliptic, the equations (82) will
give the required values of the heliocentric co-ordinates. If the co-
ordinates referred to the mean equinox and equator of the date t Q "
are to be determined, the proper corrections having been applied to
Q Q and i w the mean obliquity of the ecliptic for this date will be
employed in the determination of the auxiliary constants a, J., &c.
with respect to the equator, and the equations (83) will then give
the required values of the co-ordinates.
If we differentiate the equations (83), we obtain, by reduction,
-^ = r cos ft sin a cos ( A -f w) rr -f- sec ft sin a sin ( A -j- w) j-
-f (cos a tan ft sin a sin (A + w)) -rrt
^- = r cos ft sin b cos (B -j- w) -7- -j- sec ft g i n & sm C^ + tu) -jr
dz (84)
-f (cos b tan ft sin b sin (J5 +w)) -77.
A' =rcos/5sinccos((7-f w)- + sec ft sin c sin ( C + w) -j-
dt at
-|-(cos c tan ft sin c sin ( C + to)) -?-,
by means of which the components of the velocity of the disturbed
body in directions parallel to the co-ordinato axes may be determined.
The values of -^ and -^ will be obtained from -^ and -^ by a
single integration, and then we have
474 THEORETICAL ASTRONOMY.
dw &V / pT(l-hm) , d$w dr k\/l-\-m
,
I ~ - -3P I -^T -di' (85)
from which to find =- and :-
at at
175. EXAMPLE. In order to illustrate the calculation of the per-
turbations of r, w, and z, let us take the data given in Art. 166, and
determine these perturbations instead of those of the rectangular co-
ordinates.
In the first place, we derive from the tables of the motion of
Jupiter the values
' = 98 58' 22".7, i' = 1 18' 40".5,
which refer to the ecliptic and mean equinox of 1860.0. We find,
also, from the data given by the tables the values of u' measured
from the ecliptic of 1860.0. Then, by means of the formulae (66),
using the values of & and i given in Art. 166, we derive
N= 194 0' 49".9, N' = 301 38' 31".7, 1= 5 9' 56".4.
The value of u f is given by equation (68), and then w f and $' are
found from the equations (67). Thus we have
Berlin Mean Time. log r w = UQ log r wf ft'
1863 Dec. 12.0, 0.294084 192 4' 24".5 0.73425 14 18' 54".6 V 38".l
1864 Jan. 21.0, 0.294837 207 40 52 .2 0.73368 17 21 44 .2 18 9 .1
March 1.0, 0.300674 223 3 5 .9 0.73305 20 25 5 .2 34 39 .9
April 10.0, 0.310864 237 51 38 .3 0.73237 23 28 59 .8 51 7 .6
May 20.0, 0.324298 251 52 47 .9 0.73164 26 33 32 .1 17 29 .7
June 29.0, 0.339745 264 59 30 .0 0.73086 29 38 44 .8 1 23 43 .5
Aug. 8.0, 0.356101 277 10 24 .6 0.73003 32 44 41 .2 1 39 46 .3
Sept. 17.0, 0.372469 288 28 4 .1 0.72915 35 51 24 .6 1 55 35 .2
Oct. 27.0, 0.388214 298 57 16 .3 0.72823 38 58 57 .5 2 11 7 .5
Dec. 6.0, 0.402894 308 43 48 .7 0.72726 42 7 23 .3 2 26 20 .3
1865 Jan. 15.0, 0.416240 317 53 39 .1 0.72625 45 16 43 .9 2 41 10 .6
The values of p may be found from (76) or (78) as already given in
Art. 166.
The forces R, S , and Z may now be determined by means of the
equations (74), h being found from (70), and if we introduce the
factor (o 2 for convenience in the integration, as already explained, we
obtain the following results :
Date. 6> 2 .K u*S r u*Z
1863 Dec. 12.0, + 1".4608 + 0".1476 4 0".0009 + 0".0282
1864 Jan. 21.0. + 1 .4223 .6757 4- .0101 - .2361
NUMERICAL EXAMPLE. 475
Date.
IR
rfS r
tfZ
u\ S r dt
1864 March 1.0,
+ 1".2616
1".4512
-} 0".0190
1".3060
April 10.0,
1 .0018
2 .1226
.0273
3 .1035
May 20.0,
.6760
2 .6473
.0347
5 .5020
June 29.0,
+ .3179
2 .9988
.0406
8 .3402
Aug. 8.0,
.0452
3 .1650
.0449
11 .4378
Sept. 17.0,
.3944
3 .1437
.0470
14 .6076
Oct. 27.0,
.7180
2 .9392
.0466
17 .6640
Dec. 6.0, 1 .0097 2 .5586 .0432 20 .4273
1865 Jan. 15.0, 1 .2674 - 2 .0081 + .0362 22 .7245
The integral wl S Q r dt is obtained from the successive values of Q)*S Q r 9
by means of the formula (32).
Next we compute the values of the differential coefficients by
means of the formulae (65). For the dates 1863 Dec. 12.0 and 1864
Jan. 21.0 we may first assume dr = Q, and, by a preliminary inte-
gration, having thus derived very approximate values of dr for these
dates, the values of ^- will be recomputed. Then, commencing
anew the table of integration, we may at once derive an approximate
value of dr for the date March 1 .0 with which the last term of the
expression for 7 may be computed. Continuing this indirect pro-
cess, as already illustrated in the case of the perturbations of the rec-
tangular co-ordinates, we obtain the required values of the second
differential coefficient. In a similar manner, the values of -TZ will
be obtained. The values of =r will then be given directly by means
of the first of equations (65) ; and the final integration will furnish
the perturbations required. Thus we derive the following results :-
1863 Dec. 12.0, 0".0423 -j-l".4509 +0".0009 0".00 + 0".18 +0".00
1864 Jan. 21.0, .1086 1 .3405 .0101 .02 .17 .00
Mar. 1.0, .7162 +0 .7829 .0183 .40 1 .47 .01
Apr. 10.0, 1.61140.0455 0.0251 1.55 3.53 0.04
May 20.0, 2 .4795 .9344 .0300 3 .61 5 .54 .09
June 29.0, 3 .0807 1 .7333 .0326 6 .42 6 .62 .18
Aug. 8.0, 3 .2971 2 .3752 .0331 9 .64 5 .98 .29
Sept. 17.0, 3 .1080 2 .8533 .0311 12 .88 +2 .98 .44
Oct. 27.0, 2 .5425 3 .1872 4-0 .0265 15 .73 2 86 +0 .62
476 THEORETICAL ASTRONOMY.
1864 Dec. 6.0, 1".6443 3".4009 -f 0".0190 17".85 11".88 +0".83
1865 Jan. 15.0, .45113 .5334 -fO .007918 .9224 .29+1 .05
It has already been found that, during the period included by these
results, the perturbations arising from the squares and products of
the disturbing forces are insensible, and hence the application of the
complete equations for the forces and for the differential coefficients
is not required. The equations (83) will give, by means of the
results for w u -f- dw, r = r -f- dr, and z, the values of the helio-
centric co-ordinates of the disturbed body, and the combination of
these with the co-ordinates of the sun will give the geocentric place.
When we neglect terms of the second order, we have, according to
the equations (84),
y
dx lf = X Q cot ( A -j- w] dw -f _ dr -f z cos a,
r o
dy,, = y cot (B -f w) dw -f ?? dr -f z cos b, (86)
r o
dz n = z cot (C -{- w) dw -{- $r -\- z cos c,
r o
the heliocentric co-ordinates x 0) y 0) Z Q being referred to the same fun-
damental plane as the auxiliary constants, a, 6, A, &c. Thus, in the
case of Eurynome, to find the perturbations of the rectangular co-or-
dinates, referred to the ecliptic and mean equinox of 1860.0, from
1864 Jan. 1.0 to 1865 Jan. 15.0, we have
A = 296 34 X 37 x/ .5, B = 206 43 7 34".4, C=Q,
log cos a = 8.557354n, log cos b = 8.856746, log cos c = log cos i = 9.998590,
log X Q = 0.399807 n , log y = 9.838709, log z = 9.148170,,
w = w + 6 W = 317 53 r 20 // .2,
and hence, by means of (86), we derive
3x, = + 36".559, fy, = + 41".083, dz, = 0".588.
If we express these in parts of the unit of space, and in units of the
seventh decimal place, we obtain
dx, = + 1772.4, dy, = + 1991.8, to, = 28.5,
agreeing with the results already obtained by the method of the va-
riation of rectangular co-ordinates, namely,
9x, = + 1772.6, fy, = f 1992.3, to, = 28.2.
CHANGE OF THE OSCULATING ELEMENTS. 477
176. By using the complete formulae, the perturbations of r, w,
and z may be computed with respect to all powers of the disturbing
force, and for a long series of years, using constantly the same fun-
damental osculating elements. But even when these elements are so
accurate as not to require correction, on account of the effect of the
large perturbations of long period upon the values of dw and dr, the
numerical values of the perturbations will at length be such that a
change of the osculating elements becomes desirable, so that the
integration may again commence with the value zero for the variation
of each of the co-ordinates. This change from one system of ele-
ments to another system may be readily effected when the values of
the perturbations are known. Thus, having found the disturbed
values of r, w, and z, we have
dv* .-did* , dp* Vp(l + ro)
_ = cos . /? _ + _ == __ -- ,
p being the semi-parameter of the instantaneous orbit of the disturbed
body. In the undisturbed orbit we have
_ dv
ffo ~ dt
and hence we derive
dv*
Substituting for -5- the value above given, there results
1 ddw
J n
by means of which p may be determined. To find --^ we have
d3 1 dz ian/3 dr
~dt rcos/5 ~df r dt
We have, also,
dr kVT+^n, . &l/l-f-m . dfr
- _ e sm v = -- / e sin v + -^-,
dt i/p Vp Q
and if we put
(89)
( ;
p,
478 THEORETICAL ASTRONOMY.
this equation becomes
e sin v = e n sin v -f- ae Q sin v -f- Y. (90)
We have, further.
ecosv = - 1,
r
and, putting
*?-!+* ()
we obtain
e cos r = e cos v -}- /5 .
r o
This equation, combined with (90), gives
e sin (v v ) = ae n sin v cos v -f- f cos v $ sin v ,
p (92)
e cos (v v ) = e -f- ae sin 2 v + r sin v + - ft cos v ,
r o
by means of which the values of e and v may be found, those of the
auxiliaries a, ft f t being found from (89) and (91). Then we have
e = sin , a=p sec* ,
p = ^ 1/1 + m , tan $E = tan (45 ?) tan |v,
a^
M=E esinE,
by means of which ^, a, //, and Jf may be determined. In the case
of orbits of great eccentricity, we find the perihelion distance from
and the time of perihelion passage will be derived from e and v by
means of Table IX. or Table X.
It remains yet to determine the values of 2, i, and a> or it. Let
6 C denote the longitude of the ascending node of the instantaneous
orbit on the plane of the osculating orbit, defined by & and i Q) mea-
sured from the origin of w, and let ^ denote its inclination to this
plane. Then we have
tan fj sin (w ) = tan /?,
. , dw O dp (93)
tan i? cos (w 0^ = sec' /? -^,
and hence
CHANGE OF THE OSCULATING ELEMENTS. 479
.dfiw
g + ~df
tan (w O = ^sin 2/9 ^ , (94)
"dT
by means of which may be found. The quadrant in which is
situated is determined by the condition that sin (w ) and tan /3
must have the same sign. The value of % will be found from the
first or the second of equations (93).
If we denote by the argument of the latitude of the disturbed
body with respect to the adopted fundamental plane, we have
cos>?
(95)
and the angle must be taken in the same quadrant as w .
Then, from the spherical triangle formed by the intersection of the
planes of the ecliptic and instantaneous orbit of the disturbed body,
and the fundamental plane, with the celestial vault, we derive
cos \ i sin ( (u C) -f- A (& & )) = sin J)) cos 2^0 cos i (*o ~f~ ^o);
sin ^ i sin (^(u C) i ( & &)) = sin A0 sin ^ (t ^ ),
sin \ i cos {^(u C) ^ ( & & )) :r= cos l^o s ^ n i (*o ~f~ 7o)'
These equations will furnish the values of i, u f , and ^
hence, since and & are given, those of R> and u. The value of v
having been already found, we have, finally,
a)=U V,
and the elements are completely determined. These elements will
be referred to the ecliptic and mean equinox to which & and i Q are
referred, and they may be reduced to the equinox and ecliptic of any
other date by means of the formulae which have already been given.
The elements of the instantaneous orbit of the disturbed body may
also be determined by first computing the values of #, y,,, z, n in
reference to the fundamental plane to which & and i are to be re-
ferred, by means of the equations (83), and also those of -|p -,'-', -~
by means of (85) and (84), and then determining the elements from
the co-ordinates and velocities, as already explained.
It should be observed that when the factor w 2 , or the square of the
480 THEORETICAL ASTRONOMY.
adopted interval, is introduced into the expressions for the forces and
differential coefficients, the first integrals will be
dSr dtiw dz
* "IT "dp
and that when these quantities are expressed in seconds of arc, they
must be converted into their values in parts of the unit of space
whenever they are to be combined with quantities which are not ex-
pressed in seconds. In other words, the homogeneity of the several
terms must be carefully attended to in the actual application of the
formulae.
When the elements which correspond to given values of the per-
turbations have been determined, if we compute the heliocentric
longitude and latitude of the body for the instant to which the ele
ments belong, the results should agree with those obtained by com-
puting the heliocentric place from the fundamental osculating ele-
ments and adding the perturbations.
177. The computation of the indirect terms when the perturba-
tions of the co-ordinates r, w, and z are determined, is effected with
greater facility than in the case of the rectangular co-ordinates,
although the final results are not so convenient for the calculation of
an ephemeris for the comparison of observations. This indirect cal-
culation, which, when the perturbations of any system of three co-
ordinates are to be computed, cannot in any case be avoided without
impairing the accuracy of the results, may be further simplified by
determining, in a peculiar form, the perturbations of the mean
anomaly, the radius-vector, and the co-ordinate z perpendicular to the
fundamental plane adopted.
Let the motion of the disturbed body be, at each instant, referred
to the plane of its instantaneous orbit; then we shall have /9 = 0,
and the equations (51) and (54) become
T> ^ = far dt + kl/p^l + m),
at J ,0-
d?r dw* , &'(! +m) _
dt* T dt 3 H r* '
in which R denotes the component of the disturbing force in the
direction of the disturbed radius- vector, and S the component in the
plane of the disturbed orbit and perpendicular to the disturbed radius-
vector, being positive in the direction of the motion. The effect of
VARIATION OF POLAR CO-ORDINATES. 481
the components R and S is to vary the form of the orbit and the
angular distance of the perihelion from the node. If we denote by
Z the component of the disturbing force perpendicular to the plane
of the instantaneous orbit, the eifect of this will be to change the
position of the plane of the orbit, and hence to vary the elements
which depend on the position of this plane.
Let us take a fixed line in the plane of the instantaneous orbit,
and suppose it to be directed from the centre of the sun to a point
whose angular distance back from the place of the ascending node is
0, and let the value of a be so taken that, so long as the position of
the plane of the orbit is unchanged, we shall have
The line thus taken in the plane of the orbit may be regarded as
fixed during all changes in the position of this plane. Let denote
the angle between this fixed line and the semi-transverse axis ; then
will
* = + ', (98)
and when the position of the plane of the orbit is unchanged, we have
But if, on account of the action of the component Z, the position of
the plane of the orbit is changed, we have, according to the equations
(72) 2 , the relations
&, (99)
dn =dz + (l wai)dtt =dfc + 28in f lida.
We have, further,
v being the true anomaly in the instantaneous orbit.
The two components of the disturbing force which act in the plane
of the disturbed orbit will only vary / and the elements which deter-
mine the dimensions of the conic section. We have, therefore, in the
case of the osculating elements, for the instant t ,
Let us now suppose /I to denote the true longitude in the orbit, so
that we have
J = t;-f7r = v+a>-t-a,
31
482 THEORETICAL ASTRONOMY.
or
* = +* ('); (101)
then, since is equal to TZ when the position of the plane of the orbit
is unchanged, it follows that a & represents the variation of the
true longitude in the orbit arising from the action of the component
Z of the disturbing force. The elements may refer to the ecliptic or
the equator, or to any other fundamental plane which may be adopted.
178. For the instant t we have, in the case of the disturbed motion,
the following relations :
E e sin E= M + ft (t y,
r cos v = a cosE ae, fl 02")
r sin v = al/1 e 2 sin E,
*=*+* o a).
Let us first consider only the perturbations arising from the action of
the two components of the disturbing force in the plane of the dis-
turbed orbit, and let us put
(103)
Further, let Jf + /* (t Q -f- 8M be the mean anomaly which, by
means of a system of equations identical in form with the preceding,
but in which the values of a , e Q , are used instead of the instanta-
neous values a, 6, and , gives the same longitude ^,, so that we have
r, cos v, = a cos E, a e ,
r, sin v, = a T/l e 2 sin E f
If, therefore, we determine the value of dM so as to satisfy the con-
dition that ^, = v + , the disturbed value of the true longitude in
the orbit, neglecting the effect of the component ^of the disturbing
force, will be known. The value of r, will generally differ from that
of the disturbed radius-vector r, and hence it becomes necessary to
introduce another variable in order to consider completely the effect
of the components R and S. Thus, we may put
r = r,(l + 0, (105)
and v will always be a very small quantity. When dM and v have
been found, the effect of the disturbing force perpendicular to the
plane of the instantaneous orbit may be considered, and thus the
Complete perturbations will be obtained.
VARIATION OF CO-ORDINATES. 483
In the equations (97), ^~JT expresses the areal velocity in the in-
stantaneous orbit, and it is evident that, since the true anomaly is not
affected by the force ^perpendicular to the plane ol the actual orbit,
\r^ -jj- must also represent this areal velocity, and hence the equations
become
^ = f Srdt
,
W r \~dt}^ ~^~
179. If we differentiate each of the equations (104), we get
dr. . dv, . ,., dE,
cos v, - r,smv,-j- = a a sm E, -= t
* * ' (107)
sin v, -j- + r, cos v, - = a T/l e * cos .E, =--',
(it dt dt
~dt~' = ~di'
From the second and the third of these equations we easily derive
cosE-ar cos* sm) '.
' ' dt
it 77*
Substituting in this the values of r, sin v n r, cos v,, and ^p and re-
ducing, we get
r dr L _
or
W =
JTV
dr,
From the same equations, eliminating , we get
r, -^r = (a l^l e * r > cos v t cos ^ + a o r ^ sm v / sm -^/) -gj-'
which reduces to
+ -~i (109)
. /- , 1 ^Jf\ /1AQ> .
m v, 1 H ^T- . (108)
\ H Q dt 1
484 THEORETICAL ASTRONOMY.
Combining this with the first of equations (106), we get
from which dM may be found as soon as v is known.
The equation (105) gives
dr dr, dv
= (l+v)^ + 2^. + r .
(Li)
Differentiating equation (108) and substituting for -^ its value
already found, we obtain
~d& = ~~^ \ + v '~dr) +
and the last of the preceding equations becomes
^ = r,j H >< e cosv, ^ + ^
. . , rfv 2
___ eoSm __,__ + 2 __ + -
The equation (110) gives
2 dv 2
s , ,.
"
which is easily reduced to
~fT d?~^~ < *~dt ~^v~ ' ~dt'~dT' = l + v*
and hence we derive
The equation (109) gives
VARIATION OF CO-ORDINATES. 485
dv,\*_ ffip (l + m) I 1 d3M\*
and, since
this becomes
, m 3,0081),,,
~~
, (H3)
Combining equations (112) and (113) with the second of equations
(106), we get
^v_l + v p ,
= "-*-
From (110) we derive
and the preceding equation becomes
^
or at
which is the complete expression for the determination of v.
180. It remains now to consider the effect of the component of the
disturbing force which is perpendicular to the plane of the disturbed
orbit. Let x n y n z f denote the co-ordinates of the body referred to
the fundamental plane to which the elements belong, and x, y the
co-ordinates in the plane of the instantaneous orbit. Further, let a
denote the cosine of the angle which the axis of x makes with that
of x n and /9 the cosine of the angle which the axis of y makes with
that of y,, and we shall have
z, = ax + {3y. (116)
If the position of the plane of the orbit remained unchanged, these
486 THEORETICAL ASTRONOMY.
cosines a and /9 would be constant; but on account of the action of
the force perpendicular to the plane of the orbit, these quantities are
functions of the time. Now, the co-ordinate z, is subject to two dis-
tinct variations : if the elements remain constant, it varies with the
time; and, in the case of the disturbed orbit, it is also subject to a
variation arising from the change of the elements themselves. We
shall, therefore, have
dt ~\ dt
in which I -^ 1 expresses the velocity resulting from the constant
elements, and - that part of the actual velocity which is due
to the change of the elements by the action of the disturbing force.
But during the element of time dt the elements may be regarded as
constant, and hence the velocity -~ in a direction parallel to the
axis of z f may be regarded as constant during the same time, and as
receiving an increment only at the end of this instant. Hence we
shall have
d* L _(dz L \
dt~\dt]
Differentiating equation (116), regarding a and /9 as constant, we
get
dz, \ dz, dx dy
and differentiating the same equation, regarding x and y as constant,
we get
(118)
Differentiating equation (117), regarding all the quantities involved
as variable, the result is
d*z,_da dx d? dy d*x
Eliminating da from the same equations, we obtain, in a similar
manner,
(126)
Substituting these values in equation (124), it becomes
. _ 1 ll^dy v dx\, ,
H -- / I X- Y-rr ] dz, -f- (Fa;
A;l/l m\ dt dt]
.
dt
If we introduce the components R and S of the disturbing force, we
have
r r
and hence
r
Yx Xy =Sr.
Therefore the equation (127) becomes
tffc, & 2 (l+mV
dr\ Sr
p 5 dr\
\ r jfei/p(l+m) ' * / Z '
We have, further,
which, by means of the equations (108) and (109), gives
dr e sinv, 9 dv, . dv ^
Substituting this value in the equation (128), we obtain
VARIATION OF CO-ORDINATES. 489
Sr Iddz, 3z, (129)
\ ^ " 1 + ' '
which is the complete expression for the determination of dz t .
182. The equations (110), (115), and (129) determine the complete
perturbations of the disturbed body. The value of v must first bo
obtained by an indirect process from the equation (115), and then dM
is given directly by means of (110). The value of dz will also be
Determined by an indirect process by means of (129).
In order to obtain the expressions for the forces It, S, and Z, let w
denote the longitude of the disturbed body measured in the plane of
the instantaneous orbit from its ascending node on the fundamental
plane to which & and i are referred, it being the argument of the
latitude in the case of the disturbed motion. Let w f denote the lon-
gitude of the disturbing body measured from the same origin and in
the plane of the orbit of the disturbed body, and let $' denote its
latitude in reference to this plane. Finally, let N, N', /, and u '
have the same signification in reference to the plane of the instanta-
neous orbit that they have in reference to the plane of the undisturbed
orbit in the case of the equations (66). Then we shall have
= cos -- sn *
^Jsini (N N') = sin J (ft'
from which to determine N, N f , and /. We have, also,
tan (w r N) tan u 9 ' cos I, (131)
tan p = tan /sin (uf JV),
from which to find w f and /?', u f being the argument of the latitude
of the disturbing body in reference to the plane to which & and i
are referred.
Since, when the motion of the disturbed body is referred to the
plane of its instantaneous orbit, /? = 0, the equations (71), (72), and
(73) become
R = m'telhi* cosfl cos(w f w) - t },
* p ' (132)
8 = m'tfh r' cos ft sin (w f w),
Z =m'k 2 hr f sin/5',
490 THEORETICAL ASTRONOMY.
by means of which the required components of the disturbing force
may be found, the value of h being given by
To find p, we have
f = r' 2 + r 8 2rr' cos p cos (yf 10), (133)
or, putting
cos Y cos f? cos (w f w),
the equations
P sin w = / sin ?,
p cos n = r r' cos y.
The values of r' and u' for the actual places of the disturbing
body will be given by the tables of its motion, and the actual values
of & ' and V will also be obtained by means of the tables. The de-
termination of the actual values of r and w requires that the pertur-
bations shall be known. Thus, when dM and v have been found,
we compute, by means of the mean anomaly M Q -f- f2 (t t ) + dM
and the elements a , e 0) the values of v, and r,. Then, since
v -f- % = v, -f- TT O , we have, according to (100),
W = V, + 7T ff. (135)
We have, also,
Tn the case of the fundamental osculating elements, we have
which may be used as an approximate value of and i as well as those of Q, ' and i', shall be used
in the determination of N, N', and I for each instant. When these
have been found, it will be sufficient to compute the actual values of
N, N'j and Jat intervals during the entire period for which the per-
turbations are required, and to interpolate their values for the inter-
mediate dates. The variations of these quantities arising from the
variations of Q , i, & ', and i f may also be determined by means of
differential formulae. Thus, from the differential relations of the
parts of the spherical triangle from which the equations (130) aie
derived, we easily find
VARIATION OF CO-ORDINATES. 491
,, T , smi , ^., ~x sin .AT ,. sinN
dN := -
sini' wjfnt ~N sinJV' ., , sinA r (136)
d/ = cos JV di' cos -ZVdi -}- sin sin Nd (ft' ft ).
When i and / are very small, it will be better to use
sin i sin N' sin i f sin JV
sin/ sin(ft' ft)' sin/ sin(ft' ft)'
(137)
in finding the numerical values of these coefficients. By means of
these formulae we may derive the values of dN, dN', and dl corre-
sponding to given values of ft, di, ft', and di 1 . The formulae
by means of which da, $ft, and di may be obtained directly, will be
presently considered.
The results for dN, dN', and di being applied to the quantities to
which they belong, we may compute the actual values of w r and /9'.
The value of r will be found from the given value of v, and that of
w will be given by means of equation (135). Then, by means of
the formulae (132), the forces R, S, and Z will be obtained. The
perturbations will first be computed in reference only to terms de-
pending on the first power of the disturbing force, and, whenever it
becomes necessary to consider the terms of the second order, the
results already obtained will enable us to estimate the values of the
perturbations for two or more intervals in advance with sufficient
accuracy for the determination of the three required components of
the disturbing force; and when there are two or more disturbing
bodies to be considered, the forces for each of these may be computed
at once, and the values of each component for the several disturbing
bodies may be united into a single sum, thus using 2R, 2S, and 2Z
in place of R, S, and Z respectively. The approximate values of the
perturbations will also facilitate the indirect calculation in the deter-
mination of the complete values of the required differential coeffi-
cients.
183. When only the perturbations due to the first power of the
disturbing force are required, the osculating elements ft and i will
be used in finding N 9 N', and /, and r , w will be used instead of r
and w in the calculation of the values of R, S, and Z. The equations
for the determination of the perturbations dM, v, and dz, 9 neglecting
terms of the secotfi order, are, according to the equations (110),
(115), and (129), the following:
492 THEORETICAL ASTRONOMY
ddM 1
frq+m),
-5 dz >-
The value of v is first found by integration from the results given
by the second of these equations, and then dM is found from the first
equation. Finally, dz, is found by means of the last equation. The
integrals are in each case equal to zero for the dates to which the
fundamental osculating elements belong, and the process of integra-
tion is analogous, in all respects, to that already illustrated in the
case of the variation of the rectangular co-ordinates. It will be ob-
d*v
served, however, that the expression for -^- involves only one indi-
rect term, the coefficient of which is small, and the same is true in
. d?dz, ddM .
the case of ~^r> while j^- is given directly. When the perturba-
tions have been found for a few dates, the values for the following
date can be estimated so closely that a repetition of the calculation
will rarely or never be required ; and the actual value of r may be
used instead of the approximate value r in these expressions for the
differential coefficients. Neglecting terms of the second order, we
have
= logr, -f V,
wherein ^ denotes the modulus of the system of logarithms. We
may also use v f instead of V Q ; but in this case, since r, and v, depend
on dM, only the quantities required for two or three places may be
computed in advance of the integration.
A comparison of the equations (138) with the complete equations
(110), (115), and (129) shows that, if the values of /3' and w' are
known to a sufficient degree of approximation, we may, with very
little additional labor, consider the terms depending on the squares
and higher powers of the masses. It will, however, appear from
what follows, that when we consider the perturbations due to the
higher powers of the disturbing forces, the consideration of the effect
of the variation of z, in the determination of the heliocentric place
of the disturbed body, becomes much more difficult than when the
terms of the second order are neglected ; and hence it will be found
advisable to determine new osculating elements whenever the con-
sideration of these terms becomes troublesome.
VARIATION OF CO-ORDINATES. 493
The results may be conveniently expressed in seconds of arc, and
afterwards v and Sz, may be converted into their values expressed in
units of the seventh decimal place, or, giving proper attention to the
homogeneity of the several terms of the equations, in the numerical
operations, SM may be expressed in seconds of arc, while v and 8z f
are obtained directly in units of the seventh decimal place. It will
be advisable, also, to introduce the interval CD into the formulse in
such a manner that this quantity may be omitted in the case of the
formulae of integration.
184. In the case of orbits of great eccentricity, the mean anomaly
and the mean daily motion cannot be conveniently used in the nu-
merical application of the formula?. Instead of these we must
employ the time of perihelion passage and the elements q and e.
Thus, let T Q be the time of perihelion passage for the osculating ele-
ments for the date t , and let T + 8T be the time of perihelion pas-
sage to be used in the formulae in the place of T and in connection
with the elements q and e in the determination of the values of r,
and v, 9 so that we have
In the case of parabolic motion we have, neglecting the mass of the
disturbed body,
=s (139)
the solution of which to find v, is eifected by means of Table VI. as
already explained. To find r,, we have
r, = q sec 2 %v t .
For the other cases in which the elements M Q and fjt cannot be em-
ployed, the solution must be effected by means of Table IX. or Table
X. Thus, when Table IX. is used, we compute M from
wherein log <7 = 9.9601277, and with this as the argument we derive
from Table VI. the corresponding value of V. Then, having found
t = |rr-^ by means of Table IX. we derive the coefficients required
1 H~ e n
in the equation
v, = V + A (1000 + B (1000* + (7(1000', (140)
494 THEORETICAL ASTRONOMY.
from which v, will be determined. Finally, r, will be found from
When Table X. is used, we proceed as explained in Art. 41, using
the elements T T + ST, q w and e w and thus we obtain the required
values of v, and r,.
It is evident, therefore, that, for the determination of the pertur-
bations, only the formula for finding the value of dM requires modi-
fication in the case of orbits of great eccentricity, and this modifica-
tion is easily effected. The expression
gves
^ ft ) cos (ft h) cos (> ft ) sin (ft A) cost),
cos 6 sin (Z A) (146)
cos (A . ft ) (cos (o)=cos(ff & )cos(& A)4-sin( & )sin (& A) sin (*, & )cos(A & ) cos t' cos (>*,& ) sin (h Q & )
H-cos(/l, )sin(A ^ )(l-f-cosV) (148)
4-sin (A, & ) ((cos^ cosi ) cos (h ^ )-fsin (%
and hence the last of these equations gives
-^ = sin (A, ) (cos (ff & ) sini sin i )
cos (J, ^ ) sin ()sin(A & ) (l + cosi/)-
We have, further, from the auxiliary spherical triangle,
cos i = sin i sin >/ cos (A ^ ) cos to cos if,
from which we get
cos i cos t' = sin t* cos (A & ) sin if cos t* (1 + cos if).
We have, also,
sin (a & ) sin t = sin rj sin ( AQ & o)>
sin ( & A) sin ?' = sin i sin (Ao & ),
VARIATION OF CO-ORDINATES. 497
or
sin ( sin if.
Hence we derive
(cos i cos i ) cos (h Q> o) -f sin () cos (ho & ) cosi cos (A, & ) sin (h^ &
sin t{ Sz,
sin b =sm (A, & ) sin i^ -\ -.
r
1 COST/ r
If we multiply the first of these equations by cos(A & ), and the
second by sm(/i & ), and add the results; then multiply the
first by sin (h & )> and the second by cos (h & ), and add, we get
cosfc cos(Z ^o (& Ao))=cos(A, ^ )+sin(^o S^o) ~' *
*
-~,
T
sin b =sin (I, & ) sin ^^ '- (152)
Let us now put
y= S in( sn o
c[ = sin V cos (&o ^o) cos ^ sin to (1 cos >?')
Therefore, we have
cos ~ o> = tan *
32
498 THEORETICAL ASTRONOMY.
and, if we put F= h h , the equations (152) became
cos b cos Mo _ r) =cos ft -a J + _J^_, -
cosfcsin -- - -
sin 6 =sin (A, & ) sin i -] -
As soon as 7 1 , p', 5', and ^' are known, these equations will furnish
the exact values of I and 6, those of X, and r being found by means
of the perturbations v and dM.
186. The value of F may be expressed in terms of p f and 0 ~df~ Sm >0 ~dt'
From the equations (118) and (121), observing that
we derive, by elimination,
da _ r sin A, cos i d __ r cos A ; cos i ,
~~ ~ : ""'
500 THEOEETICAL ASTRONOMY.
Therefore we shall have
dp' = rcosi sin (A, & ) ^
* ~ Al/pCl + m)
r COS * COS (^' &o)
oy means of which p f and 5' may be found by integration, the inte-
gral in each case being zero for the date t at which the determina-
tion of the perturbations begins.
When the value of dz, has already been found by means of the
equation (129), if we compute the value of q f , that of p f will be
given by means of (154), or
and if p f is determined, q' will be given by
/ I J. f 1 /-V \ I >
If both p f and q' are found from the equations (162), dz, may be de-
termined directly from (154); but the value thus obtained will be
less accurate than that derived by means of equation (129).
Since the formula for ~ completely determines the perturbations
due to the action of the component Z perpendicular to the plane of the
instantaneous orbit, instead of determining p' and q f by an independent
integration by means of the results given by the equations (162), it
will be preferable to derive them directly from dz, and -^-'- The
equations (161) give
p' = cos & da sin & d{3, q' = sin * + cos & dp.
Substituting for a and dfl their values given by (125) and (126),
and putting
x" x cos &, + y sin , y" = x sin & + y cos & ,
we obtain
/ ,,dtz, . dx"\
r dt ' & 'W./-
VAKIATION OF CO-OKDINATES. 501
Substituting further the values
x" = r cos (J, & ), y" = r sin (J,
and also
(W, _
~dt
-\-m . _ &lp (1 -f- m) e sin v
= - 6 Sin V -
- j
1 -j- e cos v
we easily find, since X, v = %,
, O n ) d
JP = cos
ijp
, . f . ,. ^.x . . / _ v.fe, A v f
5' = + (Bin (^- ft ) + esm (/- ))-
il/p (1 + m) a*
which may be used for the determination of p f and q f . These equa-
tions require, for their exact solution, that the disturbed values e, %,
and p shall be known, but it is evident that the error will be slight,
especially when e is small, if we use the undisturbed values e Q) p 09
and = TT O . The actual values of X, and r are obtained directly from
the values of the perturbations.
When p f and q f have been found, it remains only to find cos i, and
1 cos r/, in order to be able to obtain F by means of the equation
(159). From (153) we get
p' y -f- 5" = sin 2 i sin 2 i 2q f sin i ,
and hence
cos i = 1/1 p' 2 (^ + sini ) 2 , (165)
from which cos* may be found. The equation (157) gives
1 cos if = cos i Q (cos i Q + cos i) q f sin i , (166)
by means of which the value of 1 cos rf will be obtained.
If we substitute the values of p f , q f , -Jp and -~ given by the
equations (153) and (162) in (159), it is easily reduced to
= f
J
**' Zdt, (167;
(1 cos V) kVp (1 + m)
which may be used for the determination of P. When we neglect
terms of the order of the cube of the disturbing force, in finding P
we may use p in place of p and put 1 cos rf = 2 cos 2 i 0) so that the
formula becomes
502 THEORETICAL ASTRONOMY.
Zdt. (168)
Cdz,
187. By means of the formulae which have thus been derived, we
may find the values of all the quantities required in the solution of
the equations (155), in order to obtain the values of I and b for the
disturbed motion. From r, I, and b the corresponding geocentric
place may be found. The heliocentric longitude and latitude may
also be determined directly by means of the equations (145), provided
that &, 0, and i are known; and the required formula for the deter-
mination of these elements may be readily derived. Thus, the equa-
tions (160) give, by differentiation,
da, . di dff
-TT = sin ff cos i ^ sin i cos a rr,
at at at
d/3 . di , . . dff
whence
cos ff cos i -=- sin i sm ff ,.
at at at
. . dff da, . d0
sm i -=- = cos ff -=T sm ff -=-,
at at at
. di da . dp
cosi-j- = sm ff -=- -j- cosff-j-.
dt dt dt
Introducing the values of ~rr and -=7- already found into these equa-
tions, and putting
we obtain
d** 1 ^
7- = cot i sin U,
* (169)
cos .
and also, since c? -7-, and ;? re-
eft d eft
quire, for an accurate solution, that the disturbed values i, 0, and p
shall be known, and require, besides, that three separate integrations
shall be performed, unless the perturbations are computed only in
reference to the first power of the disturbing force, in which case we
use i w p , and & in place of i, p, and ' will be
found by means of the data furnished by the tables of the motion of
the disturbing body, and the corresponding corrections for N y N f t
and I having been found by "means of the terms of (136) involving
di f and d&', there remain the corrections due to di and S&> to be
applied. These may be found in terms of the quantities p f and q'
already introduced. Thus, the equations
dp' = cos i sin (' 4-
by means of which we may compute the value of w -}- d0; then the
value of w f w required in the equations (132), and also in finding
the value of /?, will be given by
' w = (w r -j- da) (
w
and the forces R, 8 9 and Z may be accurately determined.
By thus determining the correct values of H, S, and Z from date
to date, the perturbations 3M, v, and dz, may be determined in refer-
ence to the higher powers of the disturbing forces according to the
process already explained. The only difficulty to be encountered is
that which arises from the quantities 7 1 , p' 9 and q f , required in the
determination of the heliocentric place of the disturbed body by
means of the equations (155). If an exact ephemeris for a short
period is required, by means of the complete perturbations we may
determine new osculating elements, and by means of these the required
heliocentric or geocentric places.
189. EXAMPLE. We will now illustrate the application of the
formulae for the determination of the perturbations 3M, v, and 8z, by
a numerical example; and for this purpose let it be required to
determine the perturbations of Eurynome @ arising from the action
of Jupiter from 1864 Jan. 1.0 to 1865 Jan. 15.0, Berlin mean
506
THEORETICAL ASTRONOMY.
time, the fundamental osculating elements being those given in
Art. 166.
In the first place, by means of the formulae (130), using the values
= 206 39' 5".7,
'= 98 58 22 .7,
i=4 36'52'M,
i' = l 18 40 .5,
which refer to the ecliptic and mean equinox of 1860.0, we obtain
N= 194 0' 49".9, N'.= 301 38' 31".7, 1= 5 9' 56".4.
Then, by means of the data furnished by the Tables of Jupiter, we
find the values of u f , the argument of the latitude of Jupiter in refer-
ence to the ecliptic of 1860.0, and from the equations (131) we derive
w' and f) f . The values of r' are given by the Tables of Jupiter, and
the values of r and v are found from the elements given in Art.
166. The results thus obtained are the following:
Berlin Mean Time.
Iogr
*V>
logr 1
ft
1863 Dec.
12.0,
0.294084
354 26' 18 /x .O
0.73425
14 18' 54".6
l/38".l
1864 Jan.
21.0,
0.294837
10
2 45
.7
0.73368
17 21
44 .2
18
9 .1
March
1.0,
0.300674
25
24 59
.4
0.73305
20 25
5 .2
34
39 .9
April
10.0,
0.310864
40
13 31
.8
0.73237
23 28
59 .8
51
7 .6
May
20.0,
0.324298
54
14 41
.4
0.73164
26 33
32 .1
1
7
29 .7
June
29.0,
0.339745
67
21 23
.5
0.73086
29 38
44 .8
1
23
43 .5
Aug.
8.0,
0.356101
79
32 18
.1
0.73003
32 44
41 .2
1
39
46 .3
Sept.
17.0,
0.372469
90
49 57
.6
0.72915
35 51
24 .6
1
55
35 .2
Oct.
27.0,
0.388214
101
19 9
.8
0.72823
38 58
57 .5
2
11
7 .5
Dec.
6.0,
0.402894
111
5 42
.2
0.72726
42 7
23 .3
2
26
20 .3
1365 Jan.
15.0,
0.416240
120
15 32
.6
0.72625
45 16
43 .9
2
41
10 .6
The value of w for each date is now found from
w = v. + r - = v + 197 38' 6".5,
and the cojmponents of the disturbing force are determined by means
of the formulae (132), p being found from (133) or (134), and h from
(70). The adopted value of the mass of Jupiter is
1047.879
and the results for the components R, S, and Z are expressed in units
of the seventh decimal place. The factor k* 1 r e sinv Q 20 w'P
-j^: = -- -- 1 --- T= w I $.S -- r v,
d? r r 3 J r*
,
= wZ cos l
Substituting in these the- results already obtained, and also
log fi = 2.967809, log Po = 0.371237, log e = 9.290776,
we obtain first, by an indirect process, as illustrated in the case of
the direct determination of the perturbations of the rectangular co-
ordinates, the values of w 2 -^ and a)2 ~Jp L > an ^ then, having found v,
w rr- is given directly by the first of these equations. The integra-
tion of the results thus derived, by the formulae for mechanical quad-
rature, furnishes the required values of v, dM, and dz,. The calcula-
tion of the indirect terms in the determination of v and dz,, there
being but one such term in each case, is, on account of the smallness
of the coefficient, effected with very great facility.
The final results are the following :
508
THEORETICAL ASTRONOMY.
Date.
it
dSM
dt
&v
^w
(W,
^-W
dM
V
*,
1863
Dec.
12.0,
0'
'.028
+ 36.16
+ 0.04
+ 0'
'.01
+ 4.41
4- 0.02
1864
Jan.
21.0,
.072
33.61
0.49
.01
4.31
0.04
March
1.0,
.499
22.55
0.89
.27
37.11
0.54
April
10.0,
1
.213
-f 5.58
1.21
1
.11
91.96
1.93
May
20.0,
2
.070
- 13.52
1.45
2
.75
152.22
4.52
June
29.0,
2
.902
31.59
1.53
5
.24
199.05
8.54
Aug.
8.0,
3
.546
46.65
1.60
8
.49
214.54
14.10
Sept.
17.0,
3
.858
57.88
1.52
12
.22
183.69
21.24
Oct.
27.0,
3
.723
65.19
1.28
16
.05
+ 95.29
29.90
Dec.
6.0,
3
.056
68.83
0.92
19
.49
58.00
39.82
1865
Jan.
15.0,
1
.800
69.19
+ 0.4021
.97
279.84
+50.64
Since, during the period included by these results, the perturbations
of the second order are insensible, we have, for the perturbations of
Eurynome arising from the action of Jupiter from 1864 Jan. 1.0 to
1865 Jan. 15.0,
3 M = 21".97,
= 0.00002798,
, = + 0.00000506.
It is to be observed that dz, is not the complete variation of the co-
ordinate z f perpendicular to the ecliptic, but only that part of this
variation which is due to the action of the component Z alone; and
hence the results for dz, differ from the complete values obtained
when we compute directly the variations of the rectangular co-
ordinates.
Let us now determine the heliocentric longitude and latitude for
1865 Jan. 15.0, Berlin mean time, including the perturbations thus
derived. From the equations
E, e sin E, = M,,
r, =a (l e cosJ,),
sin J (v, E,) = sin ? sin E,
we obtain
M, = 99 29' 35".51,
log r, = 0.4162304,
log r = 0.4162183,
,= 110 0'33".75,
v, = 120 15 13 .80,
J = 164 32 25 .97.
The calculation of the values of r, and v, from the values of M f9 a ,
and e , may be effected by means of the various formulae for the
NUMERICAL EXAMPLE. 509
determination of the radius-vector and true anomaly from given
elements. If we substitute these results for ^,, r, and dz, in the equa-
tions (172), we get
I = 164 37' 59".05, b = 3 5' 32".54,
which are referred to the ecliptic and mean equinox of 1860.0, and
from these we may derive the geocentric place of the disturbed body.
If the place of the body is required in reference to the equinox and
ecliptic of any other date, it is only necessary to reduce the elements
x , QQ, and i to the equinox and ecliptic of that date; and then,
having computed X, and r, we obtain by means of the equations (172)
the required values of I and 6. In the determination of the pertur-
bations it will be convenient to adopt a fixed equinox and ecliptic
throughout the calculation ; and afterwards, when the heliocentric or
geocentric places are determined, the proper corrections for precession
and nutation may be applied.
In order to compare the results obtained from the perturbations
8M, v, and dz, with those derived by the method of the variation of
rectangular co-ordinates, we have, for the date 1865 Jan. 15.0,
x = 2.5107584, y = + 0.6897713, z = 0.1406590 ;
and for the perturbations of these co-ordinates we have found
8x = + 0.0001773, 3y = + 0.0001992, dz = 0.0000028.
Hence we derive
x = 2.5105811, y= + 0.6899705, z = 0.1406618,
and from these the corresponding polar co-ordinates, namely,
log r = 0.4162182, I = 164 37' 59".05, b = 3 5' 32".54,
from which it appears that the agreement of the results obtained by
the two methods is complete.
190. When the perturbations become so large that the terms of the
second order must be retained, the approximate values which may be
obtained for several intervals in advance by extending the columns
of differences, will serve to enable us to consider the neglected terms
partially or even completely, and thus derive the complete perturba-
tions for a very long period. But on account of the increasing diffi-
culties which present themselves, arising both from the consideration
510 THEORETICAL ASTRONOMY.
of the perturbations due to the action of the component Z in com-
puting the place of the body, and from the magnitude of the numeri-
cal values of the perturbations, it will be advantageous to determine,
from time to time, new osculating elements corresponding to the
values of the perturbations for any particular epoch, and thus com-
mencing the integrals again with the value zero, only the terms of
the first order will at first be considered, and the indirect part of the
calculation will, on account of the smallness of the terms, be effected
with great facility. The mode of effecting the calculation when the
higher powers of the masses are taken into account has already been
explained, and it will present no difficulty beyond that which is in-
separably connected with the problem. The determination of F 9 p f ,
and q r may be effected from the results for -77-, -r-, and -~r by means
dt dt dt J
of the formulae for integration by mechanical quadrature, as already
illustrated, or we may find F by a direct integration, and the values
of p f and q f by means of the equations (164), ^- being found from
-^- by a single integration. The other quantities required for. the
complete solution of the equations for the perturbations will be
obtained according to the directions which have been given; and in
the numerical application of the formulae, particular attention should
be given to the homogeneity of the several terms, especially since, for
convenience, we express some of the quantities in units of the seventh
decimal place, and others in seconds of arc.
The magnitude of the perturbations will at length be such that,
however completely the terms due to the squares and higher powers
of the disturbing forces may be considered, the requirements of the
numerical process will render it necessary to determine new osculating
elements; and we therefore proceed to develop the formulae for this
purpose.
191. The single integration of the values of a) 2 -p- and co 2 ^- will
give the values of o> -57- and CD jr* and hence those of -j and -rr>
which, in connection with ^ , are required in the determination of
the new system of osculating elements. Since r 2 -^ represents double
the areal velocity in the disturbed orbit, we have
CHANGE OF THE OSCULATING ELEMENTS. 511
dv, _ TcVp(l-\-m)
dt ~ ~7
The equation (109) gives
dv, _kV Po (l+m) I 1 d9M\
'dt ~ r? \ %/ dt r
Hence, since r = r, (1 + v), we obtain
by means of which we may derive p. This formula will furnish at
once the value of p, which appears in the complete equation foi
TOT* and also in the equations (164); and the value of cost may be
determined by means of (165).
In the disturbed orbit we have
dr kyU 4- m
and the equations (108) and (111) give
dr
Therefore we obtain
dv
which, by means of (176), becomes
The relation between r and r, gives
P __ __. Po / I
1 -f e cos v 1 + e cos v,
and, substituting in this the value of p already found, we get
, cos.,,) ( 1 + 1 . ^ J (1 + v)' - 1. (178)
e cos v
512 THEORETICAL ASTRONOMY.
Let us now put
(179)
a and /5 being small quantities of the order of the disturbing force,
and the equations (177) and (178) become
e sin v = e sin v t -\- ae sin v, -f- /?,
e cos v = e cos v, -f- ae cos v, -f- a.
These equations give, observing that r, (cos v, -f- e ) =p cos .#
e sin (v, v) = a sin v, ft cos v n
e cos (v, v) = e -f cos E,-\- P sin v,,
^*/
from which e, v, v, and ?; may be found; and thus, since
* = * + (,), (181)
we obtain the values of the only remaining unknown quantities in
the second members of the equations (164). The determination of
p f and (f may now be rigorously effected, and the corresponding
value of cosi being found from (165), -- and -g- will be given by
(162). Then, having found also 1 cos 37' by means of (166), F may
be determined rigorously by the equation (159), and not only the
complete values of the perturbations in reference to all powers of the
masses, hut also the corresponding heliocentric or geocentric places
of the body, may be found.
If we put
Y* = a sin v, p cos v n
and neglect terms of the third order, the equations (180) give
r' *
v v = s s,
e, ej
in which s = 206264".8. These equations are convenient for the
CHANGE OF THE OSCULATING ELEMENTS. 513
determination of e and v, v, and hence 7. by means of (181), when
the neglected terms are insensible.
The values of p, e, and v having been found, we have
m
,
a 2
tan A E = tan (45 ?) tan % v, M= Eesin E,
from which to find the elements may be found directly, terms of the third order
being neglected.
In the case of the orbits of comets for which e differs but little
from unity, instead of dM we compute by means of the formula
(142) the value of dT, and since we have
ddT _ _!_ dSM
dt n dt
the equation for p becomes
(jxm \2
l-^f) (! + ")'; (186)
and for a we have
Then e, t/, and q will be found by means of the equations
514 THEORETICAL ASTRONOMY.
e sin (Vf v) = a sin v, /5 cos v,,
e cos (v, v) = e Q -f a (cos v, + e ) + /5 sin v,, (188)
P
=IT7
and the time of perihelion passage will be derived from e and v by
means of Table IX. or Table X.
There remain yet to be found the elements (191)
by means of which the value of & may be found. This equation
gives, when we neglect terms of the third order,
= '+ r +^ + 2^-^-.). ^2)
Substituting in this the values of ff & and i i given by (190),
we get
1 ~ 3 iri2/1 (193)
sin i cos i, sm z I Q -~~ a - " *
CHANGE OF THE OSCULATING ELEMENTS. 515
r being expressed in seconds of arc. Finally, for the longitude of;
the perihelion, we have
*=*+*, (194)'
and the elements of the instantaneous orbit are completely deter-
mined. When we neglect terms of the third order, this equation,
substituting the values given by (190) and (192), becomes
It should also be observed that the inclination i which appears in
these formulae is supposed to be susceptible of any value from to
180, and hence when i exceeds 90 and the elements are given in
accordance with the distinction of retrograde motion, they are to be
changed to the general form by using 180 i instead of i, and ;
2& TT instead of TT.
The accuracy of the numerical process may be checked by com-
puting the heliocentric place of the body for the date to which the
new elements belong by means of these elements, and comparing the
results with those obtained directly by means of the equations (155).
We may remark, also, that when the inclination does not differ much
from 90, the reduction of the longitudes to the fundamental plane
becomes uncertain, and F may be very large, and hence, instead of
the ecliptic, the equator must be taken as the fundamental plane to
which the elements and the longitudes are referred.
192. Although, by means of the formulae which have been given,
the complete perturbations may be determined for a very long period
of time, using constantly the same osculating elements, yet, on
account of the ease with which new elements may be found from dM,
. , f . f .. . , , .
v, oz,, -ji- -ji and fj-> and on account of the facility afforded in.
the calculation of the indirect terms in the equations for the differen-
tial coefficients so long as the values of the perturbations are small,
it is evident that the most advantageous process will be to compute
8M, v, and 8z, only with respect to the first power of the disturbing
force, and determine new osculating elements whenever the terms of'
the second order must be considered. Then the integration will
Hgain commence with zero, and will be continued until, on account
of the terms of the second order, another change of the elements is
required. The frequency of this transformation will necessarily de-
5io THEORETICAL ASTRONOMY.
pend on the magnitude of the disturbing force; and if the disturbed
body is so near the disturbing body that a very frequent change of
the elements becomes necessary, it may be more convenient either to
include the terms of the second order directly in the computation
of the values of dM, v y and dz n or to adopt one of the other methods
which have been given for the determination of the perturbations of
a heavenly body. In the case of the asteroid planets, the consider-
ation of the terms of the second order in this manner will only
require a change of the osculating elements after an interval of seve-
ral years, and whenever this transformation shall be required, the
equations for and ?'.
Let us now differentiate the equation
regarding the elements as variable, and we get
2rdrl_ _1_ da 2V l
r L dt J " a 2 ' dt + k 2 (1 + m) ' dt
T
dt tf (1 + m) dt
The differential coefficient is here the increment of the accele-
dt
rating force, in the direction of the tangent to the orbit at the given
point, due to the action of the disturbing force; and if we designate
the angle which the tangent makes with the prolongation of the
radius-vector by ^ , we shall have
_ = E cos 4' + S sin 4' v
dt
-Substituting this value in the preceding equation, we obtain
VARIATION OF CONSTANTS. 519
But we have, according to the equations (50) 6 ,
dv
in wnich v denotes the true anomaly in the instantaneous orbit; and
hence there results
e sn
by means of which the variation of a may be found.
If we introduce the mean daily motion ft, we shall have
^r = ^'-^p ( 199 )
and hence
? sin vR + 2- ), (200)
for the determination of dp.
The first of the equations (97) gives
and hence we obtain
d (i/p) =
(205)
from which the value of ~- may be derived.
at
If we introduce the element co, or the angular distance of the peri-
helion from the ascending node, it will be necessary to consider also
the component Z; and, since co = X,
and hence
(206)
In the case of the longitude of the perihelion, we have
dt dt dt
and therefore
dn 1 1
( p cos vR + (p
sin 2 Ji^. (207)
The first of the equations (15) 2 gives
dt*, de
in which M Q denotes the mean anomaly at the epoch, which is usually
adopted as one of the elements in the case of an elliptic orbit. Sub-
stituting for -=r and -=- the values already found, we get
dt dt
dM
L o
{ (p cot ) It
f (2 cos 2 v cos v cos E} cot y S] (t O-> />
sin v ^ dt
or
dM 1
- . = (( p cot
cos vR
d/
-f cos
Sfi.
ftjf
1863 Dec.
12.0,
+ 0'
'.01
0"
.00
+ 8".43
+ 0"
.12 -
f 0".0007
5".48
1864 Jan.
21.0,
.04
.01
8
.49
.38
.0040
+ 5 .72
March
1.0,
.24
.03
26
.78
1
.80
.0216
19 .15
April
10.0,
.66
.06
48
.01
3
.88
.0502
37 .11
May
20.0,
1
.35
.08
72
.82
6
.27
.0875
60 .91
June
29.0,
2
.28
.10
100
.83
8
.61
.1299
90 .73
Aug.
8.0,
3
.40
.09
130
.93
10
.65
.1751
125 .79
Sept.
17.0,
4
.63
.07
161
.66
12
.26
.2200
164 .79
Oct.
27.0,
5
.84
.03
191
.48
13
.48
.2624
206 .19
Dec.
6.0,
6
.93
4-0
.03
219
.27
14
.44
.3004
248 .72
1865 Jan.
15.0,
y
.81
-f
.10
244
.24
15
.37
+
.3323
+ 291 .33
Applying the variations of the elements thus obtained to the oscu-
lating elements for 1864 Jan. 1.0, as given in Art. 166, the osculating
elements for the instant 1865 Jan. 15.0 are found to be the following:
Epoch = 1865 Jan. 15.0 Berlin mean time.
M= 99 34' 48".81
TT = 44 13 7 .93
i = 4 36 52. 21
? = 11 15 35 .65
log a = 0.3880283
ft = 928".8897.
1860.0.
NUMERICAL, EXAMPLE. 531
In order to compare the results thus derived with the nerturbations
computed by the other methods which have been given, let us com-
pute the heliocentric longitude and latitude, in the case of the dis-
turbed orbit, for the date 1865 Jan. 15.0, Berlin mean time. Thus,
by means of the new elements, we find
M= 99 34' 48".81, E= 110 5' 14".15,
logr= 0.4162182, v = 120 19 18 .01,
I = 164 37' 59".04, I = 3 5 32 .54,
agreeing completely with the results already obtained by the other
methods. The heliocentric place thus found is referred to the ecliptic
and mean equinox of 1860.0, to which the elements ;r, 2, and i are
referred ; and it may be reduced to any other ecliptic and equinox by
means of the usual formulae. Throughout the calculation of the per-
turbations it will be convenient to adopt a fixed equinox and ecliptic,
the results being subsequently reduced by the application of the cor-
rections for precession and nutation.
In the determination of dM, if we denote by AM the value which
7 Ttr
is obtained when we neglect the last term of the equation for jr> we
shall have
which form is equally convenient in the numerical calculation. Thus,
for 1865 Jan. 15.0, we find
AM = + 234".74,
and from the several values of 1600^- we obtain, for the same date v
by means of the formula for double integration,
Hence we derive
dM = + 234".74 + 56".59 = + 291".33,
agreeing with the result already obtained.
If we compute the variation of the mean anomaly at the epoch, by
means of equation (209), we find, in the case under consideration,
dM = + 165".29,
532 THEORETICAL ASTKONOMY.
and since the place of the body in the case of the instantaneous orbit
is to be computed precisely as if the planet had been moving con-
stantly in that orbit, we have, for 1865 Jan. 15.0,
and hence
dM = 3M + (* 4,) 8?= + 9.91" M.
The error of this result is 0".23, and arises chiefly from the in-
crease of the accidental and unavoidable errors of the numerical cal-
culation by the factor t 1 09 which appears in the last term of the
equation (209). Hence it is evident that it will always be preferable
to compute the variation of the mean anomaly directly; and if the
variation of the mean anomaly at a given epoch be required, it may
easily be found from dM by means of the equation
If the osculating elements of one of the asteroid planets are thus
determined for the date of the opposition of the planet, they will
suffice, without further change, to compute an ephemeris for the brief
period included by the observations in the vicinity of the opposition,
unless the disturbed planet shall be very near to Jupiter, in which
case the perturbations during the period included by the ephemeris
may become sensible. The variation of the geocentric place of the
disturbed body arising from the action of the disturbing forces, may
be obtained by substituting the corresponding variations of the ele-
ments in the differential formulae as derived from the equation (1) 2 ,
whenever the terms of the second order may be neglected. It should
be observed, however, that if we substitute the value of dM directly
in the equations for the variations of the geocentric co-ordinates, the
coefficient of d/j. must be that which depends solely on the variation
of the semi-transverse axis. But when the coefficient of dp. has been
computed so as to involve the effect of this quantity during the in-
terval t t w the value of dM Q must be found from dM and substi-
tuted in the equations.
200. It will be observed that, on account of the divisor e in the
expressions for -> -TT-, and -^p these elements will be subject to large
perturbations whenever e is very small, although the absolute effect
on the heliocentric place of the disturbed body may be small ; and on
VARIATION OF CONSTANTS. 533
account of the divisor sin i in the expression for 5 the variation
of & will be large whenever i is very small. To avoid the difficul-
ties thus encountered, new elements must be introduced. Thus, in
the case of &, let us put
a" = sin i sin & , 0" = sin i cos & ; (224)
then we shall have
da" . di . d
7 = sm & cos i-j- -f- sin t cos & 7 >
at ac at
- cos a cos t- - sm * sin
Introducing the values of -^- and given by the equations (212),
and introducing further the auxiliary constants a, b, A, and B com-
puted by means of the formulae (94) x with respect to the fundamental
plane to which & and i are referred, we obtain
7 tl -j
rZsin a cos (.4 + w).
(225)
6 cos (B + w),
(1 -f m)
by means of which the variations of a." and /9" may be found. If
the integrals are put equal to zero at the beginning of the integration,
the values of da" and dp" will be obtained, so that we shall have
sin i sin & = sin i sin & -f- a",
sin i cos & = sin i cos & -f &&' *
or
sin i sin ( & ) = cos #" sin & <5y?",
sin i cos (& $^ ) = sin i + sin ^ ^a" -J- cos ^ dp", (226)
by means of which i and & & may be found.
In the case of #, let us put
y" = e sin /, C" = e cos /, (227 )
and we have
de . d x
d?' de d x
534 THEORETICAL ASTRONOMY.
Substituting for -=- and -j- the values given by the equations (203)
and (205), and reducing, we obtain
-{- m) *
-f-ersin/j/S),
d" 1 I (228)
p Sin + ^ ^ + ^ + ') CQS
ercos/j/Sl,
by means of which the values of dy" and ^f" may be found. Then
we shall have
e sin x = e sin TT O -f &/',
e cos / = e cos TT O -f- <5C",
or
e sin (/ TT O ) = cos TT O &/' sin TT O (*>
and, also,
dy m' dtf
~dt = 1-fm' ' "3T
m' fr
1 -f- m' ' eft '
If, therefore, from the elements of the orbit of the disturbing planet
we compute the auxiliary constants for the adopted fundamental
plane by means of the equations (94) x or (99) 1? and also V and U r
from
V P '
(e r sin a/ + sin w') =7' sin J7',
l m ' (e r cos / + cos ti') = F cos U',
'
Vp'
the equations (100)! and (49), in connection with (232) and (233),
give
9x = ' r , r' sin a' sin (A* + i*'), (234^
PERTURBATIONS OF COMETS. 539
dy = f r ' sin b f sin ( f -f u') t
m f
Sz = -T r r sm c sm ( (J -f- w') ;
X -J- Wi
*" m ' i' + U'), (234)
"* j,., . , ^ ~, TT f \
If we add the values of dx 9 %, z, 3-j-> J-TT and (5 -yr- to the cor-
w* wt dt
responding co-ordinates and velocities of the comet in reference to
the centre of gravity of the sun, the results will give the co-ordinates
and velocities of the comet in reference to the common centre of
gravity of the sun and disturbing planet, and from these the new
elements of the orbit may be determined as explained in Art. 168.
The time at which the elements of the orbit of the comet may be
referred to the common centre of gravity of the sun and planet, can
be readily estimated in the actual application of the formulae, by
means of the magnitude of the disturbing force. In the case of Mer-
cury as the disturbing planet, this transformation may generally be
effected when the radius-vector of the comet has attained the value
1.5, and in the case of Venus when it has the value 2.5. It should
be remarked, however, that the distance here assigned may be in-
creased or diminished by the relative position of the bodies in their
orbits. The motion relative to the common centre of gravity of
the sun and planet disregarding the perturbations produced by the
other planets, which should be considered separately may then be re-
garded as undisturbed until the comet has again arrived at the point
at which the motion must be referred to the centre of the sun, and at
which the perturbations of this motion by the planet under consider-
ation must be determined. The reduction to the centre of the sun
will be effected by means of the values obtained from (234), when the
second member of each of these equations is taken with a contrary
sign.
204. In the cases in which the motion of the comet will be referred
to the common centre of gravity of the sun and disturbing planet,
the resulting variations of the co-ordinates and velocities will be so
smalJ that their squares and products may be neglected, and, there-
540 THEORETICAL ASTRONOMY.
fore, instead of using the complete formulae in finding the new ele-
ments, it will suffice to employ differential formulae. The formulas
(100), give
dx . . f . dr . . , A dv
-Tr = 8iiia sin ( A -f u) -%- -f r sm a cos (A + u) -JT*
Hj- = sin b sin (B + u) ^ + r sin b cos ( + u) ^-, (235)
(it ut (it
dz . . , dr . f ~ . dv
-jr = sm c sm ( C + w ) -57 + r sm c cos ( (7 -f- w) -TI-
at at at
If we multiply the first of these equations by 3x, the second by dy }
and the third by dz; then multiply the first by d> the second by
7 7 Ctf
^-^Y* and the third by S-j-> and put
at dt
P= sin a sin (A -{-u) 3x -{- sin 6 sin (JB -}- w
-f- sin c sin ( C -}- w) ^2,
Q = sin a cos ( J. + w) 5 -f sin 6 cos (JB -f w
4- sin c cos ((7 -f- u) dz;
P' = sin a sin (4 + i*) *- + sin 6 sin (B + ti) d-- (236)
g' = sin a cos (J. + w) d-^ + sin 6 cos (B + w) J-~
-f- sin c cos ( C + u) djr*
we shall have, observing that - = -^= e sin t? and that -^ = -^r-
-dt -dT--dr -df- r -dt v ^t
From the equations
dr dx
_
~ df ^ dt' ^ de'
PERTUKBATIONS OF COMETS. 541
we get
which by means of (237) become
'+ ! +v (238)
=-= esinvP'H- ^ O^
Xf>
From the equation
V-W-TF *
we get
Substituting the values given by (238), observing also that P=dr,
this becomes
dk dp __ V*r re* sin 2 v esi
T' h ^ ^ J "7T-
and, since
F 8 = -(l + 2ecosv+e),
we obtain
flr-., (239,
by means of which the variation of \/p may be found.
The equation
^. = 2P_ F
a t
gives
from which we derive
542 THEORETICAL ASTRONOMY.
from which the new value of the semi-transverse axis a may be
found. To find dp we have
1 3k
dfj. = Ifiad J- /A -, (241 ")
a k
or
Next, to find 8e, we have, from p = a (1 e 2 ),
**! '<** (243 >
or
jocose smv smv~l/p' f , ^P ,
to = -j P H H r-^ 1 -P H 77- (cos v + cos E}
The equation (12) 2 gives
(2 + e cos i>) to, (245)
v
2
a 2 cos? a'cos'p
and from = 1 + e cos v we get
e sin i; re sm v re sin -y
Substituting this value of dv in (245), and reducing, we find
(24 2
the co-ordinates referred to the equator or to any other plane making
the angle e with the ecliptic, the positive axis of x being directed to
the point from which longitudes are measured in this plane; and if
we introduce also the auxiliary constants a, A, 6, B, &c., we shall
have
dx" = sin a sin A8x-\- sin b sin B dy -f sin c sin C 8z,
dy" = sin a cos A ox -f- sin b cos B dy -f- sin c cos C dz, (248)
dz" cos a dx -\- cos 6 fy -f- cos c -f sin w _
ft = - .# -f-
by means of which 8Q and 8i may be found. To find dot and d;r we
have
&y = and d-jr by means of the formulae
Ctv Ctt> Civ
(234) ; then, by means of (236) and (250), we compute P, , R, P' ,
Q f , and jR r , the auxiliary constants a, A, &c. being determined in
reference to the fundamental plane to which the co-ordinates are re-
ferred. When the fundamental plane is the plane of the ecliptic, or
that to which & and i are referred, we have
sinc = sini, 0=0.
The algebraic signs of cos a, cos 6, and cos c, as indicated by the equa-
tions (101) 17 must be carefully attended to. The formulae for the
variations of the elements will then give the corrections to be applied
to the elements of the orbit relative to the sun in order to obtain
those of the orbit relative to the common centre of gravity of the
sun and planet. Whenever the elements of the orbit about the sun
are again required, the corrections will be determined in the same
manner, but will be applied each with a contrary sign.
35
546 THEORETICAL ASTRONOMY.
Since the equations have been derived for the variations of more
than the six elements usually employed, the additional formulae, as
well as those which give different relations between the elements em-
ployed, may be used to check the numerical calculation; and this
proof should not be omitted. It is obvious, also, that these differen-
tial formulae will serve to convert the perturbations of the rectangular
co-ordinates into perturbations of the elements, whenever the terms
of the second order may be neglected, observing that in this case
8k 0. If some of the elements considered are expressed in angular
measure, and some in parts of other units, the quantity s = 206264".8
should be introduced, in the numerical application, so as to preserve
the homogeneity of the formulae.
