IMAGE EVALUATION TEST TARGET (MT-3) h // €// :/. m, ^ ^ 1.0 I.I Ui tW 125 • SO ■^™ Hl^l 1^ 1^ 1 2.2 - lis |20 Ul 1.25 1.4 \h < 6" ► V] V. -p; ^# ''^^:> '/ /A Photographic Sciences Corporation d :<\^ ^\^* OF SI'E'IAI. 1 ERTLRIIATION'.: TOilETIlEK WITH THE THK.ollV iiK THK inMUl- NATION or UIISERVATIUN.-! AND THE SaiUOD UF LKAiiT tliUAKS*. ttulith Jlumcrical Oh'amjjlcs and Suxiliarg 3'ablfs BY JAMES V. WATSON lllREl'TOIt OF THE ODSERVATOKV AT AW ARnoil A\n PROFESSOR OF ASTBOXOJIT IN VMVER.<1TY OF MH UIUAN PHTT-AT>FTiPIITA J. B. LTPPINCOTT & CO. LONDON: TRUnNER l)h iii.s which were to exhihit the etU'ct of tlie nnituul attraction of the bodies of our sy.-teni, was the development of the infinitesimal calculus; and the lal)ors of those who devoted themselves to pure aiuilysis have contril)uti'd a n\ost iiiipnriant part in the uttninnient of the liiL'h degree of perfection which character- izes the resui's of astronomical investigations. Of the earlier efforts to develop the great results following from the law of gravitation, attention of Li: f^oncrous intorcwt wliich tlu'V liavc inaiiil'cstcil in tlio |iiililicatii)ii ut' tlio work, ami aUo to Dr. !i. A. Cioii.i), ol' ('aiiil)ritlj,'<', Ma^x., aixl to Dr. Ori'oi.zinj, of Vifiiiia, fur valiiaMo sufjfj<'.>*tion.«<. For tlx' (Ictcnniiiatioii of tlic tiini> from tlu> pcriliclion aixl of tlio true aiioinalv in very ('crcnlrii' nrhits I have jijivcn tlio iiictlio of paraliolii' motion before eomplefing the solution of the general problem of linding all of the elements of the orbit by means of three observed places. The ditfercntial formuhe and the other formuhe for eorreotiiig approximate elements are given in a form convenient for appliple formula, suggested by Ciiauvknf.t, wliich fol- lows directly from the fundamental equations for the probability of errors, and which will answer for the purposes here required as well a^ the more complete criterion proposed by Pr.iitcK. In the chapter devoted to the theory of special jierturbations I have taken particular pains to develop the whole subject in a coni[)lete 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 diiTerential coefficients with respect to the eccentric anomaly, the equations betweeit this function and the mean motion will enable us to effect readily the required transformation. 4 rilKKAcK. Till' nuiiinical cxniiiph's invulvc ilalii (Urivi)! iVuin actual oliscrva- tioiis, and care lias liocii takrii to iiinkr tlicni coiiiiiK tc in every rc.<<|Kiiiiu>d that tliu rcadiT IH familiar with the eltniciitrt of Hphcrii-al aHtroiiomy, xi that it is nnnccosmiry to Htutc, in all ca-xos, whcthor tiio contro of the nphorc is taken at the centre of the earth, or at any other point in space. The preparation of the Tahlcs has cost mo n jrront nmonnt of labor, logarithms of ten decimals heinj: employed in order to he sure of the last decimal {.dven. Several of those in previous use luvvt! been recom- puted ami extended, and others here ;.'iveii for the lirst time liave been prepared w ith spec iai care. The adopted value of the constant of the solar attraction is that j;iven by ('Aiss, 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 cf spncu; but until the absolute value of the earth's mean motic n is known, it is best, for the sake of uniformity and accuracy, to retain (JAiS't's constant. The j)riparation of this wcrk has been eflected amid many interrup- tions, and with other labors constantly j»rcssin^' mc, by wliii-h the itroj,M*ess of its ])ublication has been somewhat delayed, even since the stereo- tyj)ing was commenced, so that in some eases I have been anticipated in the publication of formuhe which wouhl 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- veloj)ment of the formuhe, to preserve unitbrmity and elegance in the notation, and to elucidate the successive steps in the analysis, so that the work may be read by those who, possessing a respectable mathematical education, desire to be informed of the means by whicli astronomers are enabled to arrive at so many grand results connected with the motions of the heavenly bodies, and by which the grandeur and sublimity of creation arc unveiled. The labor of the i>re])aration of the work will have been fully repaid if it shall be the menus of directing a more general attention to the study of tlie wonderful mechanism of the hea- vens, the contemplation of which must ever serve to imi)ress ujjon the mind ihc reality of the perfection of the omnh'otent, the living GOD! Observatouy, Ann Aubob, June, 1867. CONTENTS. TIIEOllETIOAL ASTRONOMY. CIIAPTKIl I. » INVKsTKi.VTt(V V TMK ITNDAMr.NTAI, Ko(ly .".l Till' I'liits of Spaci', TiiiU', ;iii(l Mass ;i(i Motion of a liody ivlativc to the Sun :',X Kijiiations for ('iiilixtiirhcd Motion -I'i iK'tiTiniiiation of tlio Attractivo Vorco of tiu- Sun 4!t Dotcrniinaiion of tlio IMace in an I'^lliptic Oi'i)it ')',i Dctiiinination of liic I'luct in a I'araliolic OiMt oO lU'ti'i-ininiitiiPii of llic Piaco in a iry]n'i'tiolic ()ri)it ti') Mctiiodn I'or tindiiif; tlie True Anomaly and tlicTiine from tiie IVriiiuiion in tiie case of < trliits of (rivat Kcccntricity 70 Di'ti-rniination of tlie I'oHition in Spaii' SI IK'lioiviitiii.' Longitude and Latitude S3 Ri'dnction to tlio Kdiptic 85 Cii'ooentrif Li)iif{itii(le and Latitude Sfi Transformation of Spherical Co-ordinates 87 Direct Determination of the Geocentric Kiglit Ascension and Declination 90 Reduction of the Element* from one Kpoch to another 09 Numerical Kxamples 103 InterpoUtioii ^ ll'J Time of Oiiposition 114 10 CONTENTS. CHAPTER II. INVKSTIOATION OF TIIK DJFFERKNTIAL FORMt:i,-K WltlCII EXPRESS THE KELATIOJf HETWEEN THE C.EOC'EXTRIC OR IIKI KKENTRIC PI-ACJK OF A IIEAVENEY BODY AND THE VARIATIONS OF THE ELEMENTS OF ITS OKBIT. PAGE Variatii)ii of tlic Right Ascension and Declination 118 Case of Piirabolic Motion 125 Case of Hyperhoiic Motion 128 Case of Oil.its didorinj,' l)iit little from the Parabola 1.30 Tsuiiicrical Examples 135 Variation of the J^onKitude aniiroxii CHAPTER III. INVESTIGATION OF FOP.MVIwK FOR COMPl'TINO THE ORBIT OF A COMET MOVING IN A PARABOLA, AND FOR COUUIX'TINa APPROXIMATE ELEMENTS BY THE VARIATION OF THE JEOCENTRIC DISTANCE. Correcti(i.i of the Observations for Parallax 167 Fundamental Equations 169 Particular Cases 172 Ratio of Two Curtate I)istanees 178 Determination of the Curtate Distances 181 Relation between Two Radii-Vectorcs, the Chord joining their Extremities, and the Time of describing the Parabolic Arc 184 Determination of the Node and Inclination 192 Perihelion Distance 'uid 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 OB FORM OF THE CONIC SECTION. Reduction of the D.ita 220 CorrectioiiH for Parallax 223 CONTEXTS. 11 PAGE Fimdampntal Eiiuations 225 I'onnulii' for tlio Curtate Distaiurs 228 Modification of tiie Forimiiu' in Particular Cases 231 lU'teriiiinatioii of the Curtate Distance for tlie Middle ()l)servati()n 2.'!() Case of a l)oul)le Solution 2;5!