When the motion of the comet is regarded as undisturbed about
the centre of gravity of the system, the variations of the elements for
the instant t in order to reduce them to the centre of gravity of the
system, added algebraically to those for the instant t' in order to
reduce them again to the centre of the sun, will give the total pertur-
bations of the elements of the orbit relative to the sun during the
interval t f t. It should be observed, however, that the value of
dM for the instant t should be reduced to that for the instant t f , so
that the total variation of M during the interval t 1 t will be
In this manner, by considering the action of the several disturbing
bodies separately, referring the motion of the comet to the common
centre of gravity of the sun and planet whenever it may subsequently
be regarded as undisturbed about this point, and again referring it to
the centre of the sun when such an assumption is no longer admissi-
ble, the determination of the perturbations during an entire revolu-
tion of the comet is very greatly facilitated.
207. If we consider the position and dimensions of the orbits of
the comets, it will at once appear that a very near approach of some
of these bodies to a planet may often happen, and that when they
approach very near some of the large planets their orbits may be
entirely changed. It is, indeed, certainly known that the orbits of
comets have been thus modified by a near approach to Jupiter, and
there are periodic comets now known which will be eventually thus
acted upon. It becomes an interesting problem, therefore, to con-
sider the formulse applicable to this special case in which the ordinary
methods of calculating perturbations cannot be applied.
PEKTUKBATIONS OF COMETS. 547
If we denote by x f , y r , z', r', the co-ordinates and radius-vector of
the planet referred to the centre of the sun, and regard its motion
relative to the sun as disturbed by the comet, we shall have
fflaf
dP r"
f + - 1 ^ s m) ^ = mp(^-|), (260)
W + ~ ~7*~
Let us now denote by , 57, the co-ordinates of the comet referred
to the centre of gravity of the planet; then will
Substituting the resulting values of #', y f , z r in the preceding equa-
tions, and subtracting these from the corresponding equations (1) for
*-he disturbed motion of the comet, we derive
These equations express the motion of the comet relative to the centre
of gravity of the disturbing planet; and when the comet approaches
Very near to the planet, so that the second member of each of these
equations becomes very small in comparison with the second term
of the first member, we may take, for a first approximation,
(fy k* (m -f- m') I _ (262)
dt* "* >*
_ A
and, since * ^ - is the sum of the attractive force of the planet
on the comet and of the reciprocal action of the comet on the planet,
548 THEORETICAL ASTRONOMY.
these equations, being of the same form as those for the undisturbed
motion of the comet relative to the sun, show that when the action
of the disturbing planet on the comet exceeds that of the sun, the
result of the first approximation to the motion of the comet is that
it describes a conic section around the centre of gravity of the planet.
Further, since x f , y f , z f are the co-ordinates of the sun re-
ferred to the centre of gravity of the planet, it appears that the
second members of (261) express the disturbing force of the sun on
the comet resolved in directions parallel to the co-ordinate axes
respectively. Hence when a comet approaches so near a planet that
the action of the latter upon it exceeds that of the sun, its motion
will be in a conic section relatively to the planet, and will be dis -
turbed by the action of the sun. But the disturbing action of the
sun is the difference between its action on the comet and on the
planet, and the masses of the larger bodies of the solar system are
such that when the comet is equally attracted by the sun and by the
planet, the distances of the comet and planet from the sun differ so
little that the disturbing force of the sun on the comet, regarded as
describing a conic section about the planet, will be extremely small.
Thus, in a direction parallel to the co-ordinate the disturbing force
exercised by the sun is
l*_
r' 3
fa I / o I '" I /j
and when the comet approaches very near the planet this force will
be extremely small. It is evident, further, that the action of the
gun regarded as the disturbing body will be very small even when
its direct action upon the comet considerably exceeds that of the
planet, and, therefore, that we may consider the orbit of the comet to
be a conic section about the planet and disturbed by the sun, when it
is actually attracted more by the sun than by the planet.
208. In order to show more clearly that the disturbing force of the
sun is very small even when its direct action on the comet exceeds
that of the planet, let us suppose the sun, planet, and comet to be
situated on the same straight line, in which case the disturbing force
of the sun will be a maximum for a given distance of the comet from
1
the planet. Then will the direct action of the sun be , and that
m'Tc*
of the planet ^-- The disturbing; action of the sun will be
PERTURBATIONS OF COMETS. 549
which, since p is supposed to be small in comparison with r, may btj
put equal to
and hence the ratio of the disturbing action of the sun to the direct
action of the planet on the comet cannot exceed
If the comet is at a distance, such that the direct action of the sun is
equal to the direct action of the planet, we have
p* = m 'r\
and the ratio of the direct action of the sun to its disturbing action
cannot in this case exceed 21/m'. In the case of Jupiter this amounts
to only 0.06.
So long as p is small, the disturbing action of the planet is very
m'k*
nearly in all positions of the comet relative to the planet, and
hence the ratio of the disturbing action of the planet to the direct
action of the sun cannot exceed
At the point for which the value of p corresponds to R = R' y the
comet, sun, and planet being supposed to be situated in the same
straight line, it will be immaterial whether we consider the sun or
the planet as the disturbing body ; but for values of p less than this
R will be less than R f , and the planet must be regarded as the con-
trolling and the sun as the disturbing body. The supposition that
R is equal to R r gives
and therefore
p = rS/X 5 . (263;
Hence we may compute the perturbations of the comet, regarding
the planet as the disturbing body, until it approaches so near the
550 THEORETICAL ASTRONOMY.
planet that p has the value given by this equation, after which, so
long as p does not exceed the value here assigned, the sun must be
regarded as the disturbing body,
If (p represents the angle at the planet between the sun and comet,
the disturbing force of the sun, for any position of the comet near
the planet, will be very nearly
cos
and when this angle is considerable, the disturbing action of the sun
will be small even when p is greater than rl/^m' 2 . Hence we may
commence to consider the sun as the disturbing body even before the
comet reaches the point for which
and, since the ratio of the disturbing action of the planet to the
direct action of the sun remains nearly the same for all values of ^,
when p is within the limits here assigned the sun must in all cases
be so considered. Corresponding to the value of p given by equation
(263), we have
and in the case of a near approach to Jupiter the results are
P = 0.054 r, # = 0.33.
209. In the actual calculation of the perturbations of any particu-
lar comet when very near a large planet, it will be easy to determine
the point at which it will be advantageous to commence to regard the
gun as the disturbing body; and, having found the elements of the
prbit of the comet relative to the planet, the perturbations of these
elements or of the co-ordinates will be obtained by means of the
formulse already derived, the necessary distinctions being made in the
notation. When the planet again becomes the disturbing body, the
elements will be found in reference to the sun; and thus we are
enabled to trace the motion of the comet before and subsequent to it*s
being considered as subject principally to the planet. In the case of
the first transformation, the co-ordinates and velocities of the comet
and planet in reference to the sun being determined for the instant at
which the sun is regarded as ceasing to be the controlling body, we
shall have
PERTURBATIONS OF COMETS. 551
d__dx__dx' di) _ dy dif dZ __ dz cM ,
~dt~~~dt~~~dt' ~dt~~~dt~~~dt* ~dt" ~dt~~~dt'
and from , 57, , -^-, -7-, and -7% the elements of the orbit of the
comet about the planet are to be determined precisely as the elements
in reference to the sun are found from x, y y z, yp j-, and -^-> and
as explained in Art. 168. Having computed the perturbations of
the motion relative to the planet to the point at which the planet is
again considered as the disturbing body, it only remains to, find, for
the corresponding time, the co-ordinates and velocities of the comet
in reference to the centre of gravity of the planet, and from these the
co-ordinates and velocities relative to the centre of the sun, and the
elements of the orbit about the sun may be determined. As the in-
terval of time during which the sun will be regarded as the disturb-
ing body will always be small, it will be most convenient to compute
the perturbations of the rectangular co-ordinates, in which case the
values of , y, , -TJ-> -^-> and -rr will be obtained directly, and then,
having found the corresponding co-ordinates x r , y', z' and velocities
rr> -Jr 7T of the planet in reference to the sun. we have
at at at
___ _ _ \ I
~dt ~~ ~dt "" ~di' dt ~~ dt " dt ' dt " = dt h dt '
by means of which the elements of the orbit relative to the sun will
be found. If it is not considered necessary to compute rigorously
the path of the comet before and after it is subject principally to the
action of the planet, but simply to find the principal effect of the
action of the planet in changing its elements, it will be sufficient,
during the time in which the sun is regarded as the disturbing body,
to suppose the comet to move in an undisturbed orbit about the
planet. For the point at which we cease to regard the sun as the
disturbing body, the co-ordinates and velocities of the comet relative
to the centre of gravity of the planet will be determined from the
elements of the orbit in reference to the planet, precisely as the corre-
sponding quantities are determined in the case of the motion relative
to the sun, the necessary distinctions being made in the notation.
552 THEOEETICAL ASTRONOMY.
210. The results obtained from the observations of the periodic
comets at their successive returns to the perihelion, render it probable
that there exists in space a resisting medium which opposes the motion
of all the heavenly bodies in their orbits; but since the observations
of the planets do not exhibit any effect of such a resistance, it is in-
ferred that the density of the ethereal fluid is so slight that it can
have an appreciable effect only in the case of rare and attenuated
bodies like the comets. If, however, we adopt the hypothesis of a
resisting medium in space, in considering the motion of a heavenly
body we simply introduce a new disturbing force acting in the direc-
tion of the tangent to the instantaneous orbit, and in a sense contrary
to that of the motion. The amount of the resistance will depend
chiefly on the density of the ethereal fluid and on the velocity of the
body. In accordance with what takes place within the limits of our
observation, we may assume that the resistance, in a medium of con-
stant density, is proportional to the square of the velocity. The
density of the fluid may be assumed to diminish as the distance from
the sun increases, and hence it may be expressed as a function of the
reciprocal of this distance.
Let ds be the element of the path of the body, and r the radius-
vector; then will the resistance oe
K being a constant quantity depending on the nature of the body,
and =
Vp
Jc Jct/p ds
V cos (f> = -7- e sin v, Fsin ^ = - -, V = -rr>
r
\ve
have
EESISTING MEDIUM IN SPACE. 553
Substituting these values of R and 8 in the equation (205), it reduces to
= 2Ky ( - 1 sin v ds.
Now, since
we have
V= -- (I -f- 2e cos v + e,
Vp
ds = Vdt = (l + 2e cos v
and hence
' X \ f* ' V^OOJ
If we suppose the function
2e cos v -f- e*) 3 ,
( i j r 3 (1
the value of which is always positive, to be developed in a series
arranged in reference to the cosines of v and of its multiples, so that
we have
^ ( 7 ) ** C 1 + 2e cos v + e rf = A + ^ cos v + c cos 2v + &c -> ( 267 >
in which A, B, &c. are positive and functions of e, the equation (266)
becomes
2
ed% = -- ( A -j- -B cos v -f- . . . .) sin v dv.
Hence, by integrating, we derive
+ ---- ), (268)
from which it appears that / is subject only to periodic perturbations
on account of the resisting medium.
In a similar manner it may be shown that the second term of the
second member of equation (210) produces only periodic terms in the
value of dM, so that if we seek only the secular perturbations due to
the action of the ethereal fluid, the first and second terms of the
second member of (210) will not be considered, and only the secular
perturbations arising from the variation of // will be required.
Let us next consider the elements a and e. Substituting in the
554 THEORETICAL ASTRONOMY.
equations (198) and (202) the values of E and S given by (265), and
reducing, we get
2o / 1 \ 2 .. | ,
da = r- K
n\
\ r I
(270)
It remains now to make an assumption in regard to the law of the
density of the resisting medium. In the case of Encke's comet it
has been assumed that
and this hypothesis gives results which suffice to represent the obser-
vations at its successive returns to the perihelion. Substituting for F
its value in terms of r and a, the equations (270) thus become
dt r'r a
- OM TT
- LKr \J
a COS COS E
r
dt v*
l 2 *
1 ---
\ r a
by means of which djy. and d
-j- = a cos
cos v I sin v
in which s = 206264". 8, /* being expressed in seconds of arc. Com-
bining the results thus obtained with the differential coefficients of
the geocentric spherical co-ordinates with respect to r and v, as indi-
cated by the equations (42) 2 , we obtain the required coefficients of x
and y to be introduced into the equations of condition. The solution
of all the equations of condition by the method of least squares will
then furnish the most probable values of y and #, or of the secular
variations of the eccentricity and mean motion, without any assump-
tion being made in rei jrence either t ) the density of the ethereal fluid
or to the modifications of the resistance on account of the changes in
the form and dimensions of the comet, and the results thus derived
may be employed in determining the values of J!f, //, and
f
9476
9
99
50
2.15
'5
J 3
4807
40
12
13
14
15
16
17
4 40.06
5 1-85
5 23.28
5 44-33
6 4.95
6 25.14
21.79
21.43
21.05
20.62
20.19
19.72
9-999 9377
9271
9'57
935
8905
8768
106
114
122
I 3
137
144
37
10
20
30
40
50
II 3.28
4-39
5-47
6-54
7-58
8-59
.11
.08
.07
.04
I.OI
1. 00
9-999 4767
4726
4686
4645
4604
45 6 3
41
40
4 1
4 1
4 1
4'
18
19
20
21
22
23
6 44.86
7 4.09
7 22.80
7 40.99
7 58.61
8 15.66
19.23
18.71
18.19
17.62
17.05
16.44
9.999 8624
8472
8314
8149
7977
7799
165
172
I 7 8
185
38
10
20
30
40
50
ii 9.59
10.56
11.51
12.44
13-34
14.22
0.97
0.95
0.93
0.90
0.88
0.86
9-999 45"
4481
4440
4399
4358
43'7
4 1
4 1
4 1
4 1
24
25
26
27
28
8 32.10
8 47.93
9 3- 12
9 17.65
9 3 x -5
I5-83
15.19
14-53
I3-85
9-999 7614
7424
7228
7027
6820
190
196
201
20 7
39
10
20
30
40
ii 15.08
15.92
16.73
17.52
18.29
0.84
0.8 1
0.79
0.77
9-999 4276
4234
4*93
4152
4110
42
42
29
9 44.66
13.16
12.46
6608
212
216
50
19.04
0.75
0.72
4069
42
30
10
20
30
40
50
9 57.12
9 59- 12
10 I. II
3-7
5.02
6.94
2.00
1-99
1.96
1.95
1.92
1.91
9-999 6392
6 355
6319
6282
6245
6208
P
37
37
37
37
40
10
20
30
40
50
ii 19.76
20.46
21.13
21.79
22.42
23.02
0.70
0.67
0.66
0.63
0.60
9.999 4027
3985
3944
3902
3860
42
4 1
42
42
4 1
42
31
10
20
30
40
50
10 8.85
10.73
12.59
14.44
16.26
1 8.06
1.88
1.86
I.g 5
1.82
1.80
1.78
9.999 6171
6096
6059
6021
5984
I
38
3 8
41
10
20
30
40
50
ii 23.61
24.17
24.70
25.22
25.71
26.18
0.56
o-53
0.52
0.49
0.47
0.44
9-999 3777
3735
3 6 93
3651
3609
3567
42
42
42
42
42
42
32
10
20
30
40
50
10 19.84
21. 60
23-34
25.05
26.75
28.43
1.76
1.74
1.71
1.70
1.68
1.65
9-999 5946
5908
5870
5832
5794
5755
38
38
38
38
39
3 8
42
10
20
30
40
50
ii 26.62
27.04
27.44
27.82
28.17
28.50
0.42
0.40
0.38
o-35
o-33
0.30
9-999 3525
3483
3399
3357
33'5
42
42
42
42
42
33
10
20
cO
40
10 30.08
33-32
34-9J
36.48
1.63
1.61
i-59
i-57
9-999 5717
5678
5640
5601
39
38
39
39
43
10
20
30
40
ii 28.80
29.08
29-34
29.58
29.79
0.28
0.26
0.24
0.21
9-999 3273
3 270
3188
3H6
3*4
43
42
42
42
42
50
38.03
"55
1.52
5523
39
39
50
29.98
o.i 6
3062
T-*
43
34
10
20
30
40
50
10 39-55
41.06
42.54
44.00
JIU
1.51
1.48
1.46
i-44
1.42
I. -JO
9-999 5484
5445
5406
53 6 7
5327
5288
39
39
39
40
39
4
44
10
10
SO
40
50
ii 30.14
30.29
30.41
30.50
3-57
30.62
0.15
0.12
0.09
0.07
0.05
0.03
9-999 3 OI 9
2977
2935
2892
2850
2808
42
42
43
42
42
42
35
10 48.25
J 7
9-999 5248
45
ii 30.65
9.999 2766
661
TABLE I, Angle of the Vertical and Logarithm of the Earth's Eadius,
= Geocentric Latitude.
P = Earth's Eadius.
0-0'
Diff.
logp
Diff.
0-0'
Diff.
logp
Diff.
O /
45
10
20
30
40
50
II 30.65
30.65
30.63,
30.58
30.42
0.00
0.02
O.O5
0.07
0.0 9
Oil
9.999 2766
2723
2681
2639
2596
43
42
42
43
42
O /
55
10
20
30
40
50
10 49.74
48.36
46.97
45-55
2%
II
1.38
1.39
1.42
1.44
1.46
9-999 *75
0235
0195
0153
OIK
0076
40
40
40
39
40
42
1.49
39
46
10
20
30
40
50
II 30.31
30.17
30.01
29.82
29.61
29.38
0.14
0.16
0.19
0.21
0.23
O.26
9.999 2512
2470
2427
2385
2343
2300
42
43
42
42
43
42
56
10
20
30
40
50
10 41.16
39-65
38.13
36.5!
35-oJ
33-4 1
1.51
1.52
'I 7
1. 60
1.61
9-999 37
9.998 9998
9958
9919
9880
9841
39
40
39
39
39
39
4T
10
20
30
II 29.12
28.85
28.54
28.22
0.27
0.31
0.32
9.999 225*
2216
2174
2132
42
42
42
57
10
20
30
10 31.80
30.16
28.50
26.83
1.64
1.66
1.67
9.998 9802
9764
9725
9686
38
39
39
40
50
27.87
27.50
-35
0-37
0.40
2089
2047
43
42
42
40
50
23.40
1.70
1.73
1.74
9648
9610
39
48
10
20
II 27.10
26.69
26.24
0.41
0.45
9.999 2005
1963
1921
42
42
58
10
20
10 21.66
19.90
18.11
1.76
9.998 9571
9533
9495
1
30
40
50
25.78
25.29
24.78
0.40
0.49
0.51
0.54
1879
1837
'795
42
42
42
42
30
40
50
16.31
14.48
12.63
I'll
1.85
1.86
9457
9419
9382
|
49
10
20
30
40
50
ii 24.24
23.69
23.11
22.50
21.87
21.22
0.55
0.58
0.61
0.63
0.65
0.67
9-999 753
1711
1669
1627
1586
1544
42
42
42
42
42
59
10
20
30
40
50
10 10.77
8.88
6.97
5.04
3-o8
IO I. II
1.89
1.91
1.93
1.96
1.97
1.99
9.998 9344
9307
9269
9232
9158
11
37
37
37
37
50
10
20
30
40
50
II 20.55
19.85
18.39
17.63
16.84
0.70
0.72
0.74
0.76
0.79
0.82
9.999 1502
1460
1419
'377
^335
1294
42
42
42
4'
42
60
61
62
63
64
65
9 59-i*
9 46.74
9 33-65
9 19.85
9 5-36
8 50.21
12.38
13.09
13.80
14.49
15.15
15.81
9.998 9121
8902
8688
8479
8275
8077
219
214
209
204
198
51
10
20
30
40
50
II l6.02
15.19
14-33
13-45
12.55
11.62
0-.83
0.86
0.90
-93
0.95
9.999 1252
I2II
II 7
1128
1087
1046
42
4 1
4 1
66
67
68
69
70
71
8 34-40
8 17-97
8 0.92
7 43- 2 9
7 25.08
7 6.33
16.43
17.05
17.63
18.21
18.75
19.27
9.998 7884
7697
75'7
734*
7174
7013
187
1 80
III
161
J 54
52
10
20
30
II 10.67
9.70
8.71
7.69
0.97
0.99
.02
9-999 1005
0963
0922
O88l
42
72
73
74
75
6 47.06
6 27.28
6 7.03
5 46-33
19.78
20.25
20.70
9.998 6859
6713
6573
6441
146
140
132
40
50
6.66
5.60
3
.09
0840
0800
40
4 1
76
77
5 3-67
21.13
2I -53
21.90
6317
6201
Til
1 08
53
10
20
30
40
50
ii 4.51
3.40
2.27
II 1. 12
10 59.94
58.74
.11
13
.1C
.18
.20
.22
9-999 759
0718
0677
0637
0596
0556
40
4i
40
4 1
78
79
80
81
82
83
4 41-77
4 19-53
3 56.96
3 34-io
3 10.98
2 47.63
22.24
22.57
22.86
23.12
2 3-35
23.56
9.998 6093
5993
5901
5818
5676
100
ii
57
54
10
20
30
40
50
10 57.52
56.28
55.02
53-73
52.42
51.09
.29
33
35
9.999 0515
0475
435
395
355
03*5
40
40
4
40
40
4
84
85
86
87
88
89
2 24.07
2 0.33
I 36.44
I 12.43
o 4.8.3!
o 24.18
23:89
24.01
24.09
24.16
24.18
9.998 5619
5570
553
5498
5476
5463
49
40
3*
22
5
55
10 49.74
J J
9.999 0275
90
O O.OO
9-998 5458
662
TABLE II.
For converting intervals of Mean Solar Time into equivalent intervals of Sidereal
Hours.
Minutes.
Seconds.
Decimals.
Mean T.
Sidereal Time.
Mean T.
Sidereal Time.
lean T.
Sidereal Time.
Mean T.
Sidereal Time.
h
A m s
m
m s
g
1
s
s
I
I o 9.856
I
I 0.164
I
1.003
0.02
0.02O
2
2 19.713
2
2 0.329
2
2.005
0.04
0.04.0
3
3 o 29.569
3
3 0-493
3
3.008
0.06
O.OOO
4
4 o 39.426
4
4 0.657
4
4.011
0.08
0.080
c
5 o 49.282
5 0.821
5
5.014.
0.10
O. IOO
i
6 o 59-139
6
6 0.986
6
6.016
0.12
O.I 20
1
7 8.995
8 18.852
I
7 -15
8 -3H
i
7.019
8.022
O.IJ
o.i 6
0.140
O.I 60
9
9 28.708
9
9 .478
9
9.025
0.18
0.180
10
10 38.565
10
10 643
10
10.027
0.2O
0.201
ii
n 48.421
ii
ii .807
ii
I 1.030
0.22
0.221
12
12 58.278
12
12 .971
12
12.033
0.24
0.241
'3
4
13 2 8.134
14 2 17.991
14
13 2.136
I 4 2.300
13
4
I3-036
14.038
0.26
0.28
0.26l
0.28l
i5
15 2 27.847
'5
'5 2.464
15.041
0.3.0
0.301
16
16 2 37.704
if
16 2.628
1 6
16.044
O.J2
0.321
18
17 2 47.560
1 8 2 5,7.416
18
17 2.793
18 2.957
ii
17.047
18.049
0-34
0.36
0.341
0.361
19
19 3 7.273
19
19 3.121
19
19.052
0.38
0.381
20
20 3 17.129
20
20 3.285
20
20.055
0.40
0.401
21
21 3 26.986
21
21 3.450
21
21.057
0.42
0.421
22
22 3 36.842
22
22 3.614
22
22.000
o-44
0.441
23
24
23 3 46.699
*4 3 56.555
2 4
24 3-943
23
24
24.066
. f O
0.46
o. 4 S
0.461
0.481
11
25 4.107
26 4-271
11
25 ooo
26.071
0.50
0.52
0.501
0.521
oJ
27
27 4-435
27
27.074
o-54
0.541
Q C
28
28 4.600
28
28.077
0.56
0.562
EH 8
2 9
30
29 4-764
30 4.928
2 9
3
29.079
30.082
0.58
0.60
0.582
0.602
g i
3 1
31 5.092
3 1
31.085
0.62
O.622
rQ fl
32
S 2 5-257
32.088
0.64
0.642
OQ bfi
33
33 5-421
33
33.090
0.66
O.662
2 .5 o5
34
34 5-585
34
34-093
0.68
0.682
8 15
35
35 5-750
35
35.096
0.70
0.702
S fj
S 6
3 6 5-9H
3 6
36.099
0.72
0.722
ill
37
37 6.078
37
37.101
0-74
0.742
UU 4) 2
PJ rfi H
38 '
38 6.242
38
38.104
0.76
0.762
* 1 s
39
39 6.407
39
39.107
0.78
0.782
40
40 6.571
40
40.110
0.80
0.802
H a "So
4 1 6-735
41.112
0.82
0.822
*o .5 a)
42
42 6.899
42
42.115
0.84
0.842
Ii!
43
44
43 7-064
44 7.228
43
44
43.118
44.120
086
0.88
0.862
0.882
ill
O i, c3
46
45 7-392
46 7-557
Jl
45.123
46.126
0.90
0.92
0.902
0.923
.
47
47 7-721
47
47.129
0.94
0.943
Ill
48
48 7.885
48
48.131
0.96
0.963
5-1 O
O *J3 ^
49
49 8-049
49
49.134
0.98
0.983
5
50 8.214
5
50.137
1. 00
1.003
3 * i
51 8.378
5 1
51.140
<3
5*
52 8.542
52
52.142
j S
53
53 8.707
53
53-145
.s
54
54 8.871
54
54.148
<2 ^
II
55,
57
55 9-035
5 6 9 '99
57 9-364
I?
57
55-I5I
56.153
57.156
e
58
58 9.528
58
58.159
M
I 9
60
59 9.692
60 9.856
59
60
59.162
60.164
563
TABLE III.
For converting intervals of Sidereal Time into equivalent intervals of Mean Solar Time,
Hours.
Minutes.
Seconds.
Decimals.
Sid. T.
Mean Time.
Sid. T.
Mean Time.
Sid. T.
Mean Time.
Sid. T.
Mean Time.
h
Am
m
m 5
s
8
s
s
I
o 59 50.170
I
o 59.836
I
0.997
O.02
0.020
a
i 59 40.341
a
I 59.672
2
1.995
0.04
0.040
3
a 59 30.511
3
a 59.509
3
2.992
O.O6
0.060
4
3 59 20.682
4
3 59-345
4
3-989
0.08
0.080
5
4 59 IO - 8 5 2
5
4 59.181
5
4.986
O.IO
O.I 00
6
5 59 i- oa 3
6
5 59017
6
5.984
0.12
0.120
7
6 58 51.193
7
6 58.853
7
6.981
0.14
0.140
8
7 58 4i-3 6 3
8
7 58.689
8
7-97 8
0.16
0.160
9
8 58 31.534
9
8 58.526
9
8.975
0.18
0.180
10
9 58 21.704
10
9 58-362
10
9-973
0.20
0.199
ii
10 58 11.875
ii
10 58.198
ii
30.970
0.22
0.219
la
ji 5$ 2.045
la
ii 58.034
12
11.967
0.24
0.239
J 3
la 57 52.216
'3
12 57.870
13
12.964
0.26
0.259
H
13 57 42.386
4
13 57.706
H
13.962
0.28
0.279
II
H 57 3 2 -557
15 57 22.727
11
*4 57-543
15 57-379
!i
14-959
I5-956
0.30
0.32
0.299
0.319
17
16 57 12.897
17
16 57.215
17
16.954
0-34
o-339
18
17 57 3.068
18
17 57.051
18
17.953
0.36
o-359
19
18 56 53.238
'9
18 56.887
19
18.948
0.38
0:379
ao
19 5 6 43-49
20
19 56.723
20
19.945
0.40
o-399
ai
20 56 33.579
21
20 56.560
21
20.943
0.42
0.419
aa
21 56 23.750
22
21 56.396
22
21.940
o-44
0.439
a 3
22 56 13.920
a 3
22 56.232
2 3
22.937
0.46
0,459
24
43 5 6 4-09 1
24
23 56.068
2 4
a 3-934
0.48
0-479
a 5
24 55.904
2 5
24.932
0.50
0.499
26
25 55.740
26
25.929
0.52
0.519
C)
a 7
26 55.577
27
26.926
0.54
0.539
a
a8
27 55-4I3
28
27.924
0.56
0.558
EH
a 9
28 55.249
29
28.921
0.58
0-578
83 73
30
2 9 55.085
3
29.918
0.60
0-598
'o g
3 1
30 54.921
3 s
30.915
0.62
0.618
rj
3 a
3 1 54-758
32
3 I -9 I 3
0.64
0.638
o3 "OD
33
32 54.594
3.3
32.910
0.66
0.658
S C? S
34
33 54-43
34
33.907
0.68
0.678
Vn '&
3S
34 54.266
35
34.904
0.70
0.698
C 8 73
36
35 54.102
36
35-9 oa
0.72
0.718
i II
H
36 53.938
37 53-775
P
36.899
37.896
0.74
0.76
0-738
0.758
-5
50 51.645
50.861
a ~l~
5 a
51 51.481
52
51.858
*.d
53
5 a S'-S 1 ?
53
52.855
.a^g
54
53 51.153
54
53-853
3 1
55
54 50.990
55
54.850
5 S3
5 6
55 50.826
5 6
55- 8 47
.2 8
57
56 50.662
57
56.844
~ s
58
57 50-498
5 8
57.842
*^
59
5 8 50-334
59
58.839
60
59 S -^
60
59.836
564
TABLE IV.
For converting Hours, Minutes, and Seconds into Decimals of a Day.
Hours.
Decimal.
Min.
Decimal.
Min.
Decimal.
Sec.
Decimal.
Sec.
Decimal.
1
0.0416 +
1
.000694 +
31
.021527 +
1
.0000116
31
.0003588
2
833 +
2
.001388 +
32
.022222 +
2
.0000231
32
.0003704
3
.1250 +
3
.002083 +
33
.022916 +
3
.0000347
33
.0003819
4
.1666 +
4
.002777 +
34
.023611 +
4
.0000463
34
.0003935
5
.2083 +
5
.003472 +
35
.024305 +
5
.0000579
35
.0004051
6
.2500 +
6
.004166 +
36
.025000 +
6
.0000694
36
.0004167
7
0.2916 +
7
.004861 +
37
.025694 +
7
.0000810
37
.0004282
8
3333 +
8
.005555 +
38
.026388+
8
.0000925
38
.0004398
9
375 +
9
.006250 +
39
.027083 +
9
.0001042
39
.0004514
10
.4166 +
10
.006944 +
40
.027777+
10
.0001157
40
.0004630
11
.4583 +
11
.007638 +
41
.028472 +
11
.0001273
41
.0004745
12
.5000 +
12
008333 +
42
.029166 +
12
.0001389
42
.0004861
13
0.5416 +
13
.009027 +
43
.029861 +
13
.0001505
43
.0004977
14
.5833 +
14
.009722 +
44
030555+
14
.0001620
44
.0005093
15
.6250 +
15
.010416 +
45
.031250 +
15
.0001736
45
.0005208
16
.6666 +
16
.01 II I I +
46
.031944 +
16
.0001852
46
.0005324
17
7083 +
17
.011805 +
47
.032638+
17
.0001968
47
.0005440
18
.7500 +
18
.012500 +
48
033333+
18
.0002083
48
.0005556
19
0.7916 +
19
.013194 +
49
.034027 +
19
.0002199
49
.0005671
20
8333 +
20
.013888 +
50
.034722 +
20
.0002315
50
.0005787
21
.8750 +
21
.014583 +
51
.035416 +
21
.0002431
51
.0005903
22
.9166 +
22
.015277 +
52
.036111 +
22
.0002546
52
.0006019
23
0.9583 +
23
.015972 +
53
.036805 +
23
.0002662
53
.0006134
24
1.0000 +
24
.016666 +
54
.037500 +
24
.0002778
54
.0006250
25
.017361 +
55
.038194 +
25
.0002894
55
.0006366
26
.018055 +
56
.038888+
26
.0003009
56
.0006481
27
.018750 +
57
039583+
27
.0003125
57
.0006597
28
.019444 +
58
.040277 +
28
.0003241
58
.0006713
29
.020138 +
59
.040972 +
29
.0003356
59
.0006829
30
.020833 H"
60
.041666 +
30
.0003472
60
.0006944
The sign +, appended to numbers in this table, signifies that the last figure repeats to fcfinity
TABLE V.
For finding the number of Days from the beginning of the Y<
Date.
Com.
Bi8.
January o.o
February o.o
March o.o
3 1
59
11
April o.o
90
9'
May o.o
120
121
June o.o
!5'
152
July o.o
181
182
August o.o
212
2I 3
September o.o
243
244
October o.o
*73
274
November o.o
34
35
December o.o
334
335
565
TABLE VI.
For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit.
V.
1
2
3
M.
Diff. 1".
M.
Difif. 1".
M.
Diff. 1".
M.
Diff. 1".
0'
1
o.oooooo
0.010908
181.81
181.81
0.654532
0.665442
181.83
181.83
1.309263
1.320178
181.92
181.92
1.964393
1.975316
182.05
182.06
2
0.021817
181.81
0.676352
181.83
1.331093
181.92
1.986240
182.06
3
0.032725
181.81
0.687262
181.84
1.342008
181.92
1.997164
182.06
4
0.043633
181.81
0.698172
181.84
1.352923
181.92
2.008087
182.07
5
0.054542
181.81
0.709082
181.84
1.363839
181.93
2.019011
182.07
6
0.065450
181.81
0.719993
181.84
1-374755
181 93
2.029936
182.07
7
0.076358
181.81
0.730903
181.84
1.385670
181 93
2.040860
182.07
8
0.087267
181.81
0.741813
181.84
1.396586
181.93
2.051785
182.08
9
0.098175
181.81
0.752724
181.84
1.407502
181.93
2.062709
182.08
10
0.109083
181.81
0.763634
181.84
1.418418
181.94
1.073634
182.08
11
o.i 19992,
181.81
0-774545
181.84
1-4*9334
181.94
2.084559
182.08
12
0.130900
181.81
0.785456
181.84
1.440251
181.94
2.095485
182.09
13
0.141808
181.81
0.796366
181.85
1.451167
181.94
2.106410
182.09
14
0.152717
181.81
0.807277
181.85
1.462083
181.94
2.117335
182.09
15
0.163625
181.81
0.818188
181.85
1.473000
181.95
2.128261
182.10
16
0-174534
181.81
0.829099
181.85
1.483917
181.95
2.139187
182.10
17
0.185442
181.81
0.840010
181.85
1.494834
181.95
2.1501 14
182.10
18
19
0.196350
0.207259
181.81
181.81
0.850921
0.861832
181.85
181.85
1-50575 1
1.516668
181.95
181.95
2.161040
2.171966
181 ii
182.11
20
0.218167
181.81
0.872743
181.85
1.527585
181.96
2.182894
182.11
21
0.229076
181.81
0.883654
181.86
i-53 8 53
181.96
2.193820
182.12
22
0.239984
181.81
0.894566
181.86
1.549420
181.96
2.204747
182.12
23
0.250893
181.81
0.905478
181.86
1.560338
181.96
2.215674
182.12
24
0.261801
181.81
0.916389
181.86
1.571256
181.96
2.226602
182.13
25
26
0.272710
0.283619
181.81
181.81
0.927301
0.938212
181.86
181.86
1.582174
1.593092
181.97
181.97
2.237529
2.248457
182.13
182.13
27
0.294527
181.81
0.949124
181.86
1.604010
181.97
*-*593 8 5
182.14
28
0.305436
181.81
0.960036
181.86
1.614928
181.97
2.270313
182.14
29
0.316345
181.81
0.970948
181.87
1.625847
181.97 2.281242
182.14
30
0.327253
181.81
0.981860
181.87
1.636766
181.98 2.292170
182.14
31
0.338162
181.81
0.992772
181.87
1.647684
181.98 2.303099
182.15
32
0.349071
181.81
1.003684
181.87
1.658603
181.98 2.314028
182.15
33
0.359980
181.81
1.014596
181.87
1.669522
181.98 2.324957
182.15
34
0.370888
181.81
1.025509
181.87
1.680441
181.99 *-335 88 7
182.16
35
0.381797
181.81
1.036421
181.87
1.691361
181.99 2.346816
182.16
36
0.392706
181.81
1.047334
181.87
1.702280
181.99 2.357746
182.16
37
0.403615
181.81
1.058246
181.88
1.713200
181.99 2.368676
182.17
38
0.414524
181.82
1.069159
181.88
1.724120
182.00 2.379606
182.17
39
-4*5433
181.82
1.080072
181.88
1.735039
182.00 2.390536
182.17
40
0.436342
181.82
1.090985
181.88
1.745960
182.00
2.4^1467
182.18
41
0.447251
181.82
1.101898
181.88
1.756880
182.00
2.411 ^98
182.18
42
0.458160
181.82
1.112811
181.89
1.767800
182.01
2.425329
182.18
43
0.469069
181.82
1.123724
181.89
1.778 21
182.01
2.434260
182.19
44
0.479970
181.82
1.134637
181.89
1.789(41
182.01
2.445191
182.19
45
46
0.490888
0.501797
181.82
181.82
1.145550
1.156464
181.89
181.89
1.800562
1.811483
182.01
182.02
2.456123
2.467055
182.19
182.20
47
0.512706
181.82
1.167377
181.89
1.822404
182.02
2477987
182.20
48
0.523616
181.82
1.178291
181.89
1.833325
182.02
2.488919
182.20
49
0-5345*5
181.82
1.189205
181.90
1.844247
182.02
2.499851
182.21
50
0-545435
181.82
1.200119
181.90
1.855168
182.03
2.510784
182.21
51
0.556344
181.82
1.211033
181.90
1.866090
182.03
2.521717
182.22
52
0.567254
181.82
1.221947
181.90
1.877012
182.04
2.532650
182.22
53
0.578163
181.83
1.232861
181.90
1.88 79 34
182.04
*-5435 8 3
182.22
54
0.589073
181.83
i-*43775
181.91
1.898856
182.04
2.554517
182.23
55
-5999 8 3
181.83
1.254689
181.91
1.909779
182.04
2.565450
182.23
56
0.610892
181.83
1.265604
181.91
1.920701
182.04
2.576384
182.23
57
0.621802
181.83
1.276518
181.91
1.931624
182.05
2.587319
182.24
58
0.632712
181.83
1.287433
181.91
1.942547
182.05
2.598253
182.24
59
0.643622
181.83
1.298348
181.91
1-953470
182.05
2.609187
182.24
60
0.654532
181.83
1.309263
181.92
I-9 6 4393
182.05
2.620122
182.25
566
TABLE VI.
For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit.
V.
4
5
6
7
M.
Diflf. V.
If,
Diflf. 1".
M.
Diflf. 1".
M.
Diflf. 1".
2.620122
182.25
3.276651
182.50
3.934182
182.80
4.592917
183.17
1
2.631057
182.25
3.287602
182.50
3-945J5 1
182.81
4.603907
183.18
2
2.641993
182.26
3.298552
182.51
3.956119
182.82
4.614898
183.18
3
2.652928
182.26
3-309503
182.51
3.967088
182.82
4.625889
183.19
4
2.663864
182.26
3.320454
182.52
3.978058
182.83
4.636880
183.19
5
2.674800
182.27
3-33I405
182.52
3.989028
182.83
4.647872
183.20
6
2.685736
182.27
3.342356
182.53
3.999998
182.84
4.658864
183.21
7
2.696672
182.27
3-353308
182.53
4.010968
182.84
4.669857
183.21
8
2.707609
182.28
3.364260
182.54
4.021939
182.85
4.680850
183.22
9
2.718546
182.28
3,375212
182.54
4.032911
182.86
4.691843
183.23
10
2.729483
182.29
3-386165
182.55
4.043882
182.86
4-70*837
183.24
11
2.740420
182.29
3.397118
182.55
4.054854
182.87
4.713831
183.24
12
2.751358
182.29
3.408071
182.56
4.065826
182.87
4.724826
183.25
13
14
2.762295
2.773233
182.30
182.30
3.419024
3.429978
182.56
182.57
4.076799
4.087772
182.88
182.88
4.735821
4.746816
183.25
183.26
15
2.784172
182.31
3.440932
182.57
4.098745
182.89
4.757812
183.27
16
2.795110
182.31
3.45x887
182.58
4.109718
182.90
4.768809
183.27
17
2.806049
182.31
3.462841
182.58
4.120692
182.90
4.779805
183.28
18
2.816988
182.32
3-473796
182.59
4.131667
182.91
4.790802
183.28
19
2.827927
182.32
3.484752
182.59
4.14264!
182.91
4.801800
183.29
20
2.838867
182.33
3.495707
182.60
4.153616
182.92
4.812797
183.30
21
2.849806
182.33
3.506663
182.60
4.164592
182.93
4.823796
183.31
22
2.860746
182.33
3.517619
182.61
4.175568
182.93
183.32
23
2.871686
182.34
3-5*8575
182.61
4.186544
182.94
4.845794
183.32
24
2.882627
182.34
3.539532
182.61
4.197520
182.94
4.856793
183-33
25
2.893567
182.35
3.550489
182.62
4.208497
182.95
4.867793
l8 3-34
26
2.904508
182.35
3.56i447
182.62
4.219474
182.95
4.878793
183.34
27
2.915449
182.36
3.572404
182.63
4.230451
182.96
4.889794
183.35
28
2.926391
182.36
3.583362
182.63
4.241429
182.97
4.900795
183.36
29
2.937332
182.36
3-5943*
182.64
4.252408
182.97
4.911797
183.36
30
2.948274
182.37
3.605279
182.64
4.263386
182.98
4.922799
183-37
31
2.959217
182.37
3.616238
182.65
4.274365
182.99
4.933801
183.38
32
2.970159
182.37
3.627197
182.65
4.285344
182.99
4.944804
183.38
33
2.981102
182.38
3.638156
182.66
4.296324
183.00
4.955807
183-39
34
2.992045
182.38
3.649116
182.66
4.307304
183 oo
4.966811
183.40
35
3.002988
182.39
3.660076
182.67
4.318284
183.01
4.977815
183.41
36
3.013931
182.39
3.671037
182.68
4.329265
183.01
4.988820
183.41
37
3.024875
182.39
3.681997
182.68
4.340246
183.02
4.999825
183.42
38
39
3.035819
3.046763
182.40
182.40
3.692958
3.703920
182.69
182.69
4.351228
4.362210
183.03
183.03
5.010830
5.021836
183-43
18 3-43
40
3.057707
182.41
3.714*81
182.70
4-373'9*
183.04
5.032842
183.44
41
3.068652
182.41
3-7*5843
182.70
4-3 8 4 T 75
183.05
5.043849
183.45
42
43
3-079597
3.090542
182.42
182.42
3.736806
3.747768
182.71
182.71
4-395I58
4.406141
183.05
183.06
5.054856
5.065864
183.46
183.46
44
3.101488
182.43
3-75873I
182.72
4.417125
183.06
5.076872
183.47
45
1 46
; 47
48
3.112433
3.123379
3-'343*5
182.43
182.44
182.44
182.44
3.769694
3.780658
3.791622
3.802586
182.72
182.72
182.73
182.74
4.428109
4-439093
4.450078
4.461064
183.07
183.08
183.08
183.09
5.087880
5.098889
5.109898
5.120908
183.48
185.48
183.49
183.50 i
49
3.156219
182.45
3.813551
182.74
4.472049
183.10
5.131918
183.51
50
3.167166
182.45
3.824515
182.75
4-483035
183.10
5.142929
183.51
51
3.178113
182.46
3.835481
182.76
4.494022
183.11
5- I 5394
183.52
52
3.189061
182.46
3.846446
182.76
4.505008
183.12
5.164951
183-53
53
3.200009
182.47
3.857412
182.77
4-5I5995
183.12
5- J 759 6 3
183.54
54
3.210957
182.47
3.868378
182.77
4.526983
183.13
5.186975
183.54
55
3.221905
182.48
3-879345
182.78
4-53797 1
183.14
5.197988
!8 3 . 5 5
56
3,232854
182.48
3.890312
182.78
4.548959
183.14
5.209002
183.56
57
3.243803
182.49
3.901279
182.79
4^59948
183.15
5.220015
183.57
58
3.254752
182.49
3.912246
182.79
4-570937
183.15
5.231029
*83-57
59
3.265702
182.49
3.923214
182.80
4.581927
183.16
5.242044
183.58
60
3.276651
182.50
3.934182
182.80
4.592917
183.17
5-*5359
183.59
567
TABLE VI.
For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit.
V.
8
9
10
11
M.
Diff. 1".
M.
Diff. 1".
M.
Diff. 1".
M.
Diff. 1".
0'
5.253059
183.59
5.914815
184.06
6.578391
184.60
7-243997
185.19
1
5.264075
183.59
5.925859
184.07
6.589467
184.61
7.255109
185.20
2
5 275090
183.60
5.936904
184.08
6.600544
184.62
7.266222
185.21
3
5.286107
183.61
5-947949
184.09
6.611622
184.63
7-277335
185.22
4
5.297124
183.62
5.958995
184.10
6.622700
184.64
7.288449
185.23
5
5.308141
183.62
5.970041
184.11
6.633778
184.65
7.299563
185.25
6
5.319159
183.63
5.981087
184.11
6.644857
18466
7.310678
185.26
7
5 33oi77
183.64
5.992134
184.12
6.655937
184 67
7.321793
185.27
8
5.341195
183.65
6.003182
184.13
6.667017
184.67
7.332909
185.28
9
5.352214
183.66
6.014230
184.14
6.678098
184.68
7.344026
185.29
10
5.363234
183.66
6.025279
184.15
6.689179
184.69
7-355*44
185.30
11
5.374254
183.67
6.036328
184.16
6.700261
184.70
7.366262
185.31
12
13
5-385275
5.396296
183.68
183.69
6.047378
6.058428
184.17
184.18
6.711343
6.722426
184.71
184.72
7.377381
7.388500
185.32
185-33
14
S-WS 1 ?
183.69
6.069479
184.18
6.733510
184.73
7.399620
185-34
15
5-4 l8 339
183.70
6.080530
184.19
6-744594
184.74
7.410741
185.35
16
5.429361
183.71
6.091582
184.20
6.755679
184.75
7.421862
185.36
17
5.440384
183.72
6.102634
184.21
6.766764
184.76
7.432983
185.37
18
5.451407
183.73
6.113687
184.22
6.777850
184.77
7.444106
185.38
19
5.462431
183-73
6.124740
184.23
6.788937
184.78
7-455230
185.39
20
5-473455
183.74
6-135794
184.24
6.800024
184.79
7-466354
185.40
21
5.484480
183.75
6.146849
184.25
6. 811112
184.80
7.477478
185.4!
22
5-49555
183.75
6.157904
184.25
6.822200
184.81
7.488603
185.42
23
24
5-506530
5-5I7556
18376
183.77
6.168959
6.180015
184.26
184.27
6.833289
6.844378
184.82
184.83
7.499729
7.510855
185.43
185.44
25
5.528583
183.78
6.191072
184.28
6.855468
184.84
7.521982
185.46
26
27
5.539610
5.550637
183.79
183.79
6.202129
6.213187
184.29
184.30
6.866559
6.877650
184.85
184.86
7.533110
7-544239
185.47
185.48
28
5.561665
183.80
6.224245
184.31
6.888742
184.87
7-555368
185.49
29
5-572693
183.81
6.235304
184.32
6.899834
184.88
7.566497
185.50
30
5.583722
183.82
6.246363
184.32
6.910927
184.89
7.577628
185.51
31
5-594752
183.8;
6.257422
184.33
6.922021
184.90
7.588759
185.52
32
5.605782
183.8;
6.268482
184.34
6.933115
184.91
7-59989
185.53
33
5.616812
183.84
6-279543
l8 4-35
6.944210
184.92
7.611022
185.54
34
5.627843
183.85
6.290605
184.36
6.9553 5
184.93
7.622155
185.55
35
5.638874
183 86
6.301667
184.37
6.966401
184.94
7.633289
185.57
36
37
5.6499 6
5.660938
183.87
183.87
6.312729
6.323792
184.38
184.39
6.977498
6.988595
184.95
184.96
7.644423
7.655558
185.58
185.59
38
5.671971
183.88
6-334855
1 84.40
6.999693
184.97
7.666694
185.60
39
5.683004
183.89
6.345919
184.41
7.010791
184.98
7.677830
185.61
40
5.694038
183.90
6.356984
184.41
7.021890
184.99
7.688967
185.62
41
5.705072
183.91
6.368049
184.42
7.032990
185.00
7.700104
185.63
42
5.716106
183.91
6.379115
184.43
7.044090
185.01
7.711242
185.64
43
5.727141
183.92
6.390181
1 84.44
7.055191
185.02
7.722381
185.65
44
5.738177
183-93
6.401248
184.45
7.066292
185.03
7-733521
185.66
45
5.749213
183.94
6.412315
184.46
7-077394
185.04
7.744661
185.68
46
5.760250
18395
6.423383
184.47
7.088497
185.05
7.755802
185.69
47
5.771287
183.96
6.434451
184.48
7.099600
185.06
7.766943
185.70
48
5.782325
183.96
6.445520
1 84.49
7.1 10704
185.07
7.778085
185-71
49
5-793363
183.97
6.456590
184.50
7.121808
185.08
7.789228
185.72
50
51
5.804401
5.815440
183.98
183.99
6.467660
6.478731
184.51
184.52
7.132913
7.144019
185.09
185.10
7.800372
7.811516
185.73
185.74
52
5.826480
184.00
6.489802
184.52
7.155125
185.11
7.822661
185.75
53
5.837520
184.01
6.500874
184-53
7.166232
185.12
7.833807
fH
54
5.848561
184.01
6.511946
184.54
7.177340
185.13
7.844953
185.78
55
5.859602
184.02
6.523019
184.55
7.188448
185.14
7.856100
J o 5 -z 9
56
(5.870644
184.03
6.534092
184.56
7-199557
185.15
7.867247
185.80
57
5.881686
184.04
6.545166
184-57
7.210666
185.16
7.878396
185. Si
58
59
5.892728
5.903771
184.05
184.06
6.556241
6.567316
184.58
184.59
7221776
7.232886
185.17
185.18
7.889545
7.900694
185.82
185.83
60
5.914815
184.06
6.578391
184.60
7.243997
185.19
7.911845
185.84
568
TABLE VI.
For finding the True Anomaly or the Time from the Perihelior in a Paralolic Orbit.
V.
12
13
14
15
M.
Diff. 1".
M.
Diff. 1".
M.
Diff. 1".
M.
Diff. 1".
0'
7.911845
185.84
8.582146
186.56
9.255120
187.33
9.930984
188.16
1
7.922995
185.86
8.593340
186.57
9.266360
187.34
9.942274
188.18
2
7-934I47
185.87
8.604535
186.58
9.277601
187.35
9-953565
188.19
3
7.945300
185-88
8.615730
186.59
9.288842
187.37
9.964857
188.21
4
7.956453
185.89
8.626926
186.61
9.300085
187.38
9.976149
188.22
5
7.967606
185.90
8.638123
186.62
9.311328
187.40
9.987443
188.23
6
7.978761
185.91
8.649320
186.63
9.322572
187.41
9.998738
188.25
7
7.989916
185.92
8.660518
186.64
9-3338I7
187.42
10.010033
188.26
8
8.001072
l8 5-93
8.671717
186.66
9.345063
187.44
10.021329
188.28
9
8.012228
185-95
8.682917
186.67
9.356310
187.45
10.032626
188.29
10
8.023385
185.96
8.694117
186.68
9-3 6 7557
187.46
10.043924
188.31
11
8.034543
185.97
8.705318
186.69
9.378805
187.48
10.055223
188.32
12
8.045702
185.98
8.716520
186.71
9.390054
187.49
10.066523
188.34
13
8.056861
185.99
8.727723
186.72
9.401304
187.50
10.077823
188.35
14
8.068021
186.00
8.738927
186.73
9-41*555
187.52
10.089125
188.37
15
8.079181
186.02
8.750131
186.74
9.423806
187.53
10.100427
188.38
16
8.090343
186.03
8.761336
186.76
9.435058
187.54
10.1 1 1730
'^ 39
17
8.101505
186.04
8.772542
186.77
9.446311
187.56
10.123035
188.41
18
8.112668
186.05
8.783748
186.78
9-4575 6 5
187.57
10.134340
188.42
19
8.123831
186.06
8 -794955
186.79
9.468820
187.59
10.145646
188.44
20
8.134995
186.07
8.806163
186.81
9.480076
187.60
10.156952
188.45
21
8.146160
186.09
8.817372
186.82
9.491332
187.61
10.168260
188.47
22
8.157326
186.10
8.828582
186.83
9.502589
187.63
10.179568
188.48
23
8.168492
186.11
8.839792
186.84
9.513847
187.64
10.190878
188.50
24
8.179659
186.12
8.851003
186.86
9.525106
187.65
10.202188
188.51
25
8.190826
186.13
8.862215
186.87
9.536366
187.67
10.213499
188.53
26
8.201995
186.15
8.873427
186.88
9547626
187.68
10.224812
188.54
27
8.213164
186.16
8.884641
186.90
9.558888
187.70
10.236125
188.56
28
8.224334
186.17
8.895855
186.91
9.570150
187.71
10.247439
X ^- S7
29
8.235504
186.18
8.907070
186.92
9.581413
187.72
10.258753
188.59
30
31
8.246675
8.257847
186.19
186.20
8.918286
8.929502
186.93
186.95
9.592676
9.603941
187.74
187.75
10.270069
10.281386
188.60
188.62
32
8 269020
186.22
8.940719
186.96
9.615207
187.77
10.292703
188.63
33
8.280193
186.23
8 -95i937
186.97
9.626473
187.78
10.304021
188.65
34
8.291367
186.24
8.963156
186.99
9.637740
187.79
10.315341
188.66
35
8.302542
186.25
8.974376
187.00
9.649008
187.81
10.326661
188.68
36
8.313717
186.26
8.985596
187.01
9.660277
187.82
10.337982
188.69
37
38
8.324893
8.336070
186.28
186.29
8.996817
9.008039
187.02
187.04
9.671547
9.682817
187.84
187.85
10.349304
10.360627
188.71
188.72
39
8.347248
186.30
9.019262
187.05
9.694088
187.86
10.371951
188.74
40
41
8.358426
8.369605
186.31
186.32
9.030485
9.041709
187.06
187.08
9.705361
9.716634
187.88
187.89
10.383275
10.394601
188.75
188.77
42
8.380785
186.34
9.052934
187.09
9.727908
187.91
10.405927
188.78
43
44
8.391966
8.403147
186.35
186.36
9.064160
9.075387
187.10
187.12
9.739182
9.750458
187.92
18793
10.417255
10.428583
188.80
188.81
45
8.414329
186.37
9.086614
187-13
9.761734
187.95
10.439912
188.83
I 46
8.425512
186.38
9.097842
187.14
9.773012
187.96
10.451242
188.84
47
8.436695
186.40
9.109071
187.16
9.784290
187.98
10.462573
188.86
48
49
8.447879
8.459064
186.41
186.42
9.120301
9.131531
187.17
187.18
9795569
9.806849
187.99
188.00
10.473905
10.485238
188.87
188.89
50
8.470250
186.43
9.142763
187.20
9.818129
188.02
10.496572
188.90
51
52
8.481436
8.492623
186.45
186.46
9-J53995
9.165228
187.21
187.22
9.829410
9.840693
188.03
188.05
10.507907
10.519242
188.92
188.93
53
8.503811
186.47
9.176462
187.23
9.851977
188.06
10.530579
188.95
54
8.515000
186.48
9.187696
187.25
9.863261
18808
10.541916
188.97
55
8.526189
186.49
9.198931
187.26
9.874546
188.09
10-553*55
188.98
56
8-537379
186.51
9.210167
187.27
9.885832
188.10
10.564594
189.00
57
8.548569
186.52
9.221404
187.29
9.897118
188.12
10-575934
189.01
58
59
8.559761
8'570953
186.53
186.54
9.232642
9.243880
187.30
187.31
9.908406
9.919694
188.13
188.15
10.587276
10.598618
189.03
189.04
60
8.582146
186.56
9.255120
187.33
9.930984
188.16
10.609961
189.06
669
TABLE VI.
For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit.
V.
16
17
18
19
M.
Diff. I".
M.
Diff. 1".
M.
Diff. 1".
M.
Diff. I".
0'
10.609961
189.06
11.292277
190.02
11.978162
191.04
12.667850
192.13
I
2
10.621305
10.632649
189.07
189.09
11.303679
11.315082
190.03
190.05
11.989625
12.001089
191.06
191.08
12.679379
12.690908
192.15
192.17
3
10.643995
189.10
11.326485
190.07
12.012554
191.09
12.702439
192.19
4
10.655342
189.12
11.337889
190.08
12.024021
191.11
12.713970
192.21
5
10.666690
189.14
11.349295
190.10
12.035488
191.13
12.725503
192.22
6
10.678038
189.15
11.360701
190.12
12.046956
191.15
12.737037
192.24
7
10.689388
189.17
11.372109
190.13
12.058425
191.16
I2 -748573
192.26
8
10.700738
189.18
11.383517
190.15
12.069896
191.18
12.760109
192.28
9
10.712090
189.20
11.394927
190.17
12.081367
191.20
12.771646
192.30
10
10.723442
189.21
11.406337
190.18
12.092840
191.22
12.783185
192.32
11
12
10.734795
10.746149
189.23
189.24
11.417749
11.429161
190.20
190.22
12.104313
12.115788
191.24
191.25
12.794724
12.806265
192.34
192.36
13
10.757505
189.26
11.440575
190.23
12.127264
191.27
12.817807
14
10.768861
189.28
11.451989
190.25
12.138741
191.29
12.829350
192.39
15
10.780218
189.29
"4 6 3405
190.27
12.150219
191.31
12.840894
192.41
16
10.791576
189.31
11.474821
190.28
12.161698
191.32
12.852440
192.43
17
10.802935
189.32
11.486239
190.30
12.173178
191.34
12.863986
192.45
18
10.814295
189.34
11.497657
190.32
12.184659
191.36
12.875534
192.47
19
10.825655
I89-35
11.509077
190.33
12.196141
191.38
12.887082
192.49
20
10.837017
189.37
11.520497
190.35
12.207624
191,/j.o
12.898632
192.51
21
10.848380
189.39
11.531919
190.37
12.219108
191.41
12.910183
192.53
22
10.859744
189.40
"543342
190.39
12.230594
191.43
12.921736
192.55
23
10.871108
189.42
"554765
190.40
12.242080
191.45
12.933289
192.56
24
10.882474
189.43
11.566190
190.42
12.253568
191.47
12.944843
192.58
25
10.893840
189.45
11.577616
190.44
12.265057
191.49
12.956399
192.60
26
10.905208
189.47
1 1.589042
190.45
12.276546
191.50
12.967956
192.62
27
10.916576
189.48
11.600470
190.47
12.288037
191.52
12.979514
192.64
28
10.927946
189.50
11.611899
190.49
12.299529
191.54
12.991073
192.66
29
10.939316
189.51
11.623328
190.50
12.311022
191.56
13.002633
192.68
30
10.950687
189-53
11.634759
190.52
12.322516
191.58
13.014195
192.70
31
10.962059
189.55
11.646191
190.54
12.334011
191.60
13.025757
192.72
32
10973433
189.56
11.657624
190.56
12.345508
191.61
13.037321
192.74
33
34
10.984807
10.996182
189.58
189.59
11.669057
11.680492
190.57
190.59
12.357005
12.368503
191.63
191.65
13.048886
13.060452
192.76
192.78
35
11.007558
189-61
11.691928
190.61
12.380003
191.67
13.072019
192.80
36
11.018935
189.63
11.703365
190.62
12.391504
191.69
13.083587
192.82
37
11.030313
189.64
11.714803
190.64
12.403006
191.70
13.095157
192.83
38
11.041692
189.66
11.726242
190.66
12.414509
191.72
13.106727
192.85
39
11.053072
189.67
11.737682
190.68
12.426013
191.74
13.118299
192.87
40
11.064453
189.69
11.749123
190.69
12-4375 1 ?
191.76
13.129872
192.89
41
11.075835
189.71
11.760565
190.71
12.449023
191.78
13.141446
192.91
42
11.087218
189.72
11.772008
190.73
12.460531
191.80
13.153022
192.93
43
11.098602
189.74
11.783452
190.74
12.472039
191.81
13.164598
192.95
44
11.109987
189.76
11.794897
190.76
12.483548
191.83
13.176176
192.97
45
11.121372
189.77
11.806344
190.78
12.495059
191.85
i3- l8 7755
192.99
46
11.132759
189.79
11.817791
190.80
12.506571
191.87
i3-'99335
193.01
47
11.144147
189.80
11.829239
190.81
12.518083
191.89
13.210916
193.03
48
11.155536
189.82
11.840689
190.83
12.529597
191.91
13.222498
193.05
49
11.166925
189.84
11.852139
190.85
12.541112
191.93
13.234082
193.07
50
11.178316
189.85
11.863590
190.87
12.552628
191.94
13.245667
193.09
51
11.189708
189.87
11.875043
190.88
12.564145
191.96
13.257253
193.11
52
II.20IIOO
189.89
11.886496
190.90
12.575664
191.98
13.268840
193.13
53
11.212494
189.90
11.897951
190.92
12.587183
192.00
13.280428
193.15
54
11.223889
189.92
11.909407
190.94
12.598704
192.02
13.292017
I93- X 7
55
11.235284
189.93
11.920863
190.95
12.610225
192.04
13.303608
193.19
56
11.246681
189.95
11.932321
190.97
12.621748
192.06
13.315200
193.21
57
11.258078
189.97
11.943780
190.99
12.633272
192.07
13.326793
193.23
58
11.269477
189.98
191.01
12.644797
192.09
13-338387
193.25
! 59
11.280876
190.00
11.966700
191.02
12.656323
192.11
13.349982
193.27
60
11.292277
190.02
11.978162
191.04
12.667850
192.13
13.361579
193-29
570
TABLE VI.
For finding tne True Anomaly or the Time from the Perihelion in a Parabolic Orbit.
20
21
22
23
M.
Diff. 1".
K,
Diff. 1".
M.
Diff. 1".
M.
Diff. 1".
0'
13.361579
193.29
14.059591
194-51
14.762133
195.80
15.469459
197.17
1
13-373177
193-3I
14.071262
194-53
14.773882
195-83
15.481290
197.19
2
13-384776
193-33
14.082935
194-55
14.785632
195.85
15.493122
197.21
3
13.396376
193-35
14.094608
194-57
14.797384
195-87
15-504956
197.24
4
13.407977
193-37
14.106283
194.59
14.809137
195.89
15-516791
197.26
5
13.419580
193-39
14.117960
194.61
14.820891
I95-9I
15.528627
197.28
6
13.431183
193-4I
14.129637
194.64
1483*647
19594
15.540465
197-31 i
7
13.442788
193-43
14.141316
194.66
14.844403
195.96
15.552304
197.33
8
13-454394
193-45
14.152996
194.68
14.856161
195-98
I5-564H4
'97-35
9
13.466002
193-47
14.164677
194.70
14.867921
196.00
15.575986
197.38
10
13.477610
193-49
14.176360
194.72
14.879682
196.03
15.587830
197-4
11
13.489220
I93-5I
14.188044
19474
14.891444
196-05
15-599675
197-43
12
13.500831
193-53
i4-i997*9
194.76
14.903208
196.07
15.611521
197-45
13
I 3.5 12443
193-55
14.211415
194.78
I4-914973
196.09
15.623369
'97-47
14
13.524056
19357
14.223103
194.81
14.926739
196.12
15.635218
197.50
15
16
13.535671
13.547287
193-59
193.61
14.234792
14.246482
194.83
194.85
14.938506
14.950275
196.14
196.16
15.647068
15.658920
197.52
197-54
17
13.558904
193-63
14.258174
194.87
14.962045
196.18
15.670773
197-57
18
13.570522
14.269867
194-89
14.973817
196.20
15.682628
197-59
19
13.582141
193.67
14.281561
194.91
14.985590
196.23
I5-694484
197.61
20
13.593762
193-69
14.293256
194-93
14-997365
196.25
15.706342
197.64
21
22
13.605383
13.617006
I93-7I
193-73
14.304953
14.316651
194.95
194.98
15.009140
15 020917
196.27
196.30
15.718201
15.730061
197.66
197.69
23
13.628631
193-75
14.3*835
195.00
15.032696
196.32
15.741923
I97-71
24
13.640256
193-77
14.340050
195.02
15-044475
196.34
I5-753786
197-73
25
13.651883
193-79
14.351752
195.04
15.056256
196.36
15.765651
197.76
26
13-663511
193.81
14.363455
195.06
15.068039
196.39
15.777517
197.78
27
13.675140
193-83
H-375'59
195.08
15.079823
196.41
I5-789385
197.80
28
29
13.686770
13.698401
I93-85
I93-87
14.386865
14.398572
195.10
'95-13
15.091608
i5-i3394
196.43
196.45
15.801254
15.813124
197.83
197-85
30
13.710034
I93-89
14.410280
195.15
15.115182
196.48
15.824996
197.88
31
13.721668
193.91
14.421990
195.17
15.126971
196.50
15.836870
197-9
32
13-733303
193-93
14-433700
I95-19
15.1 38762
196.52
15-848744
197.92
33
13.744940
193-95
14.445412
195.21
15.150554
196.54
15.860620
197-95
34
13.756577
193-97
14.457126
195.23
15.162348
196.57
15.872498
197-97
35
36
37
13.768216
13.779856
13.791498
19399
194.01
194.03
14.468841
14.480557
14.492274
195.26
195.28
195-3
15.174142
15.185938
15.197736
196.59
196.61
196.64
r ff
I5-884377
15.896258
15.908140
198.00
198.02
198.04
38
13.803140
194-05
14.503992
195-3*
15.209535
196.66
15.920023
198.07
39
13.814784
194.07
14.515712
195-34
15.221335
196.68
15.931908
198.09
40
13.826429
194.09
14.527434
195-36
15.233137
196.70
15-943794
198.12
41
13.838075
194.11
14.539156
195-39
15.244940
196.73
15-955682
198.14
42
13.849723
194.14
14.550880
195.41
15.256744
196.75
15.967571
198.17
43
13.861372
194.16
14.562605
195-43
15.268550
196.77
15.979462
198.19
44
13.873022
194.18
14-574331.
195-45
15.280357
196.80
15-991354
198.21
45"
i 46
47
I 48
13.884673
13-8963*5
13-907979
13-919634
194.20
194.22
194.24
194.26
14.586059
14.597788
14.609519
14.621250
195-47
195.50
195.52
195-54
15.292165
15-303975
15-315786
15-3*7599
196.82
196.84
196.87
196.89
16.003248
16.015143
16.027039
16.038937
198.24
198.26
198.29
198.31
49
13.931290
194.28
14.632983
I95-56
I5-339413
196.91
16.050836
198.34
50
51
52
53
54
13.942948
13.954606
13.966266
13.977927
13.989590
194-30
194-3*
194-34
194.36
I94-38
14.644718
14.656453
14.668190
14.679929
14.691668
195.58
195.60
195-63
I95-65
195.67
15.351228
15-363045
15.374863
15.386683
15.398504
196.94
196.96
196.98
197.00
197.03
16.062737
16.074639
16.086543
16.098449
16.110355
198.36
198.38
198.41
198-43
198.46
55
14.001254
194.41
14.703409
195-69
15.410326
197.05
16.122263
198.48
56
14.012919
194-43
14-715151
195.71
15.422150
197.07
10.134173
198.51
57
58
59
14.024585
14.036252
14.047921
194.45
194.47
194.49
14.726895
14.738640
14.750386
195-74
195.76
195.78
15-433975
15.445802
15.457630
197.10
197.12
197.14
16.146084
16.157997
16.16991 1
198-53
198.56
198.58
60
14.059591
194.51
14.762133
195.80
15-469459
197.17
16.181826
198.60
571
TABLE VI.