} Position indicated liy tlie Curvature of the Observed Path of the llody 212 Fornuilfe for a Second A)i]iroxirnati')n 243 Foniudie for lindiiijf the Hatio of the Sector to the Trianjjle 247 Final Correction for Aherration 2o7 Determination of the Flenients of the Orliit 2V,) Numerical Example 204 Correction of the First llyjiothesis 278 Approxinuite Method of finding the Itatio of the Sector to the Triangle 271) CHAPTER V. DETrnjnxATiox of the onniT of a he.vvenia' body from fofu obsf-rva- TIUXS, OF WHICH THE SECOND AND THIRD MIST BE COMPLETE. Fundamental F(|uations 282 Deiermination of the Curtate Distances 28!) Successive Approximations 2!)3 Determination of the J^lementu of the Urhit 21)4 Numerical ICxaiuple 294 Method for the Final Approximation 307 CHAPTER VI. INVESTIGATION OF VARIOI'S FORMFL.F FOR THE CORRECTION OF TIIE APPROXI- MATE ELEMENTS OF THE ORHIT OF A HEAVENLY BODY. Determination of the Elements of a Circular ()rl)it Variation of Two (leoccntric Distances DiHerential Formula' Plane of the Orbit taken as the Fundamental Plane Variation of the Node and Inclination Variation of One Geocentric Distance Determination of the Elenieiii3 of 'he Orbit by means of the Co-ordinates and Velocities Correction of the Ephemeris Final Correction of the Elements Relation between Two Places in the Orbit Modification when ihe Semi-Tranavcrse Axis is very large Modification for Hyperbolic Motion Variation of the Semi-Transverse Axis and Ratio of Two Curtate Distances 311 313 318 320 324 328 332 33.-, 338 339 341 34r, 349 12 CONTEXTS. PAOR Variation of tlie Geocentric Distance aiid of the Reciprocal of the Senil-Trans- vcrse Axis ;5')2 K(|nationH of Comlitioii IJijS {)rl)it of ii Comet.. .'J55 Variation of Two Kudii-Vectoreri 357 CHAPTER VII. METHOD OF LEAST SQUARES, THEORY OP THE COMBINATION OF OBSERVATIONS, AND DETERMINATION OF THE MOST PROBABLK SYSTEM OF ELEMENTS I'UOM A SERIES OF OBSERVATIONS. Statement of tiie Problem .^fiO Fundiiniental Ivjuations for the Proliahility of Error,* .Sli'J Determination of the Form of the Function which expresses the Probability ... 303 The Measure of Precision, and the Probable Error StitJ Distriluition of the Errors 3(>7 The Mean Error, and tlie Mean of the Errors o(!8 The Probable Error of the Arithmetical Mean .370 I)eterniiiiation of the Mean and Probable Errors of t)bservations 371 Wei>.dits of ()i)served Values 372 E(| nations of Condition 376 Isoniial Iviuations ,378 Metiiod of Elimination , 380 Detcrminaticm of the Weights of the Resulting Values of the Unknown Quanti- ties 38R Separate Determination of the Unknown Quantities and of their Weights 30'J Relation between the Weiglits and the Determinants 396 Casein which the Problem is nearly Indeterminate 398 !Mean and Probable Errors of the Results 399 Cond)i nation of C)l)servations 401 Errors peculiar to certain Observations 408 Rejection of ])oid)tful Observations 410 Correction of the Elements 412 Arrangement of the Xtnnerical Operations 4Io Kumerical Example 418 Case of very Eccentric Orbits 423 CHAPTER VIII. INVESTIGATION OF VARIOUS FORMULAE FOR THE DETERMINATION OF TIIE SPECIAL rERTURBATION.S OF A HEAVENLY BODY. Fundamental Equations 426 Statement of the Problem 428 Variation of Co-ordinates 429 CONTEXTS. 13 PAQE Mcphaiiiral Quadrature 431? Tlie Interval fur (^nadratiiri' 44.'$ Mode of c'lli'c-tiiif? tin- Inti'^rration 44') lVrtiirl)ations dc-]H>iuling on tlie S Variation of the Periodic Time .VJtj Numerical Example o'i!) Forinnhe to he used when the Eccentricity or the Inclination is small '>',V,i Correction of the Assumed Value of the Disturbing Ma;Subse([uent Motion of the Comet 551 Efleet of a Resisting Medium in Space 552 Variation of the Elements on accoinit of the Resisting Medium 554 Method to be applied when no Assumption is made in regard to the Density of the Ether 55fi 14 CONTENTS. TABLES. 9 PAflE T. Angle of the Vertical and Logarithm of tlie Earth's Radiiia oGl II. For converting Intervals of Mean Solar Time into Kquivalent Intervals of Sidereal Time 5('k5 III. For coiiviTting Intervals of Sidereal Time into JMinivalent Intervals of Mean Solar Time 564 IV. For converting Hours, Miiuites, and Seconds into Decimals of a Day... 505 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 ParaholicOrhit 566 VII. For finding the True .\nomaly in a I'araholic <)ri)it when r is nearly 1S0° til 1 VII 1. For finding the Time from the Perihelion in a Paraliolic Oi-hit C12 IX. For tinding the True Anomaly or the Time from the Perihelion in Orhits of < Jreat Ivccentricity 614 X. For finding the True Anonuily <>t the Time from the Perihelion in El- liptic and IIyi)erholic Orhits 618 XI. For the Motion in a Paraholic Orhit (ill) XII. For the Limits of the Koots of tlie lC(ination sin (2' — C) ■- JHq f*''!*-'" f'-- XHl. For finding the Ratio of the Sector to the Triangle 6"J4 XIV. For finding the Ratio of tlie Sector to the Triangle G'lO X\'. For Ell ii>tic Orhits of (ireat Eccentricity (V.Vl XVI. For Ilyiierholic Orhits (1:52 XVIL For Sjiecial Pert urhat ions (V.'m XVIIl. Elements of the Orhits of the Comets which have heen ohfeerved 6;?8 XIX. Elements of the Orhits of the Minor Planets 616 XX. Elements of the Orhits of the Major Planet« 618 XXL Constants, &c 649 Explanation of the Table-s G51 ArriiNUix. — Precession 657 Nutation 658 Aberration 65!) Intensity of Light 660 Numerical Calculations 662 THEORETICAL ASTRONOMY. CHAPTER I. IXVK«TIfiATION OF THE FUNDAMKN'TAL KljrATIOXS OF MOTION', AND OF TIIK I-Oll- Mll.-K Kiill DKTKKMIMXli, I'lJoM KNltWN KLKMKNTS, I'lIK lIKMiK KNTlilc AM) (!K(R'KNTltIf ri-ACKS OF A IIKAVKM.V IIODY, AUAll'KU TO NLMKUUAI- CDMl'lTA- TIOK FOR (.ASKS OF ANY KCCEXTllK ITV WHA TKVKU. 1. Tin; Study of the motion.s of the heavenly hotlics dues not re- quire that we should know the ultiniate limit of divisibility oi' the matter of which they are composed, — whether it may \h' sulxlivided indefinitely, or whether the limit is an indivisible, im|)enetral)le atom. Nor are we concerned with tlic relations which exist between the separate atoms or molecules, except so far as they ibrm, in tiie a«fgre- gate, a definite body whose relation to other l)odies of the system it is required to investigate. On the contrary, in considering the ope- ration ol" ihe laws in obedience to which matter is aggregated into single bodies and systems of bodies, it is sufKcient to conceive simply of its divisi! '"W to a limit which may be regarded as infinitesimal compared with the finite volume of the body, and to regard the mag- nitude ol' the element of matter thus arrived at as a matlKunatieid 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 connnunicated to it. This tendency of matter to resist all changes of its existing state of rest or motion is known as incrtid, and is the fundamental law of the motion of bodies. Iv\- perience l:ivariably confirms it as a law of nature; the continuance of motion as resist^mces are removed, as well as the sensibly unchanged motion of the heavenly bodies during many centuries, affording the U 16 THEORETICAI. A8TK()X0>[Y. *| m 'HI iii|i most convinciii}; pioof of it.«i iiiiivortiality. Whoiievcr, tlicreforo, a ujutcrial point cxpc'ricnccs any clianfjje ol' its state as respects rest or motion, the eause must l)e attributed to the operation of something external to tlie element itself, and which we desij^nate by the word force. The nature of forces is. ffcnerally unknown, and wt; estimate them by the efl'ects which they produce. They are tluis 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 conununicated shall cease to act, the subsequent motion of the point, according to the law (tf inertia, nmst be rectilinear and nnlfonn, equal spaces being described in equal times. Thus, if s, r, and t represent, resi)ectivcly, the ^jtace, the vclodty, and the Iliac, the measure oi' v being the space described in a unit of time, we shall have, in this case, 8 = 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 tlue, and, the nature of the force being unknown, we must necessarily compare the velocities (lommunicated to the point by ditf'erent I'orces, in order to arrive at the relation of their elT'ects. We are thus led to regard the ibrce 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 accclenitinf/ force. In regard to the mode of operation of the force, however, we may consider it as acting absolutely witijout 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 intinitesimal 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 (Is, and the ele- ment of time by dt, the instantaneous velocity will be ds "i which will vary from one instant to another. FUNDAM KXTA L IMM XCI IM-RS. 17 may 3. Siiirc tlio force is proportional to the velocity, its nu'asurc at aiiv instant will ho tU'tcrniiiicd l)y the correspond inj; velocity. It' the accelerating force i.s constant, the motion will he nniformly accele- rated; and if we designati' the acceleration due to the force by/, the unit of/ being the velocity generated in a unit of time, we shall have V =:ft. If, however, the force be variable, we shall have, at any instant, the relation J- dt' the force being regarded as constant in its action during the elemeut of time (It. The instantaneous value of v gives, by ditferentiation, and hence we derive dv 'dt~ (/'.« w /= d'.i^ df' CD so that, in varied motion, the acceleration due to the force is mea- sured by the second ditt'erential of the space divided by the sc^uare of the element of time. 4. By the tiumi of the body we mean its absolute quantity of mat- ter. The den.slti/ is the mass of a unit of volume, and hence the entire niass is equal to the volume multiplied by the density. If it is required to compare the ibrces which act upon ditfercnt bodies, it is evident that the masses must be considered. W 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 Ibrce that which gives to a unit of mass the unit of velocity, we have f)r the measure of a force F, denoting the mass bv 31, F^Mv. This is called the quanfitj/ 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 ojjpose c-cpial and 2 1» TIIEOHKTICAL ASTKOXOMY. :l ! oontniry ri'si.shuu'o. Ilencc, as iv lu'ccssaiy ('(tnso(|ii(Mi('o of inertia, it follows that action and ivactiori aio siinnltancous, e<)nal, and contrary. If the body is acted upon hy a force su(!li that tlie motion is varied, the aeceleratinjjj forct; upon eacli element t)f its mass is represented by do dt , and tiie entire mothr force F is expressed by dv F=M dt' M being the sum of all the elejuents, or the mass of the body. Since ds v = this gives dt' 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 dt:' ' and this, therefore, expresses that part of the intensity of the motive force wliich is impressed upon the unit of mass, aud is what is usually called the accdc rating force, 5. The force in obedience to which the heavenly bodies perform their journey through space, is known as the attraction of f/rucitation; and the law of the operation of this i()rce, in it^self simple and unique, has b(!en 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 i*evolutions of binary systems of stars in the remotest regions of space proclaim the uni- versality of its operation. It unfailingly exi)lains all the phenomena observed, and, outstripi)ing observation, it has furnished the means of predicting many phenomena subsequently observed. The law of this force is that everi/ particle of matter i,s attracted by every other particle by a force which varies directly as the iiias8 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. Arrii.vcTiox op si'iiereh. 19 If wo consider two bodies the mnsscs of wliich arc hi and in', and wliosc inagnitudts are .so small, relativt^ly to their mutual distance y, that we may ri'jj;ard them as material points, aceordinjr to the law of gravitation, the aetion of in on each molecule or unit of m' will bo in and the totiil force on ni' will bo % Hi ,m m The ai'tion of m' on each molecule of in will bo expressed by — -, and its total action by m m The absolute or movinj? force with which tho masses in and m' tend toward each other is, therefore, the same on each body, which result is a necessary consequence of the ('(juality 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 la', which would be generated in a unit of time if the force remained constant, is obtained by dividing the absolute force exerted by in by the mass moved, which gives m 7 and this is, therefore, the measure of the acceleration due to the action of in at the distance f». For the acceleration due to the action of tn' we derive, in a similar manner, 6. Observation shows that the heavenly bodies arc nearly spherical in form, and we shall therefore, preparatory to finding the equations wliich express the relative motions of the bodies of the system, de- termine the attraction of a spherical mass of uniform density, or vaiying 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 I'espect to it. Let dm denote an element 20 TH KOIIETICA L AHTHONOM Y. 'i.i;n of the inushof the sphere, ami (> its dibtunce from the pouit attracted} then will dm expres.s the ivetioii of this element on the point uttrm*te clr dif> dff Let us now put and we shall have dA^ dV-. da r* cos

ut Hiuoo ft is n fuiu'tioii of ^f, if we (litVcroutiato the exprosHJon for (»• with respect to (f, we have and Iience r 008 ip dip =^ — f//», u ^^-viT'-'''-''^" Correspondinp: to tlio limits of tf we have p~-a — r, and p~a -\- r; and taking the integral with res[)oet to (t between thej^e limits, we obtain « * Integrating, finally, between tiie limits /• •- and r --■ r,, we get r — •» „ » r, being the radius of the sphere, and, if we denote its entire mass by m, this becomes m Therefore, V= A — ~ '^~~ - da a*' liilii from whioh 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 con ^nrface to the centre, we may rc;;ard them as composed of homo<;ene«.us, eoncentrie layers, the density varyinj; he indetinite. The action of each of dilferinf; but little from spheres; and the error of the assumption of an exact spherical form, so far as nilates 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, ily from one layer to another, and the numlH-r of the layers may rhatc' be the f( )f the bodv, if its di 11 ,'er may ue ine lorm oi me oody, ii its (umen.sions are .- in comparison with its distanc(^ 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 su])pose that the dimensions of the bodies are small in comparison with their mutual distances, the formation of tlie 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 resj)eetive 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 c -isider, at first, those for a simple system of bodies, considered as pf' nts and subject to their mutual actions and the action of the forces which correspond to the 2t TII EOUETK AL ASTUONOM Y. actual velocities of the ditfcrcnt |)art.s of the system for any instant. It is evident that we cannot considcT the nioti(l(! hody as free, and sul)ject only to the action of the ))riinitiNe iin|)nIsion which it has received and the accelerating foree,^ which act ni)on it; but, on the conlrary, the motion of each ly dy will de[)en(l on the force which acts upon it directly, and also on the reaction due to the other bodies of the system. The consideration, however, of the varia- tions of the motion of the several bodies of the system is reduced to the simple ease of e(juilibrium by means of *he general principle that, if we assign to the diU'erent bodies of the system motions which arc 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 existtnl, the system would be in oqui- lihrinm. We are thus enabled to form at once the eouations for the motion of a system of bodies. Let m, m', m", &(!. be the masses of the several bodies of the system, and ;(•, _//, s, x' , y', z', &c. their co- ordinates referred to any system of rectangular axes. Further, let the couiponents of the total force acting upon a unit of the mass of VI, or of the accelerating force, resolved in directions ])arallel to the co-ordinate axes, be denoted by A', V, and Z, respectively, then will mX, m Y, mZ, be the forces which act upon the body in the same directions. The velociti< s of the body m at any instant, in directions parallel to the co-ordinate axes, will bo dx 'dt' dz di' and the corresponding forces arc m dx dt' m dy dt' dz ''-dT By virtue of the action of the accelerating force, tlu?sc forces for the next instant become m —.J + mAdf, m '^J^ -]- mYdt, wi-./ + viZdt, dt which may be written respectively: MOTIOX OF A SYSTEM OF BODIES. 