For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit.
V.
24
25
26
27
M.
Diff. 1".
M.
Diff. I".
M.
Diff. 1".
M.
Diff. 1".
0'
16. 181826
198.60
16.899499
200.12
17.622747
201.70
18.351847
203.37
1
16.193743
198.63
16.911507
200.14
17.634850
201.73
18.364050
203.40
2
16.205662
198.65
16.923516
200.17
17.646954
201.76
18.376255
203.42
3
16.217582
198.68
16.935527
200.19
17.659060
201.78
18.388461
203.45 j
4
16.229503
198.70
16.947539
200.22
17.671168
201. 8l
18.400669
203.48
5
16.241426
198.73
16959553
200.24
17.683278
201.84
18.412879
203.51
6
16.253350
198.75
16.971568
200.27
17.695389
201.87
18.425090
203.54
7
16.265276
198.78
16.983585
200.30
17.707502
201.89
18.437303
203.57
8
16.277204
198.80
16.995604
200.32
17.719616
201.92
18.449518
203.59
9
16.289133
198.83
17.007624
200.35
17.731732
201.95
18.461735
203.62
10
16.301063
198.85
17.019646
200.37
17.743850
201.97
18.473953
203.65
11
16.312995
198.88
17.031669
200.40
17.755969
202.00
18.486173
203.68
12
16.324928
198.90
17.043694
200.43
17.768090
2O2.O3
18498395
203.71
13
16.336863
198.93
17.055720
200.45
17.78021 3
202.06
18.510618
203.74
14
16.348799
198.95
17.067748
200.48
17.792337
202.08
18.522843
203.77
15
16.360737
198.97
17.079777
200.50
17.804462
202.11
18.535070
203.80
16
16.372676
199.00
17.091808
200.53
17.816590
202.14
18.547299
203.82
17
16.384617
199.02
17.103841
200.56
17.828719
202.17
18.559529
203.85
18
16.396559
199.05
17.115875
200.58
17.840850
2O2.I9
18.571761
203.88
19
16.408503
199.07
17.127911
200.61
17.852982
202.22
18.583995
203.91
20
16.420448
199.10
17.139948
200.64
17.865116
2O2.25
18.596230
203.94
21
16.432395
199.12
17.151987
200.66
17.877252
202.28
18.608467
203.97
22
16.444343
199.15
17.164028
200.69
17.889389
202.30
18.620706
204.00
23
16.456292
199.17
17.176070
200.71
17.901528
202-33
18.632947
204.03
24
16.468243
199.20
17.188114
200.74
17.913669
202.36
18.645190
204.05
25
16.480196
199.22
17.200159
200.77
17.925811
202.39
18.657434
204.08
26
16.492151
199.25
17.212206
200.79
J7-937955
202.41
18.669679
204.11
27
16.504107
199.27
17.224254
200.82
17.950101
202.44
18.681927
204.14
28
16.516064
199.30
17.236304
200.85
17.962248
202.47
18.694177
204.17
29
16.528022
'99-33
17.248356
200.87
17-974397
202.50
18.706428
204.20
30
16.539983
199-35
17.260409
200.90
17.986548
202.52
18.718680
204.23
31
l6 -55 J 945
199.38
17.272464
200.93
17.998700
202.55
18.730935
204.26
32
16.563908
199.40
17.284520
200.95
18.010854
202.58
18.743191
204.29
33
l6 -575 8 73
199-43
17.296578
200.98
18.023010
202.6l
18.755449
204.32
34
16.587839
199.45
17.308637
201.00
18.035167
202.64
18.767709
204.35
35
16.599807
199.48
17.320698
2OI.O3
18.047326
202.66
18.779971
204.37
36
16.611776
199.50
17.332761
201.06
18.059487
202.69
18.792234
204.40
37
16.623747
199-53
17.344825
2OI.O8
18.071649
202.72
18.804499
204.43
38
16.635719
J 99-55
17.356891
201. II
18.083813
202.75
18.816767
204.46
39
16.647693
199.58
17.368959
201.14
18.095979
202.78
18.829036
204.49
40
16.659669
199.60
17.381028
201. l6
18.108146
202.80
18.841305
204.52
41
16.671646
199.63
17.393098
201.19
18.120315
202.83
18.853577
204.55
42
16.683624
199.65
17.405171
201.22
18.132486
202.86
18.865851
204.58
43
16.695604
199.68
17.417245
201.24
18.144658
202.89
18.878127
204.61
44
16.707586
199.70
17.429320
201.27
18.156832
202.92
18.890404
204.64
45
16.719569
J99-73
17.441397
201.30
18.169008
202.94
18.902684
204.67
46
* 6 -73 f 553
199.76
17.453476
201.32
18. 181186
202.97
18.914965
204.70
47
l6 -743539
199.78
17.465556
201-35
18.193365
203.00
18.927247
204.73
48
16.755527
199.81
17.477638
201.38
18.205546
203.03
18.939532
204.76
49
16.767516
199.83
17.489722
201.41
18.217728
203.06
18.951818
204.79
50
16.779507
199.86
17.501807
201.43
18.229912
203.08
18.964106
204.81
51
16.791499
199.88
17.513894
201.46
18.242098
203.11
18.976396
204.84
52
16.803493
199.91
17.525982
201.49
18.254286
203.14
18.988687
204.87
53
16.815488
199.94
17.538072
201.51
18.266475
203.17
19.000981
204.90
54
16.827485
199.96
17.550163
201.54
18.278666
203.20
19.01 3276
204.93
55
16.839484
199.99
17.562257
201.57
18.290859
203.23
19.025573
204.96
56
16.851484
200.01
17.574352
201.59
18.303053
203.25
19.037871
204.90
57
16.863485
200.04
17.586448
201.62
18.315249
203.28
19.050172
205.02
58
16.875488
2OO.O6
17.598546
201.65
18.327447
203.31
19.062474
205.05
59
16.887493
200.09
17.610646
201.68
18.339646
203.34
19.074778
205.08
60
16.899499
200.12
17.622747
201.70
18.351847
203.37
19.087084
205.11
572
TABLE VI,
For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit.
V.
28
29
30
31
M.
Diff. 1".
M.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. 1".
0'
19.087084
205.11
19.828747
206.94
1.313 3849
44.08
1.329 0430
42.92
I
19.099391
205.14
19 841164
206.97
.313 6493
44.06
.329 3004
42.91
2
19.111701
205.17
I9-853583
207.00
.313 9136
44.04
329 5578
42.89
3
19 124012
205.20
19.866004
207.03
.314 1778
44.02
.329 8151
42.87
4
19.136325
205.23
19.878427
207.06
.314 4419
44.00
.330 0723
42.85
5
19 148639
205 26
19 890852
207.09
1.314 7058
43.98
1.330 3293
42.83
6
19.160956
205.29
19.903279
207.13
.314 9696
43.96
.330 5862
42.81
7
19.173274
205.32
19.915707
207.16
3 X 5 2333
43-94
33 8431
42.80
8
19.185594
205-35
19.928137
207.19
.315 4969
43.92
.331 0998
42.78 j
9
19.197916
205.38
19.940569
207.22
.315 7604
43-90
33 1 3564
42.76
10
19.210240
205.41
19.953003
207.25
1.316 0237
43.88
1.331 6129
42.74
11
19 222566
205.44
19.965439
207.28
.316 2869
43.86
.331 8693
42.72
12
19.234893
205.47
19.977877
207.31
.316 5500
43-84
332 1255
42.70
13
19.247222
205.50
19.990317
207.34
.316 8130
43.82
332 3817
42.69
14
!9-259553
205-53
20.002759
207.38
.317 0759
43.80
332 6378
42.67
15
19.271885
205.56
20.015202
207.41
1.317 3386
43.78
1.332 8937
42.65
16
19.284220
205.59
20.027647
207.44
.317 6013
43.76
333 H96
42.63
17
18
19.296556
19.308894
205.62
205.65
20.040095
20.052544
207.47
207.50
.317 8638
.318 1262
43-74
43-72
333 453
-333 6609
42.61
42.59
19
19.321234
205.68
20.064995
207.53
.318 3885
43-70
333 9 l6 4
42.58
20
19.333576
205.71
20.077448
207.57
1.318 6506
43.68
1.334 1718
42.56
21
19.345920
205.74
20.089903
207.60
.318 9127
43.67
334 4271
42.54
22
19.358265
205.77
20.102360
207.63
.319 1746
43-65
334 6823
42.52
23
19 370612
205.80
20.114818
207.66
.319 4364
43.63
334 9374
42.50
24
19.382961
205.83
20.127279
207.69
.319 6981
43.61
335 1924
42-49
25
19.395312
205.86
20.139741
207.72
I -3 I 9 9597
43-59
1-335 4472
42.47
20
19.407665
205.89
20.152206
207.76
.320 2212
43-57
-335 7020
42.45
27
19.420019
205.92
20.164672
207.79
.320 4825
43-55
335 95 6 7
42.43
28
19432375
205.95
20.177140
207.82
.320 7438
43-53
.336 2112
42.41
29
1 9-4447 34
205.98
20.189610
207.85
.321 0049
435 1
.336 4656
42.40
30
19.457094
206.01
20.202082
207.88
I.32I 2659
43-49
1.336 7199
42.38
31
19.469455
206.04
20.214556
207.91
3*1 5268
43-47
.336 9742
42.36
32
19.481819
206.08
20.227032
207.95
.321 7875
43-45
-337 2283
42-34
33
19.494184
206. 1 1
20.239510
207.98
.322 0482
43-43
337 4823
42.33
34
19.506551
206.14
20.251989
208.01
.322 3087
43-41
337 73 6 2
42.31
35
19.518921
206.17
20.264471
208.04
1.322 5692
43.40
1-337 99
42.29
36
19.531292
206.20
20.276954
208.07
.322 8295
43-38
.338 2437
42.27
37
19.543664
206.23
20.289440
208. II
.323 0897
43-36
.338 4972
42.25
38
19.556039
206.26
20.301927
208.14
3*3 3498
43-34
338 757
42.24
39
19.568415
206.29
20.314416
208.17
.323 6097
43.32
339 0041
42.22
40
19.580794
206.32
20.326907
208.20
1.323 8696
43-3
i-339 2573
42.20
41
19.593174
206.35
20.339400
208.24
.324 1294
43.28
339 5 I0 5
42.18
42
19.605556
206.38
20.351895
208.27
.324 3890
43.26
339 7635
42.17
43
19.617939
206.41
20.364392
208.30
.324 6485
43.24
.340 0165
42.15
44
19.630325
206.44
20.376891
208.33
.324 9079
43.22
340 2693
42.13
45
19.642713
206.47
20.389392
208.36
1.325 1672
43.21
1.340 5221
42.11
46
47
19.655102
19.667493
206.50
206.53
20.401895
20.414399
208.39
208.43
.325 4263
.325 6854
43.19
43-17
.340 7747
.341 0272
42.10
42.08
48
19.679886
206.57
20.426906
208.46
325 9443
43-15
.341 2796
42.06
19
19.692281
206.60
20.439415
208.49
.326 2032
43-13
-341 53'9
42.04
50
19.704678
206.63
20.451925
208.52
1.326 4619
43.11
1.341 7841
42.03
51
19.717076
206.66
20.464437
208.56
.326 7205
43-09
.342 0362
42.01
52
19.729477
206.69
20.476952
208.59
.326 9790
43-07
.342 2882
41.99
53
19.741879
206.72
20.489468
208.62
.327 2374
43-05
.342 5401
41-97
54
19.754283
206.75
20.501986
208.65
3*7 4957
43.04
.342 7919
41.96
55
19.766689
206.78
20.514506
208.69
1.327 7538
43.02
1-343 43 6
41.94
56
19.779097
206.81
20.527029
208.72
.328 0119
43-0
343 2952
41.92
57
19.791507
206.84
20.539553
208.75
.328 2698
42.98
343 5467
41.90
58
19.803919
206.88
20.552079
208.78
.328 5276
42.96
343 798o
41.89
59
19.816332
206.91
20.564607
208.82
.328 7853
42.94
344 493
41.87
60
19.828747
206.94
20.577137
208.85
1.329 0430
42.92
1.344 3005
41.85
573
TABLE VI.
b or finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit.
V.
32
33
34
35
logM.
Diflf. 1".
logM.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. 1".
0'
1.344 3005
41.85
1.359 1859
40.86
1-373 7*5i
39-93
1.387 9418
39.06
1 -344 55*5
41.84
359 43*o
40.84
373 9 6 46
39.91
.388 1761
39-5
2 .344 8025
41.82
359 6 76o
40.82
-374 *4i
3990
.388 4104
39.04
3 -345 0534
41.80
359 9*9
40.81
374 4434
39.88
.388 6446
39.02
4
345 34i
41.78
.360 1657
40.79
374 6827
39.87
.388 8787
39.01
5
1.345 5548
41-77
1.360 4104
40.78
1.374 9218
39-85
1.389 1127
38-99
6
345 8053
4*-75
.360 6550
40.76
375 l6 9
39-84
.389 3466
38.98
7
.346 0558
41-73
.360 8995
40.74
375 3999
39.82
.389 5804
38.97
8
.346 3061
41.72
.361 1439
4-73
.375 6388
39.81
.389 8x42
38.95
9
.346 5564
41.70
.361 3883
40.71
.375 8776
39-79
.390 0479
38.94
10
1.346 8065
41.68
1.361 6325
40.70
1.376 1164
39-78
1.390 2815
3893
11
347 05 6 5
41.66
.361 8766
40.68
.376 3550
39-77
39 5*5
38.91
12
347 3 6 5
41.65
.362 1207
40.66
376 5935
39-75
.390 7484
38.90
13
347 55 6 3
41.63
.362 3646
40.65
.376 8320
39-74
39 9817
38.88
14
.347 8060
41.61
.362 6084
40.63
377 703
39-7*
.391 2150
38.87
15
1.348 0557
41.60
1.362 8522
40.62
1.377 3086
39-71
1.391 4482
38.86
16
348 35*
41.58
.363 0959
40.60
377 54 68
39- 6 9
.391 6813
38.84
17
.348 5546
41.56
3 6 3 3394
40.59
377 7849
39.68
.391 9143
38.83
18
.348 8040
4!-55
.363 5829
40.57
378 0230
39.66
.392 1472
38.82
19
349 53*
4'-53
.363 8263
40.56
.378 2609
39- 6 5
-39* 3801
38.80
20
1.349 3023
41.51
1.364 0696
40.54
1.378 4987
39.64
1.392 6128
38.79
21
349 5513
41.50
.364 3128
40.52
.378 7365
39.62
39* 8455
38.77
22
349 8o 3
41.48
3 6 4 5559
40.51
.378 9742
39.61
393 78i
38.76
23
.350 0491
41.46
.364 7989
40.49
379 *"7
39-59
393 3 I0 7
38.75
24
.350 2978
41.45
.365 0418
40.48
379 449*
39.58
393 543 1
38.73
25
1.350 5464
41.43
1.365 2846
40.46
1.379 6866
39-56
1-393 7755
38.72
26
35 7950
41.41
3 6 5 5*73
40.45
379 9*4
39-55
394 78
38.71
27
.351 0434
41.40
.365 7699
40.43
.380 1612
39-53
394 *4
38.69
28
.351 2917
41.38
.366 0125
40.41
.380 3983
39-5*
394 47*i
38.68
29
351 5399
41.36
.366 2549
40.40
.380 6354
39.50
394 7041
38.67
30
1.351 7880
41-35
1.366 4973
40.38
1.380 8724
39-49
1.394 9361
38.65
31
.352 0361
4'-33
.366 7395
4-37
.381 1093
39-47
.395 1680
38.64
1 32
.352 2840
41.31
.366 9817
4-35
.381 3461
39-4 6
395 3998
38-63
I 33
35* 53'8
41.30
.367 2238
40.34
.381 5828
39-45
395 6 3'5
38.61
34
35* 7795
41.28
.367 4657
40.32
.381 8194
39-43
-395 8631
38.60
35
1.353 o*7*
41.26
1.367 7076
40.31
1.382 0559
39-4*
1.396 0947
38.59
36
353 *747
41.25
3 6 7 9494
40.29
.382 2924
39.40
.396 3262
38-57
37
353 5**i
41.23
.368 1911
40 28
.382 5288
39-39
.396 5576
38.56
38
353 7 6 94
41.21
.368 4327
40.26
.382 7651
39-37
.396 7889
38.55
39
354 Ol6 7
41.20
.368 6742
40.25
.383 0013
39-3 6
.397 0201
38.53
40
1.354 2638
41.18
1.368 9157
40.23
1.383 2374
39-35
1.397 2513
38.52
41
354 5 IQ 8
41.16
.369 1570
40.21
383 4734
39-33
397 4823
38-51
42
354 7578
41.15
3 6 9 39 8 3
40.20
383 793
39.32
397 7133
38.49
43
355 4 6
41-13
3 6 9 6 394
40.18
383 94-5*
39-3
397 944*
38.48
44
355 *5'3
41.11
.369 8805
40.17
.384 1809
39-*9
.398 1751
38-47
45
1.355 49 8
41.10
1.370 1214
40.15
1.384 4166
39.27
1.398 4058
38.45
46
355 7445
41.08
.370 3623
40.14
.384 6522
39.26
398 6365
38.44
47
355 999
41.07
.370 6031
40.12
.384 8878
39-*5
.398 8671
38-43
48
.356 2373
41.05
.370 8438
40.11
.385 1232
39*3
-399 976
38.41
49
.356 4836
41.03
.371 0844
40.09
385 3585
39.22
-399 3*8i
38.40
50
1.356 7297
41.02
1.371 3249
40.08
1.385 5938
39.20
1.399 5584
38-39
51
35 6 9758
41.00
371 5 6 54
40.06
.385 8290
39-19
399 7887
38.37
52
357 **'7
40.98
.371 8057
40.05
.386 0641
39.18
.400 0189
38.36
53
357 4 6 76
40.97
37* 0459
40.03
.386 2991
39.16
.400 2491
38.35
54
357 7134
40.95
.372 2861
40.02
.386 5340
39-15
.400 4791
38.33
55
i-357 959
40.94
1.372 5261
40.00
1.386 7689
39-13
1.400 7091
38.32
56
.358 2046
40.92
.372 7661
39-99
.387 0036
39.12
.400 9390
38-31
57
.358 4501
40.90
.373 0060
39-97
-387 *383
39-"
.401 1688
.38.30
58
.358 6954
40.89
373 *458
39.96
387 47^9
39.09
.401 3985
38.28
59
.358 9407
40.87
373 4 8 55
39-94
.387 7074
39.08
.401 6282
38.27
60
1.359 1859
40.86
1.373 7251
39-93
1.387 9418
39.06
1.401 8578
38.26
574
TABLE VI.
For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit.
V.
36
37
38
39 |
logM.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. 1".
O'
1.401 8578
38.26
1.415 4930
37-5
1.428 8662
36.80
1.441 9943
36.14
1
.402 0873
38.24
.415 7180
37-49
.429 0869
36-79
.442 21 1 I
36.13
2
.402 3167
38-23
.415 9429
37-47
.429 3076
36.78
.442 4.279
36.12
3
.402 5460
38.22
.416 1678
37-46
.429 5281
36-77
.442 6446
36.11
4
.402 7753
38.20
.416 3925
37-45
.429 7488
36-75
.442 86l2
36.10
5
1.403 0045
38.19
1.416 6172
37-44
1.429 9693
36.74
1-443 0778
36 co
6
.403 2336
38.18
.416 8419
37-43
.430 1897
36.73
443 2943
36.08
7
.403 4626
38.17
.417 0664
37-4 1
.430 4101
36.72
.443 5107
36.07
8
.403 6916
38.15
.417 2909
37.40
43 6304
36-71
443 7271
36.06
9
.403 9205
38.14
417 5153
37-39
.430 8506
36-7
443 9434
36.05
10
1.404 1493
38.13
1-4*7 7396
37-38
1.431 0708
36.69
1.444 1597
36-04
11
.404 3780
38.12
4i7 963.9
37-37
431 2909
36.68
444 3758
36-03
12
.404 6067
38.10
.418 1881
37-36
.431 5109
36.66
444 5920
36.02
13
.404 8352
38.09
.418 4122
37-35
.431 7308
36.65
.444 8080
36.00
14
.405 0637
38.08
.418 6362
37-33
431 957
36.64
.445 0240
35-99
15
1.405 2921
38.06
1.418 8602
37.32
1.432 1705
36.63
1.445 2400
35-98
16
17
.405 5205
.405 7488
38.05
38.03
.419 0841
.419 3079
37-31
37.3
-432 393
.432 6100
36.62
36.61
445 4558
445 6716
35-97
35-96
18
.405 9769
38.02
419 5317
37-29
.432 8296
36.60
.445 8874
35-95
19
.406 205 i
38.01
.419 7554
37-27
.433 0491
36.59
.446 1031
35-94
20
1.406 4331
38.00
1.419 9790
37.26
1.433 2686
36.57
1.446 3187
3593
21
.406 66 1 1
37-99
.420 2026
37-25
.433 4881
36.56
446 5343
3592
22
.406 8889
37-97
.420 4260
37-24
-433 774
36-55
.446 7498
35-91
23
.407 1 1 68
37.96
.420 6494
37-23
-433 9267
36.54
.446 9652
35-90
24
407 3445
37-95
.420 8728
37.22
.434 1459
36.53
.447 1806
35-89
25
1.407 5721
37-94
1.421 0960
37-20
1.434 3651
36-52
1-447 3959
35-88
26
.407 7997
37-92
.421 3192
37-19
434 5842
36-51
.447 6112
35.87
27
.408 0272
37-91
.421 5423
37-i8
434 8032
36.50
.447 8263
35.86
28
.408 2547
37.90
.421 7654
37.17
.435 0221
36.49
.448 0415
35.85
29
.408 4820
37.89
.421 9884
37.16
435 2410
36.48
.448 2565
35-84
30
1.408 7093
37.87
1.4*2 2113
37.15
1-435 4598
36.47
1.448 4715
35-83
31
32
33
.408 9365
.409 1636
.409 3907
37-86
37.85
37.84
.422 A 3 4i
.422 6569
.422 8796
37-13
37-12
37-11
.435 6786
435 8973
43 6 H59
36-46
36.44
36.43
.448 6865
.448 9014
.449 1162
35-82
35.81
35.80
34
.409 6177
37.82
.423 1 022
37.10
43 6 3345
36-42
449 339
35-79
35
1.409 8446
37.81
1.423 3248
37-09
1.436 5530
36-41
1-449 5456
35-78
36
.410 0714
37.80
423 5473
37.08
.436 7714
36.40
449 7603
35-77
37
.410 2981
37.78
423 7697
37.06
436 9898
36.39
449 9749
35-76
38
.410 5248
37-77
.423 9920
37-05
.437 2081
36.38
.450 1894
35-75
39
.410 7514
37.76
.424 2143
37.04
437 4263
36.37
.450 4038
35-74
40
1.410 9780
37-75
1.424 4365
37-03
i-437 6445
36.36
1.450 6182
35-73
41
.411 2044
37-74
.424 6586
37.02
.437 8626
36.35
.450 8325
35-72
42
43
.411 4308
.411 6571
37-72
37-7 1
.424 8807
.425 1027
37.01
36.99
.438 0806
.438 2986
36-34
36.32
.451 0468
.451 2610
35-71
35-70
44
.411 8833
37-7
.425 3246
36-98
.438 5165
3 6 .31
.451 4752
35-69
45
1.412 1095
37- 6 9
1.425 5465
36.97
1.438 7344
36-30
1.451 6893
35.68
< 46
.412 3356
37-68
.425 7683
36.96
.438 9522
36.29
451 933
35.67
47
t s-
.412 5616
37.66
.425 9900
36.95
439 1699
36.28
.452 1173
35-66
48
.412 7875
37-65
.426 2117
36-94
439 3875
36.27
.452 3312
35-65
49
.413 0134
37-64
.426 4333
36-92
.439 6051
36.26
452 5450
35-64
50
1.413 2392
37-63
1.426 6548
36-91
1.439 8226
36.25
1.452 7588
35-63
51
.413 4649
37.61
.426 8762
36.90
.440 0401
36-24
.452 9725
35-62
52
53
54
.413 6905
.413 9161
.414 1416
37.60
37-59
37.58
.427 0976
427 3 l8 9
.427 5402
36.89
36.88
36.87
.440 2575
.440 4748
.440 6921
36.23
36.22
36.20
.453 1862
453 3998
453 6134
35.60
35-59
55
56
1.414 3670
37.56
37-55
1.427 7613
.427 9824
36.86
36-85
1.440 9093
.441 1264
36-19
36.18
1.453- 8269
.454 0403
35-57
57
.414 8176
37.54
.428 2035
36-83
441 3436
36.17
454 2537
35-56
58
59
.415 0429
.415 2680
37-53
37-51
.428 4244
.428 6453
36.82
36.81
.441 5605
441 7774
36.16
36.I5
454 4670
.454 6802
35-55
35-54
60
1.415 4930
37-5
1.428 8662
36.80
1.441 9943
36.14
1-454 *934
35-53
075
TABLE VI.
For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit.
V.
40
41
42
43
logM.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. 1".
O'
1.454 8934
35-53
1.467 5782
34-95
1.480 0627
34-41
T -492 3597
33-9 1
1
455 1065
35-51
.467 7879
34-94
.480 2691
34.40
.492 5631
33-90
2
455 3*9 6
35-51
.467 9976
34-93
.480 4755
34.40
.492 7665
3389
3
.455 5326
35-5
.468 2071
34.92
.480 6819
34-39
.492 9698
33-88
4
455 7456
35-49
.468 4166
34- 9 l
.480 8882
34-38
493 i73i
33-87
5
T -455 95 8 5
35.48
1.468 6261
34.90
1.481 0944
34-37
r -493 3764
33-87
6
.456 1713
35-47
.468 8355
34-9
.481 3006
34-36
493 5796
33.86 |
7
.456 3841
35-46
469 44 8
34.89
.481 5068
34-35
493 7827
33-85
8
.456 5968
35-45
.469 2541
34.88
.481 7129
34-34
493 9858
33-84 1
9
.456 8094
35-44
.469 4634
34.87
.481 9189
34-33
.494 1888
33-83
10
1-457 0220
35-43
1.469 6725
34.86
1.482 1249
34-33
1.494 39 l8
33.83
11
457 2346
35-42
.469 8817
34-85
.482 3308
34-32
494 5948
33-82
12
457 4470
35-41
.470 0907
34.84
.482 5367
34-31
494 7977
33-8i
13
457 6 595
35-40
.470 2998
34.83
.482 7425
34-3
.495 0005
33.80
14
457 8718
35-39
.470 5087
34.82
.482 9483
34-29
495 233
33-79
15
1.458 0841
35.38
1.470 7176
34.81
1.483 1540
34.28
1.495 46i
33-79
16
.458 2964
35-37
.470 9265
34.80
483 3597
34.28
.495 6088
33-78
17
.458 5086
35-3 6
471 1353
34-79
483 5 6 53
34-27
-495 8114
33-77
18
.458 7207
35-35
.471 3440
34-79
483 779
34.26
.496 0140
33.76
19
.458 9328
35-34
471 55*7
34.78
.483 9764
34-25
.496 2166
33-75
20
1.459 H4 8
35-33
1.471 7613
34-77
1.484 1819
34-24
1.496 4191
33-75
21
459 35 6 7
35.32
.471 9699
34.76
.484 3873
34-23
.496 6216
33-74
22
.459 5686
35-31
.472 1784
34-75
.484 5927
34-22
.496 8240
33-73
23
459 7805
35-3
472 3 86 9
34-74
.484 7980
34-22
.497 0264
33-72
24
459 99"
35-^9
472 5953
34-73
.485 0033
34.21
497 2287
33.71
25
1.460 2040
35.28
1.472 8037
34-73
1.485 2085
34.20
1.497 4310
33-71
26
.460 4156
35.27
.473 0120
34-72
485 4'37
34-19
497 6332
33.70
27
.460 6272
35.26
.473 2203
34.71
.485 6188
34.18
497 8354
33-69
28
.460 8388
35-25
473 4285
34-7
485 8239
34-17
.498 0376
33-68
29
.461 0503
35-24
473 6366
34.69
.486 0289
34.16
.498 2396
33.68
30
1.461 2617
35-23
1.473 8447
34.68
1.486 2338
34.16
1.498 4417
33.67
31
.461 4731
35-23
474 5 2 7
34-67
.486 4388
3415
.498 6437
33.66
32
.461 6844
35-22
.474 2607
34-66
.486 6436
34-14
.498 8456
33.65
33
.461 8957
35-21
.474 4686
34.65
.486 8484
34.13
499 475
33-65
34
.462 1069
35.20
474 6765
34.64
487 <>53 2
34.12
.499 2494
33-64
35
1.462 3180
35- J 9
1.474 8843
34-63
1.487 2579
34.12
1.499 4512
33-63
36
.462 5291
35-i8
.475 0921
34.62
.487 4626
34- "
499 6 53o
33.62
37
.462 7401
35- J 7
475 2998
34.61
.487 6672
34.10
499 8547
33.62
38
.462 9511
35-i6
-475 575
34.61
.487 8718
34.09
.500 0563
33.61
39
.463 1620
35-15
475 7 J 5i
34.60
.488 0763
34.08
.500 2580
33.60
40
1.463 3729
35-H
1.475 9227
34-59
1.488 2807
34.07
1.500 4595
33-59
41
4 6 3 5 8 37
35- x 3
.476 1302
34.58
.488 4852
34.07
.500 6611
33.58
42
4 6 3 7944
35- 12
.476 3376
34-57
.488 6895
34.06
.500 8625
33-58
43
.464 0051
35- 11
.476 5450
34-56
.488 8939
34.05
.501 0640
33-57
44
.464 2158
35-i
.476 7524
34-55
.489 0981
34-4
.501 2654
33-56
45
1.464 4263
35-9
1.476 9596
34-54
1.489 3023
34-03
1.501 4667
33-55
46
.464 6369
35.08
477 1669
34-54
489 5 6 5
34.02
.501 6680
33-55
47
.464 8473
35.07
477 374i
34-53
.489 7106
34.02
.501 8693
33-54
48
465 577
35.06
477 5812
34-5*
.489 9147
34.01
.502 0705
33-53
49
.465 2681
35-5
477 7883
34-5 i
.490 1187
34.00
.502 2716
33-5 a
50
1.465 4784
35-4
1-477 9953
34-5
1.490 3227
33-99
1.502 4727
33-5 1
51
.465 6886
35-4
.478 2023
34-49
.490 5266
33.98
.502 6738
33-5 1
52
.465 8988
35-3
.478 4092
34-48
49 735
33-97
.502 8748
33-5
53
.466 1090
35.02
.478 6161
34-47
.490 9343
3396
53 758
33-49
54
.466 3190
35> 01
.478 8229
34.46
.491 1381
33-95
.503 2767
33-48
55
1.466 5290
35.00
1.479 0297
34-46
1.491 3418
33-95
1.503 4776
33-48
56
.466 7390
34-99
479 2364
34-45
49 1 5455
33-94
.503 6784
33-47
57
.466 9489
34.98
.479 4430
34-44
49 * 749 *
33-93
53 8792
33-46
58
.467 1587
34-97
479 6 496
34-43
.491 9527
3392
.504 0800
33-45
59
.467 3685
34.96
.479 8562
34.42
.492 1562
33.91
.504 2807
33-44
60
1.467 5782
34-95
1.480 0627
34.41
!-49 a 3597
33.91
1.504 4813
33-44
576
TABLE VI.
For finding the True Anomaly or the Time from the Perihelion in a Parabolic Oibit.
V.
44
45
4:6
47
log M.
Diff. 1".
logM.
Diff. I".
logM.
Diff. I".
logM.
Diff. 1".
0'
1.504 4813
33-44
1.516 4390
33.00
1.528 2435
3*59
*-539 94 8
32.20
1
.504 6819
33-43
.516 6370
32.99
.528 4390
32.58
.540 0980
32.20
2
.504 8825
33 4*
.516 8349
32.98
.528 6344
32.57
.540 2912
32.19
3
.505 0830
33-4^
.517 0328
3298
.528 8299
32.57
.540 4843
32.18
4
.505 2835
33-41
.517 2306
32.97
.529 0252
32.56
.540 6774
32.18
5
6
1.505 4839
.505 6843
3340
33-39
1.517 4284
.517 6262
32.96
32.96
1.529 2206
.529 4159
3M5
3255
1.540 8705
.541 0635
32-I7
32.17
7
.505 8846
33-39
.517 8239
32-95
.529 6112
32-54
.541 2564
32.16
8
.506 0849
33-3^
.518 0216
3*94
.529 8064
32.53
.541 4494
32.15
9
.506 2852
33 37
.518 2192
32-93
.530 0016
32-53
.541 6423
32.15
10
11
1.506 4854
.506 6855
33-36
33-3 6
1.518 4168
.518 6143
32.93
3292
1.530 1967
.530 3918
32.52
32-51
1.541 8352
.542 0280
32.14
32.14
12
.506 8856
33-35
.518 8118
32.91
.530 5869
32-51
.542 2208
32.13
13
.507 0857
33-34
.519 0093
32.91
53 78*9
32.50
542 4135
32.12
14
.507 2857
33-33
.519 1067
32.90
.530 9769
32.49
.542 6063
32.11
15
1.507 4857
33-33
1.519 4041
32.89
1.531 1719
32.49
1.542 7989
32.11
16
.507 6856
33-32
.519 6014
32.89
.531 3668
32.48
542 99 l6
32.10
17
.507 8855
33-3 1
.519 79 8 7
32.88
.531 5616
32.48
543 l8 42
32.10
18
.508 0853
33-3
.519 9960
32- 8 7
53 1 75 6 5
32.47
543 376 8
32.09
19
.508 2851
33.29
.520 1932
32.86
53 1 95 J 3
32.46
543 5 6 93
32.09
20
1.508 4849
13.29
1.520 3904
3286
1.532 1460
32.46
1.543 76i 8
32.08
21
.508 6846
33.28
.520 5875
32.85
.532 3407
32-45
543 9543
32-08
22
.508 8843
33-27
.520 7846
32.84
532 5354
32-44
544 1467
32.07
23
.509 0839
33- 2 7
.54.0 9816
32.84
.532 7300
3244
544 339i
32.06
24
.509 2835
33.26
.521 1786
32.83
.532 9246
32-43
544 53*5
32.06
25
1.509 4830
33-25
1.521 3756
32.82
1.533 "92
32.43
1.544 7238
32.05
26
.509 6825
33- 1 4
.521 5725
32.82
533 3 J 37
32-42
544 9*6i
32.04
27
.509 8819
33 24
.521 7694
32.81
533 5*2
32.42
545 ' 8 3
32.04
28
.510 0813
33,23
.521 9662
32.80
533 70 2 7
32.41
545 35
32.03
29
.510 2807
33.22
.522 1630
32.80
533 8 97i
32.40
545 4927
32.03
30
1.510 4800
33,21
1.522 3598
32-79
1.534 0914
32.39
1.545 6849
32.02
31
.510 6792
33- 21
.522 5565
32.78
.534 2858
32-39
.545 8770
32.02
32
.510 8785
33,20
.522 7531
32.78
534 4801
32.38
.546 0690
32.01
33
.511 0776
33-*9
.522 9498
32.78
534 6743
32-37
.546 2611
32.00
34
.511 2768
i3 .i8
.523 1464
3*-77
.534 8685
32.37
54 6 453 1
32.00
35
1.511 4759
33.18
1.523 3429
32.76
1.535 0627
32.36
1.546 6450
3*-99
36
.511 6749
33-*7
523 5394
32.75
535 2568
32-35
.546 8370
31.98
37
.511 8739
33.16
5*3 7359
32.74
555 459
32-35
.547 0289
31.98
38
.512 0729
33- I 5
5*3 93*3
32-73
535 6450
32-34
547 2207
31.97
39
.512 2718
33-15
.524 1287
32.73
535 8 39
32.33
547 4"5
3 J 97
40
1.512 4707
33-H
1.524 3251
32.72
1.536 0330
32.33
1.547 6043
31.96
41
.512 6695
33-!3
.524 5214
32.71
.536 2270
32.32
.547 7961
31.96
42
.512 8683
33-n
.524 7176
32.71
.536 4209
32.32
547 9 8 7 8
31-95
43
.515 0670
33- 12
.524 9138
32.70
.536 6148
32.31
.548 1795
3i-94
44
.513 2657
33-"
.525 noo
32.70
.536 8086
32.30
.548 3711
3'-94
45
1.513 4644
33.11
1.525 3062
32.69
1.537 0024
32.30
1.548 5627
31-93
46
s r
.513 6630
33.10
.525 5023
32.68
537 i9 62
32.29
54 8 7543
3*-93
47
.513 8615
33.09
.525 6983
32.67
-537 3899
32.28
.548 9458
31.92
48
.514 0601
3308
.525 8944
32.67
537 5 8 3 6
32.28
549 H73
3*-9i
49
.514 2586
33-7
.526 0903
3266
537 7772
32.27
549 S 288
3 l '9 l
50
51
1.514 4570
.514 6554
33.07
33.06
1.526 2863
.526 4822
32.65
32.64
1.537 9708
.538 1644
32.26
32.26
1.549 5202
549 7"6
31.90
31.90
52
.514 8537
33-5
.526 6780
32.64
53 8 3579
32.25
549 93
31.89
53
.515 0520
33-5
.526 8739
32.63
538 55H
32.25
.550 0943
31.88
rt r>
54
5*5 2503
33-4
.527 0696
32.62
53 8 7449
32.24
.550 2856
31.88
55
1.515 4485
33-4
1.527 2654
32.62
i-53 8 93 8 3
32.23
1.550 4769
31.87
56
57
.515 6467
.515 8449
33-3
33.02
.527 4611
.527 6567
32.61
32.61
539 i3'7
539 325
32.23
32.22
.550 6681
.550 8593
31.87
31.86
58
.516 0430
33 oi
.527 8524
32.60
539 5 l8 3
32.21
.551 050A
31.86
59
.516 2410
33.01
.528 0479
32.60
539 7" 6
32.21
.551 2416
3i- 8 5
60
1.516 4390
33.00
1.528 2435
3 2 -59
1.539 9048
32.20
1.551 4326
3I-85
577
TABLE VI,
For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit.
V.
48
49
50
51
logM.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. l".
0'
1
1.551 4326
.551 6237
31.85
31.84
1.562 8360
.563 0250
HI!
1.574 1234
574 3 T 6
31.20
31.20
.585 4886
30.91
30.91
2 . 55 i 8i 4 7
3I-83
.563 2140
3 I -5
574 4977
3 I - I 9
585 6740
30.90
3 .552 0057
.563 4030
31.50
574 68 49
3 I - I 9
.585 8594
30.90
4
.552 1966
31.82
.563 5920
3M9
.574 8720
31.18
.586 0448
30.89
5
1.552 3876
31.82
1.563 7809
31.48
1.575 0590
31.18
1.586 2302
30.89
6
.552 5784
31.81
.563 9698
31.48
575 2461
31.17
.586 4155
30.89
7
.552 7693
31.80
.564 1586
3 '-47
575 4331
31.17
.586 6008
30.88
9
.552 9601
553 '5 08
31.80
5 6 4 3475
5 6 4 53 6 3
3M7
31.46
.575 6201
575 8070
31.16
31.16
.586 7859
.586 9713
30.87
30.87
10
1-553 34 l6
31.79
1.564 7250
31.46
J-575 9939
31.15
1.587 1565
30-87 ;
11
553 5323
3I-78
.564 9138
345
.576 1808
31.15
587 3417
30.86
12
553 723
31.78
.565 1025
3-45
.576 3677
3 I - I 4
.587 5268
30.86
13
553 9*3 6
3'-77
.565 2911
3i-44
.576 5546
3 I - I 4
.587 7120
30.85
14
.554 1042
31.76
.565 4798
3i-44
.576 7414
3 I - 1 3
.587 8971
30.85
15
1.554 2 94 8
31.76
1.565 6684
3i-43
1.576 9281
31.13
1.588 0821
30.84
16
554 4853
.565 8569
577 H49
31.12
.588 2672
30.84
17
554 6 75
31-75
.566 0455
31.42
-577 3 l6
31.12
.588 4522
30-83
18
554 8663
3 I -74
.566 2340
31.41
577 4 88 3
31.11
.588 6372
30.83
19
555 5 6 7
3'-74 '
.566 4225
3J-4 1 '
577 6749
31.11
.588 8222
20
1.555 2472
3 J -73
1.566 6109
31.40
1.577 8615
31.10
1.589 0071
30.82
21
555 4375
.566 7993
31.40
.578 0481
31.10
.589 1920
30.82
22
555 6279
31.72 -
.566 9877
31.39
57 8 2347
31.09
.589 3769
30.81
23
555 8182
31.71
.567 1761
3 r -39
.578 4213
31.09
.589 5618
30.81
24
.556 0084
31.71 :
5 6 7 3 6 44
31.38
.578 6078
31.08
.589 7466
30.80
25
1.556 1987
31.70 i
1.567 5527
31.38
1.578 7942
31.08
1.589 9314
30.80
26
.556 3888
31.70
.567 7409
3i-37
578 9807
31.07
.590 1162
30.79
27
55 6 579
31.69
.567 9291
3*-37
.579 1671
31.07
.590 3009
30-79
28
.556 7691
31.68
.568 1173
31.36
579 3535
31.06
59 4857
30.78
29
.556 9592
31.68
.568 3055
31.36 i
579 5399
31.06
59 6 74
30.78
30
i-557 H93
31.67 i
1.568 4936
31-35
1.579 7262
31.06
1.590 8550
30.78
31
557 3393
31.67
.568 6817
579 9 I2 5
3 I>O 5
.591 0397
3-77
32
557 5293
31.66
.568 8698
3'-34
.580 0988
31.04
.591 2243
30-77
33
557 7i93
31.66
.569 0579
3>-34
.580 2851
31.04
.591 4089
30.76
34
557 9092
3 I - 6 5
.569 2459
31-33
.580 4713
3 J -3
,59i 5935
30.76
35
1.558 0991
3*- 6 5
1.569 4338
3 J -33
1.580 6575
31.03
1.591 7780
30-75
36
.558 2890
31.64
.569 6218
31.32 i
.580 8436
.591 9625
30-75
37
558 4788
31.64
.569 8097
31.32 ,
.581 0298
31.02
.592 1470
30-75
38
.558 6686
31.63
.569 9976
.581 2159
31.02
592 33*5
30-74
39
558 8584
31.62
.570 1854
31.30 (
.581 4020
31.01
592 5'59
30.74
40
1.559 0482
31.62
i-570 3733
31.30
1.581 5.8*0
31.01
1.592 7003
30-73
41
559 2379
31.61
.570 5611
31.29
.581 7740
31.00
.592 8847
3-73
42
559 4275
31.61
.570 7488
31.29
.581 9600
31.00
.593 0690
30.72
43
559 6172
31.60
.570 9366
31.28
.582 1460
3-99
593 2534
30.72
44
.559 8068
31.60
.571 1243
31.28
.582 3319
30.99
593 4377
30-72
45
r -559 99 6 3
3'59
1.571 3119
31.28
1.582 5179
30.98
1.593 6219
30.71
46
.560 1859
3'-59
.571 4996
31.27
-582 7037
3098
.593 8062
30.71
47
5 6 3754
31.58
.571 6872
31.27
.582 8896
3-97
593 994
30.70
48
49
.560 5648
.560 7543
$*-57
3i-57
.571 8748
.572 0623
31.26
31.26
-583 754
.583 2612
30.97
30.96
594 1746
594 3588
30.70
30.69
50
1.560 9437
3 J -5 6
1.572 2499
31.25
1.583 4470
30.96
1.594 5429
30.69
51
.561 1331
31.56
572 4373
31.25
.583 6327
3-95
594 7270
30.68
52
.561 3224
3 J -55
.572 6248
31.24
.583 8184
3-95
594 9 111
30.68
53
.561 5117
355
.572 8123
31.24
.584 0041
3-94
595 952
30.68
54
.561 7010
3'-54
572 9997
31.23
.584 1898
30.94
595 2792
30.67
55
1.561 8902
3*-54
1.573 1870
31.23
1-584 3754
30.94
i-595 4633
30.67
56
57
58
.56* 0794
.562 2686
.562 4578
si-si
J-53
573 3743
573 5 6 6
573 7489
31.22
31.22
31.21
.584 5610
.584 7466
584 93 21
3-93
30-93
30.92
595 6473
595 8312
.596 0151
30 66
30.66
30.65
59
.562 6469
S'-S*
573 93 6z
31.21
.585 1176
30.92
>$g6 1990
30.65
60
1.562 8360
M-S-
1.574 1234
31.20
i-5 8 5 303 1
30.91
1.596 3829
30.65
578
TABLE VI.
For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit.
V.
52
53
54
55
logM.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. I".
logM.
Diff. 1".
O'
1.596 3829
30.65
1.607 3703
30.40
1.618 2724
30.17
1.629 959
29.96
1
.596 5668
30.64
.607 5527
30-39
.618 4534
30.17
.629 2757
29.96
2
.596 7506
30.64
.607 7350
3-39
.618 6344
30.16
.629 4554
29.96
3
59 6 9344
30.63
.607 9174
30-39
.618 8153
30.16
.629 6351
29.95
4
.597 1182
30.63
.608 0997
30.38
.618 9963
30.16
.629 8148
29.95
5
1.597 3020
30.62
1.608 2820
30.38
1.619 !77 Z
30.15
1.629 9945
29.95
6
7
597 4857
597 6694
30.62
30.62
.608 4642
.608 6465
30-38
30-37
.619 3581
.619 5390
3 -I5
30.15
.630 1742
.630 3538
29.94
29.94
8
597 8531
30.61
.608 8287
30-37
.619 7199
30.14
.630 5335
29.94
9
.598 0368
30.61
.609 0109
30.36
.619 9007
30-I4
.630 7131
29.93
10
1.598 2204
30.60
1.609 1931
30.36
1.620 0816
30.14
1.630 8927
29.93
11
.598 4040
30.60
.609 3752
3-3 6
.620 2623
30.13
.631 0722
29.93
12
.598 5876
30.59
.609 5573
30-35
.620 4431
30.I3
.631 2518
29.92
13
.598 7711
3-59
.609 7394
3-35
.620 6239
30.12
63 1 43 J 3
29.92
14
.598 9547
3-59
.609 9215
30-34
.620 8046
30.12
.631 6108
29.92
15
1.599 '382
30.58
1.610 1036
30-34
1.620 9853
30.12
1.631 7903
29.91
16
599 3217
30.58
.610 2856
3 .34
.621 1660
30.11
.631 9698
29.91
17
599 5051
3-57
.610 4676
30-33
.621 3467
30.11
.632 1492
29.91
18
.599 6885
3-57
.610 6496
30.33
.621 5274
30.11
.632 3286
29.90
19
599 8 7i9
3-57
.610 8315
30-32
.621 7080
30.10
.632 5081
29.90
20
i. 600 0553
30.56
1.611 0135
30.32
1.621 8886
30.10
1.632 6875
29.90
21
.600 2387
30.56
.611 1954
30.32
.622 0692
3P.IO
.632 8668
29.89
22
.600 4220
3-55
.611 3773
30-3 1
.622 2497
30.09
.633 0462
29.89
23
.600 6053
30.55
.611 5591
3-3'
.622 4307
3^-09
.633 22 5 C
29.89
24
.600 7886
30-55
.611 7410
30.31
.622 6108
30.09
.633 4048
29.88
25
1.600 9718
3-54
1.611 9228
30-3
1.622 7917
30.08
1.633 5841
29.88
26
.601 1551
3-54
.612 1046
30.30
.622 9718
30.08
.633 7634
29.88
27
.601 3383
3-53
.612 2864
30.29
.623 1523
30.08
.633 9427
29.87
28
.601 5214
3-53
.612 4681
30.29
.623 3327
3.0.07
.634 1219
29.87
29
.601 7046
30.52
.612 6499
30.29
-623 5131
30.07
.634 3011
29.87
30
1.601 8877
30.52
i. 612 8316
30.28
*-62 3 6935
30.06
1.634 4803
29.86
31
.602 0708
3 -52
.613 0132
30.28
.623 8739
30.06
-634 6 595
29.86
32
.602 2539
30-5 1
.613 1949
30.28
.624 0543
30.06
.634 8387
29.86
33
.602 4370
30.51
.613 3765
30.27
.624 2346
30.05
.635 0178
29.86
34
.602 6200
30.50
.613 5582
30.27
.624 4149
30,05
.635 1969
29.85
35
1.602 8030
30.50
1.613 7398
30.26
i.6z 4 5952
30.05
1.635 376o
29.85
36
.602 9860
30.50
.613 9213
30.26 ,
.624 7755
30.04
635 555i
29.85
37
38
39
.603 1690
.603 3519
.603 5348
30.49
30.49
30.48
.614 1029
.614 2844
.614 4659
30.26
30.25
30.25
.624 9557
.625 1360
.625 3162
30.04
30.04
30.03
.635 7342
.635 9132
.636 0922
29.84
29.84
29.84
40
1.603 777
30.48
1.614 6474
30.25
1.625 4964
30.03
1.636 2713
29.83
41
42
43
44
.603 9005
.604 0834
.604 2662
.604 4490
3-47
30.47
30.47
30.46
.614 8288
.615 0103
.615 1917
615 373 1
30.24
30.24
30-23
30-23
.625 6765
.625 8567
.626 0368
.626 2169
30.03
30.02
30.02
30.02
.636 4502
.636 6292
.636 8082
.636 9871
29.83
29.83
29.82
29.82
45
1.604 6317
30.46
1-615 5545
30.23
1.626 3970
30.01
1.637 1660
29.82
46
.604 8145
3-45
.615 7358
30.22
.626 5771
30.01
637 3449
29.82
47
48
.604 9972
.605 1799
3-45
3-45
.615 9171
.6i v 6 0984
30.22
30.22
.626 7571
.626 9372
30.01
30.00
.637 5238
.637 7027
29.81
29.81
49
.605 3626
3-44 ,
.616 2797
30.21
.627 1172
30.00
.637 8815
29.81
50
1.605 5452
3-44
1.616 4610
30.21
1.627 2972
30.00
1.638 0603
29.80
51
.605 7278
3-43
.616 6422
30.20
.627 4771
29.99
.638 2391
29.80
52
.605 9104
3-43
.616 8234
30.20
.627 6571
29.99
.638 4179
29.80
53
.606 0930
30.43 ,
.617 0046
30.20
.627 8370
29.99
.638 5967
29-79
54
.606 2755
30.42
.617 1858
30.19
.628 0169
29.98
.638 7754
29.79
55
56
57
58
59
i. 606 4581
.606 6406
.606 8230
.607 0055
.607 1879
30.42
3 -42
30.41
30.41
30.40
1.617 3669
.617 5481
.617 7292
.617 9102
.618 0913
30.19
30.19
30.18
30.18
30.17
1.628 1968
.628 3766
.628 5565
.628 7363
.628 9161
29.98
2998
29.97
29.97
29.97
1.638 9542
.639 1329
.639 3116
.639 4902
.639 6689
29.79
29.78
29.78
29.78
29.77
60
1.607 3703
30.40
1.618 2724
30.17
1.629 959
29.96
1.639 8475
29.77
579
TABLE VI.
For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit.
V.
56
57
58
59
logM.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. 1".
0'
1.639 8475
29.77
1.650 5336
49.60
1.661 1601
49.44
1.671 7331
4 93 o
1
.640 0262
29.77
.650 7112
49.60
.661 3368
49.44
.671 90^9
49.30
2
.640 2048
29.77
.650 8887
29.59
.661 5134
49.44
.674 0846
49.30
3
.640 3833
29.76
.651 0663
29.59
.661 6900
29.43
.674 4604
29.29
4
.640 5619
29.76
.651 4438
49.59
.661 8666
49.43
.672 4362
29.29
5
1.640 7405
29.76
1.651 4213
49.58
1.662 0432
29.43
1.672 6119
29.29
6
.640 9190
29.75
.651 5988
49.58
.662 2197
29-43
.672 7876
29.29 1
7
.641 0975
29.75
.651 7763
29.58
.662 3963
29.42
.672 9634
29.28
8
.641 2760
29.75
65 1 953 8
49.58
.662 5728
29.42
.673 1391
29.28
9
.641 4545
29.74
.652 1312
29.57
.662 7493
29.42
- 6 73 3*47
29.28
10
1.641 6329
29.74
1.652 3086
29-57
1.662 9258
49.42
1.673 4904
29.28
11
.641 8114
29.74
.652 4861
49.57
.663 1023
29.41
.673 6661
29.28
12
.641 9898
29.74
.652 6635
49.57
.663 2788
29.41
.673 8417
29.27
13
.642 1682
29.73
.652 8408
29.56
66 3 4553
29.41
.674 0174
29.27
14
.642 3466
29-73
.653 0182
29.56
.663 6317
49.41
.674 1930
29.47
15
1.642 5250
29.73
1.653 1956
49.56
1.663 8o8 2
49.40
1.674 3 686
49.47
16
.642 7033
29.72
.653 3729
29-55
.663 9846
49.40
.674 5442
49.47
17
.642 8816
29.72
.653 5502
29-55
.664 1610
49.40
.674 7198
29.26
18
.643 0599
29.72
.653 7275
29-55
.664 3374
49.40
.674 8954
29.26
19
.643 2382
29.71
.653 9048
29-55
.664 5137
29-39
.675 0709
29.26
20
1.643 4165
29.71
1.654 0821
49.54
1.664 6901
29-39
1.675 2465
29.26
21
.643 5948
29.71
.654 2593
49.54
.664 8664
29-39
.675 4220
2925
22
.643 7730
29.71
.654 4366
29.54
.665 0428
29-39
6 75 5975
29.25
23
.643 9513
29.70
.654 6138
29.54
.665 2191
29-39
.675 7730
29.25
24
.644 1295
29.70
.654 7910
29-53
.665 3954
49.38
.675 9485
29.25
25
1.644 377
29.70
1.654 9682
29-53
1.665 5717
49.38
1.676 1240
29.25
26
.644 4858
49:69
655 H54
29.53
.665 7480
49.38
.676 2995
29.24
27
.644 6640
29/69
6 55 3225
29-53
.665 9242
49.38
.676 4749
29.24
28
.644 8421
29.69
6 55 4997
49.54
.666 1005
49.37
.676 6504
29.24
29
.645 0203
29.69
.655 6768
29.52
.666 2767
29-37
.676 8258
29.24
30
1.645 J 9^4
29.68
*- 6 55 ^539
49.54
1.666 4529
29-37
1.677 0012
29.24
31
645 3765
29.68
.656 0310
49.51
.666 6291
29-37
.677 1766
29.23
32
645 5545
29.68
.656 2081
4951
.666 8053
49.36
.677 3520
29-23
33
.645 7326
29.67
.656 3852
49.51
.666 9815
29.36
.677 5274
29.23
34
.645 9106
29,67
.656 5644
49.51
.667 1577
2936
.677 7028
29.23
35
1.646 0886
49.67
1.656 7394
49.50
1.667 3338
29.36
1.677 8 7 8 I
29.23
36
.646 2 6 #6
29.67
.656 9163
49.50
.667 5100
29-35
.678 0535
29.22
37
.646 4446
2,9.66
*57 933
29.50
.667 6861
29-35
.678 2488
29.22
38
.646 6226
29.66
.657 4703
29.50
.667 8622
29-35
.678 4041
29.22
39
.646 8005
49.66
.657 4474
29.49
.668 0383
29-35
6 7 8 5794
29.22
40
1.646 9785
29.65
1.657 6242
49.49
1.668 2144
29-35
1.678 7547
29.22
41
.647 1564
29.65
.657 8011
49.49
.668 3904
29.34
.678 9300
29.21
42
6 47 3343
29.65
.657 9781
29.49
.668 5665
29.34
.679 1051
29.21
43
.647 5122
29.65
.658 '1550
29.48
.668 7425
29-34
.679 2806
29.21
44
.647 6900
29.64
.658 3318
49.48
.668 9185
29.34
.679 4558
29.21
45
1.647 8679
29.64
1.658 5087
49.48
1.669 945
29-33
1.679 6310
29.20
46
.648 0457
29.64
.658 6855
49.48
.669 2705
29-33
.679 8063
29.20
47
.648 2235
29.63
.658 8624
29.47
.669 4465
29-33
.679 9815
29.20
48
.648 4013
29.63
.659 0393
49.47
.669 6225
29-33
.680 1567
29.20
49
.648 5791
29.63
.659 2161
49.47
.669 7984
29.32
.680 3319
29.20
50
1.648 7569
29.63
1.659 3929
49.47
1.669 9744
29.32
i. 680 5070
29.19
51
.648 9346
29.62
.659 5697
49.46
.670 1503
29.32
.680 6822
29.19
52
.649 1123
29.62
.659 7465
49.46
.670 3262
29.32
.680 8574
29.19
53
.649 2901
29.62
.659 9232
49.46
.670 5021
29.32
.681 0325
29.19
54
.649 4677
29.61
.660 1000
49.46
.670 6780
29.31
.681 2076
29.19
55
1.649 6454
29.61
1. 660 2767
49.45
1.670 8539
29.31
1.681 3827
29.18
56
.649 8231
29.61
.660 4534
49.45
.671 0298
49.31
.681 5578
29.18
57
.650 0007
29.61
.660 6301
49.45
.671 2056
49.31
.681 7329
29.18
58
.650 1784
29.60
.660 8068
49.45
.671 3814
49.30
.681 9080
29.18
59
.650 3560
29.60
.660 9835
29.44
671 5573
49.30
.682 0831
29.18
60
1.650 5336
29.60
1.661 1601
49.44
1.671 7331
49.30
1.684 4581
29.17
580
TABLE VI.
For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit.
V.
60
61
62
63
log M.
Diff. 1".
logM
Diff. 1".
logM.
Diff. 1".
logM.
Diff. 1".
O'
1.682 2581
29.17
1.692 7408
29.07
1.703 1866
28.97
1.713 6006
28.89
1
2
.682. 4332,
.682 6082
29.17
29.17
.692 9152
.693 0896
29.06
29.06
.703 3604
.703 5342
28.97
28.97
713 7739
713 9473
28.89
28.89
3
.682 7832
29.17
.693 2640
29.06
.703 7080
28.97
.714 1206
28.88
4
.682 9582
29.17
6 93 43 8 3
29.06
.703 8818
28.96
.714 2939
28.88
5
1.683 1332
29.16
1.693 6127
29.06
1.704 0556
28.96
1.714 4672
28.88
6
.683 3082
29.16
.693 7870
29.05
.704 2293
28.96
.714 6405
28.88
7
.683 4832
29.16
.693 9613
29.05
.704 4031
28.96
.714 8138
28.88
8
.683 6581
29.16
.694 1356
29.05
.704 5768
28.96
.714 9870
28.88
9
.683 8331
29.16
.694 3099
29.05
.704 7506
28.96
.715 1603
28.88
10
1.684 0080
29.16
1.694 4842
29.05
1.704 9243
28.96
1.715 3336
28.88
11
.684 1830
29.15
.694 6585
29.04
.705 0981
28.95
.715 5068
28.88
12
.684 3579
29.15
.694 8328
29.04
.705 2718
28.95
.715 6801
28.87
13
.684 5328
29.15
.695 0070
2904
75 4455
28.95
7*5 8 533
28.87
14
.684 7077
29.15
.695 1813
29.04
.705 6192
28.95
.716 0266
28.87
15
1.684 8826
29.14
' 6 95 3555
29.04
1.705 7929
28.95
1.716 1998
28.87
16
.685 0574
29.14
.695 5298
29.04
.705 9666
28.95
.716 3730
28.87
17
.685 2323
29.14
.695 7040
29.04
.706 1402
28.95
.716 5462
28.87
18
.685 4071
29.14
.695 8782
29.03
.706 3139
28.94
.716 7194
28.87
19
.685 5820
29.14
.696 0524
29.03
.706 4875
28.94
.716 8926
28.87
20
1.685 7568
29.14
1.696 2266
29.03
1.706 6612
28.94
1.717 0658
28.86
21
.685 9316
29.13
.696 4008
29.03
.706 8348
28.94
.717 2390
28.86
22
.686 1064
29.13
.696 5750
29.03
.707 0085
28.94
.717 4122
28.86
23
.686 2812
29.13
.696 7491
29.03
.707 1821
28.94
7 X 7 5 8 53
28.86
24
.686 4560
29.13
.696 9233
29.02
707 3557
28.94
.717 7585
28.86
25
1.686 6308
29.13
1.697 0974
29.02
1.707 5293
28.93
1.717 9317
28.86
26
.686 8055
29.13
.697 2716
29.02
.707 7029
28.93
.718 1048
28.86
27
.686 9803
29.12
697 4457
29.02
.707 8765
28.93
.718 2780
28.86
28
.687 1550
29.12
.697 6198
29.02
.708 0501
28.93
.718 4511
28.86
29
.687 3297
29.12
.697 7939
29.02
.708 2237
28.93
.718 6242
28.85
30
1.687 5044
29.12
1.697 9680
29.02
1.708 3972
28.93
1.718 7974
28.85
31
.687 6791
29.12
.698 1421
29.01
.708 5708
2893
.718 9705
28.85
32
.687 8538
29.11
.698 3162
29.01
.708 7444
28,92
.719 1436
28.85
33
.688 0285
29.11
.698 4902
29.01
.708 9179
28.92
.719 3167
28.85
34
.688 2032
29.11
.698 6643
29.01
.709 0914
28.92
.719 4898
28.85
35
1.688 3778
29.11
1.698 8383
29.01
1.709 2650
28.92
1.719 6629
28.85
36
.688 5525
29.11
.699 0124
29.01
.709 4385
28.92
.719 8360
28.85
37
.688 7271
29.10
.699 1864
29.00
.709 6120
28.92
.720 0090
28.85
38
.688 9017
29.10
.699 3604
29.00
.709 7855
28.92
.720 1821
28.84
39
.689 0764
29.10
.699 5345
29.00
.709 9590
28.92
.720 3552
28.84
40
1.689 2510
29.10
1.699 7085
29.00
1.710 1325
28.91
1.720 5282
28.84
41
.689 4256
29.10
.699 8824
29.00
.710 3060
28.91
.720 7013
28.84
42
.689 6001
29.09
.700 0564
29.00
.710 4794
28.91
.720 8743
28.84
43
.689 7747
29.09
.700 2304
29.00
.710 6529
28.91
.721 0474
28.84
44
.689 9493
29.09
.700 4044
28.99
.710 8263
28.91
.721 2204
28.84
45
1.690 1238
29.09
1.700 5783
28.99
1.710 9998
28.91
1.721 3934
28.84
46
47
.690 2984
.690 4729
29.09
29.09
.700 7523
.700 9262
28.99
28.99
.711 1732
.711 3467
28.91
28.90
.721 5665
.721 7395
28.84
28.84
48
.690 6474
29.09
.701 looi
28.99
.711 5201
28.90
.721 9125
28.83
49
.690 8219
29.08
.701 2741
28.99
.711 6935
28.90
.722 0855
28.83
50
1.690 9964
29.08
1.701 4480
28.98
1.711 8669
28.90
1.722 2585
28.83
51
.691 1709
29.08
.701 6219
28.98
.712 0403
28.90
.722 4315
28.83
52
.691 3454
29.08
.701 7958
28.98
.712 2137
28.90
.722 6044
28.83
53
.691 5199
29.08
.701 9697
28.98
.712 3871
28.90
.722 7774
28.83
54
.691 6943
29.08
.702 1435
28.98
.712 5605
28.90
.722 9504
28.83
55
1.691 8688
29.07
1.702 3174
28.98
1.712 7339
28.90
1.723 1233
2 ^ 3
56
.692 0432
29.07
.702 4913
28.98
.712 9072
28.89
.723 2963
28.83
57
.692 2176
29.07
.702 6651
28.97
.713 0806
28.89
.723 4693
28.82
58
.692 3920
29.07
.702 8389
28.97
713 2 539
28.89
.723 6422
28.82
59
.692 5664
29.07
.703 0128
28.97
713 4*73
28.89
.723 8151
28.82
60
1.692 7408
29.07
1.703 1866
22.97
1.713 6006
28.89
1.723 9881
28.82
581
TABLE VI.
For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit.
V.
64
65
66
67
logM.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. 1".
0'
1.723 9881
28.82
1-734 3539
28.77
1.744 731
28.73
1-755 45
28.70
1
.724 1610
28.82
734 5265
28.77
744 8755
28.73
755 2127
28.70
2
724 3339
28.82
734 6 99i
28.77
745 479
28.73
755 3849
28.70
3
.724 5068
28.82
734 8718
28.77
.745 2202
28.73
755 557i
28.70
4
.724 6798
28.82
735 444
28.77
745 39 26
28.73
755 7293
28.70
5
1.724 8527
28.82
1.735 2169
28.76
1.745 5 6 5o
28.73
1.755 9 i5
28.70
6
.725 0256
28.82
735 3895
28.76
745 7373
28.73
.756 0737
28.70
7
.725 1984
28.81
.735 5621
28.76
745 997
28.73
.756 2459
28.70
8
.725 3713
28.81
735 7347
28.76
.746 0820
28.72
.756 4181
28.70
9
.725 5442
28.81
735 973
28.76
.746 2544
28.72
756 593
28.70
10
1.725 7171
28.81
1.736 0798
28.76
1.746 4267
28.72
1.756 7625
28.70
11
.725 8900
28.81
.736 2524
28.76
.746 5991
28.72
75 6 9347
28.70
12
.726 0628
28.81
.736 4250
28.76
.746 7714
28.72
757 1069
28.70
13
.726 2357
28.81
73 6 5975
28.76
.746 9437
28.72
757 2791
28.70
14
.726 4085
28.81
.736 7701
28.76
.747 1161
28.72
757 4513
28.70
15
1.726 5814
28.81
1.736 9426
28.76
1.747 2884
28.72
1.757 6235
28.70
16
.726 7542
28.81
737 "5 2
28.76
.747 4607
28.72
757 7957
28.70
17
.726 9270
28.81
737 2877
28.76
747 6330
28.72
757 9 6 79
28.70
18
.727 0999
28.80
.737 4602
28.76
.747 8054
28.72
.758 1401
28.70
19
.727 2727
28.80
.737 6328
28.75
747 9777
28.72
.758 3123
28.70
20
1-727 4455
28.80
1.737 8053
28.75
1.748 1500
28.72
1.758 4844
28.70
21
.727 6183
28.80
737 9778
28.75
.748 3223
28.72
.758 6566
28.70
22
.727 7911
28.80
738 i53
28.75
.748 4946
28.72
.758 8288
28.70
23
.727 9639
28.80
.738 3228
28.75
.748 6669
28.72
.759 ooio
28.70
24
.728 1367
28.80
738 4953
28.75
.748 8392
28.72
759 1731
28.70
25
1.728 3095
28.80
1.738 6679
28.75
1.749 "5
28.72
'759 3453
28.70
26
27
.728 4823
.728 6551
28.80
28.80
.738 8404
.739 0129
28.75
28.75
749 l8 3 8
749 35 61
28.72
28.72
759 5175
759 6897
28.70
28.70
28
.728 8279
28.80
739 1857
28.75
749 5284
28.72
.759 8618
28.69
29
.729 0006
28.79
739 3578
28.75
749 707
28.71
.760 0340
28.69
30
1.729 1734
28.79
1-739 533
28.75
1-749 873
28.71
1.760 2062
28.69
31
.729 3461
28.79
739 7 28
28.75
75 453
28.71
.760 3783
28.69
32
.729 5189
28.79
739 8753
28.75
.750 2176
28.71
.760 5505
28.69
33
.729 6916
28.79
.740 0477
28.75
75 3898
28.71
.760 7227
28.69
34
.729 8644
28.79
.740 2202
28.74
.750 5621
28.71
.760 8948
28.69
35
1.730 0371
28.79
1-74 3927
28.74
1.750 7344
28.71
1.761 0670
28.69
36
.730 2099
28.79
74 5 6 5i
28.74
.750 9067
28.71
.761 2392
28.69
37
.730 3826
28.79
.740 7376
28.74
.751 0789
28.71
.761 4113
28.69
38
73 5553
28.79
.740 9101
28.74
.751 2512
28.71
.761 5835
28.69
39
.730 7280
28.79
.741 0825
28.74
751 4234
28.71
.761 7556
28.69
40
1.730 9007
28.78
1.741 2550
28.74
1-75' 5957
28.71
1.761 9278
28.69
41
.731 0735
28.78
.741 4274
28.74
.751 7680
28.71
.762 0999
28.69
42
.731 2462
28.78
.741 5998
28-74
.751 9402
28.71
.762 2721
28.69
43
.731 4189
28.78
.741 7723
28.74
.752 1125
28.71
.762 4442
28.69
44
73 1 5915
28.78
74 1 9447
28.74
.752 2847
28.71
.762 6164
28.69
45
1.731 7642
28.78
1.742 1171
28.74
1.752 4570
28.71
1.762 7885
28.69
I 46
.731 9369
28.78
.742 2896
28.74
.752 6292
28.71
.762 9607
28.69
47
.732 1096
28.78
.742 4620
28.74
.752 8015
18.71
.763 1328
28.69
48
.732 2823
28.78
.742 6344
28.74
752 9737
a8. 7 i
.763 3050
28.69
49
732 4549
28.78
.742 8068
28.74
753 H 6
28.71
.763 4771
28.69
50
1.732 6276
28.78
1.742 9792
28.74
1-753 3 l8 2
28.71
1.763 6493
28.69
51
.732 8002
28.78
743 *5i6
28.73
753 494
28.71
.763 8214
28.69
1 52
.732 9729
28.77
743 324
28.73
753 66 27
28.71
.763 9936
28.69
53
733 "455
28.77
743 49 6 4
28.73
753 8349
28.71
.764 1657
28.69
54
733 3'82
28.77
.743 6688
28.73
754 7i
28.70
7 6 4 3379
28.69
55
1-733 49 8
28.77
1.743 8412
28.73
1-754 1794
28.70
1.764 5100
28.69
56
733 6635
28.77
.744 0136
28.73
754 35 l6
28.70
.764 6821
28.69
57
58
733 8 36i
734 8 7
28.77
28.77
.744 1860
.7^4 3584
28.73
28.73
754 5238
.754 6960
28.70
28.70
.764 8543
.765 0264
28.69
28.69
59
734 l8l 3
28.77
744 53 8
28.73
.754 8682
28.70
.765 1985
28.69
60
1-734 3539
28.77
1.744 7 3!
28.73
1.755 0405
28.70
1.765 3707
28.69
582
TABLE VI.
For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit.
V.
68
69
70
71
logM.
Diflf. 1".
logM.
Diflf. 1".
logM.
Diflf. 1".
logM.
Diflf. 1".
1.765 3707
28.69
775 6985
28.69
1.786 0284
28.70
1.796 3650
28.73
1
.765 5428
28.69
775 8706
28.69
.786 2006
28.70
.796 5374
28.73
2
.765 7150
28.69
.776 0427
28.69
.786 3728
28.70
.796 7097
28.73
3
.765 8871
28.69
.776 2149
28.69
.786 5450
28.70
.796 8821
28.73
4
.766 0592
28.69
.776 3870
28.69
.786 7172
28.70
797 0545
28.73
5
1.766 2314
28.69
.776 5591
28.69
1.786 8894
28.70
1.797 2268
28.73
6
.766 4035
28.69
.776 7313
28.69
.787 0617
28.70
797 399 2
28.73
7
.766 5756
28.69
.776 9034
28.69
787 2339
28.70
797 576
28.73
8
.766 7478
28.69
777 0755
28.69
.787 4061
28.70
797 744
28.73
9
.766 9199
28.69
777 *477
28.69
787 5783
28.70
797 9 l6 4
28.73
10
1.767 0920
28.69
1.777 4198
28.69
1.787 7506
28.70
1.798 0888
28.73 i
11
.767 2642
28.69
777 59 20
28.69
.787 9228
28.71
.798 2611
28.73
12
.767 4363
28.69
.777 7641
28.69
.788 0950
28.71
798 4335
28.73
13
.767 6084
28.69
777 93 6 3
28.69
788 2673
28.71
.798 6060
28.73
14
.767 7805
28.69
.778 1084
28.69
.788 4395
28.71
798 7784
28.73
15
1.767 9527
28.69
1.778 2806
28.69
1.788 6117
28.71
1.798 9508
28.73
16
.768 1248
28.69
.778 4527
28.69
.788 7840
28.71
.799 1232
28.74
17
.768 2969
28.69
.778 6248
28.69
.788 9562
28.71
799 2 95 6
28.74
18
.768 4691
28.69
.778 7970
28.69
.789 1284
28.71
.799 4680
28.74
19
.768 6412
28.69
778 9 6 9'
28.69
.789 3007
28.71
.799 6404
28.74
20
I. 7 68 8133
28.69
1.779 1413
28.69
1.789 4730
28.71
1.799 8128
28.74
21
.768 9854
28.69
779 3*4
28.69
.789 6452
28.71
799 9853
28.74
22
.769 1576
28.69
.779 4862
28.69
789 8175
28.71
.800 1577
28.74
23
.769 3297
28.69
779 6 57 8
28.69
789 9897
28.71
.800 3301
28.74
24
.769 5018
28.69
779 82 99
28.69
.790 1620
28.71
.800 5026
28-74
25
26
1.769 6740
.769 8461
28.69
28.69
1.780 OO2I
.780 1742
28.69
28.69
1.790 3342
.790 5065
28.71
28.71
1.800 6750
.800 8475
28.74
28.74
27
.770 0182
28.69
.780 3464
28.69
.790 6788
28.71
.801 0199
28.74
28
.770 1903
28.69
.780 5185
28.69
79 8510
28.71
.801 1924
28.74
29
.770 3625
28.69
.780 6907
28.69
.791 0233
28.71
.801 3648
28.74
30
1.770 5346
28.69
1.780 8629
28.69
1.791 1956
28.71
1.801 5373
28.74
31
.770 7067
28.69
.781 0350
28.69
.791 3678
28.71
.801 7107
28.74
32
.770 8788
28.69
.781 2072
28.69
.791 5401
28.71
.801 8822
28.74
33
.771 0510
28.69
.781 3793
28.69
.791 7124
28.71
.802 0547
28.75
34
.771 2231
28.69
.781 5515
28.69
.791 8847
28.71
.802 2271
28.75
35
1.771 3952
28.69
1.781 7237
28.69
1.792 0570
28.71
1.802 3996
28.75
36
37
.771 5673
77* 7395
28.69
28.69
.781 8959
.782 0680
28.69
28.70
.792 2293
.792 4016
28.71
28.72
.802 5721
.802 7446
28.75
28.75
38
.771 9116
28.69
.782 2402
28.70
.792 5738
28.72
.802 9171
28.75
39
.772 0837
28.69
.782 4124
28.70
.792 7461
28.72
.803 0896
28.75
40
1.772 2559
28.69
1.782 5845
28.70
1.792 9184
28.72
1.803 2621
28.75
41
.772 4280
28.69
.782 7567
28.70
793 97
28.72
.803 4346
28.75
42
.772 6001
28 69
.782 9289
28.70
793 26 3
2.8.72
.803 6071
28.75
43
.772 7722
28.69
.783 ion
28.70
793 4354
28.72
.803 7796
28.75
44
.772 9444
28.69
.783 2732
28.70
793 6 77
28.72
.803 9521
28.75
45
1.773 1165
28.69
1-783 4454
28.70
1.793 7800
28.72
1.804 1246
28.75
46
.773 2886
28.69
.783 6176
28.70
793 95 2 3
28.72
.804 2971
28.75
47
1 48
773 4607
773 6329
28.69
28.69
.783 7898
.783 9620
28.70
28.70
.794 1246
794 2 9 6 9
28.72
28.72
.804 4697
.804 6422
28.75
28.76
49
773 8 5
28.69
.784 1342
28.70
.794 4693
28.72
.804 8147
28.76
50
51
1773 977 1
.774 1493
28.69
28.69
1.784 3064
.784 4786
28.70
28.70
1.794 6416
794 8139
28.72
28.72
1.804 9877
.805 1598
28.76
28.76
52
53
.774 3214
774 4935
28.69
28.69
.784 6508
.784 8230
28.70
28.70
794 9862
795 1586
28.72
28.72
-805 3324
805 5049
28.76
28.76
54
774 66 57
28.69
.784 9952
28.70
795 339
28.72
805 6775
28.76
55
56
1-774 8 37 8
.775 0099
28.69
28.69
1.785 1674
785 339 6
28.70
28.70
1-795 533
795 6 75 6
28.72
28.72
1.805 8500
.806 0226
28.76
28.76
57
.775 1821
28.69
.785 5118
28.70
.795 8480
28.72
.806 1952
28.76
58
775 354 2
28.69
.785 6840
28.70
.796 0203
28.73
.806 3677
28.76 1
59
775 5 26 3
28.69
.785 8562
28.70
.796 1927
28.73
.806 5403
28.76
60
1.775 6 9 8 5
28.69
1.786 0284
28.70
1.796 3650
28.73
i. 806 7129
28.76
583
TABLE VI.
For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit.
V.
72
73
74
75
log It,
Difif. 1".
logM.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. 1".
0'
i. 806 7129
28.76
1.817 0765
28.81
1.827 4602
28.88
1.837 8686
28.95
1
.806 8855
28.76
.817 2494
28.81
.827 6335
28.88
.838 0423
28.95
2
.807 0581
28.77
.817 4222
28.82
.827 8068
28.88
.838 2160
28.95
3
.807 2307
28.77
.817 5951
28.82
.827 9800
28.88
.838 3898
28.95
4
.807 4033
28.77
.817 7680
28.82
.828 1533
28.88
8 3 8 5635
28.96
5
' 8 7 5759
28.77
1.817 94 10
28.82
1.828 3266
28.88
1.838 7372
28.96
6
.807 7485
28.77
.818 1139
28.82
.828 4999
28.88
.838 9110
28.96
7
.807 9211
28.77
.818 2868
28.82
.828 6732
28.88
.839 0847
28.96
8
.808 0937
28.77
.818 4597
28.82
.828 8465
28.88
.839 2585
28.96
9
.808 2663
28.77
.818 6326
28.82
.829 0198
28.89
.839 4323
28.96 ,
10
i. 808 4389
28.77
1.818 8056
28.82
1.829 1931
28.89
1.839 6060
28.96
11
.808 6116
28.77
.818 9785
28.82
.829 3665
28.89
.839 7798
28.97
12
.808 7842
28.77
.819 1515
28.83
.829 5398
28.89
.839 9536
28.97
13
.808 9568
28.77
.819 3244
28.83
.829 7131
28.89
.8^0 1274
28.97
14
.809 1295
28.77
.819 4974
28.83
.829 8865
28.89
.840 3012
28.97
15
1.809 3021
28.78
1.819 6704
28.83
1.830 0599
28.89
1.840 4751
28.97
16
.809 4748
28.78
.819 8433
28.83
.830 2332
28.89
.840 6489
28.97
17
.809 6474
28.78
.820 0163
28.83
.830 4066
28.90
.840 8227
28.97 ,
18
.809 8201
28.78
.820 1893
28.83
.830 5800
28.90
.840 9966
28.97
19
.809 9928
28.78
.820 3623
28.83
8 3 7533
28.90
.841 1704
28.98
20
1.810 1655
28.78
1.820 5353
28.83
1.830 9267
28.90
1.841 3443
28.98
21
.810 3381
28.78
.820 7083
28.83
.831 looi
28.90
.841 5182
28.98
22
.810 5108
28.78
.820 8813
28.84
8 3 ! 2 735
28.90
.841 6921
28.98
23
.810 6835
28.78
.821 0543
28.84
.831 4470
28.90
.841 8659
28.98
24
.810 8562
28.78
.821 2273
28.84
.831 6204
28.90
.842 0398
28.98
25
1.811 0289
28.78
1.821 4003
28.84
I-83 1 793 8
28.91
1.842 2138
28.98
26
.811 2016
28.78
.821 5734
28.84
.831 9672
28.91
.842 3877
28.99
27
.811 3743
28.78
.821 7464
28.84
.832 1407
28.91
842 5616
28.99
28
.811 5470
28.79
.821 9194
28.84
.832 3141
28.91
8 4* 7355
28.99
29
.811 7197
28.79
.822 0925
28.84
.832 4876
28.91
.842 9095
28.99
30
1.811 8924
28.79
1.822 2656
28.84
1.832 6611
28.91
1.843 0834
28.99
31
.812 0652
28.79
.822 4386
28.84
.832 8345
28.91
.843 2574
28.99
32
.812 2379
28.79
.822 6117
28.85
.833 0080
28.92
8 43 43*3
29.00
33
.812 4106
28.79
.822 7848
28.85
.833 1815
28.92
.843 6053
29.00
34
.812 5834
28.79
.822 9578
28.85
8 33 355
28.92
8 43 7793
29.00
35
i. 812 7561
28.79
1.823 1309
28.85
1.833 5285
28.92
i- 8 43 9533
29.00
36
.812 9289
28.79
.823 3040
28.85
.833 7020
28.92
.844 1273
29.00
37
.813 1016
28.79
.823 4771
28.85
8 33 8 755
28.92
.844 3013
29.00
38
.813 2744
28.79
.823 6502
28.85
.834 0491
28.92
8 44 4753
29.00
39
.813 4472
28.79
.823 8233
28.85
.834 2226
28.92
.844 6494
29.01
40
1.813 6199
28.80
1.823 99 6 5
28.85
1.834 3961
28.92
1.844 8234
29.01
41
.813 7927
28.80
.824 1696
28.85
.834 5697
28.93
.844 9974
29.01
42
43
8i3 9655
.814 1383
28.80
28.80
.824 3427
.824 5159
28.86
28.86
8 34 743 2
.834 9168
2893
28.93
.845 1715
.845 3456
29.01
29.01
44
.814 3111
28.80
.824 6890
28.86
.835 0904
28.93
.845 5196
29.01 1
45
1.814 4 8 39
28.80
1.824 8622
28.86
1.835 2640
28.93
1.845 6937
29.01
46
.814 6567
28.80
82 5 353
28.86
.835 4376
28.93
.845 8678
29.02
47
.814 8295
28.80
.825 2085
28.86
.835 6112
28.93
.846 0419
29.02
48
.815 0023
28.80
.825 3816
28.86
.835 7848
28.93
.846 2160
29.02
49
.815 1751
28.80
.825 5548
28.86
.835 9584
28.94
.846 3901
29.02
50
1.815 3479
28.80
1.825 7280
28.86
1.836 1320
28.94
1.846 5643
29 C2
51
.815 5208
28.81
.825 9012
28.87
.836 3056
28.94
.846 7384
2O O2
52
.815 6936
28.81
.826 0744
28.87
.836 4792
28.94
.846 9125
29.03
53
.815 8664
28.81
.826 2476
28.87
.836 6529
28.94
.847 0867
29.03
54
.816 0393
28.81
.826 4208
28.87
.836 8265
28.94
.847 2609
29.03
55
1.816 2121
28.81
1.826 5940
28.87
1.837 0002
28.94
1.847 4350
29.03
56
.816 3850
28.81
.826 7673
28.87
.837 1739
28.95
.847 6092
29.03
57
58
.816 5578
.816 7307
28.81
28.81
.826 9405
.827 1137
28.87
28.87
8 37 3475
.837 5212
28.95
28.95
.847 7834
.847 9576
29.03
29.03
59
.816 9036
28.81
.827 2870
28.87
.837 6949
28.95
.848 1318
29.04
60
1.817 0765
a8.8i
1.827 4602
28.88
1.837 8686
28.95
1.848 3060
29.04
584
TABLE VI.
For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit
76
77
78
79
logM.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. 1".
0'
1.848 3060
29.04
1.858 7769
29.14
1.869 2857
29.25
1.879 8369
29-37
1
.848 4803
29.04
858 9517
29.14
.869 4612
29.25
.880 0131
29.37
2
.848 6545
29.04
.859 1266
29.14
.869 6367
29.25
.880 1894
29.38
3
4
.848 8287
.849 0030
29.04
29.04
8 59 3 OI 4
859 4763
29.14
29.15
.869 8122
.869 9878
29.25
29.26
.880 3656
.880 5419
29.38
29.38
5
1-849 '773
29.04
1.859 6512
29.15
1.870 1633
29.26
1. 880 7182
29.38
6
849 35*5
29.05
.859 8260
29.15
.870 3389
29.26
.88c 8945
29.38
7
8
.849 5258
.849 7001
29.05
29.05
.860 0009
.860 1758
29-I5
29.15
.870 5144
.870 6900
29.26
29.26
.881 0708
.881 2471
29-39 1
29.39 1
9
.849 8744
29.05
.860 3507
29.15
.870 8656
29.26
.881 4235
29-39
10
1.850 0487
29.05
i. 860 5256
29.15
1.871 0412
29.27
1.881 5998
29.39
11
.850 2231
29.05
.860 7006
29.16
.871 2168
29.27
.881 7762
29-39
12
.850 3974
29.06
.860 8755
29.16
.871 3924
29.27
.881 9526
29.40
13
.850 5717
29.06
.861 0505
29.16
.871 5681
29.27
.882 1290
29.40
14
.850 7461
29.06
.861 2254
29.16
.871 7437
29.28
.882 3054
29.40
15
1.850 9204
29.06
1.861 4004
29.16
1.871 9194
29.28
1.882 4818
29.40
16
.851 0948
29.06
.861 5754
29.16
.872 0950
29.28
.882 6582
29.41
17
.851 2692
29.06
.861 7504
29.17
.872 2707
29.28
.882 8347
29.41
18
.851 4436
29.07
.861 9254
29.17
.872 4464
29.28
.883 0112
29.41
19
.851 6180
29.07
.862 1004
29.17
.872 6221
29.29
.883 1876
29.41
20
1.851 7924
29.07
1.862 2754
29.17
1.872 7979
29.29
1.883 3641
29.42
21
.851 9668
29.07
.862 4505
29.17
.872 9736
29.29
.883 5406
29.42
22
.852 1412
29.07
.862 6255
29.18
873 "493
29.29
.883 7171
29.42
23
.852 3157
29.07
.862 8006
29.18
.873 3251
29.29
.883 8937
29.42
24
.852 4901
29.07
.862 9756
29.18
.873 5008
29.30
.884 0702
29.42 ,
25
1.852 6646
29.08
1.863 1507
29.18
1.873 6 7 66
29.30
1.884 2468
29-43
26
.852 8391
29.08
.863 3258
29.18
.873 8524
29.30
.884 4233
29.43
27
853 OI 35
29.08
.863 5009
29.18
.874 0282
29.30
.884 5999
29.43
28
.853 1880
29.08
.863 6760
29.19
.874 2041
29.30
-884 7765
29-43
29
.853 3625
29.08
.863 8512
29.19
874 3799
29.31
884 9531
29.44
30
I-853 537
29.09
1.864 0263
29.19
1-874 5557
29.31
1.885 1297
29.44
31
853 7^5
29.09
.864 2015
29.19
.874 7316
29.31
.885 3064
29.44
32
.853 8861
29.09
.864 3766
29.19
.874 9074
29.31
.885 4830
29.44
33
.854 0606
29.09
.864 5518
29.20
.875 0833
29.31
.885 6597
29.45
34
.854 2351
29.09
.864 7270
29.20
-875 2592
29.32
.885 8364
29.45
35
1.854 4097
29.09
1.864 9022
29.20
1-875 435^
29.32
1.886 0131
29.45
36
854 5843
29.10
.865 0774
29.20
.875 6m
29.32
.886 1898
29.45 '
37
.854 7588
29.10
.865 2526
29.20
.875 7870
29.32
.886 3665
29.45
38
854 9334
29.10
.865 4278
29.20
.875 9629
29.32
.886 5432
29.46
39
.855 1080
29.10
.865 6030
29.21
.876 1389
29.33
.886 7200
29.46
40
41
1.855 2826
.855 4572
29.10
29.10
1.865 7783
.865 9536
29.21
29.21
1.876 3148
.876 4908
29-33
29.33
1.886 8967
.887 0735
29.46
29.46
42
.855 6319
29.11
.866 1288
29.21
.876 6668
29-33
.887 2503
29.47
43
.855 8065
29.11
.866 3041
29.21
.876 8428
29-33
.887 4271
29.47
44
.855 9811
29.11
.866 4794
29.22
.877 0188
29-34
.887 6039
29.47
45
1.856 1558
29.11
1.866 6547
29.22
1.877 1949
29.34
1.887 7807
29.47
46
.856 3305
29.11
.866 8301
29.22
.877 3709
29-34
887 9576
29.48
47
.856 5052
29.11
.867 0054
29.22
.877 5470
29-34
.888 1344
29.48
48
.856 6799
29.12
.867 1807
29.22
.877 7230
29-34
.888 3113
29.48 I
49
.856 8546
29.12
.867 3561
29.23
.877 8991
29-35
.888 4882
29.48
50
1.857 0293
29.12
1.867 5314
29.23
1.878 0752
29.35
1.888 6651
29.48
51
.857 2040
29.12
.867 7068
29.23
.878 2513
29.35
.888 8420
29-49
52
857 3787
29.12
.867 8822
29.23
.878 4275
29-35
.889 0189
29.49
53
857 5534
29.12
.868 0576
29.23
.878 6036
29-35
.889 1959
29.49
54
.857 7282
29.13
.868 2330
29.24
-878 7797
29.36
.889 3728
29-49
55
56
1.857 9030
858 0777
29.13
29.13
1.868 4084
.868 5839
29.24
29.24
1.878 9559
879 1321
29.36
29.36
1.889 5498
.889 7268
29-49
29.50
57
.858 2525
29.13
.868 7593
29.24
.879 3082
29.36
.889 8038
29.50
58
858 4273
29.13
.868 9348
29.24
.879 4844
29.36
.890 0808
29.50
59
.858 6021
29.13
.869 1102
29.25
.879 6606
29-37
.890 2578
29.51
60
1.858 7769
29.14
1.869 2857
29.25
1.879 8369
29-37
1.890 4349
293
32.01
31
.983 2411
31.19
994 5' 61
31-45
.005 8878
3 I -73
.017 3614
32.02
32
33
.983 4283
.983 6155
31.19
31.20
994 7048
994 8 93 6
31.46
31.46
.006 0781
.006 2685
31-73
3*-74
-017 5535
.017 7456
32.02
32.03
34
.983 8027
31.20
995 8i 3
31.46
.006 4590
3!-74
.017 9378
3*-3
35
1.983 9899
31.21
1.995 2711
3M7
2.006 6494
31-75
2.018 1300
32.04
36
.984 1772
31.21
.995 4600
31-47
.006 8399
31-75
.018 3223
32.04
37
.984 3644
31.22
.995 6488
31.48
.007 0304
31.76
.018 5145
32.05
38
.984 5517
31.22
995 8 377
31.48
.007 22 I O
31.76
.018 7068
32.05
39
.984 7391
31.22
.996 0266
31.49
.007 4116
3-77
.018 8992
32.06
40
1.984 9264
31.23
1.996 2155
3 J -49
2.007 6022
3'-77
2.019 9 I 5
32.06
41
.985 1138
31.23
.996 4045
3!-5
.007 7928
3'-77
.019 2839
32.07
42
.985 3012
31.24
.996 5935
3 l -S Q
.007 9835
3I-7 8
.019 4763
32.07
43
.985 4886
31.24
.996 7825
31.51
.008 1742
31.78
.019 6688
32.08
44
.985 6761
31.24
.996 9716
3!'5i
.008 3649
31.79
.019 8613
32.08 :
45
1.985 8636
31.25
1.997 1606
31.51
2.008 5556
31.79
2.020 0538
32.09
46
.986 0511
3 r - 2 5
997 3497
31.52
.008 7464
31.80
.020 2463
32.09
47
.986 2386
31.26
997 53 8 9
3 T -5 2
.008 9372
31.80
.020 4389
32.10
48
.986 4262
31.26
.997 7280
3 ! -53
.009 I28o
31.81
.O2O 6315
32.10
49
.986 6138
31.27
997 9 J 7*
3'-53
.009 3189
31.81
.020 8241
32.11
50
1.986 8014
31.27
1.998 1064
31-54
2.009 59 8
31.82
2.021 Ol68
32.11
51
.986 9890
31.28
.998 2956
3'-54
.009 7007
31.82
.021 2095
32.12
52
.987 1767
31.28
.998 4849
3'-55
.009 8917
3i- 8 3
.021 4022
32.12
53
.987 3644
31.28
.998 6742
3'-55
.010 0826
31.83
.021 5949
32.13
54
.987 5521
31.29
.998 8635
3'-5 6
.010 1736
31.84
.021 7877
3*-3
55
1.987 7398
31.29
1.999 0529
3i-5f
2.OIO 4647
31.84
2.021 9805
32.14
56
.987 9276
31.30
.999 2422
31.56
.OIO 6557
31.85
.022 1734
32.14
57
.988 1154
31.30
999 43' 6
31-57
.010 8468
31-85
.022 3662
32.15
58
.988 3032
31.31
.999 6211
31-57
.on 0380
31.86
.022 5591
J*.i|
59
.988 4911
31.31
999 8l 5
3'-5 8
.on 2291
31.86
.022 7521
32.16
60
1.988 6789
S 1 ^ 1
2.000 0000
31.58
2.01 1 4103
31.87
2.O22 945O
32.16
-^J
588
TABLE VI.
For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit.
V.
92
93
94
95
log M.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. 1".
1 gM.
Diff. 1".
0'
.022 9450
32.16
1-034 5797
32.48
2.046 3296
32.80
2.058 2005
33-15
1
.023 1380
32.17
34 7745
32.48
.046 5264
32.81
.058 3994
33-15
2
.023 33 II
32.17
.034 9694
32-49
.046 7233
32.82
.058 5983
33-i6
3
.023 5241
32.18
.035 1644
32.49
.046 9202
32.82
.058 7973
33.16
4
.023 7172
32.18
035 3593
32.50
.047 1172
32.83
.058 9963
31-17
5
2.023 9103
32.19
035 5543
32.50
2.047 3141
32.83
2.059 1953
33-i8
6
.024 1035
32.19
.035 7494
3^-51
.047 5111
32-84
59 3944
33-i8
7
.024 2967
32.20
035 9444
32-51
.047 7082
32.84
059 5935
33-19
8
.024 4899
32.20
.036 1395
32.52
.047 9053
32.85
.059 7927
33-19
9
.024 6831
32.21
.036 3347
32.52
.048 1024
32.85
.059 9919
33-20
10
2.024 8764
32.21
2.036 5298
32.53
2.048 2995
32.86
2.060 1911
33.21
11
.025 0697
32.22
.036 7250
32.53
.048 4967
32.87
.060 3904
33-21
12
.025 2630
32.22
.036 9202
32.54
.048 6939
32-87
.060 5897
33.22
13
.025 4564
32.23
.037 1155
32-54
.048 8912
32.88
.060 7890
33-22
14
.025 6498
32.23
.037 3108
32.55
.049 0884
32.88
.060 9884
33-23
15
2.025 8432
32.24
2.037 5061
32.55
2.049 2857
32.89
2.061 1878
33-24
16
.026 0367
32.24
.037 7015
32.56
.049 4831
32.89
.061 3872
33-24
17
.026 2301
32.25
.037 8969
32.57
.049 6805
32.90
.061 5867
33-25
18
.026 4236
32.26
.038 0923
32.57
.049 8879
32.90
.061 7862
33-25
19
.026 6172
32.26
.038 2877
32-58
.050 0753
32.91
.061 9857
33.26
20
2.026 8lo8
32.27
2.038 4832
32.58
2.050 2728
32.92
2.062 1853
33-27
21
.027 0044
32.27
.038 6787
32.59
.050 4703
32-92
.062 3849
33-27
22
.027 1980
32.28
.038 8743
32.59
.050 6679
32.93
.062 5846
33-28
23
.027 3917
32.28
.039 0699
32.60
.050 8655
32.93
.062 7842
33.28
24
.027 5854
32.29
.039 2655
32.61
.051 0631
32.94
.062 9840
33-29
25
2.027 7791
32.29
2.039 4611
32.61
2.051 2608
32.95
2.063 1837
33-3
26
27
.027 9729
.028 1667
323
32.3
.039 6568
.039 8525
32.62
32.62
.051 4585
.051 6562
32.95
32.96
.063 3835
.063 5833
33-3
33-31
28
.028 3605
32-31
.040 0482
32.63
.051 8539
32.96
.063 7832
33-31
29
.028 5544
32.31
.040 2440
32.63
.052 0517
32.97
.063 9831
33.32
30
2.028 7483
32-3*
2.040 4399
32.64
2.052 2496
32.97
2.064 l8 3i
33-33
31
.028 9422
32.32
.040 6357
32.64
.052 4474
32-98
.064 3830
33-33
32
.029 1361
32-33
.040 8316
32.65
.052 6453
32.98
.064 5830
33-34
33
.029 3301
32-33
.041 0275
32.65
.052 8432
32.99
.064 7831
33-34
34
.029 5241
32.34
.041 2234
32.66
.053 0412
33.00
.064 9832
33-35
35
2.029 7182
32.34
2.041 4194
32.67
2053 2392
33.00
2.065 l8 33
33-3 6
36
37
.029 9123
.030 1064
32.35
32.35
.041 6154
.041 8114
32.67
32.68
53 4372
.053 6353
33-oi
33.01
.065 3834
.065 5836
33-36
33-37
38
.030 3005
32.36
.042 0075
32.68
53 8 334
33.02
.065 7839
33-37
39
.030 4947
32.36
.042 2036
32.69
.054 0315
33-3
.065 9841
33-38
40
2.030 6889
32-37
2.042 3998
32.69
2.054 2297
33-03
2.066 1844
33-39
41
.030 8831
32-37
.042 5960
32.70
.054 4279
33-4
.066 3847
33-39
42
.031 0774
32.38
.042 7922
32.70
.054 6262
33-4
.066 5851
33-4
43
.031 2717
32.39
.042 9834
32.71
.054 8244
33-05
.066 7855
33-40
44
.031 4660
32.39
.043 1847
32.71
.055 0227
33-5
.066 9860
33.41
45
2.031 6604
32.40
2.043 3810
32.72
2.055 221 1
33.06
2.067 1865
33-42
46
.031 8548
32.40
043 5773
32.73
.055 4195
33.07
.067 3870
33-42
47
.032 0492
32.41
043 7737
32.73
.055 6179
33-07
.067 5875
33-43
48
.032 2437
32.41
.043 9701
32.74
.055 8l6l
33.08
.067 7881
33-43
49
.032 4382
32.42
.044 1665
32.74
.056 0148
33.08
.067 9887
33-44
50
2.032 6327
32.42
2.044 3630
32-75
2.056 2133
33-9
2.068 1894
33-45
51
52
53
.032 8272
.033 0218
.033 2164
32.43
32-43
32-44
044 5595
.044 7561
.044 9526
32-75
32.76
32.76
.056 4119
.056 6105
.056 8091
33.10
33.10
33-"
.068 3901
.068 5908
.068 7916
33-45
33-46
33-47
54
.033 4111
32-44
.045 1492
32.77
.057 0078
33-"
.068 9924
33-47
55
2.033 6058
32.45
2.045 3459
32.78
2.057 2065
33.12
2.069 1933
3348
56
.033 8005
32-45
.045 5426
32.78
.057 4052
33.12
.069 3942
33-48
57
33 9952
32.46
45 7393
32.79
.057 6040
33.13
.069 5951
33-49
58
.034 1900
32-47
.045 9360
32-79
.057 8028
33-14
.069 7960
33.50
59
.034 3848
32.47
.046 1328
32.80
.058 00l6
33-H
.069 9970
33.50
60
2.034 5797
3^.48
2.046 3296
32.80
2.058 2005
33-*5
2.070 1980
33-51
589
TABLE VI.
For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit.
V.
96
97
98
99
logic.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. 1".
logM.
Dtff. l".
0'
2.070 1980
33-51
2.082 3282
33-88
2.094 5971
34.28
2.107 0109
34.69
1
.070 3991
33-51
.082 5316
33-89
.094 8028
34.29
.107 2190
34-70
2
.070 6002
33-52
.082 7349
33-9
.095 0085
34.29
.107 4272
34.70
3
.070 8014
33-53
.082 9383
33-90
.095 2143
34-3
.107 6355
34-71
4
.071 0025
33-53
.083 1418
33-9 1
.095 4201
34.31
.107 8437
34-72
5
2.071 2037
33-54
2-083 3453
33-92
2.095 6260
34-31
2.108 0521
34-72
6
.071 4050
33-54
.083 5488
33-92
.095 8318
34-32
.108 2604
34-73 i
7
.071 6063
33-55
.083 7523
33-93
.096 0378
34.33
.108 4689
34-74
8
.071 8076
33-5 6
.083 9559
33-94
.096 2438
34-33
.108 6773
34-75
9
.072 0090
33.56
.084 1596
33-94
.096 4498
34-34
.108 8858
34-75
10
2.072 2104
33-57
2.084 3633
33-95
2.096 6558
34-35
2.109 0944
34-76
11
.072 4118
33.58
.084 5670
33-96
.096 8619
34-35
.109 3029
34-77
12
.072 6133
33-58
.084 7707
33-96
.097 0681
34-36
.109 5116
34-77
13
-.072 8148
33-59
.084 9745
33-97
.097 2742
34-37
.109 7202
34.78
14
.073 0163
33 59
.085 1783
33.98
.097 4804
34-37
.109 9289
34-79
15
2.073 2179
33.60
2.085 3822
33.98
2.097 6867
34.38
2.IIO 1377
34.80
16
.073 4195
33.61
.085 5861
33-99
.097 8930
34-39
.no 3465
34.80
17
.073 6212
33.61
.085 7901
33-99
.098 0993
34-39
"o 5553
34.81
18
.073 8229
33.62
.085 9941
34.00
.098 3057
34-4
.no 7642
34-82
19
.074 0246
33- 6 3
.086 1981
34.01
.098 5121
34-41
.no 9731
34.82
20
2.074 2264
33- 6 3
2.086 4021
34.01
2.098 7186
34-41
2. ni 1821
34.83
21
.074 4282
33- 6 4
.086 6062
34.02
.098 9251
3442
.in 3911
34.84
22
.074 6301
33- 6 4
.086 8104
34-03
.099 1316
34-43
.in 6001
34.85
23
.074 8320
33- 6 5
.087 0146
34-03
.099 3382
34-43
.in 8092
34-85
24
75 339
33.66
.087 2188
34-04
99 5449
34-44
.HZ 0184
34-86
25
2.075 2358
33.66
2.087 4231
34-05
2.099 7515
34-45
2.II2 2275
34-87
26
.075 4378
33- 6 7
.087 6274
34.05
.099 9582
34-45
.112 4368
34-87
27
.075 6399
33-67
.087 8317
34-o6
.100 1650
34-46
.112 6460
34-88
28
.075 8419
33-68
.088 0361
34-07
.100 3718
34-47
.112 8553
34.89
29
.076 0440
33-69
.088 2405
34.07
.100 5786
34.48
.113 0647
34-90
30
2.076 2462
33.69
2.088 4449
34.08
2.100 7855
34-48
2.II3 2741
34.90
31
32
.076 4484
.076 6507
33-7
33.71
.088 6494
.088 8540
34.09
34-09
.100 9924
.101 1993
34-49
34.50
.113 4835
.113 6930
34-9 1
34-92
33
.076 8529
33-71
.089 0586
34.10
.101 4063
34.50
.113 9025
34-92
34
.077 0552
33-72
.089 2632
34-n
.101 6134
34-51
.114 II2I
34-93
35
2.077 2575
33-73
2.089 4678
34-"
2.IOI 8204
34-52
2.114 3*!7
34-94
36
.077 4599
3J-73
.089 6725
34.12
.102 0276
34-52
"4 53 J 3
34-95
37
.077 6623
33-74
.089 8772
34.12
.102 2347
34-53
.114 7410
34-95
38
.077 8647
33-74
.090 0820
34-13
.102 4419
34-54
.114 9508
34.96
39
.078 0672
33-75
.090 2868
34.14
.IO2 6492
34-54
.115 1605
34-97
40
2.078 2697
33-76
2.090 4917
34-15
2.102 8564
34-55
2.115 3704
34-97
41
.078 4723
33-76
.090 6966
34-15
.103 0638
34-56
.115 5802
34-98
42
.078 6749
33-77
.090 9015
34.16
.103 2711
34.56
.115 7901
34-99
43
.078 8775
33-78
.091 1065
34-17
.103 4785
34-57
.116 oooi
35-0
44
.079 0802
33-78
.091 3115
34-17
.103 6860
34.58
.Il6 2101
35-oo
45
2.079 2829
33-79
2.091 5165
34.18
2.103 8935
34-59
2.116 4201
35-oi
46
.079 4857
33.80
.091 7216
34.19
. 1 04 i o i o
34-59
.116 6301
35-02
47
.079 6885
3380
.091 9268
34.19
.104 3086
34.60
.116 8403
35-02
48
.079 8913
33.81
.092 1319
34-20
.104 5162
34.61
.117 0505
35-03
49
.080 0942
33-8i
.092 3371
34-20
.104 7239
34.61
.117 2607
35-04
50
2.080 2971
33-82
2.092 5424
34.21
2.104 93 J 6
34.62
2.117 4710
35-05
51
.080 5000
3383
.092 7477
34-22
.105 1393
34-63
.117 6813
35-05
1 52
.080 7030
33-83
.092 9530
34-22
.105 3471
34.63
.117 8916
35.06
53
.080 9060
33-84
.093 1584
34-23
-105 5549
34-64
.Il8 1020
35-07
54
.081 1091
33-85
.093 3638
34.24
.105 7628
34-65
.Il8 3124
35.08
55
2.081 3122
33-85
2.093 5692
34.25
2.105 977
34-66
2.118 5229
35.08
56
.081 5153
33-86
.093 7747
34.25
.106 1786
34-66
.118 7334
35-09
57
.081 7185
33-87
.093 9803
34-26
.106 3866
34.67
.118 9440
35- 10
58
.081 9217
33-87
.094 1858
34-27
.106 5947
34-68
,119 1546
35- 10
59
.082 1249
33-88
.094 3914
34-27
.106 8027
34.68
.119 3652
35-"
60
2.082 3282
33-88
2.094 5971
34.28
2.107 0109
34.69
2.119 5759
35-ia
590
TABLE VI.
For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit.
V.
100
101
102
103
logM.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. 1".
logM.
Diff I''.
2.119 5759
35-12
2.132 2989
35-57
2.145 *866
36.03
2.158 2460
36.52
1
.119 7867
35-13
.132 5123
35-57
.145 4028
36.04
.158 4652
36.53
2
.119 9974
35-13
.132 7258
35.58
.145 6191
36.05
.158 6844
36.54
3
.120 2083
35-H
132 9393
35-59
H5 8354
36.06
.158 9036
36.55
4
.120 4191
35-15
.133 1529
35.60
.146 0518
36.07
.159 I22 9
36.55 '
5
2.120 6301
35.16
2-133 3 66 5
35-6i
2.146 2682
36.07
2.159 3423
36.56
6
.120 8410
35-16
.133 5802
35-6i
.146 4847
36.08
.159 5617
36-57
7
.121 0520
35-17
!33 7939
35-62
.146 7012
36.09
.159 7 8ll
36.58
8
.121 2630
.134 0076
35-63
.146 9178
36.10
.160 0006
36.59
9
.121 4741
35-19
.134 2214
35-64
.147 1344
36.11
.l6o 2202
36.60
10
2. 121 6853
35-19
2.134 4352
35-64
2.147 3510
36.11
^.160 4398
36.60
11
.121 8965
35.20
.134 6491
35.65
.147 5677
36.12
.l6o 6594
36.61
12
.122 1077
35-21
.134 8631
35-66
.147 7845
36.13
.l6o 8791
36.62
13
.122 3190
.135 0770
35-67
.148 0013
36.14
,l6l 0989
36.63
14
.122 5303
35-22
.135 2910
35-67
.148 2182
36.15
.l6l 3187
36.64
15
2.122 7416
35-23
2.135 5051
35-68
2.148 4351
36.15
2.161 5385
36.65
16
.122 9530
35-24
.135 7192
35.69
.148 6520
36.16
.161 7584
36.65
17
.123 1644
35-24
*35 9334
35.70
.148 8690
36.17
.161 9784
36.66
18
.123 3759
35-25
.136 1476
35.71
.149 0861
36.18
.162 1984
36.67
19
.123 5875
35.26
.136 3619
35-71
.149 3032
36.19
.162 4185
36.68
20
2.123 7990
35-27
2.136 5762
35-72
2.149 5203
36.19
2.162 6386
36.69
21
.124 0107
35-27
.136 7905
35-73
H9 7375
36.20
.162 8587
36.70
22
.124 2223
35-28
.137 0049
35-74
.149 9547
36.21
.163 0789
36.70
23
.124 4340
35-29
.137 2193
35-74
.150 1720
36.22
.163 2992
36.71
24
.124 6458
35-3
*37 4338
35-75
.150 3893
36-23
.163 5195
36.72
25
2.124 8576
35-3
2.137 6484
35-76
2.150 6067
36.23
2.163 7398
36.73
26
.125 0694
35-3 1
.137 8630
35-77
.150 8242
36-24
.163 9602
36.74
27
.125 2813
35-32
.138 0776
35-77
.151 0417
36.25
.164 1807
36.74
28
29
.125 4933
.125 7052
35-33
35-33
.138 2922
.138 5070
35-78
35-79
.151 2592
.151 4768
36.26
36.27
.164 4012
.164 6218
36.75
36.76
30
2.125 9173
35-34
2.138 7217
35.80
2.151 6944
36.28
2.164 8424
36.77
31
.126 1293
35-35
138 9365
.151 9121
36.28
.165 0630
36.78
32
.126 3414
35-35
.139 1514
35'8i
.152 1298
36.29
.165 2837
36.79
33
.126 5536
35-36
.139 3663
35-82
.152 3476
36.30
.165 5045
36.80
34
.126 7658
35 37
.139 5813
35-83
.152 5654
36.31
.165 7253
36.81
35
36
2.126 9780
.127 1903
35-38
35-39
2.139 7963
.140 0113
35-84
35.84
2.152 7833
.153 OO I 2
36.32
36.32
2.165 9462
.166 1671
36.81
36.82
37
.127 4027
35-39
.140 2264
35.85
.153 2I 9 2
36.33
.166 3881
36-83
38
.127 6151
35-4
.140 4415
35-86
153 4372
36.34
.166 6091
36.84
39
.127 8275
35-4 1
.140 6567
35-87
153 6552
36.35
.166 8301
36-85
40
2.128 0400
35-42
2.140 8720
35-88
2.153 8734
36.35
2.167 0513
36.86
41
42
43
44
.128 2525
.128 4650
.128 6776
.128 8903
35.42
35-43
35-44
35-45
.141 0873
.141 3026
.141 5180
.141 7334
35-88
35.89
35-9
35.91
.154 0915
.154 3097
.154 5280
.154 7463
36.36
36.37
36.38
.167 2724
.167 4936
.167 7149
.167 9362
36.87
36.87
36.88
36.89
45
46
47
48
19
2.129 1030
.129 3157
.129 5285
.129 7414
.129 9542
35-45
35-46
35-47
3548
35.48
2.141 9489
.142 1644
.142 3799
.142 5955
.142 8112
35.92
35-93
35-94
35-95
2.154 9 6 47
.155 1831
.155 4015
.155 6200
.155 8386
36.40
36.41
36-41
36.42
36.43
2.168 1576
.168 3790
.168 6005
.168 8220
.169 0436
36.90
36.91
36.92
36.93
36.93
50
2.130 1672
35-49
2.143 0269
35-96
2.156 0572
3 6 .44
2.169 2652
36.94
51
52
53
54
.130 3801
13 593 1
.130 8062
.131 0193
35-5
35-51
35-5i
35-52
.^43 2427
-143 4585
-143 6743
.143 8902
35-96
35-97
35.98
35-99
.156 2759
.156 4946
.156 7133
.156 9321
36.45
3646
36.47
.169 4869
.169 7087
.169 9304
.170 1523
36.96
36.97
36.98
55
56
57
2.131 2325
131 4457
.131 6589
35-53
35-54
35-54
2.144 1062
.144 3222
.144 5382
36.00
36.00
36.01
2.157 i5 10
.157 3699
.157 5889
36.48
36.49
36.50
1.170 3742
.170 5961
.170 8181
36.99
36-99
37.00
58
59
.131 8722
.132 0855
35-55
35-56
.144 7543
.144 9704
36.02
36-03
.157 8079
.158 0269
36.50
36.51
.171 OAOI
.171 2622
37.01
37.02
60
2.132 2989
35-57
2.145 1866
36.03
2.158 2460
36.52
2.171 4844
37.03
591
TABLE VI,
For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit.
r.
104
105
106
107
logM.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. I".
logM.
Diff. 1".
0'
1
2
2.171 4844
.171 7066
.171 9288
37.03
37-04
37-05
.184 9092
.185 1346
.185 3600
37-56
37-57
37-57
.198 5282
.198 7568
.198 9856
38.11
38.12
38.13
.212 3493
.212 5814
.212 8136
38.68
38.69
38.70
3
4
.172 1511
.172 3735
37-05
37.06
.185 5855
.185 8110
37.58
37-59
.199 2144
.199 4432
38.14
38.14
.213 0458
.213 2781
38-7I
38.72
5
2.172 5959
37.07
2.186 0366
37.60
2.199 6721
38.15
2.213 5104
38.73
6
.172 8184
37.08
.186 2622
37-6i
.199 9010
38.16
.213 7428
38.74
7
.173 0409
37.09
.186 4879
37.62
.200 1300
38.17
.213 9753
38-75
8
.173 2634
37.10
.186 7137
37.63
.200 3591
38.18
.214 2078
38-76
9
.173 4860
37.11
.186 9395
37.64
.200 5882
38.19
.214 4404
38.77
10
2.173 70 8 7
37.12
2.187 1653
37.65
2.200 8174
38.20
2.214 673
38-78
11
173 93*4
37.12
.187 3912
37-66
.201 0467
38.21
.214 9057
38.79
12
.174 1542
37-13
.187 6172
37.67
.2OI 2760
38.22
.215 1385
38.80
13
.174 3770
37-H
.187 8432
37.67
.201 5053
38.23
.215 3713
38.81
14
.174 5999
37.15
.188 0693
37.68
.201 7347
38.24
.215 6042
38.82
15
2.174 8228
37.16
2.188 2954
37.69
2.2OI 9642
38.25
2.215 8371
38.83
16
.175 0458
37-17
.188 5216
37-70
.202 1937
38.26
.216 0701
38.84
17
.175 2688
37.18
.188 7478
37.71
.202 4233
38.27
.216 3032
38-85
18
.175 4919
37.18
.188 9741
37-72
.202 6529
38.28
.216 5363
38.86
19
.175 7150
37-19
.189 2005
37-73
.202 8826
38.29
.216 7694
38-87
20
2.175 9382
37.20
2.189 4269
37-74
2.203 1123
38.30
2.217 0027
38.88
21
.176 1615
37-21
.189 6533
37-75
.203 3421
38.31
.217 2360
38.89
22
.176 3848
37-22
.189 8798
37-76
.203 5720
.217 4693
38.90
23
.176 6081
37-23
.190 1064
37-77
.203 8019
38*32
.217 7027
38.91
24
.176 8315
.190 3330
37-77
.204 0319
38.33
.217 9362
38.92
25
2.177 0550
37-25
2.190 5597
37.78
2.204 2619
38.34
2.218 1697
38-93
26
.177 2785
37.25
.190 7864
37-79
.204 4920
38.35
.218 4033
27
.177 5020
37.26
.191 0132
37.8o
.204 7222
38.36
.218 6369
38-95
28
.177 7256
37.27
.191 2401
37.8i
.204 9524
38.37
.218 8706
38-96
29
177 9493
37.28
.191 4670
37.82
.205 1826
38.38
.219 1044
38.97
30
2.178 1730
37-29
2.191 6939
37.83
2.2O5 4.129
38.39
2.219 3382
38-98
31
.178 3968
373
.191 9209
37.84
.205 6433
38.40
.219 5721
38.99
32
.178 6206
37-3 1
.192 1480
37.85
205 8737
38.41
.219 8061
39.00
33
.178 8445
37-32
.192 3751
37.86
.206 1042
38.42
.220 0401
39.01
34
.179 0684
37-33
.192 6023
37.87
.206 3348
38.43
.220 2741
39.02
35
2 179 2924
37-33
2.192 8295
37.88
2.206 5654
38.44
2.220 5082
39-03
36
.179 5164
37-34
.193 0568
37-88
.206 7961
38-45
.220 7424
39-4
37
.179 7405
37-35
.193 2841
37.89
.207 O268
38.46
.220 97^7
39-05
38
.179 9646
37.36
.193 5115
37-9
-207 2575
38.47
.221 2110
39.06
39
.180 1888
37-37
.193 7389
37-9 1
.207 4884
38.48
221 4453
39.07
40
2.l8o 4131
37.38
2.193 9664
37-92
2.207 7 J 93
38.49
2.221 6797
39.08
41
.l8o 6374
37-39
.194 1940
37-93
.207 9502
38.50
.221 9142
39.09
42
.l8o 8617
37-4
.194 4216
37-94
.208 1812
38.51
.222 1488
39.10
43
.l8l o86l
37-4 1
.194 6493
37-95
.208 4123
222 3834
39- 11
44
.l8l 3106
37.41
.194 8770
37-96
.208 6434
38-53
.222 6l8o
45
2.181 5351
37.42
2.195 1048
37-97
2.208 8746
38.54
2.222 8528
39-'3
46
.181 7597
37-43
.195 3326
37-98
.209 1058
38.54
.223 0876
39-H
47
.181 9843
37-44
.195 5605
37-99
.209 3371
38.55
.223 3224
39-'5
48
.182 1089
37-45
.195 7885
38.00
.209 5685
38.56
.223 5573
39,16
49
-182 4337
37.46
.196 0165
38.00
209 7999
38-57
.223 7923
5O
2.182 6584
37-47
2.196 2445
38.01
2.210 0314
38.58
2.224 0273
39.18
51
.182 8833
37.48
.196 4726
38.02
.2IO 2629
38.59
.224 2624
39-19
52
.183 1082
37-49
.196 7008
38-03
.210 4945
38.60
.224 4975
39.20
53
54
.183 3331
.183 5581
37-49
37.50
.196 9290
.197 1573
38.04
38.05
.210 7261
.210 9578
38.61
38.62
.224 7327
.224 9680
39.21
39-22
55
2.183 7831
37.51
2.197 3856
38.06
2.21 I 1896
38.63
2.225 2033
39-23
56
.184 0082
37-52
.197 6140
38.07
.211 4214
38.64
.225 4387
39-24
57
.184 2334
37.53
.197 8425
38.08
.211 6533
38.65
.225 6741
3925
58
.184 4586
37-54
.198 0710
38.09
.211 8852
38.66
.225 9096
39.26
59
.184 6839
37-55
.198 2995
38.10
.212 1172
38.67
.226 1452
39-27
60
2.184 9092
37.56
2.198 5282
38.11
2.212 3493
38.68
2.226 3808
39.28
692
TABLE VI.
For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit.
V.
108
109
110
111
logM.
Diff. 1".
log M.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. 1".
1
.226 3808
.226 6165
39.28
39- 2 9
.240 6314
.240 8708
39.90
39.91
.255 1099
2 55 353*
40.54
40.55
.269 8255
.270 0728
41.21
2
.226 8523
39-3
.241 1103
39-9*
.255 5965
40.56
.270 3202
41.24
3
.227 0881
39-31
.241 3498
3993
.255 8399
40.58
.270 5676
41.25
4
.227 3240
39.32
.241 5894
39-94
.256 0834
40.59
.270 8152
41.26
5
.227 5599
39-33
2.241 8291
39-95
.256 3270
40.60
.271 0628
41.27
6
.227 7959
39-34
.242 0688
39-96
.256 5706
40.61
.271 3104
41.28
7
.228 0320
39-35
.242 3086
39-97
.256 8143
40.62
.271 5582
41.29
8
.228 2681
39-36
-242 5485
39'98
.257 0580
40.63
.271 8060
41.30
9
.228 5043
39-37
.242 7884
39-99
.257 3019
40.64
.272 0538
41.32
10
2.228 7405
39.38
2.243 0*84
40.00
2.257 5458
40.65
2.272 3018
41-33
11
.228 9768
39-39
.243 2685
40.01
.257 7897
40.66
.272 5498
41-34
12
.229 2131
39-4
.243 5086
40.02
.258 0337
40.68
.272 7979
41-35
13
.229 4496
39.41
.243 7488
40.03
.258 2778
40.69
.273 0460
41.36
14
.229 6861
39.42
.243 9890
40.05
.258 5220
40.70
.273 2942
41.38
15
2.229 9 2 *6
39-43
2.144 2293
40.06
2.258 7662
40.71
2.273 54*5
41-39
16
.230 1592
39-44
.244 4697
40.07
.259 0105
40.72
.273 7909
41.40
17
.230 3959
39-45
.244 7101
40.08
.259 2548
40-73
.274 0393
41.41
18
.230 6326
39-4 6
.244 9506
40.09
.259 4992
4-74
.274 2878
41.42
19
.230 8694
39-47
.245 1912
40.10
2 59 7437
40.75
.274 5364
41.43
20
2.231 1063
39-48
2.245 4318
40.11
2.259 9883
40.76
2.274 7850
41.44
21
.231 3432
39-49
.245 6725
40.12
.260 2329
40.78
.275 0337
41.46
22
.231 5802
39-5
.245 9132
40.13
.260 4776
40.79
.275 2825
41.47
23
.231 8172
39-5 1
.246 1541
40.14
.260 7223
40.80
-275 53*3
41.48
24
.232 0543
39.52
.246 3949
40.15
.260 9671
40.81
.275 7802
41.49
25
2.232 2915
39-53
2.246 6359
40.16
2.26l 2120
40.82
2.276 0292
41.50
26
.232 5287
39-54
.246 8769
40.17
.26l 4570
40.83
.276 2783
27
.232 7660
39-55
.247 1 1 80
40.18
.26l 7020
40.84
.276 5274
41-53
28
.233 0033
39-56
*47 359 1
40.19
.26l 9471
40.85
.276 7766
41.54
29
.233 2407
39-57
.247 6003
40.21
.262 1922
40.86
.277 0258
41-55
30
2.233 4782
39-58
2.247 8416
40.22
2.262 4374
40.88
2.277 *75 2
41.56
31
233 7*57
39-59
.248 0829
40.23
.262 6827
40.89
.277 5246
41-57
32
2 33 9533
39.60
.248 3243
40.24
.262 9281
40.90
.277 7740
41.58
33
.234 1910
39.61
.248 5658
40.25
.263 1735
40.91
.278 0236
41.60
34
.234 4287
39-63
.248 8073
40.26
.263 4190
40.92
.278 2732
41.61
35
2.234 6665
39.64
2.249 0489
40.27
2.263 6645
40-93
2.278 5229
41.62
36
.234 9043
.249 2906
40.28 .
.263 9102
40.94
.278 7726
41.63
37
.235 1422
39.66
.249 5323
40.29
.264 1559
40.95
.279 0224
41.64
38
.235 3802
39-67
.249 7741
40.30
.264 4016
40.96
.279 2723
41.65
39
.235 6183
39-68
.250 0159
40.31
.264 6474
40.98
.279 5223
41.67
40
2.235 8563
39-69
2.250 2578
40.32
2.264 8933
40.99
2.279 77 2 3
41.68
41
.236 0945
39.70
.250 4998
40.34
.265 1393
41.00
.280 0224
41.69
42
.236 3327
39-71
.250 7419
40.35
.265 3853
41.01
.280 2726
41.70
43
.236 5710
39-7 2
.250 9840
40.36
.265 6314
41.02
.280 5228
41.71
44
.236 8093
39-73
.251 2262
4037
.265 8776
41.03
.280 7731
41.72
45
2.237 0478
39-74
2.251 4684
40.38
2.266 1238
4 I. OA
2.281 0235
41.74
46
.237 2862
39-75
251 7107
40.39
.266 3701
41.06
.281 2740
4*-75
47
.237 5247
39-76
.251 9531
40.40
.266 6165
41.07
.281 5245
41.76
48
.237 7633
39-77
.252 1955
40.41
.266 8629
41.08
.281 7751
41-77
49
.238 OO2O
39.78
.252 4380
40.42
.267 1094
41.09
.282 0258
41.78
50
51
52
2.238 2407
2 38 4795
.238 7284
39-79
39.80
39.81
2.252 6806
.252 9232
253 l6 59
4-43
40.44
40.46
2.267 3560
.267 6026
.267 8493
41.10
41.11
41.12
2.282 2765
.282 5273
.282 7782
41.80
41.81
41.82
53
2 3 8 9573
39.82
253 4 8 7
40.47
.268 0961
41.13
.283 0291
41.83
54
.239 1962
.253 6515
40.48
.268 3430
4 I.I5
.283 2801
41.84
55
2 - 2 39 4353
39-84
2.253 8944
40.49
2.268 5899
4I.l6
2.283 53'*
41.85
56
.239 6744
39.86
40.50
.268 8769
41.17
.283 7824
41.87
57
58
59
.239 9235
.240 1528
.240 3921
39-87
39-88
39-89
.254 3804
.254 6235
.254 8666
40.51
40.52
40.53
.269 0839
.269 3310
.269 5782
4I.I8
41.19
41.20
.284 0336
.284 2849
.284 5363
41.88
41.89
41.90
60
2.240 6314
39.90
2-255 I0 99
40.54
2.269 8255
4 I.2I
2.284 7878
41.91
38
593
TABLE VI.
finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit.
V,
112
113
114
115
logM.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. 1".
0'
2.284 7878
41.91
2.300 0067
42.64
2.315 4927
43.40
2.331 2564
44.18
1
.285 0393
41.93
.300 2626
42.65
315 7531
43.41
.331 5216
44.20
2
.285 2909
41.94
.300 5186
42.67
.316 0136
43.42
.331 7868
44.21
3
.285 5425
41-95
.300 7746
42.68
.316 2742
43-44
.332 0521
44.22
4
285 7943
41.96
.301 0307
42.69
.316 5348
43-45
33 2 3*75
44.24
5
2.286 0461
41.97
2.301 2869
42.70
2.316 7956
43.46
2.332 5830
44.25
6
.286 2979
41.99
3 01 543 1
42.72
.317 0564
43-47
.332 8485
44.26
7
.286 5499
42.00
.301 7995
42.73
3*7 3'73
43-49
333 "4i
44.28
8
.286 8019
42.01
.302 0559
42.74
.317 5782
43-5
333 3799
44.29
9
.287 0540
42.02
.302 3123
42.75
3i7 8 393
43-51
333 6456
44.31
10
2.287 3062
42.03
2.302 5689
42.76
2.318 1004
43-53
2-333 9"5
44.32
11
.287 5584
42.04
.302 8255
42.78
.318 3616
43-54
334 1775
44-33
12
.287 8107
42.06
.303 0822
42.79
.318 6229
43-55
334 4435
44-34
13
.288 0631
42.07
.303 3390
42.80
.318 8842
43.56
334 79 6
44.36
14
.288 3155
42.08
33 595 8
42.81
.319 1456
43-58
334 9758
44-37
15
2.288 5680
42.09
2.303 8528
42.83
2.319 4072
43-59
2.335 2421
44-39
16
.288 8206
42.10
.304 1098
42.84
.319 6687
43.60
335 5 8 4
44.40
17
18
.289 0733
.289 3260
42.12
42.13
.304 3668
.304 6240
42.85
42.86
.319 9304
.320 1921
43.62
43- 6 3
335 7749
.336 0414
44.41
44-43
19
.289 5788
42.14
.304 8812
42.88
.320 4540
43- 6 4
.336 3080
44.44
20
2.289 8317
42.15
2-35 J 3 8 5
42.89
2.320 7159
43-66
2.336 5747
44-45
21
.290 0847
42.16
35 3959
42.90
.320 9778
43.67
.336 8414
44-47
22
.290 3377
42.18
35 6 533
42.91
.321 2399
43-68
337 1-083
44.48
23
.290 5908
42.19
.305 9109
42.93
.321 5020
43-69
337 3752
44-49
24
.290 8440
43.20
.306 1685
42.94
.321 7642
43-70
.337 6422
44-51
25
2.291 0972
42.21
2.306 4261
42.95
2.322 0265
43.72
2.337 9093
44.52
26
.291 3505
42.22
.306 6839
42.96
.322 2889
43-73
338 1765
44-53
27
.291 6039
42.24
.306 9417
42.98
.322 5513
43-75
338 4437
44-55
28
.291 8574
42.25
.307 1996
42.99
.322 8139
43-76
.338 7111
44-56
29
.292 1109
42.26
.307 4576
43.00
.323 0765
43-77
338 9785
44-58
30
2.292 3645
42.27
2.307 7157
43.02
2.323 3391
43-79
2.339 2 46o
44-59
31
.292 6182
42.29
.307 9738
43-03
.323 6019
43.80
339 5*35
4460
32
.292 8719
4230
.308 2320
43 -4
.323 8647
43.81
339 7812
44.62
33
.293 1258
4*. 3 *
.308 4903
43.05
.324 1277
43-83
34 49
44.63
34
293 3797
42.32
.308 7486
43-7
.324 3907
43.84
.340 3168
44.64
35
2.293 6336
42.33
2.309 0071
43.08
2.324 6537
43-85
2.340 5847
44.66
36
.293 8877
4*- 3 5
.309 2656
43.09
.324 9169
43-87
.340 8527
44 h 7
37
.294 1418
42.36
.309 5242
43.10
.325 1801
43-88
.341 1207
44.69
38
.294 3960
42.37
.309 7828
43.12
3*5 4434
43-89
.341 3889
44.70 (
39
.294 6503
42.38
.310 0416
43- ' 3
.325 7068
43-91
.341 6571
44-71 j
40
2.294 9046
42.40
2.310 3004
43-H
2.325 9703
43-9*
2-341 9255
44-73
41
.295 1590
42.41
3 10 5593
43- 1 5
.326 2339
43-93
.342 1939
44-74
42
295 4135
42.42
.310 8182
43- J 7
.326 4975
43-94
.342 4623
44-75
43
.295 6680
42.43
.311 0773
43.18
.326 7612
43.96
.342 7309
44-77
44
.295 9227
42.44
3" 33 6 4
43- J 9
.327 0250
43-97
342 9995
44.78
45
2.296 1774
42.46
2.311 5956
43.21
2.327 2889
43-98
2-343 2683
44.80
46
.296 4321
42.47
.311 8549
43.22
.327 5528
44.00
343 537i
44.81
47
.296 6870
42.48
.312 1142
43.23
.327 8168
44.01
.343 8060
44.82
48
.296 9419
42.49
.312 3736
43-24
.328 0809
44.02
344 0750
44.84
49
.297 1969
42.51
.312 6331
43.26
.328 3451
44.04
344 344
44.85
50
2.297 4520
42.52
2.312 8927
43.27
2.328 6094
44.05
2-344 6132
*-
51
.297 7071
42. 5 3
.313 1524
43.28
.328 8737
44.06
.344 8824
44.88
52
.297 9623
42.54
.313 4121
43.29
.329 1382
44.08
345 MI7
44.89
53
.298 2176
42-55
.313 6719
43-3 1
.329 4027
44.09
345 4211
44.91
54
.298 4730
42-57
3i3 93 l8
43-32
.329 6672
44.10
.345 6906
44.92
55
56
2.298 7284
.298 9839
42.58
42.59
2.314 1917
.314 4518
43-33
43-35
2.329 9319
.330 1967
44.12
44- ' 3
2.345 9601
.346 2298
44-93
44-95
57
.299 2395
42.60
.314 7119
43-36
33 4615
44.14
.346 4995
44.96
58
.299 4952
42.61
.314 9721
43-37
.330 7264
44.16
346 7693
44-97
59
.299 7509
42.63
.315 2323
43.3 8
33 99 J 4
44.17
347 0392
44-99
60
2.300 0067
42.64
2.315 4927
43.40
2.331 2564
44.18
2-347 3092
45.00
694
TABLE VI.
For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit.
V.
116
117
118
119
logM.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. 1".