25 dx J dx , dx m ,. -\- md . — md . -\- mXdt, dt dt dt VI ~ -4- md - ■ md ,, + '« ydt, dt dt dt m --77- + md -,- md —rr + mZdt. dt dt dt The actual vclocitios for this iuotant arc dx . dx "dt"^'^ dt' d,, dt dy dt' + '',/. and the corrcspoiuliiig Ibrccs arc dji dx , , dx ""'dt+'^'^-Jt' ""dt+'"'^dt' dz dz 'dt+'' df' dz dz Comparing these with the preceding expressions for the forces, it appears that tiie forces which are destnjyed, in directions parallel to the co-ordinate axes, arc dx — md ,- 4- mXdt, dt — md -J- -j- m Ydt, dz — md—T- A- mZdt. dt (8) In the same manner we find for the forces which will be destroyed in the case of the body m' : dx' — m'd -J- -j- m'X'dt, — m'd^-\-m!Y'dt, It — m'd ~- -|- m'Z'dt; and similarly for the o>her bodies of the system. According to the [fcneral principle above ennnciatcd, the system nndcr the action of these forces alone, will be in c(pjilil)rinm. The conditions of e(pu- lihrium for a system of points of invariable but arbitrary form, and subject to the action of forces directed in any manner whatever, are in which X,, ¥,, Z,, denote the components, resolved parallel to the 26 THEORETICAL ASTRONOJfY. co-ordinate axes, of the forces acting on any point, and x, y, z, the co-ordinates of the i)oint. These equations are ecpially applicahle 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 (I'll dh dC d^^ ' dP '- — lmY=0, (4) / d^x dh \ V ^ T X N A " \ ^ df ~ ^ df I ~ ^-^^ ~ "^-'^ ^ ^' which are the general equations for the motions of a system of bodies. 8. Let .1',, iji, Zt, be the co-ordinates of the centre of gravity of the system, and, by differentiation of the equations for the -^o-ordinates of the centre of gravity, whit.'h are Imx X, ■ — -V, ) we get dh; Zm -— dt' I my ^4f Z, dh, df Em ' y dh dfl Introducing these values into the first three of equations (4), they become IwX df df 2m F d% df ImZ -m (0) from which it appear.^ 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 })oint, 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 Avhich result from their mutual action, we have the obvious relation : mX- m'X', mY= — m'Y', mZ- m'Z', X'^ Hence wt Ijil'i;- MOTION OP A SYSTEM OF HODIE8. and similarly for any two bodies ; and, consoquently, 2T so that equations (5) become ImZ=0] de 0, d^z, df = 0. Integrating these once, and denoting the constants of intcg>'ation by c, c', c", we find, by combining the results, dx'' + df + dz'' df c' + c" + c' ,"2, 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 motio.i of its centre of gravity is not thereby aftccted. 9. Let us now consider the last three of eciuations (4), and suppose the system to be submitted only to the mutual action of the bodies which compose it, and to a force directed toward the origin of co- ordinates. The action of m' on m, according to the law of gravita- tion, is expressed by — , in which ft denotes the distance o£m from vi'. 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 X=- m' (x^ — x) r_ m' (y' — y) Z. m' (/ — z) Further, for the components of the accelei'ating force of m on m', we have m {x — x') v — "^ <"y — y') Z' m {z z') Hence we derive m ( Yx — Xy) + m' ( Y'x' — X'tf) ^ 0, and generally (6) ■: w 28 THEORETICAL ASTRONOMY. In a similar mannoi*, Ave find Im (Xz 2»i (Zy ■ Zx) = 0, Yz) = 0. (7) These relations will not l)e altered if, m addition to their rceiproeal action, the bodies of the system are acted npon by forces directed to the orisjin of co-ordinates. Thus, in the case of a force acting npon m, and directed to the origin of co-ordinates, we have, for its action alone, Yx ^ Xy, Xz = Zx, Zy = Yz, and similarly for the other bodies. Hence these forces disappear from the erpiations, and, therefore, when the several bodi(\s 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'x dh\ . the integration of Avhich gives Im (xdy — ydx) Im (zdx — xdz ) Imiydz — zdy) cdt, ■■ c"dt. (8) c, c', and c" being the constants of integration. Now, xdy — ydx is double the area described about the origin of co-ordinates by the projection of the radius-vector, or line joining m with the origin of co-ordinates, on the plane of xy during the element of time dt; and, further, zdx — xdz and ydz — zdy are respectively double the areas described, during the same time, by the projection of the radius-vector on the planes of xz and yz. The constant c, therefore, expresses the sum of the products formed by midtiplying the arcal velocity of each body, in the direction of the co-ordinate plone xy, by its mass; and c', c", express the same sum with reference to the co-ordinate planes xz and yz respectively. Hence the sum of the aroal velocities of the several bodies of the system about the origin of co-ordinates, each multiplied by the corresponding mass, is constant; and the sura 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 svstem, this result is correct whatever may be the point in space taken as tlie 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. Wc 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 luenibers of the equations (8). Let «, [i, and y be the angles which tliis principal plane makes with the co-ordinate planes xy, .vz, and i/z, 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 clement of time dt, are S cos a dt, S cos /? dt, S cos y dt. Therefore, by means of equations (8), we have c = ^ cos a, c' = *S' cos ,3, c" = S cos y, and, since cos"a + cos^/9 + cos^;' = 1, S^ =: C' + C'' -^ C"\ Hence we derive cosa = V c' -\- d' -\- c"-' cos/?: l/c' + C" + d"' COS J' = ]/cH^'' + c' "a 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 bo the relative positions of the several bodies ; and for this reason it is called the invariable plane of the system. Its position with reference to any known plane is easily determined when the velocities, in directions parallel to the co-ordinate axes, and the masses and co-ordinates of the several bodies of the system, are known. The values of c, c', c" are given by equations (8); and 30 THE07 .ETICAL ASTRONOMY. illiilti lionee the values of «, ^9, and y, which determine the position of the invariable plane. Since the positions of the co-ordinate ]>1anes are arbitrary, we may sui)j)ose that of .ri/ to coincide with the invariable plane, which gives cos /? - and cos y = 0, and, therefore, c' =^ and c" = 0. Further, since the positions of the axes of x and y hi 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 th(! corre- sponding mass, is zero. It may also be observed that tiie value of S is constant whatever may be the position of the co-ordinate planes, and that its value is neccssni'ily greater than that of either of the quantities in the second member of the equatity. "2 excc])t when two of them are each equal to zero. It is, therefore, a maxinuun, 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 a-,, ?y„ z, be the co- ordinates of the movable origin of co-ordinates, referred to a fixed point in space taken as the origin; and let .r,,, 7/,,, 2,,, x„', yj, zj, etc. 