0'
2.347 3092
45.00
2.363 6626
45.86
2.380 3290
46.74
1.397 3210
47.66
1
347 5792
45.02
3 6 3 9378
45.87
.380 6095
46.76
.397 6070
47.68
2
347 8494
45-3
.364 2131
45.88
.380 8901
46.77
397 8931
47.70
1 3
.348 1196
45.04
.364 4885
45.90
.381 1708
46.79
.398 1794
47-71
4
348 3 8 99
45.06
.364 7639
45.91
381 45*5
46.80
398 4657
47-73
5
2.348 6603
45-07
2-3 6 5 394
45-93
1.381 7324
46.82
2.398 7521
47-74
6
.348 9308
45-09
3 6 5 3^5
45-94
.382 0133
46.83
399 3 86
47.76
7
.349 2014
45.10
-365 5907
45.96
.382 2944
46.85
-399 3252
47-77
8
349 4720
45.11
.365 8665
45-97
.382 5755
46.86
399 6119
47-79
9
349 7428
45-13
.366 1423
45-99
.382 8567
46.88
399 8987
47.81
10
2350 0136
45.14
2.366 4183
46.00
2.383 1380
46.89
2.400 1856
47-82
11
35 2845
45.16
.366 6944
46.01
.3 8 3 4i94
46.91
.400 4725
47.84
12
35 5554
45-17
.366 9705
46.03
.383 7009
46.92
.400 7596
47.85
13
.350 8265
45 i 8
.367 2467
46.04
.383 9825
46.94
.401 0468
47.87
14
.351 0977
45.20
.367 5230
46.06
.384 2642
46.95
.401 3340
47.89
15
2351 3689
45.21
2,367 7994
46.07
2.384 5460
46.97
2.401 6214
47.90
16
.351 6402
45-^3
.368 0759
46.09
.384 8278
46.98
.401 9088
47.92
17
.351 9116
45.24
.368 3525
46.10
.385 1098
46.99
.402 1964
47-93
18
.352 1831
45-25
.368 6291
46.12
.385 3918
47.01
.402 4840
47-95
19
35* 4547
45.27
.368 9059
46.13
385 6739
47.03
.402 7718
47-97
20
2.352 7263
45-28
2.369 1827
46.15
2.385 9562
47.05
2.403 0596
47.98
21
.352 9981
45-3
.369 4596
46.16
.386 2385
47.06
43 3475
48.00
22
353 2699
45.31
.369 7367
46.18
.386 5209
47.08
43 6356
48.01
23
353 54 l8
45-33
.370 0138
46.19
.386 8034
47.09
403 9237
48.03
24
353 8138
45-34
.370 2909
46.21
.387 0860
47.11
.404 2119
48.04
25
2.354 8 59
45-35
2.370 5682
46.22
2.38-7 3687
47.12
2.404 5002
48.06
2G
354 358i
45-37
.370 8456
46.24
3 8 7 6514
47.14
.404 7886
48.08
27
354 6 33
45-38
.371 1230
46.25
-3 8 7 9343
47-15
45 O77i
48.09
28
.354 9027
45.40
.371 4006
46.26
.388 2173
47-*7
45 3657
48.11
29
355 '75 1
45-41
.371 6782
46.28
.388 5003
47.18
.405 6544
48.12
30
2-355 4476
45-42
2-37' 9559
46.29
2-388 7835
47.20
2.405 9432
4 8.H
31
.355 7202
45-44
.372 2337
46.31
.389 0667
47.21
.406 2321
48.16
32
355 9928
45-45
372 5 Ij6
46.32
389 35
47-23
.406 5211
48.17
33
.356 2656
45-47
-372 7 8 9 6
4 6 -34
3 8 9 6 335
47.24
.406 8102
48.19
34
356 53 8 5
45-48
.373 0677
46.35
389 9*70
47.26
47 0993
48.20
35
2.356 8114
45-5
2-373 3459
4 6 -37
2.390 2006
47.28
2.407 3886
48.22
36
357 8 44
45-51
373 624*
46.38
.390 4843
47.29
.407 6780
48.24
37
357 3575
45-52
373 9024
46.40
.390 7681
47-3 1
.407 9674
48.25
38
357 6307
45-54
.374 1809
46.41
.391 0519
47-32
.408 2570
4*'*l
39
357 9040
45-55
374 4594
46.43
391 3359
47-34
.408 5467
48.28
40
2.358 1773
45-57
2-374 73 8
46.44
2.391 6200
47-35
2.408 8364
48.30
41
.358 4508
45.58
375 Ol6 7
46.46
".392 9042
47-37
.409 1263
48.32
42
.358 7243
45.60
375 2955
46.47
.392 1884
47-38
.409 4162
48.33
43
.358 9979
45.61
375 5744
46.49
-392 4728
47.40
.409 7063
48.35
44
359 2716
45.62
375 8 533
46.50
.392 7572
47.41
.409 9964
48.37
45
2-359 5454
45.64
2.376 1324
46.51
*-393 4 J 7
47-43
2.410 2866
48.38
46
359 8l 93
45.65
.376 4115
46.53
393 3264
47-45
.410 5770
48.40
47
.360 0933
45.67
.376 6908
46.55
393 6l11
47-46
.410 8674
48.41 ,
48
.360 3673
45.68
.376 9701
46.56
393 8959
47.48
.411 1579
48.43
49
.360 6415
45.70
377 2495
4 6.. 5 8
.394 1808
47-49
.411 4486
48-45
50
2.360 9157
45-7 1
2-377 529
46.59
2-J94 465 8
47.51
2.411 7393
48.46
51
.361 1900
45-72
.377 8086
46.60
394 759
47.52
.412 0301
48.48
52
.361 4644
45-74
.378 0883
46.62
-395 3 61
47-54
.412 3210
48.49
53
54
.361 7389
.362 0134
45-75
45-77
.378 3681
378 6479
46.64
46.65
395 3214
395 6067
47-55
47-57
.412 0120
.412 9031
48.51
48.53
55
2.362 2881
45-78
2-378 9279
46.67
2-395 8922
47-59
2.413 1944
4 s' 5 *
56
.362 5628
45.80
379 2079
46.68
.396 1778
47.60
.413 4857
48-56
57
.362 8376
45.81
379 4 881
46.70
.396 4634
47.62
.413 7771
48.58
58
59
.363 1126
.363 3876
45.82
45.84
379 76 8 3
.380 0486
46.71
46-73
39 6 7492
397 0350
47- 6 3
47.65
.414 0686
.414 3602
48-59
48.61
60
2.363 6626
45-86
2.380 3290
46.74
2.397 3210
47.66
2.414 6519
48.62
695
TABLE VI.
For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit.
V.
120
121
122
123
logM.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. 1".
2.414 6519
48.62
M-3* 3356
49.62
2.450 3868
50.67
2.468 8205
5'-75
1
.414 9437
48.64
43* 6 334
49.64
.450 6908
50.68
.469 1311
51-77
2
.415 2356
48.66
.432 9313
49.66
.450 9950
50.70
.469 4418
51-79
3
.415 5276
48.67
433 2293
49.68
45 * 2992
50.72
.469 7526
5 l.8l
4
.415 8197
48.69
433 5*74
49.69
.451 6036
5-74
.470 0634
51.82
5
2.416 1119
48.71
2-433 8257
49.71
2.451 9081
5-75
2.470 3744
51.84
6
.416 4042
48.72
.434 1240
49-73
.452 2127
50.77
.470 6856
51.86
7
8
.416 6965
.416 9890
48.74
48.76
434 42*4
.434 7209
49-74
49.76
45* 5174
.452 8222
50.79
50.81
47 99^8
.471 3081
51.88
51.90
9
.417 2816
48.77
435 i95
49.78
453 1271
50.83
.471 6196
51.92
10
2.417 5743
48.79
2.435 3182
49.80
2.453 432i
50.84
2.471 9311
51.94
11
.417 8671
48.81
435 6171
49.81
453 7372
50.86
.472 2428
5*-95
12
.418 1600
48.82
435 916
49.83
454 04M
50.88
.472 5546
51-97
13
.418 4529
48.84
.436 2150
49.85
454 3477
50.90
.472 8665
51.99
14
.418 7460
48.85
.436 5141
49.86
454 6532
50.92
473 J 7 8 5
52.01
15
2.419 0392
48-87
1.436 8134
49.88
2.454 9587
5-93
2.473 49 6
52-03
16
4i9 3325
48.89
*437 "27
49.90
.455 2644
50.95
.473 8028
52-05
17
.419 6258
48.90
.437 4122
49.92
455 57i
50.97
474 "5*
52-07
18
.419 9193
48.92
437 7ii7
49-93
.455 8760
50.99
.474 4276
52.09
19
.420 2129
48.94
.438 0114
4995
.456 1820
51.00
474 742
52.10
20
4.420 5066
48.95
2.438 3111
49-97
2.456 4881
51.02
2.475 0529
52.12
21
.420 8003
48.97
.438 6110
49.98
.456 7943
51.04
475 3 6 57
52.14
22
.421 0942
48.99
.438 9109
50.00
.457 1006
51.06
.475 6786
52.16
23
.421 3882
49.00
.439 21 10
50.02
.457 4070
51.08
475 99 l6
52.18
24
.421 6822
49.02
439 5"2
50.04
457 7135
51.09
.476 3047
52.20
25
2.421 9764
49-3
2.439 8114
50.05
2.458 0201
51.11
2.476 6180
52.22
26
.422 2707
49.05
.440 1118
50.07
.458 3268
51.13
.476 9313
52.23
27
.422 5650
49.07
.440 4123
50.09
45 8 6 337
51.15
.477 2448
52.25
28
.422 8595
49.09
.440 7129
50.1 1
.458 9406
51.17
477 55 8 4
52.27
29
.423 1541
49.10
.441 0136
50.12
459 2 477
51.18
.477 8721
52.29
30
2.423 4488
49.12
2.441 3143
50.14
2.459 5548
51.20
2.478 1859
52-3 1
31
.423 7435
49.14
.441 6152
50.16
.459 8621
51.22
.478 4998
52-33
32
.424 0384
49.15
.441 9162
50.18
.460 1695
51.24
.478 8138
52-35
33
.424 3334
49.17
.442 2173
50.19
.460 4770
51.26
.479 1280
52 37
34
.424 6284
49.19
.442 5185
50.21
.460 7846
51.28
479 4422
52.39
35
2.424 9236
49.20
2.442 8199
50.23
2.461 0923
51.29
2.479 7566
52.40
36
.425 2189
49.22
.443 1213
50.24
.461 4001
5M 1
.480 0711
52.42
37
.425 5142
49.24
.443 4228
50.26
.461 7080
5'-33
.480 3857
52.44
38
.425 8097
49.25
.443 7244
50.28
.462 0161
51-35
.480 7004
52.46
39
.426 1053
49.27
.444 0261
50.30
.462 3242
5'-37
.481 0152
52-48
40
2.426 4010
49.29
2.444 3280
50.31
2.462 6325
51.38
2.481 3301
52.50
41
.426 6967
49-3
.444 6299
5-33
.462 9408
51.40
.481 6452
52.52
42
.426 9926
49.32
.444 9320
5-35
.463 2493
51.42
.481 9604
52-54
43
.427 1886
49-34
445 2 34i
5037
4 6 3 5579
5M4
.482 2756
52.56
44
.427 5847
49-35
445 53 6 4
50.38
.463 8666
51.46
.482 5910
52-58
45
2.427 8808
49-37
2.445 8387
50.40
2.464 1754
51.48
2.482 9065
52.59
46
.428 1771
49-39
.446 1412
50.42
.464 4843
51.49
.483 2222
52.61
47
.428 4735
49.40
.446 4437
50.44
4 6 4 7933
5*-5
4 8 3 5379
52.63
! 48
.428 7700
49.42
.446 7464
50.45
.465 1024
5'-53
483 8537
52.65
49
.429 0665
49-44
.447 0492
50.47
.465 4116
51-55
.484 1697
52.67
50
2.429 3632
49.46
4.447 3521
50.49
2.465 7210
51-57
2.484 4858
52.69
51
.429 6600
49-47
447 6 55i
50.51
.466 0305
51-59
.484 8020
52.71
52
.429 9569
49-49
,447 9582
5-53
.466 3400
51.60
.485 1183
52.73
53
.430 2539
49-5 1
.448 2614
50.54
.466 6497
51.62
485 4347
52-75
54
.430 5510
49.52
.448 5647
50.56
.466 9595
51.64
.485 7513
52-77
55
2.430 8482
49-54
2.448 8681
50.58
2.467 2694
51.66
2.486 0679
52.78
56
.431 1455
49.56
449 i7i 6
50.60
.467 5794
51.68
.486 3847
52.80
57
.431 4418
49-57
.449 4753
50.61
.467 8895
51.70
.486 7016
52.82
58
.431 7403
49-59
449 779
50.63
.468 1997
51.71
.487 0186
52.84
59
.432 0379
49.61
.450 0828
50.65
.468 5101
5'-73
487 3357
52.86
60
2.432 3356
49.62
2.450 3868
50.67
2.468 8205
51-75
2.487 6529 1 52.88
596
TABLE VI,
For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit
V.
124
125
126
127
log M.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. 1".
2.487 6529
52.88
2.506 9006
54.06
2.526 5813
55-29
2.546 7135
56-57
1
.487 9702
52.90
.507 2251
54.08
526 9131
55-31
547 0530
56-59
2
.488 2877
52.92
.507 5496
54.10
.527 2450
55-33
547 3926
56.61
3
.488 6053
52.94
.507 8742
54.12
527 577 1
55-35
547 7323
56-63
4
.488 9230
52.96
.508 1990
54.14
.527 9092
55-37
548 0722
56-65
5
2.489 2408
52.98
2.508 5239
54.16
2.528 2415
55-39
2.548 4122
56.68
6
.489 5587
53.00 .508 8489 54.18
528 5739
55-41
548 7523
56.70
7
.489 8767
53.02
.509 1741
54.20
.528 9065
55-43
.549 0926
56.72
8
.490 1949
53.03
59 4993
54-22
.529 2391
55-45
549 433
56.74
9
.490 5132
53-05
.509 8247
54.24
529 579
55.48
549 7735
56.76
10
2.490 8315
53.07
2.510 1502
54-26
2.529 9048
55-50
2.550 1141
56.79
11
491 1500
53-9
.510 4758
54.28
53 2379
55-52
55 4549
56.81
12
.491 4686
53.11
.510 8016
54-30
-53 57io
55-54
55 7958
56.83
13
.491 7874
53-13
.511 1274
54-32
53 9043
55-56
.551 1369
56.85
14
.492 1063
53-15
5 11 4534
54-34
.531 2378
55-58
55 1 478i
56.87
15
2.492 4252
53-17
2.511 7795
54.36
2-53 1 5713
55.60
2.551 8194
56.90
1C
.492 7443
53-19
.512 1057
54-38
.531 9050
55.62
.552 1608
56.92
17
493 6 35
53-21
.512 4321
54.40
.532 2388
55.64
.552 5024
56.94
18
.493 3828
53-^3
.512 7586
54-42
532 5727
55-67
552 8441
56.96
19
493 7023
53-^5
.513 0852
54-44
.532 9068
55-69
553 1859
56-98
20
2.494 0218
53.27
2.513 4119
54.46
2-533 2410
55-71
2-553 5279
57.01
21
494 3415
53-29
5i3 73 8 7
54.48
533 5753
55-73
553 8700
57.03
22
494 66l 3
53-31
.514 0657
54-50
533 997
55-75
.554 2122
57.05
23
.494 9812
53-33
.514 3927
54-52
534 2443
55-77
554 5546
57.07
24
495 3i2
53-35
.514 7199
54-54
534 579
55-79
554 8971
57.10
25
2.495 62 i3
53-37
2.515 0473
54.56
2.534 9138
55.81
2-555 2398
57-12
2G
.495 9416
53-39
5*5 3747
54-58
535 2487
55.84
555 5825
57.H
27
28
29
.496 2619
.496 5824
.496 9030
53-4i
53.42
53-44
.515 7023
.516 0300
.516 3578
54.60
54- 6 3
54-65
535 5838
535 9*9
53 6 2543
55.86
55-88
55.90
555 9254
.556 2685
.556 6116
57.16
57.18
57-21
30
2.497 2238
53-46
2.516 6857
54.67
2.536 5898
55-92
2-55 6 9549
57-23
31
.497 5446
53.48
.517 0138
54.69
.536 9254
55-94
557 2984
57-25
32
.497 8656
53-5
.517 3420
54-7i
.537 2611
55-96
557 6420
57*27
33
.498 1867
53-52
.517 6703
54-73
537 5970
55-98
557 9857
57-29
34
.498 5079
53-54
.517 9987
54-75
537 9329
56.01
558 3295
57.32
35
2.498 8292
53-5 6
2.518 3273.
54-77
2.538 2690
56.03
2.558 6735
57-34
36
.499 1506
53-58
.518 6559
54-79
.538 6052
56-05
559 OI 7 6
57.36
37
.499 4721
53.60
.518 9847
54-8i
.538 9416
56.07
559 3 6lg
57.38
38
499 793 8
53.62
5*9 3*37
54.83
539 2781
56.09
559 7062
57.41
39
.500 1156
53- 6 4
.519 6427
54.85
539 6i47
56.11
.560 0507
57-43
40
2.500 4375
53.66
2.519 9719
54.87
2-539 95H
56.13
2.560 3953
57-45
41
.500 7595
53.68
.520 3012
54.89
.540 2883
5 5- I5 >
.560 7401
57-47
42
.501 0817
53-70
.520 6306
54-9 1
.540 6253
56.18
.561 0850
57-50
43
.501 4039
53-72
.520 9601
54-93
.540 9625
56.20
.561 4301
57-52
44
.501 7263
53-74
.521 2898
54-95
.541 2997
56.22
.561 7753
57-54
45
46
47
2.502 0488
.502 3714
.502 6942
53-76
53-78
53.80
2.521 6196
.521 9495
.522 2795
54-97
54-99
55.02
2.541 6371
.541 9746
542 3*23
56.24
56.26
56.29
2.562 1206
.562 4660
.562 8116
57-56
57-59
57-6i
48
.503 0170
53.82
.522 6097
55-04
542 6500
56.31
563 '574
57-63
49
.503 3400
53-84
.522 9400
55.06
.542 9880
56.33
563 5032
57.65
50
2.503 6631
53.86
2.523 2704
55.08
2-543 3260
56.35
2.563 8492
57.68
51
.503 9863
53.88
.523 6009
55.10
543 6641
56-37
564 !953
57-70
52
53
54
.504 3096
504 6331
.504 9567
53-9
53.92
53-94
.523 9316
.524 2624
524 5933
55-12
55-14
55.16
.544 0024
544 349
544 6794
56.39
56.42
56.44
564 54i6
.564 8880
565 2345
57-72
57-74
57-77
55
56
2.505 2804
.505 6042
53-96
53.98
2.524 9243
525 2555
55.18
55.20
2.545 0181
545 3569
56.46
56.48
2.565 5812
.565 9280
57-79
57-81
57
58
59
.505 9282
.506 2522
.506 5763
54.00
54.02
54.04
-525 5867
.525 9181
.526 2497
55-22
55.24
55-26
545 6959
.546 0350
.546 3742
56.50
56-52
56-55
.566 2750
.566 6221
.566 9693
57.84
57-86
57-88
6O
2.506 9006
54.06
2.526 5813
55-29
2-546 7135
5 6 -57
2.567 3166
57.90
597
TABLE VI.
For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit.
V.
128
129
130
131
logM.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. 1".
0'
2.567 3166
57.90
2.588 4112
59-3
2.610 0188
60.75
2.632 1622
62.28
1
.567 6641
57-93
.588 7670
59-3*
.610 3834
60.78
.632 5360
62.30
2
.568 0117
57-95
.589 1230
59-35
.610 7481
60.80
.632 9099
62.33
3
.568 3595
57-97
.589 4792
59-37
.611 1130
60.83
.633 2839
62.35
4
.568 7074
57-99
.589 8355
59-39
.611 4781
60.85
.633 6581
62.38
5
6
2.569 0554
.569 4036
58.02
58.04
2.590 1919
.590 5485
59.42
59-44
2.611 8433
.612 2086
60.88
60.90
2.634 0325
.634 4070
62.41
62.43
7
.569 7519
58.06
.590 9052
59-47
.612 5741
60.93
634 78l7
62.46
8
.570 1004
58.09
.591 2620
59-49
.612 9397
60.95
.635 1565
62.48
9
.570 4490
58.11
.591 6190
59-51
.613 3055
60.98
635 5315
62.51
10
2.570 7977
58.13
2.591 9762
59-54
2.613 6715
6 1. oo
2.635 966
62.54
11
12
.571 1465
571 4955
58-15
58.18
59* 3335
.592 6909
59.56
59-58
.614 0376
.614 4038
61.03
61.05
.636 2819
.636 6573
62.56
62.59
13
.571 8447
58.20
593 4 8 5
59.61
.614 7702
61.08
.637 0329
62.61
14
.572 1939
58,22
.593 4062
.615 1368
61.10
.637 4087
62.64
15
*-57* 5434
58.25
2.593 7641
59.66
2.615 5035
61.13
2.637 7846
62.67
16
.572 8929
58.27
.594 1221
59.68
.615 8703
61.15
.638 1607
62.69
17
.573 2426
58.29
594 4803
59.70
.616 2373
61.18
638 5369
62.72
18
573 59*4
58.32
594 8386
59-73
.616 6045
61.20
638 9133
62.75
19
573 94*4
58.34
595 '97
59-75
.616 9718
61.23
.639 2899
62.77
20
2.574 2925
58.36
*-595 555 6
59-78
2.617 3392
61.25
2.639 6666
62.80
21
574 6427
58-38
595 9H3
59.80
.617 7068
61.28
.640 0435
62.82
22
23
574 993 1
575 3436
58.41
58.43
.596 *73*
.596 6322
59.82
59.85
.618 0746
.618 4425
61.30
61.33 .
.640 4205
.640 7977
62.85
62.88
24
575 6943
58.45
.596 9914
59.87
.618 8105
61.36
.641 1750
62.90
25
2.576 0451
58-48
2.597 3507
59-9
2.619 *787
61.38
2.641 5525
62.93
26
.576 3960
58.50
.597 7102
59.92
.619 5471
61.41
.641 9302
62.96
27
.576 7471
58.52
.598 0698
59-95
.619 9156
61.44
.642 3080
62.98
28
577 0983
58.55
.598 4295
59-97
.620 2843
61.46
.642 6860
63.01
29
577 4496 1 58.57
.598 7894
59-99
.620 6531
61.48
.643 0641
63.04
30
2.577 8011
58.59
2.599 J 494
60.02
2.621 0220
61.51
2.643 44*4
63.06
31
.578 1528
58.62
599 59 6
60.04
.621 3911
61.53
.643 8209
63.09
32
578 545
58.64
.599 8699
60.07
.621 7604
61.56
.644 1995
63.12
33
578 8564
58.66
.600 2304
60.09
.622 1298
61.58
.644 5783
63.14
34
.579 2085
58.69
.600 5910
60.12
.622 4994
61.61
.644 9572
63.17
35
2.579 5607
58.71
2.600 9518
60.14
2.622 8691
61.63
2 - 6 45 3363
63.19
36
579 9 I 3
58.73
.601 3127
60. 1 6
.623 2390
61.66
.645 7155
63.22
37
.580 2655
58.76
.601 6738
60.19
.623 6091
61.68
.646 0949
63.25
38
.580 6181
58-78
.602 0350
60.21
6*3 9793
61.71
.646 4745
63.27
39
.580 9708
58.80
.602 3963
60.24
.624 3496
61.74
.646 8542
63.30
40
2.581 3237
58.83
2.602 7578
60.26
2.624 7* 01
61.76
2.647 2341
6 3-33
41
.581 6768
58.85
.603 1195
60.29
.625 0907
61.79
.647 6142
63.35
42
43
.582 0299
.582 3832
58-87
58.90
.603 4813
.603 8432
60.31
60.34
.625 4615
.625 8325
61.81
61.84
.647 9944
.648 3748
63.38
63.41
44
.582 8267
58.92
.604 2053
60.36
.626 2036
61.86
648 7553
63.44
45
2.583 0903
58.94
2.604 5675
60.38
2.626 5748
61.89
2.649 J 36o
63.46
46
.583 4440
58.97
.604 9299
60.41
.626 9462
61.91
.649 5168
63.49
47
5 8 3 7979
58.99
.605 2924
60.43
.627 3178
61.94
.649 8978
63.52
48
.584 1519
59.01
.605 6551
60.46
.627 6895
61.97
.650 2790
63.54
49
.584 5061
59.04
.606 0179
60.48
.628 0614
61.99
.650 6603
50
2.584 8604
59.06
2.606 3809
60.51
2.628 4334
62.02
2.651 0418
63.60
51
.585 2148
59.09
.606 7440
60.53
.628 8056
62.04
.651 4235
63.62
1 52
585 5 6 94
59.11
.607 1073
60.56
.629 1780
62.07
.651 8053
63-65
53
.585 9241
.607 4707
60.58
6*9 555
62.09
.652 1873
63.68
54
.586 2790
59.16
607 8343
60.61
.629 9231
62.12
.652 5695
63.70
55
2.586 6340
59.18
2.608 1980
60.63
2.630 2959
62.15
2.652 9518
6 1-73
56
.586 9891
59.20
.608 5618
60.66
.630 6689
62.17
653 334*
6j.70
57
587 3444
59-*3
.608 9258
60.68
.631 0420
62.20
.653 7168
63.79
58
.587 6999
59-*5
.609 2901
60.70
.631 4152
62.22
.654 0996
63.81
59
-588 0555
59-*7
.609 6544
60.73
.631 7887
62.25
.654 4826
63.84
60
2.588 4112
59-3
2.610 0188
60.75
2.632 i6i2
62.28
2.654 8657
63.87
TABLE VI.
For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit.
V.
132
133
134
135
log M.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. 1".
o
.654 8657
63.87
2.678 1547
65-53
2.702 0562
67.27
2.726 5990
69.09
1
.655 2490
63.89
.678 5480
65.56
.702 4600
67.30
.727 0137
69.12
2
3
.655 6324
.650 oioo
63.92
63-95
.678 9414
679 335
65.59
65.61
.702 8638
.703 2679
67-33
67.36
.727 4285
7^7 84^5
69.15
69.19
4
.656 3998
63.97
.679 7288
65.64
.703 6721
67-39
.728 2587
69.22
5
1.656 7837
64.00
2.680 1227
65.67
2.704 0766
67.42
2.728 6741
69.25
6
.657 1678
64.03
.680 5168
65.70
.704 4812
67.45
.729 0897
69.28
7
8
.657 5521
.657 9365
64.06
64.08
.680 9111
.681 3056
65.73
65.76
.704 8860
.705 2909
67.48
67.51
.729 5055
.729 9215
69.31
69-34
9
.658 3211
64.11
.681 7002
65-79
.705 6961
67.54
.730 3376
69.37
10
i 11
2.658 7058
.659 0907
64.14
64.17
2.682 0950
.682 4900
65.81
65.84
2.706 1014
.706 5069
67.57
67.60
2-73 7539
.731 1705
69.40
69.44
12
.659 4758
64.19
.682 8851
65.87
.706 9126
67-63
.731 5872
69.47
13
.659 8611
64.22
.683 2804
65.90
.707 3184
67.66
.732 0041
69.50
14
.660 2465
64.25
.683 6759
65.93
.707 7244
67.69
.732 4212
69-53
15
2.660 6320
64.28
2.684 0716
65.96
2.708 1307
67.72
2.732 8385
69.56
16
.661 0178
64.30
.684 4674
65.99
.708 5371
67.75
733 *559
69.59
17
.661 4037
6 4-33
.684 8634
66.01
.708 9436
67.78
733 6736
69.62
18
.661 7897
64.36
.685 2596
66.04
.709 3504
67.81
734 0914
69.66
19
.662 1760
64.38
.685 6559
66.07
709 7573
67.84
734 594
69.69
20
2.662 5623
64.41
2.686 0524
66.10
1.710 1645
67.87
2.734 9277
69.72
21
.662 9489
64.44
.686 4491
66.13
.710 5718
67.90
735 346i
69.75
22
23
663 3356
.663 7225
64.47
64.49
.686 8460
.687 2430
66.16
66.19
.710 9792
.711 3869
67.93
67.96
735 7647
73 6 1835
69.78
69.81
24
.664 1096
64.52
.687 6402
66.22
.711 7947
67.99
.736 6025
69.85
25
2.664 4968
64.55
2.688 0376
66.25
2.712 2028
68.02
2.737 0216
69.88
26
.664 8842
64.57
.688 4352
66.27
.712 6110
68.05
737 44io
69.91
27
.665 2717
64.60
.688 8329
66.30
.713 0194
68.08
737 8605
69.94
28
.665 6594
64.63
.689 2308
66.33
.713 4279
68.11
.738 2803
69.97
29
.666 0473
64.66
.689 6289
66.36
.713 8367
68.14
738 7002
70.00
30
2.666 4354
64.69
2.690 0272
66.39
2.714 2456
68.17
2.739 I 2 3
70.04
31
.666 8236
64.72
.690 4256
66.42
.714 6547
68.20
739 54o6
70.07
32
.667 2120
64.74
.690 8242
66.45
.715 0640
68.23
739 961*
70.10
33
.667 6005
64.77
.691 2230
66.48
.715 4735
68.26
.740 3819
70.13
34
.667 9892
64.80
.691 6219
66.51
.715 8832
68.29
.740 8027
70.16
35
2.668 3781
64.83
2.692 0210
66.54
2.716 2930
68.32
2.741 2238
70.20
36
.668 7672
64.86
.692 4203
66.56
.716 7031
68.35
.741 6451
70.23
37
.669 1564
64.88
.692 8198
66.59
77 "33
68.38
.742 0666
70.26
38
.669 5457
64.91
.693 2194
66.62
.717 5*37
68.41
.742 4882
70.29
39
-669 9353
64.94
.693 6193
66.65
.717 9342
68.44
.742 9101
70.32
40
2.670 3250
64.97
2.694 0193
66.68
2.718 3450
68.48
2.743 33* 1
70.36
41
.670 7149
65.00
.694 4194
66.71
.718 7560
68.51
743 7543
70-39
42
.671 1050
65.02
.694 8198
66.74
.719 1671
68.54
744 1768
70.42
43
44
.671 4952
.671 8856
65.05
65.08
.695 2203
,695 6210
66.77
66.80
.719 5784
.719 9899
68.57
68.60
.744 5994
.745 0222
70.45
70.48
45
2.672 2761
65.11
2.696 O2I9
66.83
2.720 .4016
68.63
*-745 44p
70.52
46
.672 6668
65.13
.696 4229
66.86
.720 8135
68.66
.745 8684
70.55
47
.673 0577
65.16
.696 8242
66.89
.721 2255
68.69
.746 2918
70.58
48
.673 4488
65.19
.697 2256
66.92
.721 6377
68.72
.746 7154
70.61
49
.673 8400
65.22
.697 6272
66.95
.722 0502
68.75
747 '39 1
70.65 .
5O
2.674 2314
65.25
2.698 0289
66.97
2.722 4628
68.78
2.747 5631
70.68
51
.674 6230
65.28
.698 4308
67.00
.722 8756
68. 81
747 9873
70.71
52
i 53
54
..675 0147
.675 4066
.675 7987
65.30
65-33
65-36
.698 8330
.699 2353
.699 6377
67.03
67.06
67.09
.723 2885
.723 7017
.724 1150
68.84
68.88
68.91
.748 4116
.748 8362
.749 2609
70.74
70.78
70.81
55
2.676 1909
65.39
2.700 0404
67.12
2.724 5286
68.94
2.749 6859
70.84
56
57
.676 5833
.676 9759
65.42
65.44
.700 4432
.700 8462
67.15
67.18
.724 9423
.725 3562
68.97
69.00
.750 mo
75 53 6 4
70.87
70.90
58
59
.677 3687
.677 7616
6^.47
65.50
.701 2494
.701 6527
67.21
67.24
7*5 773
.726 1846
69.03
69.00
.750 9619
.751 3876
70.94
70.97
60
2.678 1547
65-53
2.702 0562
67.27 |2.726 5990
69.09
2.751 8135
71.00
599
TABLE VI.
For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit.
V.
136
137
138
139
log Hi
Diff. 1".
logM.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. 1".
O'
2.751 8135
71.00
2.777 7322
73.01
2.804 3 8 95
75.11
2.831 8224
77-32
1
75* 2396
71.03
.778 1703
73-4
.804 8403
75-14
.832 2864
77-35
2
.752 6659
71.07
.778 6087
73.07
.805 2912
75.18
.832 7506
77-39
3
753 925
71.10
779 472
73-"
.805 7424
75.21
.833 2151
77-43
4
753 519*
7LI3
779 4 8 59
73- J 4
.806 1938
75-25
.833 6798
77-47
5
2.753 94 6 i
71.17
2.779 9249
73.18
2.806 6454
75.29
2.834 1447
77.50
6
754 3732
71.20
.780 3641
73.21
.807 0973
75-32
.834 6098
77-54
7
754 *4
71.23
.780 8034
73-24
.807 5493
75-36
8 35 0752
77-5 8
8
755 2279
71.26
.781 2430
73.28
.808 0016
75.40
.835 5408
77.62
9
755 6 55 6
71.30
.781 6828
73-3 1
.808 4541
75-43
.836 0066
77.66
10
2.756 0835
71-33
2.782 1228
73-35
2.808 9068
75-47
2.836 4727
77.69
11
.756 5116
71.36
.782 5630
73.38
.809 3597
75-5
.836 9390
77-73
12
75 6 9399
71.40
.783 0034
73.42
.809 8128
75-54
8 37 4055
77-77
13
757 3 68 3
71.43
.783 4440
73-45
.810 2662
75-5 8
.837 8722
77.81
14
757 797
71.46
.783 8848
7349
.810 7197
75.61
.838 3392
77- 8 5
15
2.758 2259
71.49
2.784 3258
73-53
2.811 1735
75-65
2.838 8064
77.89
16
.758 6549
7L53
.784 7671
73.56
.811 6275
75.69
.839 2738
77.92
17
.759 0842
71.56
.785 2085
73-59
.812 0817
75-72
.839 7414
77-96
18
759 5*37
71-59
.785 6502
73-63
.812 5362
75.76
.840 2093
78.00
19
759 9433
71.63
.786 0920
73.66
.812 9908
75-79
.840 6774
78.04
20
2.760 3732
71.66
2.786 5341
73-7
2.813 4457
75- 8 3
2.841 1458
78.08
21
22
.760 8032
.761 2335
71.69
71.73
.786 9764
.787 4189
73-73
73.76
.813 9008
.814 3561
75- 8 7
75.90
.841 6144
.842 0832
78.0
78.15
23
.761 6639
71.76
.787 8615
73.80
.814 8117
75-94
.842 5522
78.19
24
.762 0946
71.79
.788 3044
73.83
.815 2674
75.98
.843 0215
78.23
25
2.762 5255
-71.83
2.788 7476
73.87
2.815 7234
76.01
2.843 499
78.27
26
.762 9565
71.86
.789 1909
73-9
.816 1796
76.05
.843 9607
78.31
27
.763 3878
71.89
.789 6344
73-94
.816 6360
76.09
.844 4306
7 8 -35
28
.763 8192
71.93
.790 0781
73-97
.817 0927
76.12
.844 9008
78.38
29
.764 2509
71.96
.790 5221
74.01
.817 5495
76.16
.845 3712
78.42
30
2.764 6827
71.99
2.790 9662
74.04
2.818 0066
76.20
2.845 8 4*9
78.46
31
.765 1148
72.03
.791 4106
74.08
.818 4639
76.23
.846 3128
78.50
32
.765 5470
72.06
.791 8552
74.11
.818 9214
76.27
.846 7839
78.54
33
765 9795
72.09
.792 3000
74-'5
.819 3792
76.31
8 47 2553
78.58
34
.766 4121
72.13
.792 7450
74.18
.819 8371
76.34
.847 7268
78.62
35
2.766 8450
72.16
2.793 1 9*
74.22
2.820 2953
76.38
2.848 1986
78.66
36
.767 2781
72.19
.793 6356
74.25
.820 7537
76.42
.848 6707
78.69
37
.767 7113
72.23
.794 0813
74.29
.821 2123
7646
.849 1430
7 8 -73
38
.768 1448
72 26
794 5271
74.32
.821 6712
76.49
.849 6155
7 8 -77
39
.768 5784
72.29
794 9731
74- 3 6
.822 1302
76.53
.850 0882
78.81
40
2.769 0123
7^-33
2-795 4i94
74.40
2.822 5895
76-57
2.850 5612
7 !-! 5
41
.769 4464
72.36
795 86 59
74-43
.823 0491
76.60
.851 0344
78.89
42
.769 8806
72-39
.796 3126
74-47
.823 5088
76.64
.851 5079
78.93
1 43
.770 3151
72.43
.796 7595
74.50
.823 9688
76.68
.851 9816
78. 97
1 44
.770 7498
72.46
.797 2066
74-54
.824 4289
76.72
8 5 2 4555
79.01
45
2.771 1846
72.50
2-797 6539
74-5 8
2.824 88 94
76-75
2.852 9297
79.05
46
.771 6197
7^-53
.798 1015
74.61
.825 3500
76.79
.853 4041
79.08
47
.772 0550
72.56
.798 5492
74.64
.825 8108
7683
.853 8787
79.12
48
.772 4905
72.60
.798 9972
74.68
.826 2719
76.87
8 54 3535
79.16
49
.7-2 9262
72.63
799 4454
74-7
.826 7332
76.90
.854 8286
7920
: so
2.7-3 3621
72.67
2-799 8 93 8
74-75
2.827 1947
76.94
2.855 3 4o
79.24
51
773 7982
72.70
.800 3424
74-79
.827 6565
7698
8 55 7795
79.28
52
.774 2344
7 2 -73
.800 7912
74.82
.828 1185
77.01
8 5 6 2553 i 79.32
53
.774 6709
72.77
.801 2402
74.86
.828 5807
77-5
.856 73x4
79-3&
54
775 i77
72.80
.801 6895
74.89
.829 0431
77.09
.857 2077
79.40
55
2.775 544 6
72.84
2.802 1390
74-93
2.829 5058
77.13
2.857 6842
79-44
56
775 9 8l 7
72.87
.802 5886
74-96
.829 9686
77.16
.858 1609
79.48
57
.776 4190
72.90
.803 0385
75.00
.830 4317
77.20
.858 6379
79.52
58
59
.776 8565
.777 2942
72.94
72.97
.803 4886
.803 9390
75.04
75.08
.830 8951
.831 3586
77.24
77.28
.859 1151
.859 5926
79.56
79.60
60
2.777 7322
73.01
2.804 3895
75.11
2.831 8224
77-32
2.860 0703
79.64
600
TABLE VI.
]? or finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit.
V.
140
141
142
143
logM.
Diff. 1".
logM.
Diff. 1".
log M.
Diff. 1".
logM.
Diff. 1".
O'
2.860 0703
79.64
2.889 1754
82.08
2.919 1831
84.65
2.950 1420
87-37
1
.860 5482
79.68
.889 6680
82.12
.919 6911
84.70
.950 6664
87-41
2
.861 0264
79.72
.890 1609
82.16
.920 1994
84.74
.951 1910
87.46
3
.861 5048
79.76
.890 6540
82.20
.920 7080
84.78
95 1 7159
87.50
4
.861 9835
79.80
.891 1473
82.25
.921 2169
84.83
.952 2411
87-55
5
2.862 4624
79.84
2.891 6409
82.29
2.921 7260
84.87
2.952 7665
87.60
6
.862 9415
79.88
.892 1348
82.33
.922 2353
84.92
953 2 9*3
87.65
7
.863 4209
79.92
.892 6289
82.37
.922 7450
84.96
953 8183
87.69
8
.863 9005
79.96
893 I2 33
82.41
.923 2549
85.01
.954 3446
87.74
9
.864 3803
80.00
893 6l 79
82.46
.923 7650
85.05
954 8711
87.79
10
2.864 8604
80.04
2.894 1127
82.50
2.924 2755
85.10
2.955 398o
87.81
11
.865 3408
80.08
.894 6078
82.54
.924 7862
85.14
955 9*51
87.88
12
.865 8213
80.12
.895 1032
82.58
.925 2972
85.18
.956 4525
87.93
13
.866 3021
80.16
.895 5989
82.63
.925 8084
85.23
.956 9802
87.97
14
.866 7832
80.20
.896 0948
82.67
.926 3199
85.27
-957 58^
88.02
15
2.867 2645
80.34
2.896 5909
82.71
2.926 8317
85.32
2.958 0365
88.07
16
.867 7460
80.28
.897 0873
82.75
.927 3437
85.36
.958 5651
88.11
17
.868 2278
80.32
.897 5839
82.79
.927 8560
85.41
959 939
88.16
18
.868 7098
80.36
.898 0808
82.84
.928 3686
85.45
.959 6230
88.21
19
.869 1921
80.40
.898 5780
82.88
.928 8814
85.50
.960 1524
88.26
20
2.869 6746
80.44
2.899 0754
82.92
2.929 3945
85.54
2.960 6821
88.30
21
.870 1573
80.48
.899 5730
82.96
.929 9079
85.59
.961 2I2O
88-35
22
.870 6403
80.52
.900 0709
83.01
.930 4216
85.61
.961 7423
88.40
23
.871 1235
80.56
.900 5691
83.05
93 9355
85.68
.962 2728
88.45
24
.871 6070
80.60
.901 0675
83.09
.931 4497
85.72
.962 7036
88.49
25
2.872 0907
80.64
2.901 5662
83.13
2.931 9641
85.77
*-9 6 3 3347
88.54
26
.872 5747
80.68
.902 0651
83.18
.932 4788
85.81
.963 8661
88 59
27
.873 0589
80.72
.902 5643
83.22
.932 9938
85.86
.964 3978
88.64
28
873 5433
80.76
.903 0638
83.26
933 S^ 1
85.91
.964 9297
88.68
29
.874 0280
80.80
.903 5635
83.31
934 M7
85.95
.965 4620
88.73
30
2.874 5129
80.84
2.904 0635
83-35
2.934 5405
85.99
2.965 9945
88.78
31
.874 9981
80.88
.904 5637
83-39
935 5 6 5
86.04
.966 5273
88.83
32
875 4835
80.92
.905 0642
83-43
935 57^9
86.08
.967 0604
88.87
33
.875 9692
80.96
.905 5649
83.48
.936 0895
86.13
.967 5938
88.92
34
.876 4551
81.01
.906 0659
83.52
.936 6064
86.17
.968 1275
88.97
35
36
2.876 9413
.877 4277
81.05
81.09
2.906 5672
.907 0687
83.56
83.61
2.937 1236
937 6410
86.22
86.26
2.968 6615
.969 1957
89.02
89.07
37
.877 9143
81.13
.907 5704
83.65
.938 1587
86.31
.969 7303
89.12
38
.878 4012
81.17
.908 0725
83.69
.938 6767
86.35
.970 2651
89.17
39
.878 8883
81.21
.908 5748
83.74
.939 1950
86.40
.970 8002
89.21
40
*- 8 79 3757
81.25
2.909 0773
83.78
2.939 7135
86.45
2.971 3356
89.26
41
.879 8633
81.29
.909 5801
83.82
.940 2323
86.49
971 8713
89.31
42
.880 3512
81.33
.910 0832
83.87
.940 7514
86.54
.972 4073
89.36
43
.880 8393
81.37
.910 5865
83.91
.941 2708
86.58
.972 9436
89.40
44
.881 3277
81.42
.911 0901
83-95
.941 7904
86.63
973 4801
89.45
45
2.881 8163
81.46
2.911 5940
8399
2.942 1103
86.67
2.974 0170
89.50
46
.882 3052
81.50
.912 0981
84.04
.942 8305
86.72
974 5541
89.55
47
.882 7943
81.54
.912 6024
84.08
943 35io
86.77
975 9 l6
89.60
48
.883 2837
81.58
.913 1070
84.13
943 8717
86.81
975 6293
89.65
49
883 7733
81.62
.913 6119
84.17
944 39 2 7
86.86
.976 1673
89.69
50
2.884 2631
81.66
2.914 1171
84.22
2.944 9140
86.90
2.976 7056
89.74
51
.884 7532
81.70
.914 6225
84.26
945 4355
86.95
.977 2442
89.79
52
53
.885 2436
.885 7342
81.75
81.79
.915 1282
.915 6341
84.30
84-34
945 9574
.946 4795
87.00
87.04
977 7831
.978 3223
a a o
89.84
89.89
54
.886 2251
81.83
.916 1403
84.39
947 OOI 9
87.09
.978 8618
89.94
55
2.886 7162
81.87
2.916 6468
84.43
2.947 5245
87-13
1.979 4015
89.99
56
57
58
59
.887 2075
.887 6991
.888 1910
.888 6831
81.91
81.95
81.99
82.04
.917 1535
.917 6605
.918 1678
.918 6753
84.48
84.52
84.56
84.61
.948 0475
.948 5707
.949 0942
949 6180
87.18
87.23
87.27
8 7 . 3 z
979 94i6
.980 4820
.981 1226
.981 6636
90.01
90.08
90.13
90.18
60
2.889 1754
82.08
2.919 1831
84.65
2.950 1420
87-37
2.982 1048
90.*^
601
TABLE VI.
Far finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit.
V.
144
145
146
147
logM.
Diff. 1".
logM.
Diff. 1".
log M.
Diff. 1".
log M.
Diff. 1".
0'
2.982 1048
90.23
3.015 1281
93.26
3.049 2733
96.47
3.084 6070
99.87
1
.982 6463
90.28
.015 6878
93-3 1
.049 8522
96.52
.085 2064
99.92
2
.983 1882
9-33
.016 2478
93.36
.050 4315
96.58
.085 8061
99.98
3
.983 7303
qo. 3 S
.016 8082
93.42
.051 on 2
96-63
.086 4062
100.04
4
.984 2727
90.43
.017 3688
93-47
.051 5911
96.69
.087 0066
100.10 1
5
2.984 8154
90.48
3.017 9298
93.52
3.052 1714
96.74
3.087 6073
ioo. 16
6
.985 3584
9-53
.018 4911
93-57
.052 7520
96.80
.088 2085
100.22 '
7
.985 9017
90.58
.019 0526
93.62
.053 3329
96.85
.088 8099
100.28
8
.986 4453
90.63
.019 6145
93.68
.053 9142
96.91
.089 4118
100.33
9
.986 9892
90.67
.020 1768
93-73
54 4959
96.96
.090 0140
100.39
1O
i-9 8 7 5334
90.72
3.020 7393
93-7 8
3.055 0778
97.01
3.090 6165
100.45
11
.988 0779
90.77
.021 3021
93.83
.055 6601
97.07
.091 2194
100.51
12
.988 6227
90.82
.O2I 8653
93.89
.056 2427
97.I3
.091 8226
100.57
13
.989 1678
90.87
.022 4288
93-94
.056 8256
97.19
.092 4262
100.63
14
.989 7132
90.92
.022 9926
93-99
.057 4089
97.24
.093 0302
100.69
15
2.990 2589
90.97
3.023 5567
94.04
3.057 9925
97.30
3.093 6345
100.75
16
.990 8049
91.02
.024 121 I
94.10
.0.58 5765
97-35
.094 2392
I00.8I
17
.991 3512
91.07
.024 6859
94.15
.059 1608
97.41
.094 8442
100.87
18
.991 8977
91.12
.025 2509
94.20
.059 7454
97-47
.095 4496
100.93
19
.992 4446
91.17
.025 8163
94.26
.060 3304
97.52
.096 0553
100.98
20
A. 992 9918
91.22
3.026 3820
94- 3 1
3.060 9157
97.58
3.096 6614
101.04
21
993 5393
91.27
.026 9480
94-36
.061 5013
97.63
.097 2678
IOI.IO
22
.994 0871
91.32
.027 5143
94.41
.062 0873
97.69
.097 8746
101.16
23
994 6351
9'-37
.O28 o8lO
94-47
.062 6736
97-75
.098 4818
101.22
24
995 I& 35
91.42
.028 6479
94.52
.063 2602
97.80
.099 0893
101.28
25
1-995 73"
91.47
3.029 2152
94-57
3.063 8472
97.86
3.099 6972
101.34
26
.996 2812
91.52
.029 7828
94-63
.064 4345
97.91
.100 3054
101.40
27
.996 8305
91-57
.030 3507
94.68
.065 O222
97-97
.100 9140
101.46
28
997 3 801
91.62
.030 9190
94-73
.065 6101
98.03
.101 5230
101.52
29
997 93
91.67
.031 4875
94-79
.066 1985
98.08
.102 1323
101.58
30
31
2.998 4802
999 37
91.72
91.77
3.032 0564
.032 6256
94.84
94.89
3.066 7872
.067 3762
98.14
98.20
3.102 7420
.103 352O
101.64
101.70
32
999 5 8l 5
91.82
.033 1951
94-94
.067 9655
98.25
.103 9624
101.76
33
3.000 1326
91.87
.033 7650
95.00
.068 5552
98.31
.104 5732
101.82
34
.000 6840
91.93
034 3351
95.05
.069 1453
98.37
.105 1843
101.88
35
3.001 2357
91.98
3.034 9056
95.11
3.069 7357
98.42
3- 10 5 795 8
101.94
36
.001 7877
92.03
.035 4764
95.16
.070 3264
98.48
.106 4076
102.00
37
.002 3400
92.08
.036 0475
95.22
.070 9174
98.54
.107 0198
IO2.O7
38
.002 8926
92.13
.036 6190
95.27
.071 5088
98.60
.107 6324
102.13
39
.003 4456
92.18
.037 1908
95-3 2
.072 1006
98.65
.108 2454
102.19
40
3.003 9988
92.23
3.037 7629
95.38
3.072 6927
98.71
3.108 8587
102.25
41
.004 5523
92.28
3 8 3353
95-43
.073 2851
98.77
.109 4723
102.31
42
.005 1062
9L33
.038 9080
95.48
.073 8779
98.82
.no 0864
102.37
43
.005 6603
92.38
.039 4811
95-54
.074 4710
98.88
.no 7008
102.43
44
.006 2148
92.44
.040 0545
95.60
.075 0645
98.94
"' 3*55
102.49
45
3.006 7696
92.49
3.040 6282
95-65
3.075 6583
99.00
3.111 9306
102-55
46
47
.007 3246
.007 8800
92.54
92.59
.041 2023
.041 7767
95.70
95.76
.07* 2524
.076 8469
99.05
99.11
.112 5461
.113 l620
102.61
102.67
48
.ooS 4357
92.64
.042 3514
95.81
.077 4418
99.17
.113 7782
102.73
49
.008 9917
92.69
.042 9264
95.86
.078 0370
99-13
.114 3948
102.80
50
3.009 5480
92.74
3.043 5017
95-91
3.078 6325
99.28
3.115 0118
102.86
51
.010 1046
92.79
.044 0774
95-97
.079 2284
99-34
.115 6291
102.92
52
.010 6615
92.85
.044 6534
96.01
.079 8246
99.40
.116 2468
102.98
53
.on 2188
92.90
.045 2297
96.08
.080 4212
99.46
.116 8649
103.04
54
.011 7763
92.95
.045 8064
96.14
.081 0181
99.52
.117 4833
103.10
55
3.012 3342
93.00
3.046 3834
96.19
3.081 6154
99-57
3.118 1022
103.16
56
.012 8923
93-05
.046 9607
96.25
.082 2130
99.63
.118 7213
103.23
57
.013 4508
93.10
47 53 8 3
96.30
.082 8no
99.69
.119 3409
103.29
58
.014 0096
93.16
.048 1163
96.36
.083 4093
99-75
.119 9608
103.35
59
.014 5687
93.21
.048 6946
96.41
.084 0080
99.81
.120 5811
103.41
60
3.015 I28l
93.26
3.049 2733
96.47
3.084 6070
99.87
3.I2I 20l8
103.48
602
TABLE VI.
For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit.
V.
148
149
150
151
logM.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. 1".
0'
.121 20l8
103.48
.159 1767
107.31
.198 4984
111.41
.239 3820
115.77
1
.121 8228
103.54
.159 7808
107.38
.199 1671
111.48
.240 0768
115.85
2
3
.122 4442
.127 OOOO
103.60
103.66
.160 4253
.161 0702
107.45
107.51
.199 8361
.200 5056
" X -SS
111.62
.240 7722
.241 4680
115.92
116.00
4
.123 6882
103.72
.161 7154
107.58
.201 1755
111.69
.242 1642
116.08
5
.124 3107
103.79
.162 3611
107.65
.201 8459
111.76
3.242 8608
116.15
6
.124 9336
103.85
.163 0072
107.71
.202 5166
111.83
.243 5580
116.23
7
.125 5569
103.91
.163 6536
107.78
.203 1878
111.90
.244 2556
Il6.70
8
9
.126 1805
.126 8045
103.97
104.04
.164 3005
.164 9478
107.85
107.91
.203 8594
.204 5315
111.97
112.04
.244 9536
.245 6521
116.38
116.45
10
11
12
.127 4289
.128 0537
.128 6789
104.10
104.16
104.22
3-i 6 5 5955
.166 2435
.166 8920
107.98
108.04
108.11
.205 2040
.205 8769
.206 5502
112. II
112.18
112.26
3.246 3511
.247 0505
.247 7503
116.53
116.61
116.68
13
.129 3044
104.29
.167 5409
108.18
.207 2239
112.33
.248 4507
116.76
14
.129 9303
104.35
.168 1901
108.25
.207 8981
1 1 2.40
.249 1515
116.84
15
.130 5566
104.41
3.168 8398
108.31
3.208 5727
112.47
3.249 8527
116.91
16
.131 1833
104.48
.169 4899
108.38
.209 2478
112.54
.250 5544
116.99
17
.131 8103
104.54
.170 1404
108.45
.209 9232
112.61
.251 2566
117.07
18
19
.132 4377
133 6 55
104.60
104.67
.170 7913
.171 4426
108.51
108.58
.210 5991
.211 2755
112.69
112.76
.251 9592
.252 6623
117.14
117.22
20
21
3.133 6937
.134 3223
104.73
104.79
3.172 0942
.172 7463
108.65
108.72
3-2II 9522
.212 6294
112.83
112.90
3.253 3658
.254 0698
117.30
"7-37
22
.134 9512
104.86
.173 3988
108.78
.213 3070
112.97
*54 7743
117.45
23
24
135 5o5
.136 2IO2
104.92
104.98
.174 0517
.174 7051
108.85
108.92
.213 9851
.214 6636
113.05
113.12
.255 4792
.256 1846
117.53
117.60
25
3.136 8403
105.05
3-J75 3588
108.99
3.215 3425
113.19
3.256 8905
117.68
26
.137 4708
105.11
.176 0129
109.06
.2l6 0219
113.26
.257 5968
117.76
27
.138 1016
105.17
.176 6674
109.12
.2l6 7017
"3-34
.258 3036
117.84
28
29
.138 7329
.139 3645
105.24
105.30
.177 3224
.177 9777
109.19
109.26
.217 3819
.218 0626
113.41
113-48
.259 0109
.259 7186
117.91
117.99
30
3.139 9965
105.36
S-^ 8 6 335
109.33
3.218 7437
"3-55
3.260 4268
118.07
31
.140 6289
i5-43
.179 2897
109.40
.219 4252
113.63
.261 1354
118.15
32
.141 2616
105.49
.179 9462
109.46
.220 1072
113.70
.261 8446
118.23
33
34
.141 8948
.142 5283
105.55
105.62
.180 6032
.181 2606
i9-53
109.60
.220 7896
.221 4724
"3-77
113.84
.262 5542
.263 2642
118.30
118.38
35
3.143 1622
105.68
3.181 9184
109.67
3.222 1557
113.92
3.263 9747
118.46
36
143 79 6 5
105.75
.182 5766
109.74
.222 8395
113.99
.264 6857
"8.54
37
38
.144 4312
.145 0663
105.81
105.87
183 2353
.183 8943
109.81
109.87
.223 5276
.224 2082
114.06
114.14
265 3972
.266 1091
118.62
118.70
39
.145 7018
105.94
.184 5538
109.94
.224 8933
114.21
.266 8216
118.77
40
41
3.146 3376
.146 9739
106.00
106.07
3.185 2136
.185 8739
IIO.OI
110.08
3.225 5788
.226 2647
114.28
114.36
3-^7 5345
.268 2478
118.85
118.93
42
.147 6105
106.14
.186 5346
110.15
.226 9511
"4-43
.268 9616
119.01
43
.148 2475
106.20
.187 1957
110.22
.227 6379
114.51
.269 6759
119.09
44
.148 8849
106.27
.187 8572
110.29
.228 3252
114.58
.270 3907
119.17
45
3.149 5227
106.33
3.188 5192
110.36
3.229 0129
114.65
3.271 1060
119.25
46
.150 1609
106.40
.189 1815
110.43
.229 7010
114.73
.271 8217
"9-33
47
.150 7995
1 06.4(1
.189 8443
110.50
.230 3896
114.80
.272 5379
119.41
48
.151 4385
106.53
.190 5075
110.57
.231 0786
114.88
.273 2546
119.49
49
.152 0778
106.59
.191 1711
110.64
.231 7681
114.95
.273 9717
119.57
50
3.152 7176
106.66
3.191 8351
110.71
3.232 4581
115.03
3.274 6894
119.65
51
52
153 3577
.153 9983
106.72
106.79
.192 4996
.193 1644
110.77
110.84
.233 1484
.233 8392
115.10
115.17
.275 4075
.276 1261
"9-73
119.81
53
.154 6392
106.85
.193 8297
110.91
34 535
115.25
.276 84.52
119.89
54
.155 2805
106.92
.194 4954
110.98
.235 2222
"5-3*
.277 5647
119.97
55
56
57
3.155 9222
.156 5643
.157 2068
106.99
107.05
107.12
3.195 1615
.195 8281
.196 4950
111.05
III. 12
III.I9
3.235 9144
.236 6070
.237 3 OOI
115.40
115.47
"5-55
3.278 2848
.279 0053
.279 7263
120.05
120.13
120.21
58
59
.157 8497
.158 4930
107.18
107.25
.197 1624
.197 8302
III.26
111.34
.237 9936
.238 6876
115.62
115.70
.280 4477
.281 1697
120.29
120-37
GO
3-i<;Q 1367
107.31
3.198 4984
III.4I
3.239 3820
115.77
3.281 8921
120.45
603
TABLE VI.
For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit.
V.
152
153
154
155
logM.
Diff. 1".
logM.
Diff. 1".
logM.
DiF 1".
logM.
Diff. 1".
O'
3.281 8921
120.45
3.326 1448
125.46
3.372 2684
130.85
3.420 4064
136.66
1
.282 6151
120.53
.326 8978
1^5-55
373 53 8
130.94
.421 2266
136.76
2
3
.283 3385
.284 0624
120.61
120.69
.327 6513
.328 4054
125.63
125.72
373 8397
.374 6262
131.04
131.13
.422 0475
.422 8690
136.86
136.96
4
.284 7868
120.77
.329 1600
125.81
375 4133
131.22
.423 6910
137.06
5
3.285 5116
120.85
3.329 9151
125.89
3.376 2009
131.32
3.424 5137
137.16
6
.286 2370
120.93
.330 6707
125.98
.376 9890
131.41
425 3370
137.26
7
.286 9628
121.01
.331 4268
126.07
377 7778
131.50
.426 1609
'37-37 ;
8
.287 6891
121. 10
.332 1835
126.16
378 5 6 7i
131.60
.426 9854
J37-47 |
9
.288 4160
I2I.I8
.332 9407
126.24
379 3570
131.69
.427 8105
137-57
1O
11
3.289 1433
.289 8711
121 26
121.34
3-333 6 984
334 4567
126.33
126.42
3.380 1474
.380 9384
131.79
131.88
3.428 6362
.429 4626
137.67
137.77
12
.290 5993
121.42
335 2154
126.51
.381 7300
131.98
.430 2895
137.88
13
.291 3281
121.50
335 9747
126.59
.382 5221
132.07
.431 1171
I37-98
14
.292 0574
121.59
33 6 734 6
126.68
383 3H8
132.16
.431 9452
138.08
15
3292 7872
121.67
3-337 4949
126.77
3.384 1081
132.26
3.432 7740
138.18
16
293 5 I 74
121.75
.338 2558
126.86
.384 9019
132.35
433 6034
138.29
17
.294 2481
121.83
339 OI 72
126.95
385 6963
132.45
434 4334
'38.39
18
.294 9794
121.91
339 779 2
127.03
.386 4913
132.54
.435 2641
138.49
19
.295 7111
122.00
34 5417
127.12
.387 2869
132.64
43 6 953
138.59
20
3.296 4433
122.08
3-341 3047
127.21
3.388 0830
132.73
3.436 9272
138.70
21
.297 1761
I22.I6
.342 0682
127.30
.388 8797
132.83
437 7597
138.80 !
22
.297 9093
122.24
.342 8323
127-39
.389 6770
J 32-93
.438 5928
138.90 ;
23
.298 6430
122.33
343 59 6 9
127.48
-39 4749
133.02
.439 4266
139.01
24
.299 3772
122.41
344 3620
127.57
.391 2733
133.12
.440 2609
139.11
25
3.300 1119
122.49
3-345 I2 77
127.66
3.392 0723
133.22
3.441 0959
139.22
26
.300 8471
122.58
345 8 939
'27-75
.392 8719
'33-31
.441 9315
139.32
27
.301 5828
122.66
.346 6606
127.84
393 6 72o
133.41
.442 7677
13942
28
.302 3190
122.74
347 4279
127-93
394 4728
'33-5
.443 6046
'39-53
29
.303 0557
122.83
.348 1958
128.02
395 274 1
133.60
.444 4421
139-63
30
3.303 7929
122.91
3.348 9641
128.11
3.396 0760
133.70
3.445 2802
139-74
31
.304 5306
122.99
349 733
128.19
.396 8785
J33-79
.446 1189
139.84
32
.305 2688
123.08
35 524
128.28
397 6815
133-89
44 6 9583
139-95 !
33
.306 0075
123.16
.351 2724
128.37
.398 4852
1 33-99
447 7983
140.05
34
.306 7468
123.24
.352 0429
128.46
399 2894
134.09
.448 6389
140.16
35
3.307 4865
123.33
3.352 8140
128.55
3.400 0942
134.19
3.449 4802
140.26
36
.308 2267
123.41
353 585 6
128.65
.400 8996
134.28
45 3221
140.37
37
38
.308 9674
.309 7086
123.50
123.58
354 3577
355 '34
128.74
128.83
.401 7056
.402 5122
134-38
134.48
.451 1646
452 0077
140.47
14057
39
.310 4504
123.66
355 937
128.92
.403 3193
134-57
.452 8515
140.68
40
3.311 1926
123.75
3.356 6774
129.01
3.404 1270
134.67
3-453 6959
140.79
41
3 11 9354
123.83
357 45i7
129.10
.404 9354
134-77
454 541
140.90
42
.312 6786
123.92
.358 2266
129.19
45 7443
I34-87
455 3867
141.00
43
3*3 4224
124.00
.359 0020
129.28
.406 5538
134-97
.456 2330
141.11
44
.314 1667
124.09
359 778o
129.37
.407 3639
I35-07
.457 0800
141.21
45
3-3H 9H5
124.17
3.360 5545
129.46
3.408 1746
135.16
3.457 9276
141.32
46
.315 6567
124.26
.361 3316
129.56
.408 9859
135.26
458 7759
141.43 i
47
.316 4025
124.34
.362 1092
129.65
.409 7977
'35-36
.459 6248
141.54 !
48 -317 1489
1 24-43
.362 8873
129.74
.410 6102
I35-46
.460 4743
141.64
49 -3 1 ? 8 957
124.51
.363 6660
129.83
.411 4233
I35-5 6
.461 3245
141.75
50 3-3 l8 6 43
124.60
3-3 6 4 4453
129.92
3.412 2369
135.66
3.462 1753
141.86
51
.319 3909
124.68
.365 2251
130.01
.413 0512
135.76
.463 0268
141.97
52
.320 1392
124.77
.366 0055
130.11
.413 8660
135.86
.463 8789
142.07
53
.320 8881
124.86
.366 7864
130.20
.414 6815
135.96
.464 7317
I42.I8
54
.321 6375
124.94
.367 5679
130.29
4*5 4975
136.06
465 5851
142.29
55
3.322 3874
125.03
3.368 3499
130.38
3.416 3142
136.16
3.466 4392
142.40
56
.323 1379
125.11
.369 1325
130.48
.417 1314
136.26
467 2939
142.51
57
.323 8888
125.20
.369 9156
*3-57
.417 9492
136.36
.468 1492
I42.6l
58
.324 6403
125.29
.370 6993
130.66
.418 7677
136.46
.469 0052
142.72
59
.325 3923
125-37
.371 4836
130.76
.419 5867
136.56
.469 8619
142.83
60
3.326 1448
125.46
3.372 2684
130.85
3.420 4064
136.66
3.470 7192
142.94
604
TABLE VI.
For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit.
V.
156
157
158
159
logM.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. 1".
0'
3.470 7192
142.94
3.523 3875
'49-75
3-578 6154
157.17
3.636 6351
165.28
1
2
3
47' 5772
.472 4358
473 2951
'43-5
143.16
143.27
.524 2864
.525 1860
.526 0863
149.87
149.99
150.11
579 5588
.580 5030
.581 4480
157.30
'57-43
'57-56
.637 6274
-638 6202
.639 6140
'6542
165.56
4
.474 1550
I43-38
.526 9873
150.23
S 8 ^ 3937
157.69
.640 6087
165.85
5
6
3.475 0156
475 8769
143.49
143.60
3.527 8890
.528 7915
'50-35
150.47
3.583 3403
.584 2876
157.82
'57-95
3.641 6042
.642 6006
165.99
166.13
7
.476 7388
'43-7 1
.529 6947
150.59
585 2357
158.08
.643 5978
166.28
8
.477 6014
143-82
.530 5985
150.71
.586 1846
158.21
.644 5959
166.42
9
.478 4646
'43-93
53' 53'
150-83
587 '342
'58-34
645 5948
166.56
10
3.479 3285
144.04
3-532 4085
150.95
3.588 0847
158.47
3.646 5946
166.71
11
12
.480 1931
.481 0583
144-15
144.26
533 3'45
.534 2213
151.07
151.19
589 0359
.589 9880
158.61
158.74
647 5953
.648 5968
166.85
166.99
13
.481 9242
'44-37
.535 1288
151.31
.590 9408
158.87
.649 5992
167.14
14
.482 7907
144-48
536 0370
151.43
.591 8944
159.00
.650 6025
167.28
15
3.483 6579
'44-59
3.536 9459
'5'-55
3.592 8488
'59-'3
3.651 6066
167.42
16
.484 5258
144.70
537 8556
151.67
593 8040
159.26
.652 6116
167.57
17
485 3944
144-81
.538 7660
151.79
594 7600
159.40
.653 6175
167.72
18
19
.486 2636
487 '335
'44-93
145.04
539 6771
.540 5890
151.91
152.04
595 7167
596 6743
'59-53
159.66
.654 6242
655 6318
167.86
168.01
20
21
22
3.488 0040
.488 8752
489 7472
145.15
145.26
'45-37
3-54' 5'5
.542 4148
543 3289
152.16
152.28
152.40
3-597 6327
.598 5919
599 55'8
'59-79
'59-93
160.06
[.656 6403
657 6497
.658 6599
168.15
168.30
168.45
23
.490 6198
'45-49
.544 2436
152.52
.600 5126
160.19
659 6710
168.59
24
49' 493
145.60
545 '59'
152.65
.601 4742
160.33
.660 6830
168.74
25
3.492 3670
145.71
3.546 0754
152.77
3.602 4365
160.46
;.66i 6959
168.89
26
493 2416
145-82
.546 9924
152.89
.603 3997
160.60
.662 7096
169.03
27
.494 1 1 68
145-94
.547 9101
153.01
.604 3637
160.73
.663 7243
169.18
28
.494 9928
146.05
.548 8285
153-14
.605 3285
160.87
.664 7398
169.33
29
495 8695
146.16
549 7477
153.26
.606 2941
161.00
.665 7562
169.48
30
.496 7468
146.28
3.550 6677
'53.38
1-607 2605
161.14
3.666 7735
169.62
31
.497 6248
146.39
.551 5883
'53-5'
.608 2277
161.27
.667 7917
169.77
32
.498 5035
146.50
552 597
153-63
609 1957
161.41
.668 8108
169.92
33
499 3828
146.62
553 43'9
'53-75
.610 1646
161.54
.669 8308
170.07
34
.500 2629
146.73
554 3548
153.88
.611 1342
161.68
.670 8516
170.22
35
.501 1436
146.85
555 2785
154-00
1-612 1047
161.81
.671 8734
170.37
36
.502 0250
146.96
.556 2029
'54-'3
.613 0760
161.95
.672 8961
170.52
37
38
.502 9071
.503 7899
147.08
147.19
.557 1280
558 0539
154-25
.614 0481
.615 0210
162.09
162.22
.673 9196
.674 9441
170.67
170.82
39
.504 6734
147-3'
.558 9806
154.50
.615 9948
162.36
.675 9 6 94
170.97
40
.505 5576
147-42
-559 98o
154-63
.6l6 9693
162.50
.676 9957
171.12
41
.506 4425
'47-54
.560 8361
'54-75
.617 9447
162.63
.678 0228
171.27
42
.507 3280
'47-65
.561 7650
154.88
.618 9209
162.77
.679 0509
171.42
43
.508 2143
147.77
.562 6947
155.01
.619 8980
162.91
.680 0799
'71-57
44
.509 IOI2
147-88
.563 6251
'55-'3
.620 8758
163.05
.681 1098
171.72
45
.509 9889
148.00
.564 5562
155.26
.621 8545
163.18
.682 1406
171.87
46
.510 8 77 2
148.11
.565 4882
155.38
.622 8340
163.32
.683 1723
172.03
I 47
.511 7662
148.23
.566 4209
'55-5'
.623 8144
163.46
.684 2049
172.18 -
48
49
.512 6560
5 '3 5464
148.34
148.46
5 6 7 3543
.568 2885
I55-64
155.76
.624 7956
.625 7776
163.60
163.74
.685 2384
.686 2728
172-33
172.48
50
5'4 4375
148.58
.569 2235
155.89
.626 7604
163.88
.687 3082
172.64
51
5'5 3294
148.70
.570 1592
156.02
.627 7441
164.02
.688 3445
172.79
52
.516 2219
148.81
57' 0957
156.15
.628 7287
164.16
.689 3817
172.94
53
5'7 "5i
148.93
.572 0330
156.27
.629 7140
164.30
.690 4198
173.10
54
.518 0090
149.05
.572 9710
156.40
.630 7002
164.44
.691 4588
'73-25
55
.518 9037
149.17
573 9098
156.53
.631 6873
164.58
.692 4988
I7340
56
5'9 799
149.28
574 8494
156.66
.632 6751
164.72
.693 5397
173.56
57
58
59
.520 6951
.521 5918
.522 4893
149.40
149.52
149.64
575 7897
576 738
577 6727
156.79
156.92
'57-4
633 6638
634 6534
.635 6438
164.86
165.00
165.14
694 5815
.695 6243
.696 6680
'73-7'
173.87
174.02
60
523 3875
'49-75
.578 6154
I57-I7
.636 6351
165.28
.697 7126
174.18
605
TABLE VI.
For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit.
V.
160
161
162
163
log M.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. 1".
0'
3.697 7126
174.18
3.762 1519
183.99
3.830 3147
194.87
3.902 6107
207.00
1
2
.698 7581
.699 8046
'74-34
174.49
.763 2584
.764 3639
184.16
I84-34
.831 4845
.832 6554
195.06
195.25
.903 8534
.905 0973
207.21
207.43
3
.700 8520
174.65
.765 4704
184.51
833 8275
'95-44
.906 3425
207.64
4
.701 9003
174.80
.766 5780
184.68
.835 0008
195.64
.907 5890
207.86
5
3.702 9496
174.96
3.767 6867
184.86
3.836 1752
195.83
3.908 8368
208.08 i
6
.703 9999
175.12
.768 7963
185.03
.837 3508
196.02
.910 0859
208.29
7
.705 0511
175.28
.769 9070
185.20
.838 5275
196.22
.911 3363
208.51
8
.706 1032
'75-43
.771 0187
185.38
839 754
196.41
.912 5880
208.72
9
.707 1562
'75-59
.772 1315
I85.55
.840 8844
196.60
.913 8410
208.94
10
3.708 2102
'75-75
3-773 2 454
185.73
3.842 0646
196.80
3-9'5 953
209.16
11
12
13
.709 2652
.710 3211
7" 378o
175-9'
176.07
176.22
774 3 6 3
775 4762
.776 5932
185.90
186.08
186.25
.843 2460
.844 4286
.845 6133
196.99
197.19
'97-3 8
.916 3509
.917 6078
.918 8661
209.38
209.60
209.81
14
.712 4358
176.38
777 7"2
186.43
.846 7972
197.58
.920 1256
210.03
15
16
3.713 4946
7'4 5543
176.54
176.70
3.778 8303
779 955
186.60
186.78
3-847 9833
.849 1705
197.78
197.97
3.921 3865
.922 6487
210.25
210.48
17
.715 6150
176.86
.781 0717
186.96
.850 3589
198.17
.923 9122
210.70
18
.716 6766
177.02
.782 1940
187.14
.851 5486
198-37
.925 1770
210.92
19
.717 7392
177-18
783 3'74
187.31
.852 7394
198.57
.926 4432
211.14
20
21
3.718 8028
.719 8673
'77-34
177.50
3.784 4418
.785 5672
187.49
187.67
3-853 93'4
.855 1245
198.76
198.96
3.927 7107
.928 9795
211.36
211.58
22
.720 9328
177-66
.786 6938
187.85
.856 3189
199.16
93 2 497
211. 8l
23
.721 9993
177-83
.787 8214
188.03
857 5'45
199.36
.931 5212
212.03
24
.723 0668
178.00
.788 9501
188.21
.858 7112
199.56
.932 7940
212.25
25
3.724 1352
178-15
3.790 0799
188.39
3.859 9092
199.76
3.934 0682
212.48
26
27
.725 2045
.726 2749
178.31
178.47
.791 2108
.792 3427
188.57
188.75
.861 1084
.862 3087
199.96
200.16
935 3438
.936 6207
212.70
212.93
28
29
.727 3462
.728 4185
178.63
178.80
793 4757
.794 6098
188.93
189.1!
.863 5103
.864 7131
200.36
200.56
937 8989
939 '785
213.15
213.38
30
3.729 4918
178.96
3-795 745
189.29
3.865 9171
200.77
3.940 4595
213.61
31
.730 5661
179.13
.796 8812
189.47
.867 1223
200.97
.941 7418
213.83
32
.731 6413
179.29
.798 0186
189.65
.868 3287
201.17
943 02 54
214.06
33
.732 7176
'79-45
799 '57'
189.83
869 5363
201.37
944 3'5
214.29
34
733 7948
179.62
.800 2966
190.01
.870 7452
201.58
945 59 6 9
214.52
35
3.734 8730
I79-78
3.801 4372
190.20
3-871 955*
201.78
3.946 8847
214.74
36
735 95^2
'79-95
.802 5790
190.38
.873 1665
201.98
.948 1738
214.97
37
.737 0324
180.11
.803 7218
190.56
874 379'
202.19
.949 4644
215.20
38
738 H3 6
180.28
.804 8657
190.65
.875 5928
202.39
.950 7563
215.43
39
739 '957
180.45
.806 0108
190.93
.876 8078
202.60
.952 0496
215.66
40
3.740 2789
180.61
3.807 1569
191.11
3.878 0240
202.80
3-953 3443
216.90
41
42
74' 3 6 3'
.742 4482
180.78
180.94
.808 3041
.809 4525
191.30
191.48
.879 2414
.880 4601
203.01
203.22
954 6403
955 9378
216.13
216.36
43
743 5344
i8i.ii
.810 6020
191.67
.881 6800
203.42
.957 2366
216.59
44
.744 6216
181.28
.811 7525
191.86
.882 9012
203.63
958 53 6 9
216.82
45
3.745 7097
181.45
3.812 9042
192.04
3.884 1236
203.84
3-959 8385
217.06
46
.746 7989
181.61
.814 0570
192.23
.885 3473
204.05
.961 1416
217.29
47
747 8891
181.78
.815 2110
192.41
.886 5722
204.26
.962 4460
217.53
48
748 9803
181.95
.8l6 3660
192.60
887 7983
204.46
.963 7519
217.76
49
.750 0725
182.12
.817 5222
192.79
.889 0257
204.67
.965 0592
218.00
50
3.751 1657
182.29
3.818 6 795
192.98
3.890 2544
204.88
3.966 3678
218.23
51
.752 2599
182.46
.819 8379
193.16
.891 4843
205.09
.967 6779
218.47 j
52
753 355*
182.63
.820 9974
'93-35
.892 7155
205.31
.968 9895
218.70
53
.754 4514
182.80
.822 1581
193-54
.893 9480
205.52
.970 3024
218.94
54
755 5487
182.97
.823 3199
'93-73
.895 1817
205.73
.971 6168
219.18
55
56
57
3.756 6470
757 7464
758 8467
183.14
183.31
183.48
3.824 4829
.825 6470
.826 8122
193.92
194.11
'94-3
3.896 4167
.897 6529
.898 8905
205.94
206.15
206.36
3.972 9326
.974 2498
975 5684
219.42
219.66
219.90
58
59
759 948i
.761 0505
183.65
183.82
.827 9785
.829 1460
194.49
194.68
.900 1293
.901 3694
206.57
206.79
.976 8885
.978 2100
220.13
220.37
60
3.762 1539
183.99
3.830 3147
194.87
3.902 6107
207.00
3-979 533
220.61
606
TABLE VI.
For findin e the True Anomaly or the Time from the Perihelion in a Parabolic Orbit.
V.
164
165
166
167
log M.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. l".
o
1
3-979 533
-980 8574
220.62
220.86
4.061 6673
.063 0842
236.01
236.28
4.149 7198
.151 2422
253-57
253.88
4-244 5537
.246 1975
27A.I4
2
3
4
.982 1833
.983 5106
.984 8394
221.10
221-34
221.58
.064 5027
.065 9229
.067 3447
236.56
236.83
237.11
.152 7664
.154 2925
.155 8205
254.19
254.51
247 8434
.249 4916
.251 1419
274.51
5
6
7
8
9
3.986 1696
.987 5013
9*8 8345
.990 1691
.991 5051
221.83
222.07
222.31
222.56
222.80
4.068 7682
.070 1933
.071 6201
.073 0486
.074 4787
237-39
237.66
237-94
238.22
238.50
4-157 354
.158 8822
.160 4159
.161 9515
.163 4891
255.14
255.46
255.78
256.10
256.42
4.252 7944
.254 4491
.256 1061
.257 7652
.259 4266
275.60
275-97
276.34
276.71
10
11
3.992 8427
994 1817
223.05
223.29
4.075 9106
.077 3441
238.78
239.06
4.165 0285
.166 5699
256.74
257.06
4.261 0902
.262 7560
277-45
277.82
12
.995 5222
223.54
.078 7792
239-34
.168 1132
.264 4240
' L
278.20
13
14
.996 8642
.998 2077
223.79
224.03
.080 2161
.081 6546
239.62
239.90
.169 6585
.171 2056
257.70
258.02
.266 0943
.267 7669
278.95
15
16
17
3-999 5527
I..OOQ 8991
.002 2471
224.28
224-53
224.78
^.083 0948
.084 5368
.085 9804
240.18
240.46
240.75
4.172 7547
.174 3058
.175 8588
258.35
258.67
259.00
4.269 4417
.271 1187
.272 7981
279.70
280.08
18
.003 5965
225.03
.087 4257
241.03
.177 4138
259 33
.274 4797
280.46
19
.004 9474
225.28
.088 8728
241.32
.178 9707
259.65
.276 1635
280.84
20
4.006 2999
22553
..090 3215
241.60
^.180 5296
259-98
4.277 8497
281.22
21
.007 6538
225.78
.091 7720
241.89
.182 0905
260.31
279 5381
281.60
22
.009 0093
226.04
.093 2242
242.08
.183 6534
260.64
.281 2289
281.98
23
24
.010 3663
.on 7248
226.29
226.54
.094 6781
.096 1337
242.56
242.75
.185 2182
.186 7850
260.97
261.30
.282 9219
.284 6173
282.36
25
4.013 0848
226.79
4.097 5911
243.04
4.188 3538
261.63
..286 3149
283.14
26
.014 4463
22 7 .05
.099 0502
243 33
.189 9246
261.96
.288 0149
283.52
27
.015 8093
227.30
.100 5110
243.62
.191 4974
262.30
.289 7172
283.91
28
29
.017 1739
.018 5400
227.55
227.81
.101 9736
.103 4379
243-91
244.20
.193 0722
.194 6490
262.63
262.97
.291 4218
.293 1288
284.30
284.69
30
31
4.019 9077
.021 2769
228.06
228.32
.104 9040
.106 3718
244.49
244.78
4.196 2278
.197 8086
263.30
263.64
4.294. 8 3 8l
.296 5498
285.08
285.47
32
.022 6476
228.58
.107 8414
245.08
1 99 39*5
263.98
.298 2638
285.87
33
.024 0199
228.84
.109 3127
245-37
.200 9764
264.32
.299 9802
286.26
34
.025 3937
229.09
.no 7858
245.67
202 5633
264.66
.301 6990
286.66
35
.026 7691
229.35
.112 2607
245.96
4.204 1523
265.00
4.303 4201
287.05 '
36
37
.028 1460
.029 5245
229.62
229.88
JI 3 7374
.115 2158
246.26
246.55
205 7433
.207 3363
265-34
265.68
.305 1436
.306 8695
287.45
287.85
38
.030 9045
230.14
.116 6960
246.85
.208 9314
266.02
.308 5978
288.25
39
.032 2861
230.40
.118 1780
247.15
.2IO 5286
266.37
.310 3285
288.65
40
.033 6693
230.66
.119 6618
247.45
.212 1278
266.71
.312 0616
289.05
41
.035 0540
230.92
.121 1474
247-75
.213 7291
267.06
.313 7971
289-45
42
.036 4404
231.18
.122 6348
248.05
.215 3325
267.40
3'5 535
289.86
43
.037 8283
231.45
.124 1239
248-35
.216 9379
267-75
3'7 2753
290.26
44
.039 2177
231.71
.125 6149
248.65
.218 5455
268.10
.319 0181
290.67
45
.040 6088
231.97
.127 1077
248-95
.220 1551
268.44
.320 7633
291.07
46
.042 0015
232.24
.128 6023
249.25
.221 7668
268.79
.322 5110
291.48 ;
47
1 48
43 3957
.044 7915
232.51
232.77
.130 0988
.131 5970
249.56
249.86
.223 3806
.224 9965
269.14
269.50
.324 2611
.326 0137
291.89 I
292.30
49
.046 1890
233.04
.133 0971
250.17
.226 6146
269.85
.327 7688
292.71
50
.047 5880
233-3 1
.134 5990
25047
.228 2347
270.20
.329 5263
293-13
51
.048 9887
233-57
.136 1028
250.78
.229 8570
270.55
.331 2863
293-54
52
.050 3909
233.84
.137 6084
251.08
.231 4814
270.91
333 4 8 7
293-95
53
.051 7948
234.11
.139 1158
25I-39
.233 I0 79
271.27
334 8137
294.37
54
053 2003
234-38
140 6251
251.70
.234 7366
271.62
336 5812
294.79
55
054 6074
234.65
.142 1362
252.01
236 3674
271.98
.338 3511
295.20
56
056 0161
234.92
143 6492
252.32
238 0003
272.34
.340 1236
295.62
57
057 4264
235-19
145 1641
252.63
239 6354
272.70
.341 8986
296.04
58
058 8384
235-46
146 6808
252.94
241 2727
273.06
.343 6762
296.47
59
060 2520
235.73
I 4 8 I 994
253-25
242 9121
273.42
345 4562
296.89
6O
..061 6673
236.01
I 49 7198 253.57
244 5537
273-78
347 *388
297.31
607
TABLE VI.
For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit.
V.
168
169
170
171
logM.
Diff. I".
logM. | Diff. 1".
logM.
Diff. 1".
logM.
Diff. 1".
O'
4-347 *388
297.31
4.459 1242
325.07
4.581 9445
358.31
4.717 9835
398.87 j
1
2
.349 0240
.350 8117
*97-74
298.16
.461 0761
.463 0311
3*5-57
326.08
.584 0962
.586 2516
358.92
359-53
.720 3790
.722 7790
399.62 ;
400.38
3
.352 6019
298.59
.464 9891
326.59
.588 4106
360.15
7*5 1835
401.14
4
354 3948
299.02
.466 9501
327.10
59 5734
360.76
.727 5926
401.90
5
4.356 1902
299.45
4.468 9142
327.61
4.592 7398
361.38
4.730 0063
402.66
6
.357 9882
299.88
.470 8814
328.12
594 910
362.00
.732 4245
403-43
I 7
.359 7888
300.31
.472 8517
328.64
597 0838
362.62
734 8474
44 ^9 i
8
.361 5919
300.75
474 8250
3*9-i5
599 *6i5
3 6 3-*5
737 *749
404.96 .
9
.363 3977
301.18
.476 8015
329.67
.601 4428
363.88
739 77
405.74
1O
4.365 2061
301.62
4.478 7811
330.19
4.603 6280
364.50
4.742 1438
406.52
11
.367 0171
302.05
.480 7637
33 -7i
.605 8169
365-14
744 585*
407.30
12
.368 8308
302.49
.482 7495
33'-*3
.608 0096
365-77
.747 0314
408.08
13
.370 6470
302.93
484 7385
331-75
.610 2061
366.40
749 4822
408.87
14
.372 4659
303.37
.486 7306
332.28
.612 4064
367.04
751 9378
409.66
15
4-374 2 8 75
303.81
4.488 7258
33*-8i
4.614 6106
367.68
4-754 398i
410.45
16
.376 1117
304.26
.490 7242
333-33
.616 8186
368.32
75 6 8632
411.24
17
377 93 86
304.70
.492 7258
333.86
.619 0304
368.96
759 333
412.04
18
379 7 68 i
35-!5
494 73 6
334-4
.621 2461
369.61
.761 8077
412.84
19
.381 6003
35-59
.496 7386
334-93
.623 4657
370.26
.764 2872
4I3-65
20
4-383 435*
306.04
4.498 7498
335-46
4.625 6892
370.91
4.766 7715
414.46
21
385 *7*8
306.49
.500 7642
336.00
.627 9166
37I-56
.769 2606
415.27
22
-387 "3 1
306.94
.502 7818
336.54
.630 1480
372.21
771 7547
416.08
23
.388 9561
37-39
.504 8026
337.08
.632 3832
37*-87
774 *53 6
416 90
24
.390 8019
307.85
.506 8267
337.62
.634 6224
373-53
776 7574
417.72
25
4.392 6503
308.30
4.508 8541
338.16
4.636 8656
374-19
4.779 2662
418.54 I
26
394 5 OI 5
308.76
.510 8847
338.71
.639 1127
374-86
.781 7799
4^9-37
27
39 6 3554
309.21
.512 9186
339.26
.641 3639
375-5*
.784 2986
420.20
28
.398 2121
309.67
5'4 9558
339.80
.643 6190
376.19
.786 8222
421.03
29
.400 0715
310.13
.516 9962
34-35
.645 8781
376.86
.789 3509
421.86
30
4-401 9337
310.59
4.519 0400
340.91
4.648 1413
377-53
4.791 8846
422.70
31
.403 7986
311.06
.521 0871
341-46
.650 4085
378.21
794 4*33
4*3-54
32
.405 6667
311.52
.523 1376
342.02
.652 6798
378-89
.796 9671
424.39
33
.407 5368
3H-99
.525 1913
34*-57
.654 9552
379-57
.799 5160
425.24
34
.409 4102
311-45
.527 2484
343-J3
.657 2346
380.25
.802 0700
426.09
35
4.411 2863
312.92
4.529 3089
343-69
4.659 5182
380.93
4.804 6291
426.95
36
.413 1652
3'3-39
.531 3728
344.26
.661 8059
381.62
807 1934
427.81
37
.415 0469
313-86
533 44
344.82
.664 0977
382.31
.809 7628
428.67
38
.416 9315
3'4-33
535 5 106
345-39
.666 3936
383.00
.812 3374
4*9-53
39
.418 8189
3H-8o
537 5 8 4 6
345-95
.668 6937
383.70
.814 9172
430.40
40
4.420 7091
315-28
4-539 6620
346.52
4.670 9980
38439
4.817 5022
431.28
41
.422 6022
3'5-75
.541 7429
347-09
.673 3064
385-09
.820 0925
432.15
42
.424 4982
316.23
543 8272
347-67
.675 6191
385.80
.822 6881
433.03
43
.426 3970
316.71
545 9H9
348.24
677 936
386.50
.825 2889
433-9 1
44
.428 2987
317-19
.548 0061
348.82
.680 2571
387.21
.827 8950
434.80
45
4.430 2033
317.67
4.550 1007
349.40
4.682 5825
387-9*
4.830 5065
435.69
46
.432 1108
318.16
.552 1989
349.98
.684 9121
388.63
833 !*34
436.59
47
.434 0212
318.64
554 35
350-56
.687 2460
389-34
835 7456
437.48
48
435 9345
3 I 9- I 3
.556 4056
35i.i5
.689 5842
390.06
838 373*
438-38
49
437 8507
319.61
55 8 5H3
351-73
.691 9268
390.78
.841 0062
439.29
50
4-439 7698
320.10
4.560 6264
35*-3*
4.694 2736
391.50
4.843 6446
440.20 :
51
.441 6919
320.59
.562 7421
352.91
.696 6248
392.23
.846 2886
441.11
52
443 6l6 9
321.08
.564 8614
353-5
.698 9803
392.96
.848 9380
442.03
53
445 5449
321.58
.566 9842
354-io
.701 3402
393-68
.851 5929
442.95
54
447 4758
322.07
.569 1106
354-69
.703 7046
394.42
8 54 *533
443.87
55
4.449 4097
322.57
4.571 2405
355-*9
4.706 0733
395-15
4.856 9193
444.80
56
.451 3466
323.06
573 374i
355-89
.708 4464
395.89
.859 5909
44 IH
57
453 *865
323.56
575 5 IJ 3
356.49
.710 8240
396-63
.862 2680
446.66
58
.455 2294
324.06
577 6521
357-io
.713 2060
3973
.864 9508
447.60
59
457 753
324.56
579 79 6 5
357.70
7i5 59*5
398.12
.867 6392
448.54
60
4.459 1242
3*5-07
4.581 9445
358.31
4.717 9835
398.87
4.870 3333
449-49
TABLE VI.
For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit.
V.
172
173
174
175
logM.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. 1".
logM.
Diff. 1".
0'
4-870 3333
449-49
5.043 3285
5H-47
5.243 3165
601.00
5.480 1373
722.00
1
.873 0331
450.44
.046 4191
5I5-7I
.246 9276
602.69
.484 4765
7*4-4*
2
.875 7386
451-39
.049 5171
516.96
.250 5488
604.38
.488 8304
726.87
3
.878 4499
45*-35
.052 6226
518.21
.254 1802
606.08
.493 1989
7*9-33
4
.881 1668
453-31
055 7356
5I9-47
.257 8218
607.80
497 5823
731.80
5
6
4.883 8896
.886 6182
454.28
455-*5
5.058 8562
.061 9843
5*o-73
522.00
5.261 4738
.265 1361
609.53
611.26
5.501 9806
.506 3939
734-3
736.81
7
.889 3526
456.23
.065 1202
523.28
.268 8089
613.00
.510 8223
739-33
8
.892 0929
457.20
.068 2637
524.56
.272 4922
614.75
.515 2659
741-87
9
.894 8391
458.19
.071 4149
5*5-85
.276 i860
616.52
519 7*4
744-44
10
4.897 5912
459-17
5.074 5738
5*7-14
5.279 8904
618.29
5.524 1992
747.02
11
.900 3492
460.16
.077 7406
528.44
.283 6055
620.08
.528 6890
749.61
12
.903 1132
461.16
.080 9151
5*9-75
.287 3313
621.87
533 1946
75*-*3
13
.905 8831
462.16
.084 0976
531.06
.291 0680
623.67
537 7158
754.86
14
.908 6591
463.16
.087 2879
53*-38
.294 8154
625.49
54* *5*9
757-51
15
4.911 4411
464.17
5.090 4862
533-71
5.298 5738
627.31
5.546 8060
760.18
16
.914 2291
465.18
.093 6924
535-04
.302 3432
629.15
55i 375i
762.87
17
.917 0233
466.20
.096 9067
536-38
.306 1237
631.00
555 9605
765.58
18
.919 8235
467.22
.100 1290
537-73
39 915*
632.85
.560 5621
768.31
19
.922 6299
468.25
i3 3594
539.08
313 7179
634.72
.565 1802
771.05
20
21
4-9*5 44*5
.928 2612
469.28
470. 3 i
5.106 5980
.109 8447
540.44
541.81
5.317 5319
.321 3571
636.60
638.49
5.569 8148
574 4661
773-8*
776.61
22
.931 0862
471-35
.113 0997
543-18
3*5 1938
640.39
579 1341
779-41
23
933 9174
472.39
.116 3629
544.56
.329 0418
642.30
.587 8190
782.24
24
.936 7549
473-44
.119 6344
545-95
33* 9014
644.23
.588 5210
785.08
25
4-939 5987
474-49
5.122 9143
547-34
5.336 7726
646.16
5.593 2401
787-95
26
.942 4489
475-55
.126 2026
548.74
340 6554
648.11
597 9764
790.84
27
945 353
476.61
.129 4992
550-15
344 5499
650.07 .602 7302
793-75
28
.948 1682
477-68
.132 8044
551-57
.348 4562
652.04 .607 5014
796.68
29
95* 375
478.75
.136 1181
55*-99
35* 3744
654.02
.612 2903
799.63
30
4-953 9i3*
479-83
5.139 4403
554-4*
5-35 6 345
656.01
5.617 0970
802.60
31
956 7954
480.91
.142 7711
555-86
.360 2466
658.02
.621 9216
805.60
32
.959 6841
481.99
.146 1 1 06
557-3
.364 2007
660.04
.626 7642
808.62
33
.962 5793
483.08
.149 4588
558.75
.368 1671
662.07
.631 6250
811.66
34
.965 4811
484-18
.152 8157
5 6'o. 2 1
.372 1456
664.11
.636 5041
814.72
35
36
4.968 3894
.971 3044
485.28
486.38
5.156 1813
159 5558
561.68
563.16
5.376 1364
.380 1396
666.17
668.24
5.641 4017
.646 3179
817.81
820.92
37
.974 2260
487.49
.162 9392
564.64
384 1553
670.32
.651 2528
824.05
38
977 1543
488.61
.166 3315
566.13
.388 1834
672.41
.656 2065
827.21
39
.980 0893
489.73
.169 7328
567-63
.392 2242
674.52
.661 1793
830.39
40
41
4.983 0311
985 9795
490.85
491.98
5-173 *43i
.176 5624
569.13
570.65
5.396 2777
.400 3439
676.64
678.77
5.666 1713
.671 1825
833.60
836.83
42
.988 9348
493.12
.179 9908
572.17
.404 4229
680.92
.676 2132
840.08
43
.991 8970
494.26
.183 4284
573-70
.408 5149
683.08
.681 2635
843.36
44
994 8659
495.40
.186 8752
575-*4
.412 6199
685.25
.686 3336
846.67
45
4.997 8418
496.55
5.190 3312
576.78
5.416 7379
687.44
5.691 4236
850.00
46
5.000 8246
497-71
.193 7966
578.34
.420 8692
689.64
.696 5337
853-36
47
.003 8143
498.87
.197 2713
579.90
4*5 OI 3 6
691.85
.701 6640
856.75
48
19
.006 8m
.009 8148
500.04
501.21
.200 7554
.204 2489
581.47
583-05
.429 1714
433 34*7
694.08
696.33
.706 8147
.711 9860
860.16
863.60
50
5.012 8256
502.39
5.207 7520
584.64
5-437 5*74
698.59
5-717 1779
867.06
51
.015 8435
503-57
.211 2646
586.23
.441 7258
700.86
.722 3908
870.56
52
.018 8685
* J J ' f
504.76
.214 7868
587.84
445 9378
703.15
.727 6247
874.08
53
.021 9006
55-95
.218 1l86
589.45
.450 1636
705.45
-73* 8798
I 77 ' 6 *
54
*4 9399
.221 86O2
591-07
.454 4032
707.77
.738 1563
881.21
55
5.027 9864
508.36
5.225 4116
59*-7i
5.458 6568
710.10
5.743 4544
884.82
56
.031 0402
509-57
.228 9727
59435
.462 9244
712.45
748 774*
888.46
57
.034 1013
510.79
.232 5437
596.00
.467 2062
714.81
754 "59
892.13
58
.037 1697
512.01
.236 1247
597.66
.471 5022
717.19
.759 4798
895-81
59
.040 2454
513.24
.239 7156
599.32
475 81*5
719.59
.764 8659
899.56
60
5.043 3285
5I4-47
5.243 3165
601.00
5.480 1373
722.00
5.770 2745
903.31
609
TABLE VI.
I or finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit.
V.
176
177
178
179
logM.
Diff. 1".
logM.
Diff. V.
logM.
Diff. 1".
logM.
Diff. 1".
0'
5.770 2745
903.3
6.144 6 *89
1205.3
6.672 5724
1808. 8
7-575 464
3619
1
775 75 8
907.1
.151 8807
I2I2.0
.683 4709
1824.0
597 359 6
3680
2
781 1599
910.9
59 1733
1218.8
.694 4613
1839.5
.619 6295
3744
3
.786 6370
914.8
.166 5070
1225.7
75 5454
1855.3
.642 2868
3809
4
.792 1374
918.7
.173 8823
1232.7
.716 7248
1871.3
.665 3452
3877
5
5.797 6612
922.6
6.181 2997
1239.8
6.728 ooio
1887.5
7.688 8192
3948
6
.803 2086
926.6
.188 7597
1246.9
739 3758
1904.1
.712 7239
4021
7
.808 7798
930.6
.196 2628
1254.1
.750 8509
1921.0
737 0756
4097
8
.814 3751
934.6
.203 8095
1261.4
.762 4279
1938.2
.761 8913
4176 i
9
.819 9946
938.6
.211 4002
1268.8
774 I0 9
1955.6
.787 1889
4257
10
5.825 6386
942.7
6.219 354
1276.3
6.785 8958
1973-4
7.812 9876
4343
11
83 1 373
946.8
.226 7158
1283.8
797 794
1991.5
839 375
443i
12
.837 0008
951.0
.234 4419
1291.5
.809 7946
2OIO.O
.866 1702
4524
13
.842 7195
955-2
.242 2142
1299.2
.821 9106
2028. g
.893 5986
4620
14
.848 4634
959-5
.250 0333
1307.1
.834 1404
2048.0
.921 6170
4720
15
5.854 2329
963.7
6.257 8997
1315.0
6.846 4863
2067.5
7.950 2513
4825
16
.860 0282
968.0
.265 8139
1323.0
.858 9503
2087.3
7.979 5292
4935
17
.865 8495
972.4
.273 7766
1331.1
.871 5348
2107.6
8.009 4802
5050
18
.871 6970
976.8
.281 7884
J339-4
.884 2422
2128.3
.040 1361
5170
19
.877 5710
981.2
.289 8499
1 347-7
.897 0749
2149.4
.071 5309
5296
20
5.883 4717
985-7
6.297 9617
1356.2
6.910 0353
2170.9
8.103 7 TI
5428
21
.889 3993
990.2
.306 1244
1364.7
.923 1261
2192.8
.136 6857
5568
22
.895 3542
994.8
3 J 4 33 8 7
J 373-3
.936 3498
2215.2
.170 5274
57H
23
.901 3365
999-4
.322 6052
1382.1
949 793
2238.0
.205 2717
5869
24
.907 3465
1004.0
33 9*47
1391.0
.963 2073
2261.4
.240 9679
6032
25
5.913 3845
1008.7
6-339 2977
1400.0
6.976 8466
2285.2
8.277 6700
6204
26
.919 4507
1013.4
.347 7249
1409.1
6.990 6304
2309.6
.315 4361
6387
27
925 5454
1018.1
.356 2072
1418.3
7.004 5616
2334-3
354 3 2 9 8
6580
28
.931 6688
1022.9
.364 7451
1427.6
.018 6437
2359-7
394 4205
6786
29
937 8213
1027.8
373 3395
H37-I
.032 8796
2385-7
.435 7842
7004
30
5.944 0030
1032.7
6.381 9910
1446.7
7.047 2729
2412.2
8.478 5044
7238
31
.950 2144
1037.6
.390 7005
1456.4
.061 8271
2439.4
.522 6731
7488
32
.956 4556
1042.6
399 4 68 7
1466.2
.076 5458
2467.1
.568 3920
7755
33
.962 7269
1047.7
.408 2965
1476.2
.091 4329
2495.4
.615 7739
8042
34
.969 0287
1052.9
.417 1846
1486.4
.106 4921
2524.5
.664 9442
8352
35
5-975 3613
1058.0
6.426 1337
1496.7
7.121 7276
2554.2
8.716 0431
8686
36
.981 7249
1063.2
435 J 449
1507.0
137 1434
2584.6
.769 2286
9048
37
.988 1198
1068.4
.444 2191
1517.6
.152 7440
2615.8
.824 6779
944 i
38
5.994 5464
1073.7
453 35 6 9
1528.3
.168 5336
2647.6
.882 5925
9870
39
6.001 0050
1079.1
.462 5594
1539.2
.184 5171
2680.4
.943 2018
10340
40
6.007 4958
1084.5
6.471 8275
1550.2
7.200 6993
2713-9
9.006 7690
10857
41
.014 0192
1089.9
.481 1620
1561.3
.217 0850
2748.3
.073 5974
11429
42
.020 5756
1095.4
.490 5641
1572.6
.233 6796
2783.5
.144 0401
12064
43
.027 1652
IIOI.O
.500 0346
1584.1
.250 4884
2819.7
.218 5102
12773
44
.033 7885
1106.7
.509 5746
1595.8
.267 5170
2856.8
297 4963
13572
45
6.040 4457
1112.4
6.519 1850
1607.7
7.284 7712
2894.8
9.381 5820
14476
46
.047 1372
1118.1
.528 8669
1619.6
.302 2571
2934-1
.471 4711
15510
47
.053 8634
1123.9
.538 6216
1631.8
.319 9810
2974.2
.568 0247
16704
48
.060 6246
1129.8
.548 4499
1644.2
337 9494
3015.6
.672 3106
18096
49
.067 4212
"35-7
558 353
1656.8
.356 1692
3058.1
.785 6758
19741
50
6.074 2535
1141.7
6.568 3320
1669 6
7.374 6475
3101.7
9.909 8535
21715
51
.081 1219
1147.7
.578 3881
1682.4
393 39 l8
3146.8
10.047 1256
24127
52
.088 0269
1153.8
.588 5227
1695.6
.412 4099
3193.0
.200 5829
27144
53
.oyj. 9687
1 160.0
.598 7368
1708.9
.431 7097
3240.7
374 5584
31023
54
.101 9479
1166.3
.609 0317
1722.6
.451 2999
3289.9
575 3986
36197
55
6.108 9647
1172.6
6.619 486
1736.4
7.471 1892
334-3
10.812 9421
4345
56
.116 0196
1179.0
.629 8689
1750.3
.491 3870
3392.6
11.103 6719
57
.123 1131
1185.4
.640 4141
1764.5
.511 9029
3446.5
11.478 4880
58
.130 2455
1192.0
.651 0455
1779.0
.532 7472
3502.1
12.006 7617
59
137 4173
1198.6
.661 7645
1793-8
553 935
3559-6
12.909 8516
60
6.144 6289
1205.3
6.672 5724
1808. 8
7-575 4640
3618.7
610
TABLE VII.
For finding the True Anomaly in a Parabolic Orbit when v is nearly 180.
w
AO
Diff.
w
AO
Diff.
w
AO
Diff.
/
/ //
n
f
i n
II
O 1
/ //
II
155
5
10
15
20
25
3 23-9
19.74
16.43
13.17
9-95
6.77
3-35
3-3'
3.26
3.22
3.18
3.14
160
5
10
15
20
25
I 6.70
5-33
3-97
2.64
i-33
0.04
:%
33
3'
2 ?
26
165
10
20
30
40
50
o 15.85
14.98
14.16
13.38
12.63
11.91
0.87
0.82
0.78
0.75
0.72
0.69
155 30
35
40
45
50
55
3 3- 6 3
0.54
2 57-49
54.48
5i-5i
48.58
3.09
3-5
3.01
2.97
2.93
2.89
160 30
35
40
45
50
55
o 58.78
57-54
56-3 1
55-"
53-93
52.77
.24
23
.20
.18
.16
.14
166
10
20
30
40
50
11.22
10.57
9-95
9.36
8.80
8.26
0.65
0.62
0.59 1
0.56 ;
0.54 '
0.51
156
5
10
15
20
25
2 45.69
42.84
40.03
37.26
34-53
3i-83
2.85
2.8 1
2.77
2.73
2.70
2.66
161
5
10
15
20
25
o 51.63
50.50
49.40
48.32
47.26
46.21
13
.10
.08
.06
.05
.02
167
10
20
30
40
50
o 7.75
l:ll
6.37
5.96
5-57
0.48
0.46
0.44
0.41
0-39
0.37
156 30
35
40
45
50
55
2 29.17
26.55
23-97
21.43
18.92
16.44
2.62
2.58
2.54
2.51
2.48
2.44
161 30
35
40
45
50
55
o 45.19
44.18
43-*9
42.22
41.26
4-33
1. 01
0.99
0.97
0.96
o-93
0.92
168
10
20
30
40
50
o 5.20
4.84
4.51
4.20
3-9
3-62
0.36
-33
0.31
0.30
0.28
0.26
157
5
2 14.00
I I.'sO
2.41
162
5
o 39.41
^8.ci
o 90
169
10
o 3-36
3.11
0.25
10
15
20
25
11 Ox
9.22
6.89
4 .58
2.31
2.37
2-33
2.31
2.27
2.23
10
15
20
25
J 3
37.62
3 6 -75
35-9
35.06
0.89
0.87
0.85
0.84
0.82
20
30
40
50
2.88
2.66
2.46
2.27
0.23
0.22
0.20
0.19
0.18
157 30
35
40
45
50
55
2 0.08
i 57- 8 9
55-71
53-57
51.46
49-39
2.19
2.17
2.15
2. II
2.07
2.04
162 30
35
40
45
50
55
o 34.24
33-43
32.64
31.86
31.10
3-35
0.8 1
0.79
0.78
0.76
0.75
0.73
170
10
20
30
40
50
o 2.09
1.92
1.76
1.62
1.48
1.35
0.17
0.16
0.14
0.14
0.13
O.IZ
158
5
10
15
20
25
i 47-35
45-34
43-35
41-39
39-47
37-57
2.01
1-99
1.96
1.92
I.gO
1.87
163
5
10
15
20
25
o 29.62
28.90
28.20
27.51
26.83
26.16
0.72
0.70
0.69
0.68
0.67
0.65
171
10
20
30
40
50
o 1.23
1. 12
1.02
0.93
0.84
0.76
O.I I
O.IO
0.09
0.09
0.08
0.08
158 30
35
40
45
50
55
i 3570
33.87
32.06
30.28
28.52
26.80
/
1.8 3
1.81
1.78
1.76
1.72
1.70
163 30
35
40
45
50
55
o 25.51
24.88
24.25
23.64
23.04
22.45
0.63
0.63
0.61
0.60
0.59
0.57
172
10
20'
30
40
50
o 0.68
0.6 1
0.55
0.49
0.44
0.39
0.07
0.06
0.06
0.05
0.05
0.04
159
5
10
15
20
25
i 25.10
23-43
21.78
20.16
18.57
17.00
1.67
1.65
1.62
1.59
i-57
I.CC
164
5
10
15
20
25
o 21.88
21.31
20.76
20.22
I9.6 9
I9.l8
0.57
0.55
0.54
-53
0.51
0.51
173
10
20
30
40
50
o 0.35
0.31
0.27
0.24
0.21
0.19
0.04
0.04
0.03
0.03
O.O2
0.03
159 30
35
40
45
50
55
i 15.45
13.94
12.44
10.97
9-53
8.10
J J
1.51
1.50
1.47
1.44
1.43
1.40
164 30
35
40
45
50
55
o 18.6-7
18.17
17.69
17.21
16.75
16.29
0.50
0.48
0.48
0.46
0.46
0.44
174
175
176
177
178
179
o o.i 6
0.07
O.O2
O.OI
0.00
0.00
0.09
0.05
O.OI
O.OI
O.OO
o.oc
160
i 6.70
165
o 15-85
180
O.OO
fill
TABLE Vfll.
For finding the Time from the Perihelion in a Parabolic Orbit.
V
log N
Diff.
V
log J\"
Diff.
V
log^T
Diff.
/
/
O /
30
1
30
2
30
0.025 5763
.025 5749
.025 5707
.025 5638
.025 5542
.025 5418
J?
96
I2 4
152
30
30
31
30
32
30
0.020 7913
.O2O 6368
.020 4802
.O2O 3215
.020 1607
.019 9979
'545
1566
1587
1608
1628
1649
60
30
61
30
62
30
0.008 8644
.008 6458
.008 4277
.008 2103
.007 9934
.007 7774
2186
2181
2174
2169
2160 I
2153
3
, 30
4
30
5
30
0.025 5*66
.025 5087
.025 4881
.025 4647
.025 4386
.025 4097
179
2OO
*34
261
289
316
33
30
34
30
35
30
0.019 833
.019 6662
.019 4974
.019 3267
.019 1540
.018 9795
1668
1688
1707
1727
1745
1765
63
30
64
30
65
30
0.007 5621
.007 3477
.007 1343
.006 9220
.006 7108
.006 5008
2144
2134
2123
2112
2IOO
2086
6
30
7
30
8
0.025 378 1
02 5 3437
.025 3066
.025 2668
.025 2243
344
371
398
425
36
30
37
30
38
0.018 8030
.018 6248
.018 4448
.018 2629
.018 0794
1782
1800
1819
JJJ35
66
30
67
30
68
0.006 2922
.006 0849
.005 8792
.005 6750
.005 4725
2073
2057
2042
2025
2008
30
.025 1791
452
30
.017 8941
1869
30
.005 2717
1988
9
30
0.025 i5 1 1
.025 0805
506
39
30
0.017 7072
.017 5186
1886
69
30
0.005 07*9
.004 8760
1969
10
.025 0271
534
40
.017 3283
1903
70
.004 68 i i
1949
30
.024 9711
560
30
.017 1365
1918
30
.004 4884
1927
11
.024 9124
1*7
41
.016 9432
'933
71
.004 2980
1904
- O O _
30
.024 8510
614
641
30
.016 7483
1949
1963
30
.004 1 1 oo
I ooO
1855
12
30
13
80
14
30
0.024 7869
.024 7201
.024 6507
.024 5786
.024 5039
.024 4266
668
694
721
747
773
800
42
30
43
30
44
30
0.016 5520
.016 3542
.016 1550
.015 9545
.015 7526
.015 5495
1978
1992
2005
2019
2031
2045
72
30
73
30
74
30
0.003. 9*45
.003 7416
.003 5613
.003 3839
.003 2094
.003 0380
1829
1803
1774
1745
1714
1682
15
30
16
30
17
30
0.024 3466
.024 2641
.024 1789
.024 091 1
.024 0008
.023 9079
825
852
878
9 3
929
954
45
30
46
30
47
30
0.015 3450
.015 1394
.014 9326
.014 7247
.014 5157
.014 3057
2056
2068
2079
2090
2100
2110
75
30
76
30
77
30
O.002 8698
.002 7049
.002 5433
.002 3854
.002 2311
.002 0806
1649
1616
1543
1505
1465
18
30
19
30
20
30
0.023 8125
.023 7145
.023 6140
.023 5109
.023 4054
.023 2973
980
1005
1031 !
I0 55
1081 I
1105
48
30
49
30
50
30
0.014 0947
.013 8827
.013 6698
.013 4561
.013 2416
.013 0263
2120
2129
2137
2145
2153
2IDO
78
30
79
30
80
30
o.ooi 9341
.001 7917
.001 6535
.001 5196
.001 3903
.001 2656
1424
1382
1339
1293
1247
1198
21
30
22
0.023 1868
.023 0738
.022 9584
1130
1154
51
30
52
0.012 8103
.012 5936
,OI2 7764
2167
2I 7 2
81
30
82
o.ooi 1458
.001 0309
.000 9211
1149
1098
30
23
30
.022 8405
.022 72O2
.022 5975
1179
1203
1227
30
53
30
J / T
.012 1585
.on 9402
.on 7215
2179
2183
2187
30
83
30
.000 8l66
.000 7175
.000 6240
1045
991
935
xvn
1251
2191
oy D
24
30
25
O.O22 4724
.022 3449
.022 2151
1298
54
30
55
o.on 5024
.on 2829
.on 0632
2195
2197
84
30
85
o.ooo 5364
.000 4546
.000 3790
818
756
30
.022 0829
1322
30
.010 8432
22OO
30
.000 3096
h
26
.021 9484
I3 tl
56
.010 6231
22OI
86
.000 2468
O2S
30
.021 8ll6
1368
30
.010 4029
22O2
30
.000 1906
562
1390
2202
493
27
0.021 6726
57
o.oio 1827
87
o.ooo 1413
30
28
30
29
30
.O2I 5312
.021 3876
.021 2418
.021 0938
.020 9436
1414
I43 6
1458
1480
1502
30
58
30
59
30
.009 9625
.009 7424
.009 5225
.009 3028
.009 0834
22O2
2201
2I 99
2197
2I 94
2190
30
88
30
89
30
.000 0990
.000 0639
.000 0363
.000 0163
.000 0041
423
276
200
122
4 1
30
0.020 7913
60
0.008 8644
90
O.OOO 0000
612
TABLE VIII.
For finding the Time from the Perihelion in a Parabolic Orbit.
V
log N'
Difif.
V
log N'
Diflf.
V
log N'
Diff.
O 1
1
o t
90
30
91
0.000 0000
9.999 9876
999 957
124
3 6 9
120
30
121
9.963 1069
.962 0074
,960 8971
10995
11103
150
30
151
9.889 0321
.887 8738
.886 7259
11583
11479
30
92
30
999 88 93
999 8 39
999 6 944
614
854
1095
J 33 X
30
122
30
959 7764
.958 6454
.957 5046
1 1 207
11310
11408
11504
30
152
30
.885 5887
.884 4627
.883 3481
11372
11260 .
11146
11026
93
30
94
30
1 95
30
9-999 5613
999 44 6
.999 2246
.999 0215
99 8 7955
.998 5468
1567
1800
2031
2260
2487
2711
123
30
124
30
125
30
9.956 3542
955 '945
954 0*58
.952 8483
.951 6624
.950 4684
11597
11687
11775
11859
11940
12018
153
30
154
30
155
30
9.882 2455
.881 1552
.880 0775
.879 0129
.877 9616
.876 9242
10903
10777
10646
10513
10374
10232
96
30
97
30
98
30
9.998 2757
.997 9824
.997 6669
997 3*97
.996 9708
.996 5906
2933
3155
337^
3589
3802
4015
126
30
127
30
128
30
9.949 2666
948 0573
.946 8408
945 6174
944 3875
943 15*3
12093
12165
12234
12299
12362
12421
156
30
157
30
158
30
9.875 9010
.874 8922
873 8984
.872 9198
.871 9569
.871 0099
10088
9938
9786
9629
9470
9307
99
30
100
30
101
30
9.996 1891
.995 7666
995 3*34
994 59 6
994 3755
993 8712
4225
J$
4841
543
5242
129
30
130
30
131
30
9.941 9092
.940 6615
939 4 8 5
.938 1506
.936 8881
935 6213
12477
12530
12579
12625
12668
12707
159
30
160
30
161
30
9.870 0792
.869 1652
.868 2683
.867 3886
.866 5266
.865 6827
9140
8969
8797
8620
8439
8257
103
30
103
30
104
30
9-993 347
.992 8031
.992 2397
.991 6570
.991 0553
99 4347
5439
5 6 34
5827
6017
6206
6391
132
30
133
30
134
30
9-934 35 6
933 0763
.931 7987
.930 5183
.929 2353
.927 9501
12743
12776
12804
12830
12852
12871
162
30
163
30
164
30
9.864 8570
.864 0500
.863 2620
.862 4932
.861 7439
.861 0145
8070
7880
7688
7493
7294
7092
105
30
106
30
107
30
9.989 7956
.989 1380
.988 4622
.987 7685
987 0571
.986 3281
6576
6758
6937
7114
7290
7462
135
30
136
30
137
30
9.926 6630
925 3745
.924 0848
.922 7943
.921 5035
.920 2126
12885
12897
12905
12908
12909
12906
165
30
166
30
167
30
9.860 3053
.859 6164
.858 9482
.858 3010
.857 6750
.857 0704
6889
6682
j>47
O20O
6046
582 9
108
30
109
30
110
30
9.985 5819
.984 8186
.984 0385
.983 2418
.982 4288
.981 5996
7633
7801
7967
8130
8292
84.0
138
30
139
30
140
30
9.918 9220
.917 6321
.916 3433
.915 0559
.913 7703
.912 4870
12899
12888
12874
12856
12833
12808
168
30
169
30
170
30
9.856 4875
.855 9266
855 3878
854 8714
854 3775
.853 9065
!
5164
4939
4710
4481
111
30
112
30
113
30
9.980 7545
979 8938
979 0177
.978 1264
.977 2202
.976 2993
TJ
8607
8761
8913
9062
9209
Q"K1
141
30
142
30
143
30
9.911 2062
.909 9283
.908 6538
.907 3831
.906 1164
.904 8542
12779
12745
12707
12667
12622
12 573
171
30
172
30
173
30
9.853 4584
853 335
.852 6319
.852 2538
.851 8994
.851 5687
4249
4016
378i
3544
3307 '
3067
114
30
115
30
116
30
9-975 3 6 4
974 4H5
973 45io
972 4739
.971 4833
.970 4796
7 J J 3
9495
9635
9771
9906
10037
10167
144
30
145
30
146
30
9.903 5969
.902 3449
.901 0985
.899 8582
.898 6243
897 397 2
12520
12464
12403
12339
12271
12198
174
30
175
30
176
30
9.851 2620
.850 9794
.850 7209
.850 4868
.850 2770
.850 0917
2826
2585
2341
2098
1851 -
1608 j
117
30
118
30
119
30
9.969 4629
.968 4337
.967 3920
.966 3382
.965 2726
.964 1954
10292
10417
'538
10656
10772
10885
147
30
148
30
149
30
9.896 1774
.894 9652
.893 7610
.892 5652
.891 3782
.890 2004
I2I22
J2042
II958
II870
JI778
n6?3
177
30
178
30
179
30
9.849 9309
.849 7948
.849 6833
.849 5966
.849 5346
.849 4974
1361
1115 ;
867
620
37* '
124
120
9.963 1069
150
9.889 0321
180
9.849 4850
613
TABLE IX.
F>r finding the True Anomaly or the Time from the Perihelion in Orbits of great eccentricity
X
A
Difif.
B
Difif.
c
B'
Difif.
C'
n
n
it
If
II
II
It
O.OO
0.000
0.000
o.ooo
o.ooo
1
O.OO
O.OO
0.000
o.ooo
0.000
0.000
2
0.0 1
O.OI
0.000
0.000
0.000
o.oco
3
0.05
0.04
o.ooo
o.ooo
o.ooo
o.ooo
4
0.12
0.07
0.000
o.ooo
0.000
o.ooo
O.I I
5
0.23
o. 1 6
o.ooo
0.000
0.000
o.ooo
6
0.35
0.000
o.ooo
o.ooo
0.000
7
0.02
0.23
o.ooo
0.000
0.000
o.ooo
8
o-93
0.31
0.000
o.ooo
o.ooo
o.ooo
1 9
1.77
0.40
o.ooo
0.000
0.000
o.ooo
J J
0.49
1 10
1.82
o.ooo
0.000
0.000
o.ooo
11
2.42
o.6c
0.000
0.000
0.000
0.000
12
7.14
0.72
o.ooo
0.000
o.ooo
o.ooo
13
J T^
3-99
0.85
0.000
0.000
o.ooo
0.000
14
4-99
.00
.14
O.OO I
0.000
0.001
o.ooo
15
6.13
0.001
o.ooo
O.OOI
o.ooo
16
7.4-1
.30
O.OO2
.001
0.000
0.00 1
.000
o.coo
17
/TO
8.90
f
0.002
.000
o.ooo
O.OO2
.001
o.ooo
18
19
10.55
12.40
6 5
.85
2.05
0.003
0.004
.001
.001
.001
0.000
o.ooo
0.002
0.003
.000
.001
.001
0.000
o.oco
20
21
14.45
l6.7O
2.25
0.005
0.006
.001
0.000
o.ooo
0.004
0.005
.001
0.000
o.ooo
22
I9.l8
2.48
0.008
.002
0.000
0.006
.001
0.000
23
21.89
2.71
O.OIO
.002
o.ooo
0.008
.002
o.ooo
24
X
24.81
2.94
O.OI 2
.002
0.000
O.OIO
.002
0.000
T J
3.20
.002
.002
25
28.07
0.014
0.000
0.012
o.ooo
26
27
m ** J
31.48
35.20
3-45
3-72
T.
0.017
0.020
.003
.003
0.000
0.000
0.014
0.017
.002
.003
o.ooo
0.000
28
29
39-'9
43-47
3-99
428
4-57
0.025
0.030
.005
.005
.005
0.000
0.000
O.O2O
0.024
.003
.004
004
o.ooo
0.000
30
31
32
33
34
48.04
52.91
58.09
6 3-59
69.42
4.87
5.18
5-5
5.83
6.15
0.035
0.041
0.047
0.055
0.064
.006
.006
.008
.009
.009
0.000
0.000
0.000
0.000
o.ooo
0.028
0.033
0.039
0.045
0.052
.005
.006
.006
.007
.008
o.ooo
0.000
o.ooo
0.000
0,000
35
36
37
38
75-57
82.07
88.92
96.12
6.50
6.85
7.20
0.073
0.084
0.096
0.109
.01 1
.012
.013
0.000
o.ooo
0.000
o.ooo
0.060
0.068
0.078
0.088
.008
.010
.010
o.ooo
0.000
o.ooo
0.000
39
103.68
7.56
7-93
0.123
.014
.016
0.000
O.1OO
.012
.013
o.ooo
40
41
43
43
44
in. 61
119.92
128.62
137.70
147.18
8.31
8.70
9.08
9.48
0.139
0.156
0.175
0.196
0.218
.017
.019
.021
.022
0.000
o.ooo
o.ooo
o.ooo
o.ooo
0.113
0.127
0.142
0.159
0.177
.014
.015
.017
.018
0.000
o.ooo
0.000
0.000
o.ooo
9.87
.025
.020
45
46
47
48
49
157.05
167.34
178.04
189.16
200.71
10.29
10.70
II. 12
11-55
11.98
0.243
0.269
0.298
0.328
0.361
.026
.029
.030
.033
.036
o.ooo
0.000
o.ooo
0.000
o.ooo
0.197
0.219
0.242
0.267
0.294
.022
.023
.025
.027
.029
0.000
o.ooo
0.000
o.ooo
o.ooo
5O
51
52
53
54
212.69
225 ic
237-95
251.25
265.01
12.41
12.85
13.30
13.76
14.20
Q-397
0.436
0.477
0.521
0.567
39
.041
.044
.046
.050
o.ooo
0.000
O.OOJ
0.00 1
O.OO I
0.323
0-354
0.388
0.424
0.462
.031
.034
.036
.038
.040
0.000
o.ooo
o.ooo
0.000
o.ooo j
55
56
57
58
59
279.21
293.88
309.02
324.62
340.70
14.67
I5.I 4
15.60
16.08
16.56
0.617
0.671
0.727
0.787
0.851
.054
.056
.060
.064
.068
O.OO I
0.002
O.OO2
0.002
O.OO2
0.502
0.546
0.592
0.641
0.693
.044
.040
.049
.052
.056
o ooo
0001
O OOI
C OOI
O.OOI
60
357-^6
0.919
0.003
0.749
O.OO2
614
TABLE IX.
For finding the True Anomaly or the Time from the Perihelion in Orbits of great eccentricity.
X
A
Diff.
B
Diff.
c
B 1
Diff.
C'
n
,/
n
II
n
//
n
60
61
62
357-26
374-3
391.84
17.04
17-54
18.02
0.919
0.990
.066
.071
.076
0.003
0.003
0.003
0.749
0.807
0.869
.058
.062
.066
0.002
0.002
O.OO2
63
64
409.86
428.38
18.52
19.02
'45
-230
SI
.088
0.004
0.004
0.935
1.004
.069
73
0.002
O.OO2
65
66
67
68
69
447.40
466.92
486.96
507.51
528.58
19.52
20.04
2-55
21.07
21.59
.318
.411
.510
.613
.721
.093
.099
.103
.108
.114
0.004
0.005
0.005
0.006
0.006
.077
.154
235
.321
.411
.077
.081
.086
.090
.094
0.003
o 003
0.003
0.004
0.004
70
71
72
73
74
550.17
572.29
59494
618.12
641.85
22.12
22.65
23.18
23.73
24.28
.835
954
2.078
2.209
2-345
.119
.124
'.\\6
H3
0.007
0.007
0.008
0.009
0.009
.505
.605
.709
.819
934
.100
.104
.110
.115
.121
0.004
0.005
0.005
0.006
0.006
75
666.13
_
2.488
0.0 1
2.055
,-f.
0.007
76
690.96
24.83
, f O
2.637
.149
O.OII
2.181
.1 2O
0.007
77
78
79
716.34
742.29
768.81
25.38
25.95
26.52
27.09
2-793
2.956
3.125
.163
.169
.177
0.012
0.013
0.014
2.314
2-453
2-599
139
.146
153
0.008
0.008
0.009
80
81
82
83
84
795-9
8 5 i'.8 4
880.70
910.16
27.67
28.27
28.86
29.46
30.07
3.302
3.486
3.678
3.878
4.087
.184
.192
.200
.209
.216
0.015
0.016
0.017
0.018
0.020
2.752
2.912
3-79
3-255
3-439
.l6o
S3
.184
.192
0.010
O.OII
O.OI2
0.013
0.014
85
86
87
88
89
940.23
970.92
1002.24
1034.20
1066. 81
30.69
31.96
32.6l
33-27
4-33
4.529
4.764
5.008
5.262
.226
235
244
.254
.265
0.021
0.023
0.024
O.O26
0.028
3.631
3-833
4.044
4.266
4.498
.Z02
.211
.222
.232
.243
0.015
0.016
0.018
0.019
0.021
90
1100.08
5 527
0.030
4.741
ire 1
0.023
91
92
93
94
1134.02
1168.64
1203.95
~ 1239.97
33-94
34.62
35-31
36.02
3 6 -75
5.801
6.087
6.385
6.694
.298
39
.322
0.032
0.034
0.036
0.038
4.996
5.263
5-544
5.838
255
.267
.281
.294
39
O.O25
0.027
0.029
0.032
95
98
97
1276.72
1314.21
37-49
38.24
7.016
7.350
7.698
334
.348
0.041
0.044
0.047
6.147
6.471
6.812
.324
341
"9 C" A
0.035
0.038
0.041
98
1391.46
39.01
8.060
.362
0.050
7.171
359
178
0.045
99
1431.27
40.61
8.437
377
392
0.0 5 3
7-549
3/
397
0.049
100
30
101
30
102
30
1471.88
1492.50
I534-38
1555.64
1577.12
20.62
20.83
21.05
21.26
21.48
21.70
8.829
9.032
9.238
9-449
9.664
9-883
S3
.211
.215
.219
.225
0.056
0.058
0.060
0.062
0.064
0.066
7.946
8.152
8.364
8.582
8.805
9.035
.206
.212
.218
.223
.230
0.053
0.055
0.058
O.o6o
0.063
O.o66
i 103
1598.82
10.108
0.068
9.271
0.069
30
104
1620.75
1642.91
21.93
22.16
10.337
10.570
.229
233
0.070
0.072
9-5I3
9.761
:^8
.2;6
0.072
0.075
30
105
30
1665.30
1687.93
1710.80
22.39
22.63
22.87
23. 12
10.809
11.053
11.302
239
.244
.249
0.074
0.077
0.079
10.017
10.280
10.550
.2%
.270
.2 7 8
0.078
O.o82
0.085
106
30
, 107
30
1 108
30
I733-92
1757.28
1780.90
1804.77
1828.90
1853.30
23.36
23.62
23.87
24.13
24.40
24.67
"557
11.817
12.083
12.354
12.632
12.916
.260
.266
.271
.2 7 8
.284
.291
0.082
0.084
0.087
0.093
o 096
10.828
11.114
1 1.408
11.711
12.022
12-343
.286
.294
33
.311
.321
.330
0.089
0.093
0.098
0.102
O.IO7
O.I I 2
109
1877.97
_f i
13.207
0.099
12.673
O.II7
615
TABLE IX.
For tindm fe the True Anomaly or the Time from the Perihelion in Orbits of great eccentricity
A
Diff.
B
Diff.
c
Diff.
B'
Diff.
C"
Diff.
/
n
n
n
n
11
n
II
n
109
30
110
30
111
30
1877.97
1902.91
1928.13
1953.64
1979.44
2005.54
24.94
25.22
25.51
25.80
26.10
26.40
13.207
13.504
13.808
14.119
14.438
14.764
297
-34
311
.319
.320
-333
0.099
0.102
0.106
0.109
0.113
0.116
.003
.004
.003
.004
.003
.004
12.673
13.013
13-363
I3-724
14.095
14.478
34
.350
.361
.371
.383
.396
0.117
0.122
0.128
0.134
0.141
0.148
,005
.006
.006
.007
.007
.007
112
30
113
2031.94
2058.64
2085.66
26.70
27.02
15.097
15-439
15-789
.342
35
O.I 20
0.124
0.128
.004
.004
14.874
15.282
15.702
.408
.420
0.155
O.l62
0.170
-.1:1
30
114
30
2113.00
2140.66
2168.66
27-34
27.66
28.00
28.34
16.148
16.515
16.892
359
.367
:]ll
0.132
0.137
0.142
.004
.005
.005
.005
16.135
16.583
17.045
433
.448
.462
-477
0.178
0.187
o 196
.008
.009
.009
.010
115
30
116
3D
117
30
2197.00
2225.69
2254.73
2284.13
2313.91
2344.06
28.69
29.04
29.40
29.78
3-l5
3-54
17.278
17.674
18.080
18.496
18.924
19-363
.396
.406
.416
.428
.439
45
0.147
0.152
0.157
0.162
0.168
0.174
.005
.005
.005
.006
.006
.006
17.522
18.015
18.524
19.050
19-594
20.156
-493
59
.526
-544
.562
.582
0.206
0.2 1 6
0.227
0.239
0.251
0.264
.010
.Oil
.012
.012
.013
.013
118
30
119
30
120
30
2374.60
2405.54
2436.88
2468.64
2500.83
2533-45
3-94
31-34
31.76
32.19
32.62
33-o6
19.813
20.276
20.751
21.240
21.742
22.258
4 6 3
475
489
.502
.516
531
0.180
0.186
0.193
O.200
0.207
O.2I4
.006
.007
.007
.007
.007
.008
20.738
21-339
21.962
22.606
23.273
23.964
.601
.623
.644
.667
.691
.716
0.277
0.291
0.306
0.322
0-339
0-357
.014
.015
.Ol6
.017
.018
.019
121
30
122
30
123
30
2566.51
2600.03
2634.02
2668.49
2703.46
2738.93
33-52
3399
34-47
34-97
35-47
35-98
22.789
23.336
23.898
24.477
25.073
25.687
-547
.562
579
.596
.614
6 33
0.222
0.230
0.239
0.248
0.258
0.268
.008
.009
.009
.010
.010
.010
24-680
25.422
26.191
26.988
27.815
28.673
.742
.769
-797
.827
.858
.891
0.376
0.396
0-417
0-439
0.463
0.488
.020
.O2I
.022
.024
.025
.027
124
30
125
30
126
30
2774.91
2811.43
2848.50
2886.13
2924.33
2963.12
36.52
37.07
37-63
38.20
38.79
39-4 1
26.320
26.973
27.646
28.341
29.057
29.797
.653
.673
.695
.716
.740
-765
0.278
0.289
0.300
0.312
0.325
0.338
.Oil
.Oil
.012
.013
.013
.014
29.564
30.489
3!-45o
3M48
33.485
34-563
.925
.961
.998
037
.078
.ri2
-5i5
0.544
0.606
0.640
0.676
.029
.030
.032
034
.036
.039
127
30
128
30
129
30
3002.53
3042.56
3083.23
31*4-57
3166.59
3209.31
40.03
40.67
4L34
42.02
42.72
43-45
30.562
31-351
32.167
33-on
33.885
34-789
.789
.816
.844
-874
.904
93 6
0-352
0.367
0.382
0.398
0-415
0-433
.015
.015
.Ol6
.017
.018
.019
35.685
36.852
38.067
39-33 1
40.649
42.022
.167
.215
.264
. 3 !8
-373
.430
0-715
0-757
0.800
0.846
0.896
0.949
.042
.043
.046
.050
-053
.050
130
20
40
131
20
40
3252.76
3282.13
3311-85
3341.90
3372.31
3403.09
29.37
29.72
30.05
30.41
30.78
3i.!4
35-725
36.367
37.025
37.699
38.389
39.097
.642
.658
.674
.690
.708
.725
0.452
0.465
0.479
0-493
0.508
0.523
.013
.014
.014
.015
.015
.Ol6
43.452
44-439
45-455
46.500
47-575
48.682
0.987
.016
.045
075
.i7
.138
.005
045
.087
130
175
.223
.040
.042
043
-045
.048
.050
132
20
40
, 133
20
40
3434-23
3465.74
3497-63
3529.91
3562.60
3595- 6 9
31-51
31-89
32.28
32-69
33-09
33-51
39822
40.564
41-326
42.108
42.910
43-733
.742
.762
.782
.802
.823
.843
0.539
0-555
0-572
0.590
0.609
0.629
.Ol6
.017
.018
.019
.020
.O2O
49 820
50.992
52.199
53-442
54-723
56.042
.172
.207
:3?
319
359
.273
-325
379
.436
495
558
.052
054
.057
.059
.063
.005
134
20
40
135
20
40
3629.20
3663.13
3697.50
3732.31
3767.58
3803.31
33-93
34-37
34.81
35.27
35-73
36.21
44.576
45.442
46-331
47.245
48.183
49.147
.866
.889
.914
.964
.991
0.649
0.669
0.691
o.7H
0-738
0763
.020
.022
.023
.024
.025
.025
57.401
58.802
60.247
61.736
63.273
64.857
.401
445
.489
.537
r 4
.634
.623
.692
-764
839
.917
2.000
.069
.072
.075
.078
.083
.087
136
3839.52
50.138
0.788
66.491
2.087
616
TABLE IX,
f 01 finding the True Anomaly or the Time from the Perihelion in Orbits of great eccentricity
X
A
Diff.
B
Diff.
C
Diff.
B'
Diff.
C"
Diff.