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 a; =r= .T, -f- .To, y = y,-\-y^, Z = S, + 2„, 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 "'I rf<» + dC J ' 0, 0, The condition of uniform rectilinear motion of the movable origin gives d\ di? 0, •— '^ = di' ' bodies, c MOTION OF A SOLID BODY. and the preceding equations become 31 ImX = 0, Im '^y^ - 1 iW ■mr=o, (9) at' Substituting the same values in the last throe of equations (4), ob- serving that the co-ordinates .r,, y,, z, are the same for all the bodies of the system, and reducing the resulting equations by means of equations (9), we get 2,)n I .r - - a-0 - ^^.;- 1 — -"I (AX — Zi-o) 0, t' I (10) » (W I»i(Zy,-Yz,)=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 dm, the equations (5) for the motion of the centre of gravity of the body become d% -jXdm, T =frdm, d% =fZdm, (11) the summation, or integration with reference to dm, being taken so as to include the entire mass of the body, from which it appears that the centre of gravity of the body moves in space as if the entire mass were concentrated m\ 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 rofei'cnce to this origin, by Xq, y^, z,,, we have, by means of the equations (10), 32 TIIEORETICAI. ASTRONOMY. Ill /( -'o 'Jl; - 2/0 -1^^ ) dm -/( Kr„ - ^„) dm = 0, / ( ^0 ti? - ^0 '2^ ) dm -f(A\ - Zj-J dm = 0, (12) the into^ratioii Avitli I'osju'ct to dm boiiig takon so as to include the entire iua.«s of the body. These equations, therefore, determine tiie motion of rotation of the body around its centre of tjravitv rey-arded as fixed, or as having a uniform rectilinear motion in spacr. Equa- tions (11) determine the position of the centre of gravity for any instant, and hence for the successive instants at intervals equal to the ('(t-«>nliii:iti's in rrleroncc to the centre of gravity of the system as origin, we imve Imxg — 0, i'/H^o = 0, and the prceeding equation reduces to -»i2o = 0' 'I In a similar manner, we find y» Im ,mz -»i The second members of tlicse equations are the expressions for the total accelerating force due to the action of the bodits if the system on JA resolved })arullcl to the co-ordinate axes respectively, when wo consider the several masses to be collected at the centre of gravity of the system. Hence we conclude that when an clement 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, tlu; form of the equations remains unchanged; ar.d 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 m; sses were thus united. 13. The results already obtained are sufficient to enable us to form the cijuations 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 SYSTF,>f OF nODIES. a5 each on mi oloincut of tlio mass <»f' another would bo the same ns if llu' entire mass of the attraetinuf body were concentrated at its centre of f^ravity. The s)ij>ht (h'viation from this hiw, arisinj^ from the ellipsoidal form of the heavenly bodies, is compensated by tlu; majj;- nitiide of their mntual distances; and, besides, these mntual distances are so great that the a<'tion of the attracting' body on the entire mass of the ijody 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 corresj)onding to their respective centres of gravity, the masses of which, however, are eipiivalent to those of the corres|)()nding bodies. The mutual distances of the bodies comjiosing the secondary systems of planets attended with satellites are so small, in comparison with the distances of the ditt'erent systems from each other and from the other planets, that they act upon these, and are reci[)roeally acted upon, in nearly the sanie manner as if the masses of the secondary systems were united at their I'ommon centres of gravity, respectively. The motion of the centre of gravity of a system consisting of n planet and its satellites is not aft'ectcd by the reeiiirocal action of the bodies of that system, and hence it may be considered inde[)endently of this action. The ditference of the action of the other planets on a i)lanet and its satellites will simply produce inecpialitics in the relative motions of the latter bodies as determined by their mutual action alone, and Avill not affect the motion of their common centre of gravity. Hence, in the formation of the ecj^uations for the motion of translation of the centres of gravity of the several planets or secondary systems which compose the solar system, wc have sim])ly 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 i' acting at the same distance, we have the relation m TO' > ' and h be talv Tntegn a state The ac measnn generat of time and hen W(»uld 1 a unit o this tiim in the ni wliich tl will dep general, constant move, w if the fo tance in Let th to the foi and/ tilt and k' h mass is, t as k- is c day as tli ^vhich wc lt' Tnteffratijjji; this, rejijarding/as constant, and the point to move from a state of rest, we get The acceleration in tiie case of a variahh' force is, at any instant, measured l»y the vehjcity which the force acting at that instant would generate, il" suppitsed to remain constant in its action, during a vniit of time. The last e(juation gives, when / - - i, and hence the acceleration is also measured by double the space whirli would be described by a material point, from a state of rest, dui-ing a unit of time, the force being supposed constant in its action during this tiuR', 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 Ibree, will depend on the duration which is chosen for the unit of time. In general, therefore, wc regard as the unit of mass that which, acting constantly at a distance c(pial to unity on a material ])oint 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 solav day; A^ the ac(!eleration due to the force exerted by the mass of the sun at the unit of distance; and /the acceleration corresponding to the distance /•; then will and Jc^ 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 Ic- 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 k'; by Avhieh we arc 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 coramunieated to the point a velocity which, if 38 Til KOUKTK A I, ASTIU ).\()M Y. Ili:^ the Wn'va did not tlicrcai'tcT iu'f, would vixusv. it to doscriho, in a unit o'" time, tiu' spaco i:X|)n'ss((l l»y Ir. Tlic a<'('(!U'ratiou duf to tiio action of the sun at the unit of'distanc*.' is designated by Ir, since the scjiian; root of this (juantity ai>[)ears frer|uently in t\w lornuda) wliich will be derived. H' \V(i take arbitrarily the mass of the sun as th(( unit of mass the unit of time must be (hiferniined. Let / denote the luimber of mean solar days which nuist i)e taken for the nuit of time when tlu; unit oi' mass is tiu mass of the sun. Tiie space which the fon^e (hie to this n.ass, actinjj; constantly on a material point ut a distance e(jual to the mean distance* of the earth from the sun, would clause the point to describe in the time /, is, accordin<>; to ecpiation (l-'J), But, since t ex[)re!ises the nund)er of mean solar days in the unit of time, the measure of the acceleration corresponding tu this unit is 2.s, and this being the unit of force, we have and henee kH^ == 1 ; 1 t — . ii'l i ill 1'!'^ Therefore, if the mass of the sun is r<>garded as the unit of mass, the number of n»eau solar days in the unit of time will be equal to unity divided by the s((nare root of the acceleration due to the force exerted by this mass at the unit of flistance. The munericd value of h will be subsefpiently found in be <).()172()21, wliich gives 58.1.'}244 mean sohir days for the unit of time when the inas;S of the sun is taken as the unit of mass. 15. JiCt .)•, I/, z he the co-ordinates of a lujavenly body referred to tlie centre of gravity of the sun as the origin of co-ordinates; /• its radiiiK-vccfor, or distance from this origin; and let m denote the quotien*^ obtained by dividing it'i ma.>-s by that of the sun; tluMi, taking the mean solar day as the unit of lime, the mass of \\u' sun is oxj»rcsse the acceleration due to the condn'ned and sinudtanc^us action of the several bodies of the system on the sun, resolveil par- allel to the co-ordinate axes, will be /• ■ >.'".'