1
if
n
II
n
n
136
20
40
137
20
40
3839-5*
3876.21
3913.41
3951.12
3989.35
4028.11
36.69
37.20
37-71
38.23
38.76
39-3 1
50.138
51.156
52.203
53.280
54.388
55.528
1.018
1.047
.077
.108
.140
.174
0.788
0.815
0.843
0.873
0.904
0.936
.027
.028
.030
.031
.032
33
66.491
68.178
69.920
71.718
73-575
75-493
.687
74*
798
.857
.918
.982
2.087
2.178
2.274
*-375
2.480
2.591
.091
.096
.101
.105
.Hi
.117
138
20
40
139
20
40
4067.42
4107.28
4147.72
4188.75
4230.38
4272.63
39-86
40.44
41.03
41.63
42.25
42.89
56.702
57.910
59-'54
60-436
61.757
63.119
.208
.244
.282
.321
.362
.404
0.969
.004
.041
.079
.119
.161
.035
.040
.042
.044
77-475
79-5*3
81.641
81.830
86.094
88.436
2.048
2.118
2.189
2.264
2.342
2.424
2.708
2.831
2.960
3.096
3.239
3-39
.123
.129 ,
.136
.143
.151
.159
140
20
40
141
20
40
4315.52
4359.06
4403.26
4448.15
4493-73
4540.03
43-54
44.20
44.89
45.58
46.30
47.04
64.523
65.971
67.465
69.007
70.599
72.243
.448
494
54*
.592
.644
.698
.205
.251
.299
350
.404
.460
.046
.048
.051
.054
.056
.058
90.860
93-369
95.967
98.657
101.443
104.331
2.509
2.598
2.690
2.786
2.888
2.993
3-549
3.717
3-893
4.080
4.277
4-484
.168
.176
.187
.197
.207
.220
142
10
20
30
40
50
4587.07
4610.88
4634.88
4659.07
4683.46
4708.05
23.81
24.00
24.19
24.39
24.59
24.79
73-941
74.811
75-695
76-595
77.509
78.439
0.870
0.884
0.900
0.914
0.930
0.946
.518
549
.580
.612
.645
.679
.031
.031
.032
.033
34
35
107.124
108.861
110.427
I I 2. 022
113.646
II5.30I
595
.624
655
.685
4.704
4.819
4.936
5-057
5.181
5.309
.115
.117
.121
.I2A
.128
.131
143
10
20
30
40
50
4732.84
4757.84
4783.05
4808.46
4834.10
4859-95
25.00
25.21
25.41
25.64
25.85
26.07
79-385
80.347
81.325
82.321
83-333
84.363
0.962
0.978
0.996
.012
.030
.048
.714
749
.786
.823
.862
.901
035
.037
.037
39
.039
.041
116.986
118.704
120.452
122.233
124.049
125.899
.718
.748
.781
.816
.850
.886
5.440
5-575
5-7I5
5-858
6.005
6.157
.140
'43
.147
'\\l
144
10
20
30
40
50
4886.02
4912.31
4938.81
4965.58
4992.56
5019.78
26.29
26.52
26.75
26.98
2.7.22
85.411
86.478
87.564
88.668
89-793
90.938
.067
.086
.104
.125
.145
.165
94*
.984
2.026
2.070
2.116
2.162
.042
.042
.044
.046
.046
127.785
129.707
131.666
133.661
135.698
137-774
.922
959
997
035
.076
.116
6.313
6.473
6.639
6.809
6.984
7.165
.l6o
.166
.170
!i86
145
10
20
30
40
50
5047.23
5074.93
5102.88
5131.08
5*59-53
5188.24
27.70
27.95
28.20
28.45
28.71
28.97
92.103
93.290
94.498
95.729
96.982
98.259
.187
.208
.231
*53
*77
.300
2.210
2.259
2.309
2.361
2.414
2.469
.049
.050
.052
053
.055
.057
139.890
142.048
144.249
146.494
148.784
151.120
2.158
2.2OI
2.245
2.290
2.316
2.383
7-351
7-543
7.740
8.'?5 3 3
.192
.197
.203
.210
.216
.223
146
10
20
30
40
50
5217.21
5246.45
5*75-95
535-73
5335-79
5366.13
29.24
29.50
29.78
30.06
3-34
10.61
99-559
100.884
102.234
103.610
105.012
106.441
3*5
350
376
.402
4*9
.456
2.526
2.584
2.643
2.704
2.767
2.833
.058
.059
.061
.063
.066
.067
I53-503
155-934
158.415
160.947
163.531
I66.I68
2.431
2.4I
2.512
2.584
2.637
2.692
8.592
8.822
9.060
9.304
9-555
9.815
.230
.238
.244
! 2 68
147
10
20
30
40
50
5396.76
5427.67
545888
5490.39
5522.20
5554-33
j j
30.91
31.21
3 /.8i
32.13
32.44
107.897
109.182
110.896
112.439
114.013
115.619
.485
.514
543
637
2.900
2.969
3.040
3- IJ 3
3.188
3.266
.069
.071
073
.075
.078
.080
168.860
171.608
174.414
177.280
180.206
183.194
2.748
2.806
2.866
2.926
2.988
3.052
10.083
10.359
10.645
10.940
11.244
11.558
.276
.286 -
.295
.304
.314
3*5
148
10
20
30
40
50
149
5586.77
5619.52
5652.60
5686.01
57I9-75
5753-83
5788.26
33-o8
33-74
34.08
34-43
117.256
118.926
120.631
122.370
124.144
125.955
127.804
.670
.705
739
774
.811
.849
3-346
3-4*8
3-5I3
3.601
3.691
3.881
.082
.085
.088
.090
.093
.097
186.246
189.364
192.549
195.804
199.130
202.528
2O6.OO2
3.118
3-185
3-*55
3-3*6
3-398
3-474
11.883
12.218
12.564
12.921
13.291
13.673
14.067
335
.346
-357
.370
.382
394
617
TABLE X.
For finding the True Anomaly or the Time from the Perihelion in Elliptic and Hyperbolic Orbits.
A
Ellipse.
Hyperbola.
log B
Diff.
log
log I. Diff.
log
half II. Diff.
logJB
Diff.
logtf
log I. Diff.
log
half II. Diff.
0.000
0.000
o.oo
oooo
7
0.000 OOOO
4.23990
1.778
oooo
o.ooo oooo
4.2 3 98 2 n
1.771
.01
0007
/
27
.001 7432
.24286
.783
0007
27
9.998 2688
.23686
.767
.02
0030
3
77
.003 4985
.24583
.788
0030
"3
77
.996 5493
23392
.762
.03
0067
31
e-i
.005 2659
.24885
794
0067
3 /
r I
.994 8414
.23098
758
.04
0120
68
.007 0457
.25190
799
0118
66
993 J 45
.22807
753
&.05
0188
84.
0.008 8381
4.25497
1.805
0184
81
9.991 4599
4.225i8 M
1.748
.06
0272
T
00
.010 6432
.25806
.811
0265
989 7859
.22230
-743
.07
0371
y s
I 14
.012 4613
.26116
.816
359
IOQ
.988 1231
.21943
739
.08
.09
0485
0615
*-t
130
147
.014 2924
.016 1367
.26427
.26741
.821
.827
0468
0591
* W 7
123
.986 4711
.984 8298
.21659
.21376
734
.730
O.IO
0762
162
0.017 9945
4.27057
I-833
0728
I C2
9.983 1992
4-21094*
1.725
.11
0924
178
.019 8659
.27376
839
0880
I6 5
.981 5791
.20815
.720
.12
1 102
IQ4.
.021 7511
.27697
845
1045
979 9 6 94
.20537
.716
'3
.14
1296
1507
7T
211
227
.023 6503
.025 5637
.28020
.28344
.851
.857
1223
1416
193
2O6
978 3699
.976 7805
.20260
.19986
.706
^
1734
1977
243
26l
0.027 49 r 6
.029 4340
4.28670
.28999
1.863
.869
1622
1842
220
277
9.975 201 1
973 6316
4.19712,,
.19440
1.700
695
.19
2238
2515
2809
277
294
3 II
03 1 39*3
.033 3636
.035 3511
.29331
.29665
.30001
.875
.882
.888
2075
2321
2581
v
246
260
273
.972 0719
.970 5218
.968 9813
.19170
.18901
.18633
.690
.685
.679
O.2O
.21
3120
3448
328
0.037 3542
.039 3730
4-3339
.30679
1.895
.901
2854
3140
286
9.967 4502
.965 9285
4.18367,
.18102
1.672
.666
.22
3793
4156
363
78l
.041 4077
.043 4585
.31022
.31368
.908
.915
3439
375i
312
.964 4159
.962 9124
.17840
17579
.661
.655
.24
4537
J" *
398
.045 5259
.31716
.922
4076
338
.961 4180
.17319
.649
'H
4935
535 1
4 l6
4.74.
0.047 6099
.049 7109
4.32066
.32418
1.929
936
4414
4765
767
9.959 9324
958 455 6
.16803"
1.643
637
.27
5785
T JT
4.^2
.051 8290
32773
943
5128
376
95 6 9875
.16547
.631
.28
6237
TO
4.71
.053 9646
.951
554
955 5281
.16292
.625
.29
6708
TV
488
.056 1179
33492
.958
5893
401
954 77i
.16038
.618
0.30
7196
0.058 2893
4.33856
1.966
6294
9.952 6346
4-i5785n
1.613
TABLE X, Part II,
r
Ellipse.
Hyperbola.
T
Ellipse.
Hyperbola.
A
Diff.
A
Diff.
A
Diff.
A
Diff.
o.oo
.01
.02
.03
.04
O.OOOOO
.00992
.01969
.02930
.03877
992
977
961
947
971
o.ooooo
.01008
.02033
.03074
.04132
1008
1025
1041
1058
1077
0.20
.21
.22
.24
0.17266
.18008
.18740
.19462
.20174
742
732
722
712
704
0.23867
.25309
.26779
.28280
.29813
1442
1470
1501
1533
1564
0.05
.06
.09
0.04808
.05726
.06630
.07521
.08398
oo rh M r^ w
H O ON t^so
ON ONOO OO OO
0.05209
.06303
.07417
.08550
.09702
1094
1114
"33
1152
"73
0.25
.26
.27
.28
.29
0.20878
21573
.22258
.22935
.23604
695
685
677
669
661
0.31377
O.IO
.11
.12
.I 4
0.09263
.10116
.10956
.11783
.12599
853
840
827
816
805
0.10875
.12069
.13285
.14522
.15782
1194
1216
1237
1260
1285
0.30
32
33
34
0.24265
.24917
.25561
=26198
.26826
652
644
637
628
621
O.I|
.16
17
.18
0.13404
.14198
.14981
J 5753
794
783
772
0.17067
18375
.19709
.21068
1308
1334
1359
T 1%f\
|
0.27447
.28061
.28668
.29268
614
607
600
i .19
.16515
762
75 1
.22454
1413
39
.29860
592
586
0.20
0.17266
0.23867
0.40
0.30446
618
TABLE XL
For the Motion in a Parabolic Orbit.
1?
log/*
Diff.
,
lOg /A
Diff.
lOg/4
Diff.
0.000
.001
0.000 OOOO
.OOO OOOO
0.060
.061
o.ooo 0652
.000 0674
22
0.120
.121
o.ooo 2617
.000 2661
44
.002
.000 0001
I
.062
.000 0697
23
.122
.000 2705
44
.oo 3
.000 0002
I
J
.063
.000 0719
22
.123
.000 2750
45
.004
.000 0003
I
.064
.000 0742
24
.124
.000 2795
4 6*
O.005
.oo6
o.ooo 0004
.000 0006
2
0.065
.066
o.ooo 0766
.000 0790
24
0.125
.126
o.ooo 2841
.000 2886
45
.007
.008
.000 0009
.000 0012
3
3
.067
.068
.000 0814
.000 0838
24
24
.127
.128
.000 2933
.OOO 2Q7Q
46
.009
.000 0015
3
3
.069
.000 0863
25
.129
S I 7
.OOO 3O26
s
O.OIO
o.ooo 0018
0.070
o.ooo 0888
0.130
o.ooo 3074
T
.Oil
.OOO OO22
4
.071
.000 0914
tft
'3 1
.000 3121
47
.012
.000 0026
4
5'
.072
.000 0940
20
.132
.000 3169
48
.0! 3
.000 0031
.073
.000 0966
20
.000 3218 49
.014
.000 0035
I
.074
.000 0993
27
27
'34
.000 3267
49
49
O.OI5
o.ooo 0041
0.075
0.000 1020
o 135
o.ooo 3316
.Ol6
.000 0046
I
.076
.000 1047
27
-0
.136
.000 3365
49
.017
.000 0052
.077
.000 1075
2o
-0
.137
.000 3415
5
.018
.000 0059
7
5
.078
.000 1103
2o
.138
.000 3466
5 1
.019
.000 0065
7
.079
.000 1132
29
29
'39
.000 3516
5
51
0.020
o.ooo 0072
0.080
o.ooo 1161
0.140
o.ooo 3567
J
.021
.022
.000 0080
.000 0088
8
.081
.082
.000 1190
.000 1219
29
29
.141
.142
.000 3619
.000 3671
5 2
5*
.023
.000 0096
g
.083
.000 1249
3
.000 3723
5 2
.024
.000 0104
9
.084
.000 1280
3 1
3 1
.144
.000 3775
53
0.025
o.ooo 0113
0.085
o.ooo 1311
0.145
o.ooo 3828
.026
.000 0122
9
.086
.000 1342
3 1
.146
.000 3882
54
.027
.000 0132
IO
.087
.000 1373
3 1
.147
.000 3935
53
.028
.029
.000 0142
.000 0152
IO
10
II
.088
.089
.000 1405
.000 1437
3 2
3 2
33
.148
.149
.000 3989
.000 4044
54
55
cc
0.030
o.ooo 0163
I I
0.090
o.ooo 1470
A ->
0.150
o.ooo 4099
J J
.031
.000 0174
.091
.000 1502
3 2
.151
.000 4154
55
.032
.000 0185
1 1
1 2
.092
.000 1536
34
-
.152
.000 4209
11
33
.000 0197
93
.000 1569
33
J 53
.000 4265
5
34
.000 0209
I 2r
'3
.094
.000 1603
34
35
.154
.000 4322
56
0.035
O.OOO O222
T ^
0.095
o.ooo 1638
*
0.155
o.ooo 4378
.036
.000 0235
J 3
.096
.000 1673
35
.156
.000 4435
-O
037
.038
.000 0248
.000 0262
14
.097
.098
.000 1708
.000 1743
35
35
'57
.158
.000 4493
.000 4551
5 s
$
.039
.000 0275
3
.099
.000 1779
g
.159
.000 4609
5 8
15
3
59
0.040
o.ooo 0290
0.100
o.ooo 1815
0.160
o.ooo 4668
.0
.041
.000 0304
J 4
I
.101
.000 1852
37
.161
.000 4726
r
.042
.000 0320
o
.102
.000 1889
37
.162
.000 4786
Oo
(\r\
.043
.000 0335
*5
.103
.000 1926
11
.163
.000 4846
OO
fin
.044
.000 0351
I O
16
.104
.000 1964
3
38
.164
.000 4906
OO
60
0.045
.046
.047
o.ooo 0367
.000 0383
.000 0400
16
17
0.105
.106
.107
0.000 2002
.000 2040
.000 2079
3
39
0.165
.166
.167
o.ooo 4966
.000 5027
.000 5088
61 I
61
.048
.000 0417
Q
.108
.000 21 1 8
39
.168
.000 5150
ft
.049
.000 0435
15
18
.109
.000 2158
4
4
.169
.000 5212
62
0.050
o.ooo 0453
I
O.IIO
o.ooo 2198
0.170
o.ooo 5274
fi
.051
.000 0471
I o
.III
.000 2238
4
.171
.000 5337
?3
.052
53
.054
.000 0490
.000 0509
.000 0528
19
19
20
.112
.113
.114
.ooc 2279
.000 2320
.000 2361
H IH M f<
.172
.173
.174
.000 5400
.000 5464
.000 5518
i*
64
6 4
0.055
o.ooo 0548
0.115
o.ooo 2403
0.175
o.ooo 5592
,
.056
.000 0568
20
.116
.000 2445
42
.176
.000 5657
57
.000 0589
21
.117
.000 2487
42
.177
.000 5722
ftC
.000 0610
21
.118
.000 2530
43
.178
.000 5787
05
ftfi
.059
.000 0631
21
21
.119
.000 2573
43
44
.179
.000 5853
OO
66
0.060
o.ooo 0652
0.120
o.ooo 2617
0.180
o.ooo 5919
619
TABLE XI.
For the Motion in a Parabolic Orbit.
log/*
Diff.
n
>o g .
Diff.
,
*
Diff.
o.i8o
.181
o.ooo 5919
.000 5986
6?
0.240
.241
o.ooi 0603
.001 0693
90
0.300
.301
o.ooi 6733
.001 6848
"5
.182
.000 6053
07
.242
.001 0784
9 1
.302
.001 6963
115
1 1 6
.183
.184
..ooo 6120
.000 6l88
68
68
243
.244
.001 0875
.001 0966
9 1
91
92
33
.304
.001 7079
.001 7195
116
117
0.185
.186
.187
.188
.189
o.ooo 6256
.000 6325
.000 6393
.000 6463
.000 6532
69
68
7
69
7
0.245
.246
.247
.248
.249
o.ooi 1058
.001 1150
.001 1242
.001 1335
.001 1429
92
92
93
94
93
0.305
.306
37
.308
.309
o.ooi 7312
.001 7429
.001 7546
.001 7664
.001 7783
117
117
118
119
118
0.190
o.ooo 6602
0.250
o.ooi 1522
n e
0.310
o.ooi 7901
.191
.192
193
.194
.000 6673
.000 6744
.000 6815
.000 6887
72
72
.251
.252
253
.254
.001 1617
.001 1711
.001 1806
.001 1901
95
94
95'
96
3"
.312
3'3
.314
.001 8020
.001 8140
.001 8260
.001 8381
119
120
120
121
121
0.195
o.ooo 6959
0.255
o.ooi 1997
06
-3'5
o.ooi 8502
121
.196
.000 7031
mm
.256
.obi 2093
yu
.316
.001 8623
I 22
.197
.198
.199
.000 7104
.000 7177
.000 7250
73
73
73
74
.257
.258
.259
.001 2190
.001 2287
.001 2384
97
97
98
.317
.318
.319
.001 8745
.001 8867
.001 8989
122
122
124
0.200
.201
o.ooo 7324
.000 7399
75
0.260
.261
o.ooi 2482
.001 2580
98
0.320
.321
o.ooi 9113
.001 9236
123
.202
.000 7473
74
.262
.001 2679
99
.322
.001 9360
124
.203
.000 7548
11
.263
.001 2778
99
.001 9484
124
.204
.000 7624
70
76
.264
.001 2877
99
100
3*4
.001 9609
125
125
O.205
o.ooo 7700
76
0.265
o.ooi 2977
IOO
0.325
o.ooi 9734
126
.206
.207
.208
.000 7776
.000 7853
.000 7930
/ u
77
77
*7*t
.266
.267
.268
.001 3077
.001 3178
.001 3279
101
101
I 02
.326
3*7
.328
.001 9860
.001 9986
.002 0113
126
127
.209
.000 8007
78
.269
.001 3381
101
3*9
.002 0240
127
O.2IO
.211
.212
o.ooo 8085
.000 8163
.000 8242
78
79
0.270
.271
.272
o.ooi 3482
.001 3585
.001 3688
103
103
T Of
0.330
33 1
33*
O.OO2 0367
.002 0495
.002 0624
128
129
.213
.214
.000 8321
.000 8400
79
11
.273
.274
.001 3791
.001 3894
103
103
T r*A
333
334
.002 0752
.002 0882
130
oO
104
129
O.2I5
.216
o.ooo 8480
.000 8560
80
0.275
.276
o.ooi 3998
.001 4103
I0 5
T C\A
0-335
.336
0.002 1 01 1
.002 1141
I 3
.217
.000 8641
T
.277
.001 4207
104
337
.002 1272
131
.218
.000 8722
o I
81
.278
.001 4313
.338
.002 1403
131
.219
.000 8803
o I
82
.279
.001 4418
"I
339
.002 1534
132
O.220
.221
.222
o.ooo 8885
.000 8967
.000 9050
82
ll
0.280
.281
.282
o.ooi 4524
.001 4631
.001 4738
107
107
T 0*7
0.340
.341
.342
O.OO2 l666
.002 1799
.002 1971
133
132
T * A
.223
.224
.000 9132
.000 9216
oZ
84
84
.283
.284
.001 4845
.001 4953
107
108
108
343
344
.002 2065
.002 2198
1 34
133
0.225
.226
.227
.228
o.ooo 9300
.000 9384
.000 9468
.000 9553
84
8 4
5 <
0.285
.286
.287
.288
o.ooi 5061
.001 5169
.001 5278
.001 5388
1 08
109
no
o-345
34 6
347
348
0.002 2333
.OO2 2467
.002 2602
.002 2738
134
'35
136
.229
.000 9638
ll
.289
.001 5497
109
in
349
.002 2874
l\6
O.230
o.ooo 9724
86
0.290
o.ooi 5608
I JO
0.350
0.002 3010
.231
.232
.233
.234
.000 9810
.000 9897
.000 9984
.001 0071
3
ll
.291
.292
.293
.294
.001 5718
.001 5829
.001 5941
.001 6053
in
112
112
112
35 1
352
353
354
.002 3147
.002 3284
.OO2 3422
.002 3560
137
138
138
139
0.235
.236
.237
o.ooi 0159
.001 0247
.001 0335
88
88
0.295
.296
.297
o.ooi 6165
.001 6278
.001 6391
II 3
"3
*$
357
0.002 3699
.002 3838
.002 3977
139
139
.238
.239
.001 0424
.001 0513
89
9
.298
.299
.001 6505
.001 6619
II 4
114
358
359
.002 4117
.002 4258
141
141
0.240
o.ooi 0603
0.300
o.ooi 6733
0.360
0.002 4399
620
TABLE XI.
For the Motion in a Parabolic Orbit.
,
log/*
Diff.
,
*,
Diff.
,
log ft
Diff.
o n co tj-
so so so so so
CO CO CO to CO
d
0.002 4399
.002 4540
.002 4682
.002 4824
.002 4967
141
142
142
143
143
0.420
.421
.422
4*3
.424
0.003 37 20
.003 3890
.003 4061
.003 4232
.003 4404
170
171
171
I 7 2
172
0.480
. 4 8l
.482
483
484
0.004 4858
.004 5061
.004 5263
.004 5467
.004 5670
203
202
204
20 3
205
0.365
.366
O.002 5IIO
.002 5254
144
0.425
.426
0.003 4576
.003 4749
173
0.485
.486
0.004 5875
.004 6080
20 5
.367
368
.369
.002 5398
.002 5543
.002 5688
144
'45
'45
146
.427
.428
.429
.003 4923
.003 5096
.003 5271
173
175
J 74
.487
.488
489
.004 6285
.004 6492
.004 6698
205
20J
206
208
0.370
0.002 5834
0.430
0.003 5445
0.490
0.004 6906
371
.372
.002 5980
.002 6l26
146
43 1
.432
.003 5621
.003 5797
! 7 6
f nf\
.491
.492
.004 7113
.004 7322
207
209
373
.002 6273
Hg
433
003 5973
170
493
.004 7531
209
374
.002 6421
434
.003 6150
177
177
494
.004 7740
209
211
0-375
376
0.002 6568
.002 6717
149
y A f\
o-435
.436
0.003 6327
.003 6505
I 7 8
1 7%
o-495
496
0.004 7951
.004 8161
210
377
.378
.002 6866
.002 7015
1 49
149
rr\
437
438
.003 6683
.003 6862
178
III
497
498
.004 8373
.004 8585
212
379
.002 7165
150
150
439
.003 7042
I oO
180
499
.004 8797
212
213
O M N COrJ-
OO 00 00 00 00
CO CO CO CO CO
d
0.002 7315
,OO2 7466
.002 7617
.OO2 7769
.002 7921
151
151
152
152
152
0.440
.441
.442
443
444
0.003 7 22a
.003 7402
.003 7583
.003 7765
.003 7947
180
181
182
183
0.500
5 1
.52
53
54
0.004 9010
.005 1173
.005 3397
.005 5681
.005 8029
2163
2224
2284
2348
2412
0.385
.3^6
387
388
389
0.002 8073
.002 8226
.002 8380
.002 8574
.002 8689
'53
154
'55
155
o-445
446
447
.448
449
0.003 8130
.003 8313
.003 8496
.003 8680
.003 8865
183
183
184
185
185
59
0.006 0441
.006 2919
.006 5464
.006 8079
.007 0765
2478
*545
2615
2686
2760
0.390
.391
0.002 8844
.OO2 8999
J 55
0.450
451
0.003 95
.003 9236
186
186
0.60
.61
0.007 3525
.007 6361
2836
201 1
39*
393
394
.002 9155
.002 9311
.002 9468
^56
157
158
.452
453
454
.003 9422
.003 9609
003 9797
187
188
187
.62
I 3
.64
.007 9274
.008 2268
.008 5345
*7 5
2994
377
3163
o-395
396
397
398
O.OO2 9626
.002 97 g 4
.002 9942
.003 oioi
158
158
i59
0-455
45 6
457
458
0.003 9984
.004 0173
.004 0362
.004 0551
189
189
189
0.65
.66
6 7
.68
0.008 8508
.009 1759
.009 5103
.009 8542
3*51
3344
3439
3C 1 O
399
.003 0260
*59
160
459
.004 0741
191
.69
.010 2081
33V
3642
0.400
0.003 0420
0.460
0.004 9 3 2
0.70
o.oio 5723
17CO
.401
.402
43
.404
.003 0580
.003 0741
.003 0903
.003 1064
161
162
161
163
.461
.462
.463
.464
.004 1123
.004 1315
.004 1507
.004 1700
192
192
193
193
.72
73
74
.010 9473
.on 3336
.on 7316
.012 1419
*->
3863
3980
4103
4*33
0.405
.406
.407
.408
.409
0.003 1227
.003 1389
.003 1553
.003 1716
.003 1881
162
164
163
'65
164
[467
.468
.469
0.004 1893
.004 2087
.004 2281
.004 2476
.004 2672
194
194
'95
196
196
0.75
.76
77
.78
79
0.012 5652
.013 0022
.013 4536
.013 9202
.014 4031
4370
45 H
4666
4829
5002
0.410
.411
.412
413
.414
0.003 2045
.003 221 I
.003 2376
.003 2543
.003 2709
166
165
167
166
168
0.470
.471
.472
473
474
0.004 2868
.004 3064
.004 3261
.004 3459
.004 3657
196
197
198
198
199
0.80
.81
.82
0.014 9033
.015 4219
.015 9603
.Ol6 5202
.017 1033
5186
5384
5599 '
5831 |
6087
0.415
.416
.417
.418
.419
0.003 ^877
.003 3044
.003 3213
.003 3381
.003 3550
167
169
168
169
170
0-475
476
477
478
479
0.004 3856
.004 4055
.004 4255
.004 4456
.004 4657
199
200
201
201
201
0.85
.86
87
.88
.89
O.OI7 7I 10
.Ol8 3486
.019 0165
.019 7195
.020 4629
6366
6679
7030
7434
7900
0.420
0.003 37 20
0.480
0.004 4858
0.90
0.021 2529
621
TABLE XII,
9
t
if
i
.
.4
3'
4
A
S
log wij
log TO 2
m l
m,
m,
TO!
i
m 3
7H 2
m i
00
0.0000
90 o
90 o
180 o
180 o
180 o
1
4.2976
9.9999
2 2 3
90 20
90 20
178 40
178 40
179 o
359 o
359 5'
2
3-395
9.9996
44 6
90 40
90 40
177 20
177 20
178 o
358 o
358 9
3
2.8675
9.9992
7 8
91 o
91 o
176 o
176 o
177 o
357 o
357 H
4
2.4938
9.9986
9 3 2
91 20
91 20
174 40
174 40
176 o
356 o
356 18
5
2.2044
9.9978
" 55
9 I 41
91 41
173 19
173 19
175 o
355
355 Z 3
6
.9686
9.9968
14 19
9 2 I
9 2 I
171 59
171 59
174 o
354 o
354 28
7
.7698
9-9957
1 6 42
92 22
92 22
170 38
170 38
172 59
353 *
353 3^
8
.S98i
9-9943
19 7
92 42
92 42
169 18
169 18
171 59
35* i
35 2 37,
9
4473
9.9928
ai 32
93 3
93 3
167 57
167 57
170 58
35i 2
35 1 4a;
10
.3130
9.9911
23 57
93 2 5
93 *5
166 35
166 35
169 57
35 3
35 47!
11
.1922
9.9892
26 23
93 46
93 46
165 14
165 14
1 68 55
349 4
349 5 2 ;
12
.0824
9.9871
28 50
94 8
94 8
163 52
163 52
167 54
348 6
348 56:
13
0.9821
9.9848
3 1 i?
94 3 1
94 3i
162 29
162 29
166 51
347 8
348 i
14
0.8898
9.9823
33 46
94 53
94 53
161 7
161 7
165 48
346 ii
347 6:
15
0.8045
9.9796
36 15
95 *7
95 '7
'59 43
'59 43
164 44
345 H
346 ii
16
0.7254
9.9767
38 46
95 40
95 4
158 20
158 20
163 40
344 17
345 '6
17
0.6518
9.9736
41 18
96 5
96 5
156 55
156 55
162 34
343 ai
344 21
18
0.5830
9.9702
43 5i
96 30
96 30
155 30
'55 3
161 27
342 27
343 2 7
19
0.5185
9.9667
46 26
96 56
96 56
154 4
154 4
160 19
341 32
342 32
20
0.4581
9.9629
49 2
97 23
97 a3
"5a 37
'Sa 37
'59 9
340 38
34i 37
21
0.4013
9.9588
51 41
97 5
97 5
151 10
151 10
157 58
339 45
34 43>
22
0.3479
9-9545
54
98 19
98 19
149 41
149 41
156 45
338 53
339 49
23
0.2976
9-9499
57 5
98 49
98 49
148 ii
148 ii
155 29
338 o
338 54|
24
0.2501
9-945 1
59 5 1
99 20
99 20
146 40
146 40
154 ii
337 9
338 o
25
0.2053
9.9400
62 40
99 53
99 53
H5 7
H5 7
152 50
336 19
337 6'
26
0.1631
9-9345
6 5 33
100 28
100 28
143 32
H3 3 2
I5 1 2 5
335 28
336 13
27
0.1232
9.9287
68 30
ioi 5
ioi 5
'4i 55
HI 55
149 56
334 38
335 '9
28
0.0857
9.9226
7i 33
101 45
ioi 45
140 15
140 15
148 22
333 49
334 25
29
0.0503
9.9161
74 4i
102 27
102 27
138 33
138 33
146 42
333 i
333 3 2
30
0.0170
9.9092
77 58
I0 3 I 3
I0 3 I 3
136 46
136 46
'44 55
332 12
33* 39
31
9-9857
9.9019
8x 23
104 4
104 4
134 5 6
134 5 6
142 59
331 24
331 46
32
9.95 6 5
9.8940
85 o
105 i
105 i
'3 2 59
'3* 59
140 51
33 37
330 54
33
9.9292
9.8856
88 54
106 6
106 6
'3 54
13 54
138 27
3 2 9 49
33 2
34
9.9040
9-8765
93 "
IO7 22
107 22
128 38
128 38
135 39
329 2
329 10
35
9.8808
9.8665
98 7
108 58
108 58
126 2
126 2
132 13
328 14
328 19
36
9.8600
9-8555
104 20
III 13
III 13
122 47
122 47
127 29
327 27
327 28
36 52.2
9- 8 443
9.8443
116 34
116 34
116 34
116 34
116 34
116 34
3*6 45
3^6 45
This table exhibits the limits of the roots of the equation
sin (/ C) = mo sin 4 z',
when there are four real roots. The quantities m l and w 2 are the limiting
values of ra , and the values of z/, z 2 ' 9 z/, and z/, corresponding to ea(;h of
these, give the limits of the four real roots of the equation.
622
TABLE XII,
Y
z
i'
z
/
1
8
*
/
4
b
log TO!
log ?M
m,
m,
m l
ro,
n 2
!
mi
m 3
+ 00
00
0.0000
O
90 o
90
180 o
180 o
180 o
1
4.2976
9-9999
I
I 20
I 20
89 40
89 40
177 37
180 55
181 o!
2
3-395
9.9996
2 O
2 4
2 40
89 20
89 20
175 14
181 51
182 o'
8
2.8675
9.9992
3 o
4 o
4 o
89 o
89 o
172 52
182 46
183 o
4
2.4938
9.9986
4 o
5 20
5 20
88 40
88 40
170 28
183 4*
184 o
5
2.2044
9.9978
5 o
6 41
6 41
88 19
88 19
168 5
184 37
185 o 1
6
.9686
9.9968
6 o
8 i
8 i
87 59
87 59
165 41
185 32
186 o
7
.7698
9-9957
7 i
9 22
9 22
87 38
87 38
163 18
186 28
186 59,
8
.5981
9-9943
8 i
10 42
10 42
87 18
87 18
160 53
187 23
187 59
9
4473
9.9928
9 *
i* 3
12 3
86 57
86 57
158 28
188 18
188 58!
10
.3130
9.9911
10 3
13 25
13 25
86 35
86 35
i5 6 3
189 13
189 57
11
.1922
9.9892
ii 5
14 4 6
14 46
86 14
86 14
153 37
190 8
19 56:
12
.0824
9.9871
12 6
16 8
16 8
85 52
85 52
151 10
191 4
191 54;
13
0.9821
9.9848
13 9
17 31
17 3 1
85 29
85 29
148 43
191 59
192 52
14
0.8898
9.9823
14 12
18 53
18 53
85 7
85 7
146 14
192 54
193 49
15
0.8045
9.9796
15 16
20 17
20 17
84 43
84 43
H3 45
193 49
194 46
16
0.7254
9.9767
I 6 20
21 40
21 40
84 20
84 20
141 H
194 44
195 43,
17
0.6518
9.9736
17 26
^3 5
2 3 5
83 55
83 55
138 42
195 39
196 39 J
18
0.5830
9.9702
18 33
24 30
24 30
83 3
83 30
136 9
196 33
197 33i
11)
0.5185
9.9667
19 41
25 56
25 56
83 4
83 4
133 34
197 28
198 28
20
0.4581
9.9629
20 51
27 2 3
27 23
82 37
82 37
130 58
198 23
199 22
21
0.4013
9.9588
22 2
28 50
28 50
82 10
82 10
128 19
199 17
200 15
22
0-3479
9-9545
23 15
3 19
30 19
81 41
8 1 41
125 38
200 II
201 07
23
0.2976
9.9499
24 31
3i 49
3i 49
81 ii
81 ii
122 55
201 6
202
24
0.2501
9-945 i
25 49
33 ^o
33 20
80 40
80 40
120 9
202
202 51
25
0.2053
9.9400
27 10
34 53
34 53
80 7
80 7
117 20
202 54
20 3 41
26
0.1631
9-9345
28 35
36 28
36 28
79 3 2
79 3 2
114 27
203 47
204 32
27
0.1232
9.9287
3 4
38 5
38 5
78 55
78 55
III 30
204 41
205 22
1 28
0.0857
9.9226
3i 38
39 45
39 45
78 15
78 15
108 27
205 35
206 II
29
0.0503
9.9161
33 18
4i 27
4i 27
77 33
77 33
i5 19
206 28
206 59
30
0.0170
9.9092
35 5
43 1 3
43 n
76 47
76 47
102 3
207 a i
207 48
1 31
9-9 8 57
9.9019
37 i
45 4
45 4
75 5 6
75 5 6
98 37
208 14
208 36
32
9.95 6 5
9.8940
39 9
47 *
47 i
74 59
74 59
95
209 06
209 23
33
9.9292
9.8856
4i 33
49 6
49 6
73 54
73 54
91 6
209 58
210 II
34
9.9040
9.8765
44 21
5 I 22
51 22
72 38
72 38
86 49
210 50
210 58
35
9.8808
9.8665
47 47
53 58
53 58
71 2
71 2
81 53
211 41
211 46
36
9.8600
9.8555
5 2 3i
57 13
57 13
68 47
68 47
75 40
212 32
212 33
+36 52.2
9- 8 443
9.8443
63 26
63 26
63 26
63 26
63 26
63 26
213 15
2I 3 15
This table exhibits the limits of the roots of the equation
sin (d C) = m sin* 2',
when there are four real roots. The quantities m l and m^ are the limiting
values of m , and the values of z/> z z'> z s> and z *> corresponding to each of
these, give the limits of the four real roots of the equation.
623
TABLE XIII,
For finding the Ratio of the Sector to the Triangle.
n
log*S
Diff.
logs*
Diff.
i
to*
Diff.
o.oooo
.0001
.0002
.0003
.0004
O.OOO 0000
.000 0965
.000 1930
.000 2894
.000 3858
965
965
964
964
963
0.0060
.0061
.0062
.0063
.0064
0.005 7*98
.005 8243
.005 9187
.006 0131
.006 1075
945
944
944
944
944
0.0120
.OI2I
.0122
.0123
.0124
o.on 3417
.on 4343
.on 5268
.on 6193
.on 7118
926
925
925
925
925
0.0005
o.ooo 4821
0.0065
0.006 2019
O.OI25
o.on 8043
.0000
.0007
.0008
.0009
.000 5784
.000 6747
.000 7710
.000 8672
9 6 3
963
963
962
962
.0066
.0067
.0068
.0069
.006 2962
.006 3905
.006 4847
.006 5790
943
943
942
943
942
.0126
.0127
.0128
.0129
.on 8967
.01 1 9890
.012 0814
.012 1737
924 !
923
924
923
923
O.OOI
.001 1
.0012
.0013
.0014
o.ooo 9634
.001 0595
.001 1556
.001 2517
.001 3478
961
961
961
961
960
0.0070
.0071
.0072
.0073
.0074
0.006 6732
.006 7673
.006 8614
.006 9555
.007 0496
941
941
941
941
940
0.0130
.0131
.0132
.0133
.0134
0.012 2660
.012 3583
.012 4505
.012 5427
.012 6348
9*3
922
922
921
921
0.0015
.0016
.0017
o.ooi 4438
.001 5398
.001 6357
960
959
0.0075
.0076
.0077
0.007 *436
.007 2376
.007 3316
940
94
0.0135
.0136
.0137
0.012 7269
.012 8190
.012 9III
921
921 i
.0018
.0019
.001 7316
.001 8275
959
959
.0078
.0079
.007 4255
.007 5194
939
939
n^ n
.0138
.0139
.013 0032
.013 0952
921
920
n T n
959
y3:/
y*y
O.OO2O
.0021
.0022
.0023
.0024
o.ooi 9234
.002 0192
.002 1150
.002 2107
.002 3064
958
958
957
957
957
0.0080
.0081
.0082
.0083
.0084
0.007 6133
.007 7071
.007 8009
.007 8947
.007 9884
93 8
93 8
93 8
937
937
0.0140
.0141
.0142
.0143
.0144
0.013 I ^7 I
.013 2791
.013 3710
.013 4629
.013 5547
920
919
9*9 !
918
918
O.0025
.0026
.0027
.O028
.OO29
0.002 4021
.002 4977
.002 5933
.002 6889
.002 7845
956
956
956
956
955
0.0085
.0086
.0087
.0088
.0089
0.008 0821
.008 1758
.008 2694
.008 3630
.008 4566
937
936
936
936
936
0.0145
.0146
.0147
.0148
.0149
0.013 6465
.013 7383
.013 8301
.013 9218
.014 0135
918
918
917
917
917
O.0030
.0031
.0032
0033
.0034
0.002 8800
.002 9755
.003 0709
.003 1663
.003 2617
955
954
954
954
953
0.0090
.0091
.0092
.0093
.0094
0.008 5502
.008 6437
.008 7372
.008 8306
.008 9240
935
935
934
934
934
0.0150
.0151
.0152
153
.0154
0.014 I0 5 2
.014 1968
.014 2884
.014 3800
.014 4716
916
916
916
916
0.0035
0.003 3570
n f *
0.0095
0.009 OI 74
O -5 A
0.0155
0.014 563!
.0036
.0037
.0038
.0039
.003 4523
.003 5476
.003 6428
.003 7380
y.>3
953
952
952
952
.0096
.0097
.0098
.0099
.009 i i 08
.009 2041
.009 2974
.009 3906
y 53-
933
933
932
932
.0156
!o! 5 8
.0159
.014 6546
.014 7460
.014 8374
.014 9288
914
914
914
914
0.0040
0.003 8332
A ? 1
O.OIOO
0.009 4838
C\ 1 1
0.0160
0.015 202
.0041
.003 9284
952
.0101
.009 5770
y^- 6
.0161
.015 1115
9 3
.0042
.004 0235
95 1
.0102
.009 6702
932
r\ * T
.0162
.015 2028
n T -9
.0043
.0044
.004 i i 86
.004 2136
950
95
.0103
.0104
.009 7633
.009 8564
93 1
931
93 I
.0163
.0164
.015 2941
015 3 8 54
y 1 3
913
912
0.0045
.0046
.0047
.0048
.0049
0.004 3086
.004 4036
.004 4985
.004 5934
.004 6883
950
949
949
949
949
0.0105
.0106
.0107
.0108
.0109
0.009 9495
.010 0425
.010 1355
.010 2285
.010 3215
93
93
93
93
929
0.0165
.0166
.0167
.0168
.0169
0.015 4766
.015 5678
.015 6589
.015 7500
.015 8411
912
911
911
911
911
0.0050
.0051
.0052
.0053
.0054
0.004 7832
.004 8780
.004 9728
.005 0675
.005 1622
948
948
947
947
947
O.OIIO
.0111
.0112
.0113
.0114
o.oio 4144
.010 5073
.010 6001
.010 6929
.010 7857
929
928
928
928
928
0.0170
.0171
.0172
.0173
.0174
0.015 9322
.016 0232
.016 1142
.016 2052
.016 2961
910
910
910
909
909
0.0055
.0056
.0057
.0058
.0059
0.005 ^569
005 3515
.005 4461
.005 5407
.005 6353
946
946
946
946
945
O.OII5
.0116
.0117
.0118
.OII9
o.oio 8785
.010 9712
.on 0639
.on 1565
.on 2491
927
927
926
926
926
0.0175
.0176
.0177
.0178
.0179
0.016 3870
.016 4779
.016 5688
.016 6596
.016 7504
909
909
908
908
908
O.0060
0.005 7298
0.0120
o.on 3417
0.0180
0.016 8412
624
TABLE XIII
For finding the Eatio of the Sector to the Triangle.
n
logs*
Diff.
1?
log si
Diff.
>?
logs*
Diff.
0.0180
.0181
.0182
0.016 8412
.016 9319
.017 0226
907
907
0.0240
.0241
.0242
0.022 2330
.022 3220
.022 4109
890
889
oo'
0.0300
.0301
.0302
0.027 5 2 *8
.027 6091
.027 6964
873
873
.0183
.0184
.017 1133
.017 2039
907
906
906
.0243
.0244
.022 4998
.022 5887
009
889
889
.0303
.0304
.027 7836
.027 8708
872
872
872
0.0185
.0186
.0187
0.017 ^945
.017 3851
.017 4757
906
906
0.0245
.0246
.0247
0.022 6776
.022 7664
.022 8552
888
888
ooo
0.0305
.0306
.0307
0.027 9580
.028 0452
.028 1323
872
I 71
.0188
.0189
.017 5662
.017 6567
95
905
904
.0248
.0249
.022 9440
.023 0328
oofl
888
887
.0308
.0309
.028 2194
.028 3065
871
871
871
0.0190
.0191
.0192
.0193
.0194
0.017 7471
.017 8376
.017 9280
.018 0183
.018 1087
905
904
903
904
93
O.O25O
.0251
.0252
.0253
.0254
0.023 I2I 5
.023 2IO2
.023 2988
.023 3875
.023 4761
887
886
887
886
886
0.0310
.0311
.0312
.0313
.0314
0.028 3936
.028 4806
.028 5676
.028 6546
.028 7415
870
870
870
869
869
0.0195
.0196
.0197
.0198
.0199
0.018 1990
.018 2893
.018 3796
.018 4698
.018 5600
93
903
902
902
901
0.0255
.0256
.0257
.0258
.0259
0.023 5 6 47
.023 6532
.023 7417
.023 8302
.023 9187
885
885
885
885
884
0.0315
.0316
.0317
.0318
.0319
0.028 8284
.028 9153
.029 0022
.029 0890
.029 I 75 8
869
869
868
868
868
0.0200
.0201
.0202
.0203
.0204
0.018 6501
.018 7403
.018 8304
.018 9205
.019 0105
902
901
901
900
900
0.0260
.026l
.0262
.0263
.0264
0.024 7 I
.024 0956
.024 1839
.024 2723
.024 3606
885
883
884
883
883
0.0320
.0321
.0322
.0323
.0324
0.029 2626
.029 3494
.029 4361
.029 5228
.029 6095
868
867
867
867
866
0.0205
.0206
0.019 1005
.019 1905
900
0.0265
.0266
0.024 4489
.024 5372
883
00^
0.0325
.0326
0.029 ^961
.029 7827
866
O /I
.0207
.0208
.0209
.019 2805
.019 3704
.019 4603
QOO
8 99
8 99
899
.0267
.0268
.0269
.024 6254
.024 7136
.024 8018
o o 2
882
882
882
.0327
.0328
.0329
.029 8693
.029 9559
.030 0424
oDO
866
865
866
0.0210
.0211
0.019 5502
.019 6401
III
0.0270
.0271
0.024 8900
.024 9781
881
OO f
0.0330
.0331
0.030 1290
.030 2154
864
Of. \
.0212
.019 7299
oQo
8n,
.0272
.025 0662
OO 1
00,
.0332
.030 3019
SOS
Of.
.0213
.019 8197
090
807
.0273
.025 1543
oo 1
880
333
.030 3883
a 04
864
.0214
.019 9094
7 /
898
.0274
.025 2423
880
334
.030 4747
864
0.0215
.0216
.0217
.0218
.0219
0.019 999 2
.020 0889
.020 1785
.020 2682
.020 3578
897
896
897
896
896
0.0275
.0276
.0277
.0278
.0279
0.025 333
.025 4183
.025 5063
.025 5942
.025 6821
880
880
879
879
879
0.0335
0336
0337
.0338
339
0.030 5611
.030 6475
.030 7338
.030 8201
.030 9064
861
86 3
863
86 3
862
O.0220
0.020 4474
o_
O.O28O
0.025 7700
9.Tn
0.0340
0.030 9926
8fi-7
.O22I
.0222
.0223
.0224
.020 5369
.020 6264
.020 7159
.020 8054
895
895
895
8 95
894
.0281
.0282
.0283
.0284
.025 8579
.025 9457
.026 0335
.026 1213
079
878
878
878
877
.0341
.0342
343
.0344
.031 0788
.031 1650
.031 2512
03 1 3373
OO2
862
862
861
861
O.0225
.0226
.0227
.0228
.0229
0.020 8948
.020 9842
.021 0736
.021 1630
.O2I 2523
8 94
894
894
893
893
0.0285
.0286
.0287
.0288
.0289
0.026 2090
.026 2967
.026 3844
.026 4721
.026 5597
877
877
877
876
876
0.0345
.0346
347
.0348
.0349
0.031 4234
.031 5095
031 5956
.031 6816
.031 7676
861
861
860
860
860
0.0230
.0231
.0232
0233
.0234
0.021 3416
.O2I 4309
.021 5201
.O2I 6093
.021 6985
893
8 9 2
892
892
891
0.0290
.0291
.0292
.0293
.0294
0.026 6473
.026 7349
.026 8224
.026 9099
.026 9974
876
875
875
875
875
0.0350
.0351
.0352
353
.0354
0.031 8536
.031 9396
.032 0255
.032 1114
.032 1973
860
859
859
Hi
0.023 5
.0236
.0237
.0238
.0239
0.021 7876
.021 8768
.021 9659
.022 0549
.022 1440
8 9 2
891
890
891
890
0.0295
.0296
.0297
.0298
.0299
0.027 0849
.027 1723
.027 2597
.027 3471
.027 4345
874
874
874
874
873
o-0355
.0356
.0357
.0358
.0359
0.032 2831
.032 3689
.032 4547
.032 5405
.032 6262
858
858
858
857
8 5 8
0.0240
0.022 2330
0.0300
0.027 5218
0.0360! 0.032 7120
40
625
TABLE XIII,
For finding the Ratio of the Sector to the Triangle.
i
log>
Diff.
*,
Diff.
<
log*
Diff.
0.0360
.0361
.0362
.0363
.0364
0.032 7120
.032 7976
.032 8833
.032 9689
.033 0546
856
857
856
857
855
0.060
.061
.062
.063
.064
0.052 5626
.053 3602
.054 1556
.054 9488
055 7397
7976
7954
7932
7909
7888
0.120
.121
.122
.123
.124
0.096 8849
.097 5692
.098 2520
.098 9331
.099 6127
6843
6828
6811
6796
6780
mvO t^-OO ON
CO CO CO CO CO
O O O O O
'
0.03^3 1401
.033 2257
.033 3112
.033 3967
.033 4822
856
855
855
855
855
0.065
.066
.067
.068
.069
0.056 5285
.057 3150
.058 0994
.058 8817
.059 6618
7865
7844
7823
7801
7780
0.125
.126
.127
.128
.129
o.ioo 2907
.100 9672
.101 6421
.102 3154
.102 9873
6765 >
6749
6733
6719 ,
6703
0.0370
.0371
.0372
0373
.0374
0.033 5677
.033 6531
.033 7385
.033 8239
.033 9092
854
854
854
853
854
0.070
.071
.072
.073
.074
0.060 4398
.061 2157
.061 9895
.062 7612
.063 5308
7759
7738
7696
7676
0.130
.131
.132
'33
'34
O.IO3 6576
.104 3264
.104 9936
.105 6594
.106 3237
6688
6672 !
6658 !
6643
6628
0.0375
.0376
.0377
0378
0379
0.033 9946
.034 0799
.034 1651
.034 2504
.034 3356
852
853
852
852
0.075
.076
.077
.078
79
0.064 2984
.065 0639
.065 8274
.066 5888
.067 3482
7655
7635
7614
7594
7575
0.135
.136
137
.138
'39
0.106 9865
.107 6478
.108 3076
.108 9660
.109 6229
6613
6598
6584
6569
6 554
O M O rorj-
oo oo oo oo oo
CO CO CO CO CO
q q q q q
d
0.034 4208
.034 5059
.034 5911
.034 6762
.034 7613
851
852
851
851
851
0.080
.081
.082
.083
.084
0.068 1057
.068 8612
.069 6146
.070 3661
.071 1157
7555
7534
7496
7476
0.140
.141
.142
143
.144
o.no 2783
.no 9323
.III 5849
.112 2360
.112 8857
6540
6526
6511
6497
6483
vnvo r--oo ON
00 00 00 00 00
CO CO CO CO CO
O
0.034 8464
34 93H
.035 0164
.035 1014
.035 1864
850
850
850
850
849
0.085
.086
.087
.088
.089
0.071 8633
.072 6090
073 3527
.074 0945
74 8345
7457
7437
7418
7400
738o
0.145
.146
.147
.148
.149
0.113 534
.114 1809
.114 8264
.115 4704
.116 1131
6469
6455
6440
6427
6413
0.0390
.0391
0.035 2713
035 35 6 2
849
0.090
.091
0-075 5725
.076 3087
7362
0.150
.151
0.116 7544
"7 3943
6399|
.0392
.0393
0394
035 44"
035 5259
.035 6108
848
849
848
.092
93
.094
.077 0430
.077 7754
.078 5060
7343
7324
7306
7288
.152
153
'54
.118 0329
.118 6701
.119 3059
6372
6358
6 345 i
0.0395
.0396
397
.0398
399
0.035 6956
.035 7804
.035 8651
.035 9499
.036 0346
848
847
848
847
846
0.095
.096
.097
.098
.099
0.079 2348
.079 9617
.080 6868
.081 4101
.082 1316
7269
7251
7233
7215
7197
0.155
.156
'57
.158
'59
0.119 944
120 5735
.121 2053
.121 8357
.122 4649
6331
6318
6304 |
6292
6278 |
0.040
.041
.042
43
44
0.036 1192
.036 9646
037 8075
.038 6478
.039 4856
8454
8429
8403
8378
8353
O.I 00
.101
.102
.103
.104
0.082 8513
.083 5693
.084 2854
.084 9999
.085 7125
7180
7161
7H5
7126
7110
0.160
.161
.162
.163
.164
0.123 927
.123 7192
.124 3444
.124 9682
.125 5908
6265
6252
6238
6226
6213
0.045
.046
.047
.048
049
0.040 3209
.041 1537
.041 9841
.042 8121
.043 6376
8328
8304
8280
8255
8231
0.105
.IO6
.107
.108
.109
0.086 4235
.087 1327
.087 8401
.088 5459
.089 2500
7092
7074
7058
7041
7023
0.165
.166
.167
.168
.169
0.126 ai2i
.126 8321
.127 4508
.128 0683
.128 6845
6200 ;
6187 !
6175
6162
6149
0.050
.051
.052
.053
.054
0.044 4 6 7
.045 2814
.046 0997
.046 9157
.047 7294
8207
8183
8160
8113
O.IIO
.III
.112
"3
.114
0.089 9523
.090 6530
.091 3520
.092 0494
.092 7451
7007
6990
6974
6957
6940
0.170
.171
.172
.173
.174
0.129 2994
.129 9131
13 5255
.131 1367
.131 7466
6137
6124
6112
6099
6087
0.055
.056
.057
.058
.059
0.048 5407
049 3496
.050 1563
.050 9607
.051 7628
8089
8067
8044
8021
7998
0.115
.116
.117
.118
.119
0.093 439 1
.094 1315
.094 8223
.095 5114
.096 1990
6924
6908
6891
6876
6859
0.175
.176
.177
.178
.179
0-132 3553
.132 9628
.133 5690
.134 1740
134 7778
6062
6050
6038
6026
O.O60
0.052 5626
0.120
0.096 8849
0.180
0.135 3804
626
TABLE XIII.
For finding the Ratio of the Sector to the Triangle.
*
logs*
Diff.
*
logs*
Diff.
i
toff*
Diff.
o.i8o
.181
.182
.183
.184
0.135 3804
.135 9818
.136 5821
.137 1811
.137 7789
6014
6003
5990
5978
5966
O.240
.241
.242
*43
.244
0.169 509*
.170 0470
.170 5838
.171 1197
.171 6547
5378
5368
5359
5350
534
0.300
.301
.302
.303
34
0.200 2285
.200 7157
.201 2021
.201 6878
.202 1727
4872
4864
4857
4849
4842
0.185
.186
.187
.188
.189
0-138 3755
.138 9710
!39 5 6 53
.140 1585
.140 7504
5955
5943
593*
5019
5908
0.245
.246
.247
.248
.249
0.172 1887
.172 7218
.173 2540
.173 7853
.174 3156
15322
|53'3
15303
5*95
0.305
.306
37
.308
.309
O.2O2 6569
.203 1403
.203 6230
.204 1050
.204 5862
4834
4827
4820
4812
4805
0.190
.191
0.141 3412
.141 9309
5897
c88 c
0.250
.251
0.174 8451
.175 3736
5285
O.3IO
.311
0.205 0667
.205 5464
4797
.192
193
.194
.142 5194
.143 1068
.143 6931
5874
5863
5851
.252
*53
.254
.175 9013
.176 4280
.176 9538
5*77
5267
5258
5*5
.312
3'3
3'4
.206 0254
.206 5037
.206 9813
4790
4783
4776
4768
0.195
.196
.197
.198
.199
0.144 2782
.144 8622
.145 4450
.146 0268
.146 6074
5840
5828
5818
5806
5795
0.255
.256
.257
.258
.259
0.177 4788
.178 0029
.178 5261
.179 0484
.179 5698
5*3*
5**3
5214
5*5
-3'5
.316
:\\l
3*9
O.207 4581
.207 9342
.208 4096
.208 8843
.209 3582
4761
4754
4747
4739
4733
O.200
.201
0.147 *86g
.147 7653
5784
0.260
.261
0.180 0903
.180 6100
5*97
0.320
3*i
0.209 8315
.210 3040
47*5
.202
.203
.204
.148 3427
.148 9189
.149 4940
5774
5762
575 1
574*
.262
.263
.264
.181 1288
.181 6467
.182 1638
5188
5*79
5171
5162
.322
3*3
3*4
.210 7759
.211 2470
.211 7174
4719
4711
4704
4697
O.20C
0.150 0681
0.265
0.182 6800
0.325
O.2I2 1871
.206
.150 6411
5730
.266
.183 1953
5153
.212 6562
4691
.207
.208
.209
.151 2130
.151 7838
'5* 3535
57 X 9
5708
5697
5687
.267
.268
.269
.183 7098
.184 2235
.184 7363
5*45
5'37
5128
5120
3*7
3*8
3*9
.213 1245
.213 5921
.214 0591
4676
4670
4662
0.210
.211
.212
.213
.214
0.152 9222
.153 4899
.154 0565
.154 6210
.155 1865
5677
5666
5655
5645
5634
0.270
.271
.272
*73
.274
0.185 2483
.185 7594
.186 2696
.186 7791
.187 2877
5111
5102
5086
5078
0.330
33*
33*
333
334
0*214 5*53
.214 9909
.215 4558
.215 9200
.216 3835
4656
4649
4642
4635
4629
O.2I5
.216
.217
.218
.219
o-i55 7499
.156 3123
.156 8737
.157 4340
'57 9933
5624
5614
5603
5593
5583
0.275
.276
.277
.278
.279
0.187 7955
.188 3024
.188 8085
.189 1138
.189 8183
5069
5061
553
545
537
0-335
336
337
338
339
0.216 8464
.217 3085
.217 7700
.218 2308
.218 6910
4621
4615
4608
4602
0.220
.221
.222
.223
.224
0.158 5516
.159 1089
.159 6652
.160 2204
.160 7747
5573
5563
555*
5543
553*
0.280
.281
.282
.283
.284
0.190 3220
.190 8249
.191 3269
.191 8281
.192 3286
5029
5020
5012
5005
4996
0.340
341
34*
343
344
0.219 1505
.219 6093
.220 0675
.220 5250
.220 9818
4588
4582
4575
4568
4562
0.225
.220
.227
.228
.229
0.161 3279
.161 8802
.162 4315
.162 9817
.163 5310
55*3
5513
5502
5493
5483
0.285
.286
.287
.288
.289
0.192 8282
.193 7271
.193 8251
.194 3224
.194 8188
4989
4980
4973
4964
4957
0-345
.346
347
.348
349
0.221 4380
.221 8915
.222 7483
.222 8025
,223 2561
4555
4548
4542
4536
0.230
.231
.232
*33
*34
0.164 0793
.164 6267
.165 1710
.165 7184
.166 2628
5474
5463
5454
5444
5435
0.290
.291
.292
.293
.294
0.195 3145
.195 8094
.196 3035
.196 7968
.197 2894
4949
494 1
4933
(.926
0.350
35'
35*
353
354
0.223 7090
.224 1613
.224 6130
.225 0640
.225 5143
45*3
45^7
4510
453
4497
0.235
.236
0.166 8063
.167 3488
54*5
0.295
.296
0.197 7811
.198 2721
1.910
0-355
356
0.225 9 6 4
.226 4131
449 >
4484
*37
.167 8903
r^nfi
.297
.198 7624
I-903
A 9 nA
357
.226 8615
.238
.168 4309
400
.298
.199 2518
1.094
l888
358
.227 3093
AA72
.239
.168 9705
5396
387
.299
.199 7406
4879
359
.227 7565
4--j- / *
44 66
0.240
0.169 59*
0.300
0.200 2285
0.360
0.228 2031
627
TABLE XIII,
For finding the Katio of the Sector to the Triangle.
*
logss
Diff.
<
log "
Diff.
*
logs*
Diff.
0.360
. 3 6i
.362
.363
3 6 4
0.228 2031
.228 6490
.229 0943
.229 5390
.229 9831
4459
4453
4447
4441
4434
0.420
.421
.422
4*3
4*4
-*53 9 T 53
.254 3269
*54 7379
.255 1484
*55 5584
4116
4110
4105
4100
4095
0.480
. 4 8i
.482
.483
.484
0.277 7*7*
.278 1096
.278 4916
.278 8732
*79 *543
3824
3820 i
3816
3811
3806
U^SO t^OO O
SO so SO so VO
CO CO CO CO CO
6 *
0.230 4265
.230 8694
.231 3116
.231 7532
.232 1942
4429
4422
4416
4410
444
0.425
.426
4*7
.428
4*9
0.255 9679
.256 3769
.256 7853
.257 1932
.257 6006
4090
4084
4079
4074
4069
0.485
.486
.487
.488
489
0.279 6349
.280 0151
.280 3949
.280 7743
.281 1532
3802
379 8
3794
3785
3784
0.370
.371
37*
0.232 6346
*33 743
*33 5135
4397
439*
Atxn
0430
431
43*
0.258 0075
.258 4139
.258 8198
4064
4059
0.490
.491
.492
0.281 5316
.281 9096
.282 2872
378o
3776
373
374
.233 9521
.234 3900
4300
4379
4374
433
434
.259 2252
.259 6300
4054
4048
4044
493
494
.282 6644
.283 0411
377*
3767
3762
0.375
376
377
.378
379
0.234 8274
.235 2642
.235 7003
.236 1359
.236 5709
4368
4361
4356
435
0.435
436
437
438
439
0.260 0344
.260 4382
.260 8415
.261 2444
.261 6467
4038
4033
4029
4023
o-495
496
497
.498
499
0.283 4173
.283 7932
.284 1686
.284 5436
.284 9181
3759
3754 !
375 i
3745
4344
4019
374*
O M M COrJ-
OO 00 00 00 00
CO tO CO CO CO
o
0.237 0053
.237 4391
.237 8723
.238 3050
.238 7370
4338
433*
43 2 7
4320
0.440
44 1
44*
443
444
0.262 0486
.262 4499
.262 8507
.263 2511
.263 6509
4013
4008
4004
3998
0.500
.501
.502
.503
.504
0.285 *9*3
.285 6660
.286 0392
.286 4121
.286 7845
3737
373* I
37^9
37^4 .
43* 5
3994
3720
tJ->sO t--OO O
oo oo oo oo oo
CO CO CO tO tO
6 *
0.239 *685
*39 5993
.240 0296
.240 4594
.240 8885
4308
433
4298
4291
4286
0.445
446
447
448
449
0.264 53
.264 4492
.264 8475
.265 2454
.265 6428
3989
3983
3979
3974
39 6 9
0.505
.506
507
.508
.509
0.287 ^65
.287 5281
.287 8992
.288 2700
.288 6403
3716
37" ;
3708 i
3703
3699 1
0.390
391
39*
393
394
0.241 3171
.241 7451
.242 1725
.242 5994
.243 0257
4280
4*74
4269
4*63
4*57
0.450
45 *
45*
453
454
0.266 0397
.266 4362
.266 8321
.267 2276
.267 6226
3965
3959
3955
395
3945
0.510
.511
.512
.514
0.289 OIO2
.289 3797
.289 7487
.290 1174
.290 4856
3695
3690 ;
3687 i
3682
3679
0.395
396
397
398
399
0.243 45H
.243 8766
.244 3012
.244 7252
.245 1487
4*5*
4246
4240
4*35
4229
0.455
456
457
458
459
0.268 0171
.268 4111
.268 8046
.269 1977
.269 5903
3940
3935
393 1
3926
3921
0.515
.516
517
.518
.519
0.290 8535
.291 2209
.291 5879
.291 9545
.292 3207
3674
3670
3666
3662 ;
3657
0.400
.401
.402
403
0.245 57 J 6
.245 9940
.246 4158
.246 8371
4224
4218
0.460
.461
.462
463
0.269 9824
.270 3741
.270 7652
.271 1559
39*7
39"
3907
0.520
.521
.522
5*3
0.292 6864
.293 0518
.293 4168
.293 7813
3654
365 :
3645
404
.247 2578
4207
4201
.464
.271 5462
3898
5*4
.294 1455
3642
3637
0.405
.406
.407
.408
409
0.247 6779
.248 0975
.248 5166
.248 9351
.249 3531
4196
4191
4*85
4180
4*74
0.465
.466
467
.468
.469
0.271 9360
.272 3253
.272 7141
.273 1025
.273 4904
mi
3884
3879
3874
0-5*5
.526
5*7
.528
5*9
0.294 59*
.294 8726
.295 2355
.295 5981
.295 9602
3634
36291
3626
3621 ;
3618
0.410
.411
.412
.413
.414
0.249 775
.250 1874
.250 6038
.251 0196
.251 4349
4169
4164
4158
4*53
4*47
0.470
.471
47*
473
474
0.273 8778
.274 2648
.274 6513
*75 374
.275 4230
3870
3865
3861
3856
385*
0-53
531
53*
533
534
0.296 3220
.296 6833
.297 0443
*97 449
.297 7650
3613
3610 i
3606
3601
3598
0.415
.416
.417
.418
.419
0.251 8496
.252 2638
.252 6775
.253 0906
.253 5032
4142
4*37
4131
4126
4121
0.475
476
477
.478
479
0.275 8082
.276 1929
.276 5771
.276 9609
.277 3443
3847
3842
3838
3834
3829
0-535
536
537
.538
539
0.298 1248
.298 4842
.298 8432
.299 2018
.299 5600
3594
359
3586
358*
3578
0.420
0.253 9*53
0.480
0.277 7*7*
0.540
0.299 9*78
628
TABLE XIIL
For finding the Katio of the Sector to the Triangle.
1?
logs*
Diff.
*
log*a
Diff.
*
log
Diff.
0.540
54
54*
543
0.199 9178
.300 2752
.300 6323
.300 9890
3574
35 6 7
0.560
.561
.562
563
0.306 9938
37 3437
.307 6931
.308 0422
3499
3494
0.580
.581
.582
583
0.313 9215
.314 26^.1
.314 6064
.314 9481
34*6
34*3
3419
544
.301 3452
3559
.564
.308 3910
*484
584
.315 2898
3412
0-545
b$4*
547
.548
549
0.301 7011
.302 0566
.302 4117
.302 7664
.303 1208
3555
355 1
3547
3544
3540
0.565
.566
.567
.568
.569
0.308 7394
.309 0874
.309 4150
.309 7823
.310 1292
3480
3476
3473
3469
3466
0.585
.586
.587
.588
.589
0.315 6310
3'5 9719
.316 1124
.316 6525
.316 9923
3409
3405
3401
3398
3395
0.550
55 1
55*
553
554
0-303 4748
.303 8284
.304 1816
.304 5344
.304 8869
353 6
353*
35*8
35*5
0.570
571
57*
573
574
0.310 4758
.310 8220
.311 1678
3 11 5i33
3462
3458
3455
345 i
3447
0.590
.591
59*
593
594
0.317 3318
.317 6709
.318 0096
.318 3480
.318 6861
338 7
3384
3381
3377
0-555
556
557
558
559
0.305 2390
.305 5907
.305 9420
.306 2930
.306 6436
35'7
35*3
3510
3506
3502
0-575
576
577
.578
579
0.312 2031
3 1 * 5475
.312 8915
.313 2352
3'3 5785
3444
3440
3437
3433
343
0.595
596
597
598
599
0.319 0238
.319 3612
.319 6983
.320 0350
3*o 37H
3374
33 6 7
3364
3360
0.560
0.306 9938
0.580
0.313 9215
0.600
0.320 7074
TABLE XIV.
For finding the Ratio of the Sector to the Triangle.
X
t
X
f
Ellipse.
Diff.
Hyperbola.
Diff.
Ellipse.
Diff.
Hyperbola.
Diff.
o.ooo
0.000 0000
o.ooo oooo
I
0.030
o.ooo 0523
76
o.ooo 0506
11
.001
.000 0001
1
.000 0001
J
.031
.000 0559
3
.000 0539
76
.002
.OOO OOO2
.OOO OOO2
.032
.000 0596
8
.000 0575
11
.003
.000 0005
3
.000 0005
3
033
.000 0634
3
.000 06 1 1
3 U
MM
.004
.000 0009
4
5
.000 0009
4
5
.034
.000 0674
4
40
.000 0648
3?
O.005
o.ooo 0014
o.ooo 0014
0.035
o.ooo 0714
o.ooo 0686
.006
.OOO OO2I
7
.OOO OO2O
.036
.000 0756
42
.000 0726
40
.007
.000 0028
7
.000 0028
037
.000 0799
43
.000 0766
4
.008
.009
.000 0037
.000 0047
9
10
II
.000 0036
.000 0046
10
II
.038
039
.000 0844
.000 0889
45
45
47
.000 0807
.000 0850
4 1
43
44
O.OIO
.Oil
o.ooo 0058
.000 0070
12
o.ooo 0057
.000 0069
12
0.040
.041
o.ooo 0936
.000 0984
48
o.ooo 0894
.000 0978
.012
.013
.000 0083
.000 0097
i!