/ ^••^2' ,W2 The motion of the centre of gravity of the sun, relative to tho fixed origin, will, therefore, be detern)ined by the equations ^, mx A ,.,v'".'/ /t'2''"'-. (14) Let f) denote the distance of m from to'; // its distance fron. /;(", adding an accent for eacii suc:;essive body considered ; then will the action of the boilies /u', la", tte. on m be •red to /• its )te the then, sun is l^'or a oni the of the W SlUl jdanes ■ill the K - -, of which th(! three components p;iraiiel to the co-ordinate axes, re- spectively, are yr,2V../ )n ic^^n^y-^, k'l',"'^ ■III The action of the sun on vi, resolve;! in the same man'U'r, is expressed by k\v khj p2 v r" which arc negative, since the force tends to diminish the co-ordinates a, y, and s. The three cf)mponents of the total action of the other bodies of the system ou vi are, therelbrc, 40 THEOKETICAL ASTRONOMY. vl!l .11 1 ! m' (x' — x) —^3 + ^'-" .,3 -, f- K - i ■, and, since the co-ordinates of m referred to the fixed origin are f + a;, 7i-\-y, C + 2, the equations which determine the absohite motion are -"^ ^ rf<» ^ ^ r* ifc^i: m' (a;' — a;) rf<' (15) the symbol of summation in the second members relating simply to the masses and co-ordinates of the several bodies which act on m, exclusive of the sun. Substituting for -^, -r^, and ~- their values given by equations (14), we get de ^ ^ 1* \ y (16) Since x, y, z are the co-ordinates of m relative to the centre of gravity of the sun, these equations determine the motion of m relative to that point. The second members may be put in another form, which greatly facilitates the solution of some of the problems relating to the motion of m. Thus, let us put n _ J«L / 1 xx'^-yy'-]-zz' \ m" / 1 i + m\p r" f'^l-{-m\p' to" / 1 xx"+ yy"+ zz" »"3 J+....&C., (17) and we shall have for the partial differential coefficient of this with respect to x, \ dp^ ■"' dx r \dxj l+m\ p''dx r"'/'^l-j-TO\ p" dx ,-'" / "^ " ' ' "' MOTION RELATIVE TO THE SUN. 41 But, since v;c have P' = (x' - xY + (2/ - y)' + (/ - zy, P"= ix" - xf + {f -^ yy + (z" - zy, dx dp!_ dx a!'—x and hence we derive /(?fl\_ ?ft' l aS — x x' \ m" lx"—x x" \ + *&c. We 1^ a also, in the same manner, for the partial differential coeffi- r\rm.-, \,' ;'i respcct to y and z, The cc^uations (16), therefore, become *: + ..(! +„.), 4 = .. (I +»)('!). (18) ;J,H-^ni+m)i. = P(l 1 «0(f)< It will ^ u uo '), from which we derive r = P 1 + e cos (y — w)' wliich is the polar equation of a conic section, the pole being at the focus, p being the semi-parameter, e the eccentricity, and v — id the angle at the focus between the radius-vector and a fixed line, in the plane of the orbit, making the angle to with the semi-transverse axis a. If the ang ,v — co \% counted from the perihelion, we have fo' = 0, and P \ -\- e cos V (25) The angle v is called the true anomaly. Hence we conclude that the orbit of a heavenly body revolvinr/ around the sun is a conic section with the sun in one of the foci. 01)servation 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 >\e shall have occasion to notice hereatter, greatly 4Q TiiEonirncAL astkonojiy. lessons the amount of labor in many computations respecting their motion. Introducing into equation (23) the values of h and 4/- alrcadv found, we obtain , l/« rdr which may be written or dt^ ky'l + m^ ^^-m yj^-i-)- the integration of which gives t ki/ f^(--(''-~)-«V'-("-'f)+« (2erihelion to the aphelion, 3 2^ = --^rrrr TT, ky'l -\- VI T 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 t: the semi-circumference of a circle whose radius is unity. Therefore we shall have ■An" k' (1 + my (28) MOTION REr.ATIVK TO THE SUN. 47 For a second planet, we shall have ' -4'. ,j J^ (1 + m') ' and, consequently, between the mean distances and periodic times of any two planets, we have the relation (1 4^)1)7^ (1 + j^'K'^ (29) If the masses of the two planets in and m' are very nearly the same, we may take \-\-m=^l-\- m' ; and hence, in this oxxi^^, it follows that the HquarcH of the periodic times arc to each other as the cuIhv of the mean iJistanccfi from the ,mn. The same result may bo stated in another form, which is sometimes mox'c convenient. Thus, since rrah is the area of the ellipse, a and b representing the semi-axes, we shall have - — =/=- areal velocity; and, since P = a^ (1 — c^), Ave have /= SI 1 4 — Kcr a- (1 — e") ^ Tta^ ^/p which becomes, by substituting the value of r already found. f=:Ul/p(l+Vl). In like manner, for a second planet, Ave have (30) /'==U/y(l+»0; and, if the masses are such that avc may take 1 -\- m sensibly equal to 1 + wi', it folloAVS 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 noAV 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 xi/, A\diich 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 Avith the plane of xy; then Avill the angles i and SI determine the position of the plane of 48 THKORfrrirAL astkoxomy. li ::|v < .i:i; the orbit in ^pace. The constants c, c', and c", involved in the equation cz — c'y -1- c"« = 0, are, res])octivc}y, doul)lo the projections, on tlic co-ordinate planes, xy, xz, and yz, of the areal velocity/; and iience we shall have c = 2/ cos i. The projection of 2/ on a plane passing through the intersection of the plane of the orbit with the plane of xy, and perpendicular to the latter, i8 2/sini; and the projection of this on the plane of xz, to which it is inclined at an angle equal to Si, gives c' = 2/sinicos ft. Its projection on the plane of yz gives c" = 2/sintsin ft. Hence we derive a cos i — y sin i cos ft + *' sin i cm ft = 0, (31) which is the equation of the plane of tlie orbit; and, by means of the value of / in terms of p, and the values of c, c', c", wo derive, also, dx ^~dt~ ^•''Tt ^ ^"^^ (1 +"0 cosi, dz dz dx z-jr = k i/p (1 -}- m) cos ft sin i, (32) dy it = k \/p (1 + m) sin ft sin i. These equations will enable us to determine ft , i, and j9, when, for any instant, the mass and co-ordinates of m, and the comj^onents of its velocity, in directions i)arallel to the co-ordinate axes, are known. The constants a and c are involved in the value of ji, and hence four constants, or dements, are introduced into these equations, two ot which, a and e, relate to the form of the orbit, and two, ft and ^, to the position of its plane in space. If we measure the angle v — m from the point in which the orbit intersects the plane of xy, the con- stant io will determine the position of the orbit in its own plane. Finally, the constant of integration C, in equation (26), is the time of passage of thf bo(i undisiurbi Let V i equation (; At the pe equation, \ aphelion, 1 In the p which will It will be c of r, in an since n is orbit is still the velocit}- direction of described. If the po and magniti will enable But since v of the prim the aid of therefore, b unknown el gate those a centric and be known, the problem by observati 18. To de system, we h MOTION UKLATIVE TO THE SUN. 49 of pnpsnpjo through the perihelion; and this dctornuncs tho position of thf body in its orbit. When these six constants arc known, tho undisturbed orbit of the body is completely deternnned. Let V denote the velocity of tho body in ita orbit; then will equation (liOj become At tho 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 tho parabola, a = oo, and hence V=kVl +m^-, which will determine tho velocity at any instant, when r is known. It will be observed that the velocity, corresponding to the same value of )•, in an elliptic orbit is less than in a parabolic orbit, and that, since a is negative in the hyperbola, the velocity in a hyj)crboHc 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 tho species of conic section described. If the position of a heavenly body at any instant, and the direction and magnitude of its velocity, arc given, tho 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 tho primitive impulse comnuinicated to the body, it is only by the aid of observation that these elements can be derived; and therefore, l)efore considering the formula) necessary to deteruiine 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), ifc = 2jr a 1/1 + m to THKOUKTICAT. ASTRONOMY. Tn the case of the earth, « == 1, and therefore 2rr tkI 4- TO In rcdueing this formula to numbers we shouUl properly use, for r, the alisolute length of the sidereal year, whieh is invariable. The ctfet't of the aetion of the other bodies of the system on tiie earth is to produee a very small secular change in its mean longitude corre- sponding to any fixed date taken as the epoch of the elements; and a correction corresponding to this secular variation should be apj)lied to the value of t derived from observation. The effect of this cor- rection is to slightly increase the observed value of r; but to deter- mine it with precision recpiires an exact knowledge of the masses of all the bodies of the system, and a complete theory of their relative motions, — a j)roblem 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 t 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 for the exp Since, in it will be ( tuotion of 1 In the n that it may have For the old piession (3? 19. Let I being at the If we reprc! and a line shall have m = 354710" 'II Hence we have log r = 2.5625978148 ; log i/l + m == 0.0000006122 ; log 2;: = 0.7981798684; and, consequently, log ;fc = 8.2355814414. If wo 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 exp'^essed in seconds of arc in a circle whose radius is unity, or on the orbit of ^he earth supposed to be circular. The value of k in seconds is, therefore, log /t = 3.5500065746. The quantity — expresses the mean angular motion of a planet in a meat solar day, and is usually designated by fi. We shall, therefore, have and, since c which we ha The angle

e of the tlioy are ei(^li helion, provi helion to thr Tlie same n wo regard, ii As soon as Motion and i (37) los in- out('/+|(l+5 CG2 2il/)+| (13 cos33/+3 cosiV)+ .... (51) Equation (22) gives dv = ^ 2fdt and, siuce/=p]/p(l + m), we have r (52) or rf„=^Ki+j!^«Vr=7'd^. a* But 5 = n, and therefore a dv ^^^l — e'-, iidt = Vl — e' -, dM. By expanding the factor ^/l — e^, we obtain T/r^=^=-= 1 — V — ^e*— . . . , and hence dv = (\~ le" .) -, dM. a Substituting for — its value from equation (51), and integrating, we get, since w =^ when M=^ 0, V— l/=2e sin iJf+^6' sin 2J/+^ (13 sin 3ilf— 3 sin J/) + , (53) Mhieh is the expression for the equation of the centre to terms involving c^. In the same manner, this series may be extended to higher powers of c When the eccentricity is very small, this series converges very rapidly; and the value of v — If for any planet may be arranged in a table with the argument M. For the .purpose, however, of com[)uting the places of a heavenly body from the elements of its oi'bit, it is proCorable to solve the equations which give v and J5^ directly; and when the eccentricity is 58 THEORETICAL ASTRONOMY. lif' 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 fi'om 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 circul.!". or else very eccentric. If, in equation (50), we put F{E) = E, and consequently F{M) = 31, Ave shall have, performing the operations indicated and reducing, ^ := J/ + e sin J/ + .Je^ sin 231 + &c. (54) Let us now denote the approximate value of E computed from this equation by Eq, then will E,+ £.E,=^E, in which a£^, is the correction to be applied to the assumed value of E. Substituting this in equation (39), we get 3I== E^-{- ^E^ — esinE^ — e cos ^'o^-Eo ; and, denoting by J/„ the value of 31 corresponding to j%, we shall also have 3f^=E^ — e sin Eg. Subtracting this equation from the preceding one, we obtain ilf — 3fo — e cos En A^.. m It remains, therefore, only to add the value of i^Eg found from this formula to the first assumed value of E, or to £"„, 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 31 for any instant of time, we may readily deduce from it, by th» formuhe 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 com])ut manner sider lie for dete lor el lip and e = PLACE IN THE ORBIT. 59 conipnting r and v, for 'my instant, is somewhat difforont. The manner of proceeding in the computation in such cases we shall con- sider hereafter; and we will now proceed to investigate the formuhc for determining r and v, when the orbit is a parabola, the formulte lor elliptic motion not being applicable, since, in the parabola, a =^ gc , and c = 1. 22. Observation shows that the masses of the comets are insensible in comparison with that of the sun; and, consequently, in this case, m = and equation (52), putting for p its value 2q, bef'omes or kV2q dt = ?-Mi', ^ cos* .Vy which may be written hdt V2f 1(1 + tan'' y) see' ^vdv = (14- tan' iv) d tan Av. Integrating this expression between the limits T and t, we obtain k(t-T) V2 =; tan 2^+3 tan' Iv, (55) wliich is the expression for the relation between the true anomaly and the time from the perihelion, in a parabolic orbit. liCt us now represent by Tq the time of describing the arc of a parabola corresponding to v = 90° ; then we shall have V2n' 4 3' or l/2~~^o Sk Now, — >■- is constant, and its logarithm is 8.5621876983; and if we I take 7 = 1, which is equivalent to supposing the comet to move in a ^)arabola whose perihelion distance is equal to the semi-transverse axis of the earth's orbit, we find log- dayi 2.03987229, or r^ = 109.61558 days ; that is, a comet moving in a parabola whose perihelion distance ^^mmmmmmmm 60 THEORETICAL ASTKONOMY. is equal to the mean distance of the earth from tlie sun, requires 109.61558 days to describe an arc corresponding to v ^^ 90°. E(juation (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 Ji? being the unknown quantity. If we put X = tan it', this equation becomes 3? -\- Sx — a = 0, in which a is the known quantity, and is negative before, and positive after, the perihelion passage. According to the general pri iciple in the theory of equations that in every equation, whether con plete or incom})lete, the number of positive I'oots 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'^, the time which corresponds to the same anomaly in the table. We shall have the anomaly v corresponding 3 to the time t — ^T by dividing t — Thy q^, 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 ^v + 25 tan» ^v. Let us now put M=^ 75 tan J,v + 25 tan« Av, PLACE IN THE ORBIT. 61 aiul CI ' V2 , which is a constant qnantity; then ■will The value of Q is Again, let us take ^(t—T) = M. log C„ = 9.9601277069. C 2* which is called the mean daily motion in the parabola ; then will M= m (t — T) = 75 tan ^v + 25 tun'^v. If Ave now compute the values of 31 corresponding to successive values of v from r = 0° to v = 180°, and arrange them in a table witli the argument v, we may derive at once, from this table, for the time {t — T) either 3/ when v is known, or v when 3I=^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 i' obtained from the table. Table VI. gives the values of 3/ for values of v from 0° to 180°, with ditfcrences 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 k(t—T) 1/2 «t = ]tan':Vy (1 +3cot'U'); and, multiplying and dividing the second member by (1 + cot^ Jr)^ wo shall have kit—T) V2q^ itan.A.(l+cot4.)'Ji±:|^. 