.000 0082
.000 0096
*3
14
.042
.043
.000 1033
.000 1084
49
51
.000 0984
.000 1031
$
i .014
.000 0113
ID
17
.OOO 01 I I
11
044
.000 1135
53
.000 1079
49
0.015
! .Ol6
o.ooo 0130
.000 0148
18
o.ooo 0127
.000 0145
IJB
0.045
.046
o.ooo 1 1 88
.000 1242
It
o.ooo 1128
.000 1178
5
; .017
.018
.019
.000 0167
.000 0187
.000 0209
19
20
22
22
.000 0164
.000 0183
.000 0204
19
19
21
22
.047
.048
.049
.000 1298
.000 1354
.000 1412
56
56
58
59
.000 1229
.000 1281
.000 1334
5 1
5*
53
55
0.020
.021
.022
.023
.024
o.ooo 0231
.000 0255
.000 0280
.000 0306
.000 0334
24
\\
28
28
o.ooo 0226
.000 0249
.000 0273
.000 0298
.000 0325
23
24
27
27
0.050
.051
.052
053
.054
o.ooo 1471
.000 1532
.000 1593
.000 1656
.000 1720
OS ON ON ON ON
in 4* OJ HI M
o.ooo 1389
.000 1444
.000 1500
.000 1558
.000 1616
VNVOOOOO ON
w> 10 u-> 1/1 iri
0.025
.026
.027
.028
.029
o.ooo 0362
.000 0392
.000 0423
.000 0455
.000 0489
3
31
32
34
34
o.ooo 0352
.000 0381
.000 0410
.000 0441
.000 0473
29
*9
31
3*
33
0.055
.056
.057
.058
059
o.ooo 1785
.000 1852
.000 1920
.000 1989
.000 2060
6 7
68
69
o.ooo 1675
.000 1736
.000 1798
.000 i860
.000 1924
61
62
62
64
64
0.030
o.ooo 0523
o.ooo 0506
0.060 | o.ooo 2131
o.ooo 1988
629
TABLE XIV.
For finding the Katio of the Sector to the Triangle.
X
*
X
e
Ellipse.
Diff.
Hyperbola.
Diff.
Ellipse.
Diff.
Hyperbola.
Diff.
o.o6o
.061
.062
o.ooo 2131
.000 2204
.000 2278
73
a
o.ooo 1988
.000 2054
.000 2 1 21
66
67
0.120
.121
.122
o.ooo 8845
.000 8999
.000 9154
154
155
o.ooo 7698
.000 7822
.000 7948
124
126
_ f
.063
.064
.000 2354
.000 2431
70
.OOO 2189
.000 2257
Do
68
70
.123
.124
.000 9311
.000 9469
J 57
158
'59
.000 8074
.000 8202
120
128
128
0.065
.066
.067
.068
.069
o.ooo 2509
.000 2588
.000 2669
.000 2751
.000 2834
79
81
82
83
84
o.ooo 2327
.000 2398
.oco 2470
.000 2543
.000 2617
71
72
73
74
74
0.125
.126
.127
.128
.129
o.ooo 9628
.000 9789
.000 9951
.001 0115
.001 0280
161
162
164
'65
167
o.ooo 8330
.000 8459
.000 8590
.000 8721
.000 8853
129
131
132
133
0.070
.071
.072
o.ooo 2918
,000 3004
.000 3091
86
87
o.ooo 2691
.000 2767
.000 2844
76
7 7 8
0.130
.131
.132
o.ooi 0447
.001 0615
.001 0784
168
169
o.ooo 8986
.000 9120
.000 9255
134
.073
.074
.000 3180
.000 3269
89
9 1
.000 2922
.000 3001
1
79
80
J33
.134
.001 0955
.001 1128
171
173
J 73
.000 9390
.000 9527
*35
137
0.075
.076
o.ooo 3360
.000 3453
93
o.ooo 3081
.000 3162
Si
82
.136
o.ooi 1301
.001 1477
176
o.ooo 9665
.000 9803
138
077
.000 3546
93
.000 3244
137
.001 1654
III
.000 9943
140
.078
.000 3641
95
.000 3327
0^
.138
.001 1832
178
IJ?0
.001 0083
140
.079
.000 3738
97
97
.000 3411
85
.139
.OOI 2OI2
i oo
181
.001 0224
141
142
0.080
.081
o.ooo 3835
.000 3934
99
o.ooo 3496
.000 3582
86
8-
0.140
.141
o.ooi 2193
.001 2376
183
o.ooi 0366
.001 0509
143
.082
.000 4034
IOO
.000 3669
y
88
.142
.001 2560
184
.001. 0653
144
.083
.084
.000 4136
.000 4239
I 02
103
104
.000 3757
.000 3846
89
90
.143
.144
.001 2745
.001 2933
188
188
.001 0798
.001 0944
'45
146
147
0.085
.086
.087
.088
o.ooo 4341
.000 4448
.000 4555
.000 4663
105
107
108
o.ooo 3936
.000 4027
.000 4119
.000 4212
9 1
92
93
0.145
.146
.147
.148
o.ooi 3121
.001 3311
.001 3503
.001 3696
190
192
193
o.ooi 1091
.001 1238
.001 1387
.001 1536
147
149
149
.089
.000 4773
no
in
.000 4306
94
95
.149
.001 3891
195
196
.001 1686
150
152
0.090
.091
o.ooo 4884
.000 4996
112
o.ooo 4401
.000 4496
95
0.150
.151
o.ooi 4087
.001 4285
198
o.ooi 1838
.001 1990
152
.092
93
.094
.000 5109
.000 5224
.000 5341
"5
117
117
.000 4593
.000 4691
.000 4790
97
98
99
IOO
.152
.154
.001 4484
.001 4684
.001 4886
199
2OO
2 O2
204
.001 2143
.001 2296
.001 2451
*53
153
155
156
0.095
.096
.097
.098
.099
o.ooo 5458
.000 5577
.000 5697
.000 5819
.000 5942
119
120
122
I2 3
124
o.ooo 4890
.000 4991
.000 5092
.000 5195
.000 5299
101
101
103
104
104
0.155
.156
J 57
.158
.159
o.ooi 5090
.001 5295
.001 5502
.001 5710
.001 5920
205
207
208
2IO
211
o.ooi 2607
.001 2763
.001 2921
.001 3079
.001 3238
156
158
158
III
O.IOO
.101
.102
.103
.104
o.ooo 6066
.000 6192
.000 6319
.000 6448
.000 6578
126
I2 7
I2 9
130
131
o.ooo 5403
.000 5509
.000 5616
.000 5723
.000 5832
1 06
107
107
109
109
o.i 60
.161
.162
.163
.164
o.ooi 6131
.001 6344
.001 6559
.001 6775
.001 6992
213
215
216
217
219
o.ooi 3398
.001 3559
.001 3721
.001 3883
.001 4047
161
162
162
164
164
O.I05
.106
.107
,108
o.ooo 6709
.000 6842
.000 6976
.000 7111
133
134
135
o.ooo 5941
.000 6052
.000 6163
.000 0275
III
III
112
0.165
.166
.167
.168
o.ooi 7211
.001 7432
.001 7654
.001 7878
221
222
224
o.ooi 4211
.001 4377
.001 4543
.001 4710
166 '
166
167
_ fa
.109
.000 7248
137
138
.000 6389
114
.169
.001 8103
225
227
.001 4878
IOO
169
O.I 10
.III
o.ooo 7386
.000 7526
140
o.ooo 6503
.000 66l8
115
0.170
.171
o.ooi 8330
.001 8558
228
o.ooi 5047
.001 5216
i 9
.IJ2
"3
.114
.000 7667
.000 7809
.000 7953
141
I 4 2
144
45
.000 6734
.000 6851
.000 6969
1 1 6
117
118
119
.172
.173
.174
.001 8788
.001 9020
.001 9253
230
232
233
234
.001 5387
.001 5558
.001 5730
' / *
171
172
173
0.115
.116
o.ooo 8098
.000 8245
147
_ . 6
o.ooo 7088
.000 7208
120
0.175
.176
o.ooi 9487
.001 9724
237
o.ooi 5903
.001 6077
*74
.117
.118
.119
.000 8393
.000 8542
.000 8693
148
149
152
.000 7329
.000 7451
.000 7574
121
122
123
124
.177
.178
.179
.001 9961
.002 0201
.002 0442
237
240
241
343
.001 6252
.001 6428
.001 6604
176
176
178
0.120
o.ooo 8845
o.ooo 7698
o.i 80
0.002 0635
o.ooi 6782
630
TABLE XIY.
For finding the Katio of the Sector to the Triangle.
X
S
X
Ellipse.
Diff.
Hyperbola.
Diff.
Ellipse.
Diff.
Hyperbola.
Diff.
0.180
.181
.182
.183
.184
0.002 0685
.002 0929
.002 1175
.002 I42Z
.002 1671
244
246
247
249
"7 C T
o.ooi 6782
.001 6960
.001 7139
.001 7319
.001 7500
178
179
180
181
181
0.240
.241
.242
*43
.244
0.003 8289
.003 8635
.003 8983
.003 9333
.003 9685
346
348
35
35*
0.002 8939
.002 9166
.002 9394
.002 9623
.002 9852
227
228
229
229
451
354
231
0.185
.186
.187
.188
.189
O.002 1922
.OO2 2174
.002 2428
.OO2 2683
.002 2941
252
254
255
2 5 8
2 5 8
o.ooi 7681
.001 7864
.001 8047
.001 8231
.001 8416
183
183
184
185
186
0.245
.246
.247
.248
.249
0.004 39
.004 0394
.004 0752
.004 ii ii
.004 1472
355
358
359
361
363
0.003 83
.003 0314
.003 0545
.003 0778
.003 ion
231
231
*33
233
234
0.190
.191
.192
.193
.194
O.OO2 3199
.002 3460
.OO2 3722
.002 3985
.002 4251
26l
262
263
266
267
o.ooi 8602
.001 8789
.001 8976
.001 9165
.001 9354
187
187
189
189
190
0.250
.251
.252
*53
.254
0.004 1835
.004 2199
.004 2566
.004 2934
.004 3305
364
36?
368
371
372
0.003 i*45
.003 1480
.003 1716
.003 1952
.003 2189
235
236
236
*37
238
0.195
.196
.197
.198
.199
0.002 45l8
.002 4786
.002 5056
.002 5328
.002 5602
268
2 7
272
274
275
o.ooi 9544
.001 9735
.001 9926
.002 0119
.002 0312
191
191
'93
193
J 95
0.255
.256
.257
.258
.259
0.004 3677
.004 4051
.004 4427
.004 4804
.004 5184
374
376
377
380
382
0.003 2427
.003 2666
.003 2905
.003 3146
.003 3387
239
239
241
241
241
0.200
.201
.202
.203
.204
OOO2 5877
.002 6154
.002 6433
.002 6713
.002 6995
277
279
280
282
283
0.002 0507
.002 0702
.002 0897
.OO2 1094
.OO2 1292
'95
195
197
198
198
0.260
.261
.262
.263
.264
0.004 5566
.004 5949
.004 6334
.004 6721
.004 7111
383
385
387
390
391
0.003 3628
.003 3871
.003 4114
.003 4158
.003 4603
243
243
244
245
245
0.205
.206
.207
.208
.209
0.002 7278
.002 7564
.002 7851
.002 8139
.002 8429
286
287
288
290
293
O.O02 1490
.002 1689
.002 1889
.OO2 2090
.002 2291
199
200
201
2OI
203
0.265
.266
.267
.268
.269
0.004 7502
.004 7894
.004 8289
.004 8686
.004 9085
392
395
397
399
400
0.003 4848
.003 5094
.003 5341
.003 5589
.003 5838
246
247
248
249
249
O.2IO
.211
.212
.213
.214
O.002 8722
.002 9015
.002 9311
.002 9608
.002 9907
a 93
296
297
299
300
0.002 2494
.002 2697
.OO2 2901
.002 3106
.OO2 3311
20 3
204
205
205
20 7
0.270
.271
.272
.273
.274
0.004 9485
.004 9888
.005 0292
.005 0699
.005 1107
43
404
407
408
410
0.003 6087
.003 6337
.003 6587*
.003 6839
.003 7091
250
250
252
252
*53
O.2I5
.216
.217
.218
.219
0.003 0207
.003 0509
.003 0814
.003 III9
.003 1427
302
305
35
308
309
O.002 3518
.002 3725
.002 3932
.OO2 4142
.002 4352
207
207
2IO
210
210
0.275
.276
.277
.278
.279
0.005 1517
.005 1930
.005 2344
.005 2760
.005 3178
413
414
416
418
420
0.003 7344
.003 7598
.003 7852
.003 8107
.003 8363
254
254
*55
256
257
O.22O
.221
.222
.223
.224
0.003 1736
.003 2047
.003 2359
.003 2674
.003 2990
3"
312
3'5
316
318
O.OO2 4562
.002 4774
.002 4986
.002 5199
.OO2 5412
212
212
213
2I 3
215
0.280
.281
.282
.283
.284
0.005 3598
.005 4020
.005 4444
.005 4870
.005 5298
422
4*4
426
428
430
0.003 8620
.003 8877
.003 9135
.003 9394
.003 9654
%
III
260
O.225
.226
.227
.228
.229
0.003 3308
.003 3627
.003 3949
.003 4272
.003 4597
3*9
322
3*3
3*5
327
O.002 5627
.002 5842
.OO2 6058
.002 6275
.002 6493
"I
216
217
2Ig
218
0.285
.286
.287
.288
.289
0.005 5728
.005 6160
.005 6594
.005 7030
.005 7468
432
434
436
438
44
0.003 9914
.004 0175
.004 0437
.004 0700
.004 0963
261
262
263
263
264
0.230
.231
.232
.233
234
0.003 4924
.003 5252
.003 5582
.003 5914
.003 6248
328
330
33*
334
336
0.002 6711
.OO2 6931
.002 7151
.002 7371
.002 7593
22O
220
220
222
223
0.290
.291
.292
.293
.294
0.005 7908
.005 8350
.005 8795
.005 9241
.005 9689
44*
445
446
448
45
0.004 1227
.004 1491
.004 1757
.004 2023
.004 2290
26^
266
266
^ \
267
0.235
.236
*37
.238
.239
0.003 65^4
.003 6921
.003 7260
.003 7601
.003 7944
337
339
341
343
7 AC
0.002 7816
.002 8039
.002 8263
.002 8487
.002 8713
223
224
224
226
226
0.295
.296
.297
.298
.299
0.006 0139
.006 0591
.006 1045
.006 1502
.006 1960
452
454
457
458
461
0.004 2557
.004 2826
.004 3095
.004 3364
.004 3635
269
269
269 \
271 ,
271
0.240
0.003 8289
J i J
0.002 8939
0.300
0.006 2421
0.004 3906
631
TABLE XV.
For Elliptic Orbits of great eccentricity.
c or 3
log .B or log .B '
Diff.
log^V
Diff.
eorS
logj&oorlog.Bo'
Diff.
log N
Diff.
o.ooo oooo
o.ooo oooo
30
o.ooo 0066
o.ooo 6400
A >&
1
.000 oooo
o
Q
.000 0007
7
2 1
31
.000 0075
9
J J
.000 6836
436
2
.000 OOOO
.000 0028
tft
32
.000 0086
.000 7286
450
3
.000 oooo
o
o
.000 0064
3 b
40
33
.000 0097
I I
12
.000 7750
464
4
.000 OOOO
o
.000 0113
64
34
.000 0109
13
.000 8229
493
5
o.ooo oooo
o
o.ooo 0177
78
35
0.000 0122
15
o.ooo 8722
508
6
.000 OOOO
o
.000 0255
Q2
36
.000 0137
.000 9230
J
7
.000 OOOO
o
.000 0347
y
107
37
.000 0153
18
.000 9753
537
8
.000 oooo
.000 0454
1 20
38
.000 0171
IQ
.001 0290
J J /
5r 2
9
.000 OOOI
o
.000 0574
39
.000 0190
* 7
20
.001 0842
*
567 1
10
O.OOO OOOI
o
o.ooo 0709
149
40
O.OOO O2 1
22
o.oo i 1409
581
11
12
.000 OOOI
.000 0002
I
.000 0858
.000 I 02 I
163
178
41
42
.000 0232
.000 0255
11
.001 1990
.001 2586
596
611
13
.OOO OOO2
I
.000 1199
43
.000 0281
27
.001 3197
626
14
.000 0003
I
.000 1390
206
44
.000 0308
/
29
.001 3823
640
15
16
o.ooo 0004
.000 0005
I
2
o.ooo 1596
.000 1816
220
27C
45
46
o.ooo 0337
.000 0368
3 1
77
o.oo i 4463
.001 5118
655
670
17
.000 0007
2
.000 2051
JJ
248
47
.000 0401
36
.001 5788
685
18
.000 0009
2
.000 2299
Tr
263
48
.000 0437
38
.001 6473
J
700
19
.000 00 1 1
2
.000 2562
277
49
.000 0475
J
40
.001 7173
/ w
715
20
21
22
23
24
o.ooo 0013
.000 0016
.000 0019
.000 0023
.000 0027
3
3
4
4
5
o.ooo 2839
.000 3131
.000 3437
.000 3757
.000 4091
292
306
320
334
349
50
51
52
53
54
o.ooo 0515
.000 0558
.000 0604
.000 0652
.000 0703
3
48
51
54
o.ooi 7888
.001 86l8
.001 9362
.002 0122
.002 0897
73
744
760
775
79
25
26
27
28
o.ooo 0032
.000 0037
.000 0043
.000 0050
1
7
7
o.ooo 4440
.000 4803
.000 5181
.000 5573
367
378
392
4 7
55
56
57
58
o.ooo 0757
.000 0815
.000 0875
.000 0939
64
68
0.002 1687
.002 2493
.002 3313
.002 4149
806
820
836
851
29
.000 0057
9
.000 5980
420
59
.000 1007
.002 5000
866
30
o.ooo 0066
o.ooo 6400
60
o.ooo 1078
0.002 5866
TABLE XVI.
For Hyperbolic Orbits.
m or ?!
log Q or log Qf
log I. Diff.
log half II. Diff.
m orra
log Q or log Q'
log I. Diff.
log half II. Diff.
0.00
.01
.02
.04
0.000 OOOO
9-999 9 8 7
999 9479
.999 8828
999 79 J 7
2.41597*
2.71675*
2.89259*
3.01741*
2-1149*
2.1146*
2.1130*
O.IO
.11
.12
'3
.14
9.998 7021
.998 4308
.998 1342
997 8123
997 4654
3.41256*
3.45326*
3.49028*
3.52423*
3-55547
2.1046*
a. 1 025*
2.1003*
2.0978*
2.0952*
0.05
.06
.07
.08
.09
9 999 6746
999 53i6
.999 3628
.999 1682
.998 9479
3.19290*
3.31687*
3-3 6 745
2.II2I*
2.1 1 IO*
2.1097*
2.I082*
2.1065*
0.15
.17
.19
9.997 0936
.996 6971
.996 2760
995 8305
995 3 6 8
3-5 8 453
3.61154*
3.67679*
3.66048*
3.68276*
2.0923*
2.0892*
2.0860*
2.0826*
2.0790*
O.IO
9.998 7021
3.41256*
2.1046*
O.2O
9.994 8671
3-737 8 n
2.0"C2 n
632
TABLE XVII,
For special Perturbations.
q, "
For positive values of the Argument.
For negative values of the Argument.
log/
Diff.
log/', log/"
Diff.
log/
Diff.
log/', log/"
Diff.
0.0180
.0181
.0182
.0183
.0184
0.457 9499
457 8454
457 749
457 6 3 6 5
457 53* 1
1045
1045
1044
1044
0.285 6702
.285 5864
.285 5026
.285 4188
* 8 5 335
838
838
838
Q~Q
0.497 0554
.497 1684
.497 2814
497 3944
497 575
1130
1130
1130
1131
0.316 9530
.317 0431
.317 1332
.317 2234
3*7 3*35
901
901
902
901
1044
o ^o
902
0.0185
.0186
.0187
.0188
.0189
0-457 4*77
457 3*33
457 *i8g
.457 1146
457 OI0 3
1044
1044
1043
1043
10 43
0.285 2512
.285 1675
.285 0838
.285 oooo
.284 9163
837
837
838
837
837
0.497 6206
497 7337
.497 8 46 8
497 9600
.498 0732
1131
1131
1132
1132
"3*
0.317 4037
3*7 4939
.317 5841
.317 6744
.317 7646
902
902
93
902
903
0.0190
.0191
.0192
.0193
.0194
0.456 9060
.456 8017
.456 6975
45 6 5933
.456 4891
1043
1042
1042
1042
1042
0.284 8326
.284 7490
.284 6653
.284 5817
.284 4981
836
837
836
836
836
0.498 1864
.498 2996
.498 4129
.498 5262
.498 6395
1132
"33
"33
"33
"33
0.317 8549
3*7 945*
.318 0355
.318 1259
.318 2162
93
93
904
93
904
0.0195
.0196
.0197
.0198
.0199
0.456 3849
.456 2808
.456 1767
.456 0726
455 9 68 5
1041
1041
1041
1041
1041
0.284 4'45
.284 3309
.284 2473
.284 1637
.284 0802
836
836
836
835
835
0.498 7528
.498 8662
.498 9796
499 0930
499 * 6 4
"34
"34
"34
"34
"35
0.318 3066
.318 3970
.318 4874
.318 5778
.318 5683
904
904
904
905
905
O.O2OO
.0201
.O2O2
.0203
.0204
0.455 8644
.455 7604
455 6564
455 55*4
.455 4484
1040
1040
1040
1040
1040
0.283 9967
.283 9132
.283 8297
.283 7462
.283 6627
835
835
835
835
834
0.499 3199
499 4334
.499 5469
.499 6604
499 774
"35
"35
1135
1136
1136
0.318 7588
.318 8492
.318 9398
.319 0303
.319 1208
904
900
905
905
906
0.0205
.O2O6
.0207
.0208
.0209
0.455 3444
455 *4Q5
455 '366
455 3*7
.454 9288
1039
1039
1039
1039
1039
0.283 5791
.283 4958
.283 4124
.283 3290
.283 2456
834
834
834
833
0.499 8876
.500 OOI2
.500 1149
.500 2286
.500 3423
1136
"37
"37
"37
"37
0.319 2114
.319 3020
.319 3926
.319 4832
3'9 573 8
906
906
906
906
97
0.0210
.0211
.O2I2
.0213
.0214
0.454 8 *49
.454 7211
454 6173
454 5*35
454 4097
1038
1038
1038
1038
1037
0.283 1623
.283 0789
.282 9956
.282 9123
.282 8290
834
833
833
8 33
833
0.500 4560
.500 5697
.500 6835
.500 7973
.500 9111
"37
1138
1138
1138
"39
0.319 6645
-V9 755*
.319 8459
.319 9366
.320 0273
907
907
907
907
908
O.02I5
.02l6
.0217
.0218
1 .0219
0.454 3060
454 **3
.454 0986
453 9949
453 8912
1037
1037
1037
1037
1036
0.282 7457
.282 6624
.282 5792
.282 4959
.282 4127
833
832
8 33
832
832
0.501 0250
.501 1389
.501 2528
.501 3667
.501 4807
"39
"39
"39
1140
1140
t>.32o 1181
.320 2088
.320 2996
.320 3904
.320 4813
907
908
908
909
908
0.0220
.0221
.0222
.0223
.O224
0.453 7876
453 68 4
453 5 8 4
453 478
453 3733
1036
1036
1036
I0 35
0.282 3295
.282 2463
.282 1631
.282 0800
.281 9968
832
832
831
832
831
0.501 5947
.501 7087
.501 8227
.501 9368
.502 0509
1140
1140
1141
1141
1141
0.320 5721
.320 6630
3* 7539
320 8448
3*o 9357
909
909
909
909
909
0.0225
.0226
.O227
.0228
.0229
0.453 * 6 9 8
453 l66 3
.453 0628
45* 9593
45* 8558
1035
1035
I0 35
1035
1034
0.281 9137
.281 8306
.281 7475
.281 6644
.281 5814
831
8 3 !
830
831
0.502 1650
.502 2791
.502 3933
.502 5075
.502 6217
1141
1142
1142
1142
"43
0.321 0266
.321 1176
.321 2086
.321 2996
.321 3906
910
910
910
910
910
O.0230
.0231
.0232
o*33
.0234
0.452 7524
.452 6490
45* 5456
.452 4422
45* 3389
"034
I0 34
i34
1033
1033
0.281 4983
.281 4153
.281 3323
.181 2493
.281 1663
830
830
830
830
830
0.502 7360
.502 8503
.502 9640
53 0789
.503 1932
"43
"43
"43
"43
"44
0.321 4816
.321 5727
.321 6637
.321 7548
.321 8460
9"
910
9"
912
9"
0.0235
.0236
o*37
.0238
.0239
.0240
0.452 2356
45* 13*3
.452 0290
.451 9258
.451 8226
.451 7194
1033
1032
1032
1032
0.281 0833
.281 0004
.280 9174
.280 8345
.280 7516
.280 6687
829
830
829
829
829
0.503 3076
.503 4220
53 53 6 4
.503 6508
503 7653
503 8798
"44
"44
"44
"45
"45
0.321 9371
.322 0282
.322 1194
.322 2106
.322 3018
.322 3930
9"
912
912
912
912
636
TABLE XVII.
For special Perturbations.
q, t^. t-^
O\ O\ O\ ON ON ONONONONON
oo
>* ON IN
O t^
d co to O * O
vo covo co f> oo co IN NVn t^rt-ON O\OO
^- VOON OooOvnTf- vnMvnOco IN r^ H d O
d d M d co co
t^d^ OtnONdON
O M vn t~x rj- ON ONOO
vnQco Mt^-ddO
MdcoM ddcoMM
gQOOOO ON-*
SOOOOO ooco
TH O?
TH C u-> m <$ o vo O vo <* m <*- O O t^ t->
wrowosf-s COMVOMM Os m Os ^ O
so Os rj- Os m ON O
ON O Osoo Os
OO so O t^ Os
x Osso O O
ro O
M to 00 O so t^ 00 Os Osso "6 t-. voso rt 00
sovo rx i-~ rt m ON *> r-~. ON O m
ON M rt O O ^ O Osso f^oo oo
o ONSO TJ-OO M r-x c< m r~* t-^ M m m M OsO^hrort
O > l~> t~~ t^. ON vo rt C^vOO OO t^.OO SO Oj\ O O^^ ^ O^OO
ot^osONON osoo'ososON o^ONO^ONO* O\O*O\ONO\
00
O vo O t^
f^oo rt f
Os Os Os rt
rt Osso oo
q q q q q a\ q q q q q q q q q vo o> cr\ ON q q q q q q q q o o o o q o o
O O
^-^-
0000
OO OO J- VO
_ OsOOst~xM t-^i-it-^ONrO MSOt^fOOs HOt^covo MONrtMON M
"so rtoovoMrtmt^rortt^rtoot^ SOMOOSO covortMso rf
a
Os f- vo rf o oo m vooo M M oo m ^so to rt M tooo O O so t^ M
O rt cosO SO CO ^ ON fxOO OO OOrtONOstOsOf^vo t^sO vo Os M \O rt
rt MMCO rt rt 1-1 rt rt rt M rt to rt rt f
OOOoo voOOO 1 '*
>O oo r~vO O M oo tooo vo On
O vo M t^ t^ O M HM mt^
H rtMrtmco MM
rtrtMsooo
sOOrt tooo rtONM rt
sot>.M comr-t^rt ^h
vort votovoco
^
*s|^I f|J5 tsjili f
| 'B . - T2 . d'B
S . 3 3 P c S c3 g
llflH-IFlllllilllll
.-3045 3 o> .J3 o t^coo vovoclooN
M M HI H M M
T}- rj- ^ vo
1VO VO CO ON
d co M
ON O vo t^OO
vo c4
VO H M CO CO
$
VOOO Ht^-CO COON^-T^-IO COM Tj-VO OO rj- O vo t^ t^
MfxdvOO VOCOQOOM rJ-vO'tj-voO OMt-^MOO
rHrH C<1 r-l Cq rH
Illil
rj "HI bo t> *S rj -S bb -S
"^ Q^lrJ r-s o CL.^ A 3 5
SO PH-O;
C.
00
1C
OS O O rH >> > >
$&%&&
O O rH r- 1 CO
oo oo oo oo oo
I l>. IN. i i>. i>i>. c> !> QO QCaDCCQOQID OO 00 00 QO Ct
640
otctot
ooooo
TABLE XV1IL
Elements of the Orbits of Comets which have been observed.
"a
1
.S .S
*-'! s'l
Encke.
Gibers.
//
//
Burckhardt.
Wahl.
//
Burckhardt.
Gibers.
Gauss.
c5 I'-
ll 1|1
Bessel.
Triesnecker.
Argelander.
Nicolai.
Encke.
"H
1 fa* i
Hill
Motioo.
I 1-
r 'c\r -vO
M M VO VO "4-
O w oo O t>
ON M ONVO M
ON O* ON ON O*
vo
^ i ON O ON
O vooo r-^ r>
CO M T^- COOO
Tj-00 M VO M
H ON H OO ON
VO *H t^x VO OO
ON O* ON ON ON
t~s CO ^- M OO
cooo O vo vo
** r}-oo O oo
^}- vo r-x ON ON
H ON M co H
ON f^ rj- O O
ON ON ON O o"
00 O 00
VO Tj- ON vo
O vo M CO ON
H r-- TJ- o o
CO VO CO >H ON
vo a\ o oo vo
ON ON O* ON ON
oo ON vo
O vo t^ vo ON
t^ vo M co ON
OO CO M C ^-
CO ON VO ON O
OO oo M ON ON
t~~ ON O M OO
ON ON o' O' ON
ON t ON
l-^ M O ON O
vo c M vo VO
vo ONOO t~- t
r}- ^ co 10 vo
^ M o ^* r^ to
t- l^rh
co^-c* *
* 3-
O <* cooo f*
ON vo M M to
CO C< VOVO Tt"
s.
"*" M H CO Tj- VO
d VO t^ W M
VO Tj- C< tO
t> rj- O 'J-VO
rt vo O O oo
co-1o"2?
OO M cJ t^ C^
MM M VO
co rt ON vo H
3~ M VO t^^
VO co -O VO
M M COVO rj-
tovo r-to^
c* oo ^ ** tJ
^ vo covo c^ t^
CO ^- rt CO
^ M vo rj- O 00
ON^-Ot,
M M * ON O
d VO t~x M O
H w co rt co
cooo O ONOO
M CO VO
ONVO M VO
iT rt ^ Jo **
4- vo T}-VO O
f M CO VO
CS
VO VO H VO
VO VO H t t^
O o vo vo t-^vo
to O cooo c<
CO ONOO O
M Cl 1-1
^J- t~ ON rt ON
CO M c< C^ ^"
to co M tJ
ONVO Tj- O vo
ON r> 'i- M f-
CO CO M
Tj- M f* VO t<
co vo r* vo r)
to 6) co rJ co
^h O O to to
H HH rj- ON vo
CO M C4
O c* co co O
VO TJ-00 H vo
It
ft ON ts. Tj- vo 1-.
VO Tj- CO
TOO ON Tj- rj-
t ON ci oo w
o ON rt vo M
t^ cl vo O to
C< CO CO
co O ?< f* rj-
h t* vo Tt- vo
COVO VOOO 11
rt" ^ nT c* "^
8 cooo t^ l-^
M VO tl
VO H ON VO rj"
vo ON T|- O to
VO vo O -^- w
00 rt ON ON r-
CO rl rj- T}-
to O co ^ oo
ONOO co rj-
M M CO M
t-^00 co rt- t*
<*- H 10 M
vo ON t-* O ON
o O ON t^vo
VO T- CO rt 4-
oo ^ O i~^oo
CO ^" H ^
vo vo r-s r}- ON
00 00 VO CO O
VO CO
vo to M to t^
vo ^J- co
ON t-~ ON f- vo
vo ON Tj-vo ON
*> ONVO ON ON ON
CO C CO
S 00 vo VOOO VO
rj- vo ON O 1
VO M H
vo o r^* rt vo
n H O vo co
vo t~ w OO f^
vo vo O vo vo
CO rj- VO
O to co covo
O O H ON rj-
10 M VO
ON M ON co r*
VO CO CO M CO
O to O M
O 1-- d O O
oo M oo oo vo
< ON t~ f>.OO f^
vo t l^ O w>
O ON t M co
VOOO CO M TJ-
H, f* f t- t>
5j- rt VO co t^
t O coao O
^ f^\ & l~5 ^
CO 00 00 OS OS
CO rH CO CO
CO CO CO CO
OS OS OS OS
CO CO
io o t^ co co
OS OS OS OS OS
iMslM
OS OS O O O
t~- t^ CO CO OO
jllft
CO OO CO 00 OO
S co SSi5
00 CO CO OO OO
lllll
CO CO lO CO CO
oo oo co no co
OOOO'H
i-l d W *# i
222i
SUSSw
gJSSSw
1-1 SI 95 rfl
CO CO CO CO CO
09 WCO CO-*
iHlHrHlHiH
41
641
TABLE XVIIL
Elements of the Orbits of Comets which have been observed.
"
ver.
geland
mbart.
Heilig
1= rs^ -g 5 s^
o-3O > owo^w
R
ONOO vo
f-oo O
co co cooo to vo ON to O "<$
ooootodoo OddONt-~
r~ d d cooo tovo O co d
O ON to tooo ON ON t-~- to ON
O" ON ON ON ON ONOO' ON ON O*N
tovo l~- ONVO *3-oo O d M to d
OdQvOi-i d^i-oooooo t^ d M
co r~-oo Tj-vo ON co w o to d tooo
OO O t^ d ON M vO t^ Tj" vo T$- OO tr
OQtOMMOO VO t-^ CO to M ^- O O
^- co ON to O
ON to O O\ co
O' ON ON O' ON ON ON O>' ON O* ON ONOO* ON ON
t-.00 d d to t- rj- M
rj- to co *J- d O O f--vO O ON
ONOOOvDOQONVOdro covo
$ CO to rj- ONOO 1-1 M OO OO O
t^ ONOO ** ON to co ^vo M d
**. ^ ^t" ^ ^ M . *"? ^ . *? . ^^ D . ^ ***
ONO>ONONON o'oNoVoNO*
to M
M co oo ro M
'i- O o rj- d
00 ON rj- VO O
to M t^ T*- co
OO to VO TJ- VO
rl- to OO TJ- ON
OOooOt-^voOOooO ONOO
oo oo M ON co rj- T^-
oo d O to t-~- d M
oo rj-vo ON d vo ^
TJ- tovo oo ON r*- to
TJ- ON rj- O ON ^ rj-
OOOONt--O OOOOO ONOO O O 00
MMOMO OMMOU OMMMM MOOOM MMMMM O O M O MQMMM
CO W
u-i o
O t-~ ON ONVO t~^ ^*
M to to ^J- d
ON M T$-VO M MM
** t^ o c
CO d co
Ovotod ooMoovod
d M T^" d M M M tO
Tj-vO t^-VO rj- ON co d O ON ONOO
TJ- rj- to o d ^*- co
coO co O to to M rj-coo\ tovo NO
T^- M M Tt~ d d tO ^" M M tO
to ^J-OO ON rJ-VO f^
cooo M rj- co O co
rl M co M to d co
MdCOl-IT^- MtOCOw
tOt^-ONtOO OOONt-^MCO O Tj-VO M O Th TJ-VO OO M
d covo r~*J- t^ONdt^ON MTj-o^hco O r^oo ON t^
to t~> to co t-x
co to M co ON
co d
Ot^-dOt-^ f^ONTj-vovo
totoMMto dOdt~^O
d HI d co M d M d d d
Soooodvo cod
M co to
T*- oo v
d co
M co r^-vo vo O co
eo s cTi-rcrcr CO"O > O N OO* V ^"
(N i I (M CO i-lrHr-(r--O?OCD
rl r-l ^ Jli ^H ^-t CM CNJ (M CN (M CN ^ CM C^ (N CN CM CM CM
cooooocooo oooooocooo ooooooooco c-oooooooco
^ ^^1
^_ ^
>* -*4 > " >!H ^
? s^ o
CO CO CO CO CO CO CO
COOOOOOOOO OOOOOOOOOO OOOOOOCO'CO
642
TABLE XVIII,
Elements of the Orbits of Comets which have been observed.
Encke.
Westphalen.
Encke.
Peters and O. S
Plantarnour.
g.S
Dir
ii
n
Re
Di
Direct.
Retrograde.
//
Direct.
ONTj-ioj^coooOHivotovodooTj-vo * O ooo
OO ONOO n vo -^h t---vo Qvo M co O *j- d vococoQOO
O HI o O to ON O co\O r-~ covo covo HI vo co d O vo
M covo cooo O vooo voco to^-dnON r-. tooo M T^-
t^oo vo M o rf-Ot---dcoooooTt-dO voooroONO
covo co ONOO l~x r^ co O 3- O d l-- co o vo ON O d t-~
tor~.toi>.q oo M to t^. t--. d d O ON * O^ O vo to M
o\ 6\ o\ o\ d o\ o* o\ ON t>. o o" o o\ o\ o\ o o\ o\ o'
H ON co
ON d M d ON toodnto
OOOOOOON oo vo O vo o
t^ ON ON d d oo c~^ co r*** o vo d
ddnoorj- wOOONiOt^. MOO
co d cooo o M d d -* r^oo
OjvOO oo HI HI oo ONVO co d M vO
ON ON ON O O* ON O\oo' O O O ON ON o"s ON
t^
vo
r^ ^- r^
w o rt
vo ON to O vo vo T*-
vo O l-^ to co d O
CO ON r^ O OO to ON
O co M d oo oo t^
to r^. to O r^ ON T}-
TJ-VO Tj- O ON vo ^J-
oo ONOO O ON O ONOO O
O* O O HI* o* M O* O* HI"
t^ oo t--. ON co O
vo ON ON cooo
HI r^ ON to O
ON HI oo vo vo
ON O vo t-^ ON
ON O to HI ON
ON O vovo ON
co
d d vo O O w
rj-vo d vo oo ON
t>- co o O oo oo
oo >* T!- vo co o
ON t-^ d vo co d
OO Tj- ON vo ONVO
O O ONOO ON t~- t^ ON O
00
to ON t^ ON
X3 oo d d
CO CO M HI
co ON co ON
HI ON co ON
d OO ON ON ON ON t^
t^ ON ON 0-. O ON ONO>
vo
rj- O H
COVO CO
-J- U1 H,
O HiOOOni nHiQOO OOO>iO OOOHiQ OOMHiO
vo vooo d O d covo t^ ON vo co O M t^ ONVO ON
HI Hivod voddcoco rj-covo T}- cocovo
^ M to M IO CO Hlt^.
S c^g- $
d MTj-tocod wvocodto
n 10 to to
a
* vo Tj- ONVO VO 00
O COlOCOHi CO OO^t-
co co M d HI d
VOQVO COM CO^-COCOM
Hid HI cocococOHt
4- 10
U-, w
CO ^ M
dcodHiM Hicod
vo o vo ONOO d d to d t^
HICOMM CO MCOd
M ON d ONVO HI d d
00 * ^- t-^ ON ON ONVO
dMoodd t^vooNONON covo vo to d ON r>-
CO COrJ-COdVO CO CO^-COHITJ-HICO
ONMVO OOOVO^-VO OONO>MO
co o vo to cf
^*i
_OO OOC^C^CO COCO^Tflrtl lOiO>OvOO COCOOCO?O COCOr^t~-t-> h'.t^.t^-OOQO
OO OO CO OO CO CO CO GO 0000000000 GO 00 00 OO CO CO 00 00 00 00 CO CO CO 00 QO 00 00 CO 00 00
v*G*C6'*#*6 S3t-
GCQCaDODOO QCQCQCQOO
?Df>.Gt CtO
QCQCQCQCiO C&CtCtC^Ci C6 d Gi Ct O
-,- , ^_ __ fHfHrHTHOl
ooooo
TABLE XYIII.
Elements of the Orbits of Comets which have been observed.
* s
S | iiia-s *:s
3 w2w
53 00 . g J O> 03
( i C JH r* . ^ Q} t ^C2
-S '3^^ r =^c;
CTli^ g 1^3^ ^^,5^^ | 1=J=
b^^rCO L-3 ^4 2 X K*i h"! 7-5^ O"! h> ^ ^^i^T-S^r^M
-^ H
isfcstst ftfeftfci +>
Re
Di
/
Re
H vovo
ON w o
Th oo
f-^OO vo
O <* to 6 c5
T*- c* vo c i-
O HI ONOO HI cocOt>vot> OO vo rj- t>-v
rf-OO OO HI Tfr- IH rt OO vo HI to t-~ HI ON
ON
O
6. .. .. rt t>. O
H * <* HI M H
CJ ^" HI rj- rt O ON CO cJ OO O "^ VO OO OO r^ t^OclHl C)C)HIHH t^^ O r^ t^ vo CO
OO VO vo CO vo tOVO ONVOt* VOtOONtOvooOCOi-iTj-i-i OcOr}-votJ ONOO ONVO t~^
ONOONOt^- rJOONHivo O\O\OOON ^-t)to rj-oo ONi- 00
Tj- ONVO
_._._. VOVOON ,
O 0\ ON ON O vovo ON O OO O t>- ON ON ON
HI* O* O" O* HI' o o' o' w o* H* o' O' O' O*
vo HI O ON ^t- vo ON
H ^i" vovo cJ ON rj- oo
r^ O vo HI M w ti cl
vo vooo ON TJ- to TJ- H
r-- H VOOO O ON ON HI
vo vo w ONOO H O
VO M O O ON T-VO i*- -4- vooo <$- vo
Tj- T^- vo r^-vo vo T}- rt-oo HI to ON co co
M O OO ONOO vowONONr- HiONOOO
cOTt-wONt^ HHHONCO ON ONOO ON >-i
CO>votOC^ t^ONrtOOO VDVOONO'i-
VO O * ON ON O ONOO ON ON vo M vo
O O O O ON 0\ Ov ONOO ON OsOO ON ON ON C^vO OO
M M w* O* O* O* O' O* O* O* O* 0* O* O* O* O* O*
tO O ON rt- CO 00 00
00 ON O O
vo HI
HI coox) coco ONHOOt^ HiOvOc^Hi O^-HICOCO Ol~^ ONOO t vo t-~- co rj- o
M M tO f^ HI ^* M T^ H VO VO VO VO OO t^ ^ M VO 61 HI hH OO tJ vo CO VO CO -( vo IM
HI OOOOVOf-. t COC4 COCO Clvo^cJ^J- COVOCO^J-H COCOwHHI TJ-Tl- vo
t>^ co O
ON t~ ON to
co in
O vo covo r^oo
c*
co ON ON rt O 00 .
^f^HICOVO t^O't'VOO HI
COCOCOM C M gi>H
fe
Ret
Direct.
Retrog
O
vo
Tj- <* ON VO d
O co co TJ- vo ONVO O
r^OooOd d vo M o t-^
vo oo d rj- M ^t-ddt^vo
d vo d d ON TJ- cooo vo vo
oo covo co d vo O f-- w vo
O t^- t*^ vo d M co o w ^J*
o" c?\ o\ o\ o" o oV o" o* o\
i O VO oo ON r-N
oo ^~ cooo oo T^ vo ^ vo
M t^% M ^- M vo t-^ co O
vo M Ooo co oo ^-<4-dvo
ONTJ-VOCOM M COTJ-OOO
t^vo i-i d co ONOO O O d
OvONONOjNVO ONOJNO\O\O
ON ON ON ON ON ON ON ON ON O
vo M T- - TJ- f^ ON OOvOMwd VO ONVO OO OO OO VO M ONOO
TJ-rj-Tj-coTi- cp co cp cp T}- r p rr| '*"'^"' : J" rj- co cp cp co c P T ^" T }" T i" r P
66666 66666 66666 66666 66666
ON f^ r^- co
ON CO vo -rt- O OOOMVOOO
covo OMd oo *!- d vo ON
d rj-oo d ON vo oo M >H ON
rJ-Ocor^co co d r~- ON r>
d t^ *}- O t- ON M vo d t-~
^J" cp TJ- TJ- cp r J" T *" T t" r t" T l"
66666 66666
VO
oo vo vo O
M VD OO OO d
T}- ONVO CO vo
d O O coo
q oo q vo vo
M ONTJ- "
vooooooo vodMTl-
d vo ON ONOO vo t^-vo d O
d O O oo M co T$- ONOO vo
CO OO CO CO II VO CO r*.* O VO
Ovococoro M oo oo d ^h
co oo oot~ vor~>oooooo rj-oo t^
covo oo t^. r^ cooo r>--i-d vorj-d^j-co OOONONCOM
O VO ONOO OO rl- COOO COCO ^-OdTj-Tj- OVOVOVOTJ-
vo vo ** t^ vo vo M o vo d oo ON d co t*^ vo t^*co vooo
r^* OMd vo vo vo M f^oo vo ON M co d oo vovo oo t^
M l-^. vo
ON N vo d -4-
COVO OO VO CO
OsONO ONVO
T$- T}-
c> ON
CO VO
vo vo d
ON ONOO OO OO
ONd O O oo
O w d COrJ-
t^ ONO ON ON
CO rj- COVO CO
CO 11 CO CO VO
ON f~-oo vo ON
ONVO VOONVO codMVoO
MCOVOVOt^ COVOCOQOO
OO ON t^OO O> VO OO t^-OO VO
OO VOOO COM OOCNjCOrl-Tl- MONM
d t^ vovo 6 4- O vo d vo
M d CO CO
CO IO VO
OO
O d d d co 4-oo oo d 06
M d COM ^4- M VOVO
VO CO CO CO I
COOO Tj- M
Q
co rj- vo O M co ON t-^ vo o d Tj- ON O
i O O ON O O co d
d d ONOO vo
vo T}-VO M cp
ON l^-vo d vo
t--OO CO CO O\
^cT^d 5^"
q q > vo TJ- vo vo vo co
t^ co O I s * "*J"
co d co ^
<1-00 vo ONM
CO vo O M O
VOVOM
d vo oo M co cooo vo d oo
t^ vovo d OO t^ vooo vo w
vocodM Mvorhco
I -4- t~~ <* ON ONVO t~- ON t^ d
cp vo Cfr d d
ON M ON O ON
ON vo ON vooo
. vo O d vo r^oo M O ON M oo vooo M
O vo O w vo
vo d vo M O
O vo r^vo <*
00 vo VO CO w
d
VO CO rj-VO OO
rl- ON rt- vo O
M CO CO
M O ONTj-VO
co d oo vo
d M co
tJ-OO ONVO w
00 VO M 00 00
M cod -*
i- q q oo vo
M VO VOVO 6
co d M
vocoti-coO T}-ONVOI
d w co vo co
vo rf- vo t~~- ON
rj- M VO VO Tj-
00 vo M vd CO
OVOVOOOON vOMt^dO
to vo ONOO ON
OO d rj- ON vo
VO M N M VO
W TJ- CO t^ CO
H
vo O vo O oo
M VO n CO ON
d
t^OO coO d
d vod rj-0
tO M IH CO
vo t^ d vo M
coco d vo co
d M
CO r}- d O M
ON ON T*- voO
M CO M d
t^ O ONOO vo
VOO*NOO vod
co TJ- d co
_ t^ M ONVO Tj-
O m M d M co
vo q vo t^ q
6 6* co 4- d
H VO 11
MONCpclvo vOMQcpd MO_NCO t^oo vo cooo cp O O O
vo ON M ON M vo 4" l"^ co 4" vovo vo O M vo M r^ d vo d 4"
llCOrl- CO T}-dCO rj-
rh d T}- vo vo O t r}- vo co
= 1 e-*
il -sf.pl HI'L
1 3 m'j 0^ "S'djajJ
ifq PL|HS^S H^OHHPn
lMl.2 Ii
II Is
3 S* tn
P- S ^> 3
3 c o .-
Ol^0004O r*
646
J. xi JJ J_LU WiYV i
Elements of the Orbits of the Minor Planets.
i 1: 1 1 I aa a a d . a a a I *i . d a
-g^SS-g -goc-g-g a ri S $ i! j 4 I IB i 1 Ss" c -3 "3 g.'o
TO O C 5 OT O O X W O M Jg TO 0> O 3 TO . * 0. O ( - g
2-52 2 Softs 2 So:222? 25213 2:2:2 S ** Seg-gg SS)5|2
00000 0000
nJOi-^Oi
CO T-l
,d
t- CO iO OS OS T*
iii-31 SSJrS* 9fttt
O O i-s fM S S ^ <; S >-s - d vo
f^oo t^ O
OQ M t-^ N
U~,
O
VO CO T^-
vo O oo r- d
<* o vo vo vo
COTJ-O M t-.
d r-- co rj- T|-
rt-00 rt-00 CO
TJ- CO CO CO 5J-
66666
vo vovO ON O
SON N OO CO
vo ONOO d
r}-vo d ONOO d M vo d
ON M M vo vo M ON ONOO ON
dd^J--^-ON ONTj-ddO
<4- ON M co t~- Tt- vo ONOO Tj-
M ON co cooo t~- ON t^ d CO
; . , , T *" T *"i" r *' t r T *"i" r r' t *" i o
ddddd ddddd ddddd
o o
M t^OO ^ ^J-
vo d O O oo
T}" M M O M
co rj- rj- co r>
r4 oo T*- t M
<* "*
d c d d d
VOMCOVOOOVO ^ONON
vo oo O oo d oo f-- vovo d oo t^ vooo co
CO <$- O vo CO vo vovo ON d COOO ON O **
ft ON 4 vo vo vo ONO co vo co vo ON d d **
VOVOdOO-^- ONONONCOTj- VOTJ-MVOON
ONVO dOON ONO4-MO covot^-voci
t^ d oo t^- co vo cooo xj- ON oo d TJ- vo d
r-^oo t^-r^O t^ONOONt^. oo t-~vo vo oo
TJ- t^ M VO VO Tj-VO CO O M
cooo O d M cooo co f>- t^
M o oo co d covo vo ON T!-
O O d O O
VOMTj-QVO MON OO
MONQVOt^ MOCOQVO
oovodOr^ dONdcoo
O 00 co cooo ON Tj- covo ON
i d ON
.VOCOONS. TJ-COONONVO oo^J-voOvb d -"t-vo ON co
ONVO OOt>t^ OOVOt^t^ON vovo ONOO vo OO ON f-vo OO
ON cooo
ON ON vo
O r^oo O MM
-)vo d rj-'
M r-~ d vo r--vo ooONd t-^t^-ONcovo
VO M c< COOO
. O oo vo d vo co ONOO tj- ON f^ ^- covo
ONVO OO co vo
to co cp ON rt-
ON 4 VOOO VO VO 00 vo 00 I
VO Tj- VO CO Tj- VO <$- CO
M Tj-M d VO
ONO OOO
00 CO CO M 00 MO vpvo O
4-dvovo4 - dcodt~-vo
MMd VO MCO^t-VOd
M vo d 'T rj- co TJ- TJ-
ONVO dVOVO VOVpCpOOO MONONdd
d oo co cooo vo r-'-vo d ON vo vo 4 ON 6
rf- VO ^4 M d M M VO M CO CO M CO
^ d
Tj-
00 COVO O
VOOO CO COVO d VOVO COd ONl*VOMt^ OOVO VOOO CO OOdvOMCO COVO t^OO M
847
1
jj -^ ** a *>
9
IUJI
O c- 3 o
a s i S is 9
I/ate of
)iscovery.
CO OS i OS C0 01
ec 0 OrH <*
r- r-t
O CO CO CO CO
CO CO CO OO CO
CO CO CO CO CO
co co oo co co
CO CO CO CO CO
oo co oo co co
CO CO CO CO CO CO
co oo co oo co co
8
S
VD rh vovo OO
ON r~- vooo N
M N ONVO vo
O vo vr> covo
Tfr-voN rj- N
O O O O O*
00 M fx ONTj-
CO TJ- t~^ COOO
O N vo vo N
ONOO t-^ ONO
ONVO OO t^. M
N N M OO vo
VO Tt- Tj- CO CO
0*
co N co t^. t^.
O vooo TJ- t^
oo ON co N oo
vo O vo co co
vo r)-OO t--. N
T|- -^- co co TJ-
66666
ONN ON ON M N
vo vo O O vo vo
O ** vo M co ON
O co N vo ON r
ON T$- T^ O ON O
6 o o o o o
1
a.
O O 00
ONOO CO O fx
N VO CO ON M
corl- -4- co vo
5 t-. M OO -^ ON
vo vo
vo ON M vo -4-
O O cooo O
vo O vo N oo
O rt- COM vo
-4- c?
r}-Hi VON d
O t-- TJ- VO t>
CO
00 O M vo
^- O >H VD ON t^
oo oo vo vo vooo
ON vo vovo co O
1
vo O -i-vo d
t^ ^J" M VO M
I--* O oo t-,oo
ON N vo ON ON
vo M CO N HI
vooo OO ON O
vo co t~- t^ O
cot^ co f^ N
t>- t^ONONOO
N co ON N d t^
vo Tj-vo r^ covo
VO vo t^OO VO 00
s
o
M ON VO M M
O ^h vo M vo
VOM 00 00 M
vovo f-x N t^ N
o
1
U-i
e.
5. vo vo O ON N
vo N r)- vo
- Ovo d co t^
O M MM
N O ^ co vo
i^ N co vo
ONOO M rj- CO
rt- ^- vo M co
t
5
8
.
"* vo vo co vo
v 00 CO Tj-00 O
M N Cj VO
O co vo N co vo
O vo ONVO M
vo vo co TJ- vo
M t^oo vo vo
N CO CO CO
O VOM vo N
VO M N O CO
vo vo N vo
t^. C voONM
rj- N OO OO vo vo
rj- N vo CO N N
r- HI ^- N t^. ON
TJ- VO M CO M
rj- O vovo N H
9
*s
03
O vo M co^f
Tj-VD rt-vO O
vo vo M vo co
vo O oo voO *
, Element
CS
% f ON ON ON CO
vo M vo rj-
^ ON T}- T}-OO VO
M ^" CO VO VO
_ VO t-^ t^ t-x ON
M O ONVO
co N M co
M t^oo N vo
00 ON VON HI
VO vo ^- CO
N N covo oo
M CO O M
cJ co N N
M M ON M CO
vo co
N vo r}- N H
CO VO CO N VO
dvo t^ t-x co
O c d O
coN
ON ON fx. t^ Tj- O
Tj- VO VO M
vo co T}- M o ON
\fl N *^- co M
^xVO f>. M M M
N co
Element?
M
M
W
t-q
pq
h
5. vo ON M O vo
v "4- ^ ON N fx
vo vo N d
N O vo co
N co co
00 N t~-00 N
vo CO rj- vo M
CO M N t^* VO
COOO M T$- vo
ON vo N r}- vo
M CO
t^ ON O O 00
cooo OO N N
co M N M co
COM 00 ONN
TJ- COOO CO N
M M CO CO
cooo O ON f^ co
TJ- coN N
ON M vo O covo
co N vo co M
00 f-00 ON T- vo
C* CO O r}- ON t^
co co CON
M
M
H
i--l
pq
00 CO 00 CO 00 CO
|
.
o
I
III II
i
.2 a .%
vVQoT'H <3
j
i* s* n ** *n
OJf-QOOO
sssss
Cf) 00 00 3D Q 05
e
Tf-vo O co o M o O
OOMONvovDOO
ON co O ON C-^oo ON vo
O coOvD t~~t^covo
t^ co O co N oo N O
OO N O N O cooo t^
cot^O voN VOM O
OOMHIVOONONO
M CO
<
t* vo ON ONVO O HI
OO r^- vo M ro rt- N
co N co O O N vo
O VO rf- ONVO M N
O O O O HI co O
6666666
+ 1 1 ++ 1 1
m
CO f-AO O M VO TJ- N
8O co t^~ N O CNVO
vo OO O vo vo t-^ ON
vooo r-.coM MVO ^i-
vovO vo COOO VO vo 00
O O M ONrJ-VOrJ-O
NOOOOOOO
OOOOOOOO
M vo covo vo M
<
% oo rj- O d vo co
M N M
++ 111+
vo vo N co r^ o t-
% r}-oo vo vooo M
N vo CO N
^ O co M oo ONVO f^
N VOM N 5*-Tj-
Ot^CO M M N O M
a
<
N HI ON M N ON
5 M tl- N d vo
COM 00 vo H ON
M C" CO N CO VO
II 1 1 1 1 1
cl ON vo ON !- cooo
a
4 O M cooo I--VOM
rl M M co co co
^ooTh OVOVOONI>.
vo vo N vo vo
^VOTj" OOOOMNO
^'t^ Tj-ONMt^CO
N
<
^ -<*-00 M N vo t^ O
* 't-N T-N H,
* O^ ^ ONVO M N rt-
M N M CO
ts
NniOvovoooMco
* t^ co vovo 4- oXvo d
covo vo co N M co
M COO COOO ONM t^
c <* CON <*">HI
_ Tj-oo ON N HI O 1 * t^ co
t-NONcoMOOVO'<4-
M CO M
NONvoOvoOON
^
< CO CO O VO COVO COOO
T$- M VON c ro
O COONN VOOOO VO
co co N M d ^
vOMOrJ-Nvor^vo
o vo ** O vo ** co t 1 ^ co
M M M HI M CO
Epoch and Mean
Equinox.
Greenwich Mean Time.
558 8S5 |
i I
6
to
l^4iil
6 Si Si 1-1 Z 8-
S>HS^PJ25
1 ^^^^^^^^^^^M
648
TABLE XXI, Constants, &c,
log
of Naperian logarithms e 2.71828183 0.43429448
Modulus of the common logarithms . . A = 0.43429448 9.63778431 10
Radius of a Circle in seconds r = 206264.806 5.31442513
'/ '/ a // minutes r = 3437.7468 3.53627388
" // // degrees r = 57.29578 1.75812263
Circumference of a Circle in seconds .... 1296000 6.11260500
" // // whenr=l. . . . TT = 3.14159265 0.49714987
Sine of 1 second 0.000004848137 4.68557487
Equatorial horizontal parallax of the sun, according to
Encke ' 8".57116 0.9330396
Length of the sidereal year, according to Hansen and
Olufsen 365.2563582 days 2.56259778
Length of the tropical year, according to Hansen and
Olufsen 365.2422008 // 2.56258095
This value of the length of the tropical year is for 1850.0. The annual variation is
0/0000000624.
Time occupied by the passage of light over a distance
equal to the mean distance of the earth from the
sun, according to Struve 497/827 2.6970785
Attractive force of the sun, according to Gauss . k 0.017202099 8.23558144 10
// // // // // // // in se-
conds of arc 3548.18761 3.55000657
Constant of Aberration, according to Struve 20".4451
// a Nutation, // // Peters 9".2231
Mean Obliquity of the ecliptic for 1750 -f t,
according to Bessel .... 23 28' 18".00 0".48368* 0".00000272295<
Mean Obliquity of the ecliptic for 1800 + t,
according to Struve and Peters . . 23 27' 54".22 0".4738< 0".0000014* 2
General Precession for the year 1750 + t, according to Bessel 50".21129 + 0".00024429f>6<
/, // u it 'i ii Struve 50".22980 + 0".000226*
MASSES OF THE PLANETS, THE MASS OP THE SUN BEING THE UNIT.
TO , Jupiter
_ 1
4865751
~ 1047.879'
1
1
VWlllH
, Saturn
390000
* * " ' 3501.6
1
1
TT.artTi
354936
* * * ' ' 24905
me
1
1
Mars . . .
3107713
* - 18780
649
EXPLANATION OF THE TABLES.
TABLE I. contains the values of the angle of the vertical and of the
logarithm of the earth's radius, with the geographical latitude as the
argument. The adopted elements are those derived by Bessel. De-
noting by p the radius of the earth, by being expressed in parts of the equatorial radius as the unit. These
quantities are required in the determination of the parallax of a
heavenly body. The formula for the parallax in right ascension and
in declination are given in Art. 61.
TABLE II. gives the intervals of sidereal time corresponding to
given intervals of mean time. It is required for the conversion of
mean solar into sidereal time.
TABLE III. gives the intervals of mean time corresponding to
given intervals of sidereal time. It is required for the conversion
t)f sidereal into mean solar time.
TABLE IV. furnishes the numbers required in converting hours,
minutes, and seconds into decimals of a day. Thus, to convert
I3h 19m 43.5s into the decimal of a day, we find from the Table
13A =0.5416667
19m =0.0131944
43s = 0.0004977
0.5s = 0.0000058
Therefore ISA 19m 43.5s = 0.5553646
861
652 THEORETICAL ASTRONOMY.
The decimal corresponding to 0.5s is found from that for 5s by
changing the place of the decimal point.
TABLE V. serves to find, for any instant, the number of days from
the beginning of the year. Thus, for 1863 Sept. 14, 15h 53m 37.2s,
we have
Sept. 0.0 = 243.00000 days from the beginning of the year.
Ud 15/i 53m 37.2s = 14.66224
Required number of days = 257.66224
TABLE VI. contains the values of M = 75 tan %v -f 25 tan 3 \v for
values of v at intervals of one minute from to 180. For an ex-
planation of its construction and use, see Articles 22, 27, 29, 41,
and 72.
In the case of parabolic motion the formulae are
m== f M mO T\
wherein log C = 9.9601277. From these, by means of the Table, v
may be found when t T is given, or t T when v is known. From
v = 30 to v = !SO the Table contains the values of log M.
TABLE VII., the construction of which is explained in Art. 23,
serves to determine, in the case of parabolic motion, the true anomaly
or the time from the perihelion when v approaches near to 180.
The formulae are
8 /200
=Y-
w being taken in the second quadrant. The Table gives the values
of A O with w as the argument. As an example, let it be required to
find the true anomaly corresponding to the values t T= 22.5 days
and log g = 7.902720. From these we derive
log M = 4.4582302.
Table VI. gives for this value of log M, taking into account the
second differences,
i> = 16859'32".49;
but, using Table VII., we have
w = 168 59' 29".ll, A O = 3".37,
EXPLANATION OF THE TABLES. 653
and hence
v = w + A O = 168 59' 32".48,
the two results agreeing completely.
TABLE VIII. serves to find the time from the perihelion in the
case of parabolic motion. For an explanation of its construction
and use, see Articles 24, 69, and 72.
TABLE IX. is used in the determination of the true anomaly 01
the time from the perihelion in the case of orbits of great eccen-
tricity. Its construction is fully explained in Art. 28, and its use in
Art. 41.
TABLE X. serves to find the value of v or of t T in the case of
elliptic or hyperbolic orbits. The construction of this Table is ex-
plained in Art. 29. The first part gives the values of log B and
log C 9 with A as the argument, for the ellipse and the hyperbola.
In the case of log C there are given also log I. Diff. and log half II.
Diif., expressed in units of the seventh decimal place, by means of
which the interpolation is facilitated. Thus, if we denote by log (C)
the value which the Table gives directly for the argument next less
than the given value of A, and by &A the difference between this
argument and the given value of A, expressed in units of the second
decimal place, we have, for the required value,
log 0= log (0) + Aj. X I. Diff. + AJ. 2 X half II. Diff.
For example, let it be required to find the value of log C correspond-
ing to A = 0.02497 944, and the process will be:
(1) (2)
Arg. 0.02, log (C) = 0.0034986 log I. Diff. =4.24585 log half II.Diff. = 1.778
^ _ 8770.6 log* A =9.69718 21ogA4 9.394
A4 = 0.497944, (2)= 14.8 3.94303 1.172
log = 0.0043771
The second part of the Table gives the values of A corresponding
to given values of r.
TABLE XI. serves to determine the chord of the orbit when the
extreme radii-vectores and the time of describing the parabolic arc
are given. For an explanation of the construction and use of this
Table, see Articles 68, 72, and 117.
654 THEORETICAL ASTRONOMY.
TABLE XII. exhibits the limits of the real roots of the equation
sin (/ C) = m sin* /.
The construction and use of this table are fully explained in Articles
84 and 93.
TABLES XIII. and XIV. are used in finding the ratio of the
sector included by two radii-vectores to the triangle included by the
same radii-vectores and the chord joining their extremities. For an
explanation of the construction and use of these Tables, see Articles
88, 89, 93, and 101.
TABLE XV. is used in the determination of the chord of the part
of the orbit described in a given time in the case of very eccentric
elliptic motion, and in the determination of the interval of time
whenever the chord is known. For an explanation of its construc-
tion and use, see Articles 116, 117, and 119.
TABLE XVI. is used in finding the chord or the interval of time
in the case of hyperbolic motion. See Articles 118 and 119 for an
explanation of the use of the Table, and also the explanation of
Table X. for an illustration of the use of the columns headed log I.
Diff. and log half II. Diff.
TABLE XVII. is used in the computation of special perturbations
when the terms depending on the squares and higher powers of the
masses are taken into account. For an explanation of its construc-
tion and use, see Articles 157, 165, 166, 170, and 171.
TABLE XVIII. contains the elements of the orbits of the comets
which have been observed. These elements are: T } the time of peri-
helion passage (mean time at Greenwich) ; TT, the longitude of the
perihelion; &, the longitude of the ascending node; i, the inclina-
tion of the orbit to the plane of the ecliptic; e, the eccentricity of the
orbit; and q, the perihelion distance. The longitudes for Nos. 1, 2,
12, 16, 91, 92, 115, 127, 138, 155, 156, 159, 160, 162, 171, 173-175
180, 181, 185, 191, 192, 195-199, 201, 203, 204, 207, 208, 212-215,
217-219, 221-228, 230, 233, 234, 237-248, 251-258, 261-267,
269-275, 277-279, are in each case measured from the mean equinox
of the beginning of the year. In the case of Nos. 134, 146, 172,
182, 189, 190, 205, 231, 232, 236, 259, and 268, the longitudes are
EXPLANATION OF THE TABLES. 655
pleasured from the mean equinox of the beginning of the next year.
The longitudes for Nos. 19 and 27 are measured from the me*\n
equinox of 1850.0; for No. 186, from the mean equinox of July 3;
for No. 187, from the mean equinox of Nov. 9; for No. 200, from
the mean equinox of July 1 ; for No. 202, from the mean equinox
of Oct. 1 ; for No. 206, from the mean equinox of Oct. 7 ; for No. 211,
from the mean equinox of 1848.0; for No. 216, from the mean equi-
nox of Feb. 20 ; for No. 229, from the mean equinox of April 1 ; foi
No. 250, from the mean equinox of Oct. 1 ; and for No. 276, from
the mean equinox of 1865 Oct. 4.0.
Nos. 1, 2, 11, 12, 20, 23, 29, 41, 53, 80, and 177 give the elements
for the successive appearances of Halley's comet; Nos. 104, 116, 126,
143, 149, 157, 167, 170, 176, 178, 183, 194, 210, 220, 235, 249, and
260, those for Encke's comet, the longitudes being measured from the
mean equinox for the instant of the perihelion passage. Nos. 92,
127, 159, 172, 196, and 222 give the elements for the successive ap-
pearances of Biela's comet; Nos. 187, 216, 250, and 276, those for
Faye's comet; Nos. 197 and 238, those for Brorsen's comet; Nos.
217 and 243, those for D' Arrest's comet; and Nos. 145 and 245,
those for Winnecke's comet. For epochs previous to 1583 the dates
are given according to the old style.
This Table is useful for identifying a comet which may appear
with one previously observed, by means of a similarity of the ele-
ments, its periodic character being otherwise unknown or at least un-
certain. The elements given are those which appear to represent the
observations most completely. For a collection of elements by vari-
ous computers, and also for information in regard to the observations
made and in regard to the place and manner of their publication,
consult Carl's Repertorium der Cometen-Astronomie (Munich, 1864),
or Galle's Cometen-Verzeichniss appended to the latest edition of
Olbers's Meihode die Bahn eines Cometen zu berechnen.
TABLE XIX. contains the elements of the orbits of the minor
planets, derived chiefly from the Berliner Aslronomisches Jahrbiivh
fur 1868. The epoch is given in Berlin mean time ; M denotes the
mean anomaly,
, r ,
\