62 THEORETICAL ASTRONOMY. But 1 + cot" \V ^ Sill V tan \v k it — T) ^ and consequently 8 1 + 8 cot' ^t' Bsiu'v* (.l-j-cot';iy)''" Now, when v approaches near to 180°, cot Av will be very small, .iiid the second factor of the second member of tliis 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 Ibllowing equation : Ay being a very small quantity. We shall therefore have 8 sin' to =n 3 tan j (io + \) + tan' i {w + A„), and, putting tan |w = d, and tan ^ a^ — x, we get, from this equation, tf' ~~ l — Ox'^ 0- — Oxf Multiplying this through by ^■' (1 — ^.i;)^ expanding and reducing, there results the following equation : 1 + 3tf' = SO (1 + 40' -\-20*^0»)x — SO^ (1 + AO" + 2fl* + tf«) a^ 4- ^^^ (2 + QO' + 80* 4- fl") 3^. Dividing through by the coefficient of x, we obtain _ 1 + 3fl* 3^tf(r+4^' + 2tf*-f fl«) Let us now put : X — Ox" + -r , , (P{% + &0''-\-2,0'-\-(!^)a? 3 (1 + 4 = ii; + Ao, 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 — Tis required, the table Avill give, by inspection, an approximate value of A, using v as argument, and then IV is given by 64 THEORETICAL AHTROXOMY. The exact valuo of a„ is tlion foiiiid irom the table, and hence \vc derive that of «' ; and finally t — T from t-T^ 200 J C'o aWiv 24. The problem of finding the time t — T when the true anomaly is given, may also bo solved conveniently, and especially so when v is small, by the following process: — Equation (55) is easily transformed into from which we obtain, since q^=r cos^^v. 2/2 _ /Hin^\_ /sinjtt\» ~ \ 1/2 / \ 1/2 / Let us now put and we have Con'^cciuently, Uit-T) 2r^ sin X = -rrz., 1/2 3 sin a; — 4 sin' a; = sin 3a;. t 2 f . o 3^r sm3a,-. which admits of an accurate and convenient numerical solution. To facilitate the calculation we put ,, sin 3a; sinv the values of which may be tabulated with the argument v. When t, =_- 0, we shall have iV= fV 2, and when v = 90, we have N=\; from which it appears that the value of iV changes slowly for values of V fi-om 0° to 90°. But when ?'=^180°, we shall have N=cc; and hence, when v exceeds 90°, it becomes necessary to introduce an auxiliary different from N. We shall, therefore, put in this case, from win wlien V - and, whei in which 1 when V is Tal)Ie \ tion, for vi for those 25. We from the ei formulttj fo only that c tive or im auxiliary q the two, an For this When V =^ nominator \ 180°-i^ an for the max vanish for tl either case, i In the hj quontly, we We have, ah Let us now ] N' = Nsinv = sin 3a;; PLACE IN THE ORniT. 60 from wliich it appears tlmt N' -'-■ 1 when r 90°, and that N'^^Vi when V - 180°. Tlicrcforc we have, finally, when v is less than DO", t-T. and, when v is greater than 90°, t—T-^ U Nr sin v, 2 - N'r in which log - - = 1.. '3883272995, from which t — 3' is easily derived when V is known. Tai)le VIII. gives the valncs of N, with differences for interpola- tion, for vail (•; of V from v =- 0° to v -- 90°, and the values of N' for those of v from v :^ 90° to v = 180°. 25. We shall now consider the ease of the hyperbola, which differs from the ellipse only that e is greater than 1 ; and, consequently, the formuhe for elli])tic and hyperbolic motion will differ from each other only that certain quantities which are positive in the ellipse are nega- tive or iniMginary in the hyperbola. We may, however, introduce auxiliary quanuiies whicli will serve to preserve the analogy between the two, and j c' u> mark the necessary distinctions. For this purpose, let ris I'esume the equation r = p cos •4« 2 cos A (v -f- ■4') cos ^ (y — 4.) When V = 0, the factors cos J(i; + -J/) and cos|(y — ^l/) in the de- nominator will be equal; and since the limits of the values of v are 180°— '4/ and —(180° — -v/.), it follows that the first factor will vanish for the maximum positive value of r, and that the second factor will vanish for the maximum negative value of v, and, therefore, that, in cither case, /• = 00. In the hyperbola, the semi-transverse axis is negative, and, conse- quently, we have, in this case, ^j = a(e' — 1), or a =jj cot'^-. We have, also, for the perihelion distance, q = a(e — 1). Let us now put taniF = tan -^vy e — 1 (56) :l ee THEORETICAL ASTRONOMY, which is analogous to the formula for tlic eccentric anomaly ^ in an ellipse : and, since c — — - . we shall have * ' ' cos 4. and, consequeiitly, e — 1 1 — C0S4 , ., , , — -~- = — = tan' ^4, e -j- 1 1 -j- cos 4 tan A-F = tan ^ V tan A4. We shall now introduce an auxiliary quantity a, such that 1 -f tin hF (57) ff=:Un(Ao° + } ioooo (i + tf^j* s, 10000(1 + tfO* S84rt7 I 97r>-Jin» I 37 S i 8/Jll _, lfi48/in I 17fi8/Jl5 I 181 fllT fi __ 3 13" T^ 253;-.'^^ r l I 7 r. ^_ i^ T57?>'^ ^7H7r,'^ T-fH7 5" „ 1000000(1 + ^■■')« ' 8fl7 I 5 2 8 ft/1» I 2 « 3 8 1 flU I 4 1 /ll3 i 5 12 8/JI5 I 9 4 fllT pi i" T2B3d'^ i^ 14 175 ^ i^ 3 1.5^ r 7g7ft'^ n- ^^^7^" 1000000 (1 + oy wherein s expresses the number of seconds corresponding to the length of arc equal to the radius of a circle, or logs = 5.31442513. Wo shall, therefore, have : — When X = V, v= F+ ^(lOOi) + 5(1000'+ C(lOOi)'; 76 THEORETICAL ASTRONOMY. niul, when x = v, V=v-A (1000 + £'(100/)'- C'dOOO'. Tabic IX. gives the vahies of A, B, B', C, and C for consecu- tive values of x from re =^ 0° to x ■— 149°, with differences for inter- polation. When the value of v has been found, that of r may be derived from the formula \ -\- e cos V Similar expressions arranged in reference to the ascending powers of (1 — e) or of I I zr-—- 1 — 1 I may be derived, but they do not con- verge with sufficient rapidity ; for, although I I ■ I — 1 I is less than /, yet the coefficients are, in each case, so much greater t'laii those of the corresponding jiowers of /', that three terms wil' lot afford the same degree of accuracy as the same number of teru.- in the expressions involving i. 29. Equations (73) and (74) will serve to determine v or t — T in neai'ly all cases in which, with the oi'dinary 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 formuloe 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 groat before the d' ippearance 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 h{t—T) = E — e sin£', which, bint kit- 2 If we put we shall ha k{t- Let us no and then we hav( When B is k be derived di a" and then fron to find the va of.l. Now, we hi and if we put sin We have, al E= nsccu- intor- erivod powers ot con- 31' than V\\} not erni> in -Tin cs, the erably rcji lilts exact, lase that impos- jO°,the wheu e ich the y groat general to de- estigate t wheu DOse, let s PLACE IK THE ORBIT. 77 which, >-incc q = «(! — c), may be written -^ ^ = j^ {dE 4- sm £) + j^ . -j-^ (^ — siu E). If wc put we shall luivo 15 j:—»mE ^E+ainE' i , 1 1+de A ■ \)E 4- sin E~ "*" 3 ■ 6(1 — e) Lot us now put B = » i«. — tan' ^t<; 9Jg 4- s in .E 20l/I l+9e then we have 5(1 -e) A; r~T~ " p- = tan Aw) + .J tan' |i«. 1/25'-^ -** (75) When B is known, the value of w may, according to this equation, be derived directly from Table VI. with the argument 75^«-T) v/T'a(l+9iy I 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 .1. Now, we have 2 tan IE svaE-- I and if we put tan^ \E == r, we get 2Ti 1 + tau'AJS' sin E = We have, also, 1+^: 2tJ(1— T + T» — T» + &C.). E=1 tan ' T^= 2t^ (1 — ^r + ^t» - ^t^ + &c.). 78 TIIEOKETICAI. ASTRONOMY. Thoroforc, 15(JS;— 8m£) = 2r^(10T-7T'-{- 7r'-igo-i + &c.), ami 9E + sin E = 2t^ (10 — 'j^r + '/t' — V^' + V^ — &c.). Hcneo, by division, IP sin 7? ^^9£+"8iui; """*"" o^+35^ 20Sa^ +74.1376- 1 OHOnr, 8 8r« 4- Arp • 2T80ffS76~ T^ **'^'» and, inverting this series, we get A ^ — ■•• 5-^ T^ T75-^ n- 525-^ i^ 138S7S^* T^ TiiaSliiS-^ ^ *^'^-> which converges rapidly, and from which the value of — may l)e found. Let us now put then the values of C may be tabulated with the argument A; and, besides, it is evident that as long as A is small C'^ will not differ much from 1 + ^A. Next, to find JS, we have ^ — ^ Ki- — S^ -r 175^ — 525^ i^ TDTDsljB^ *^^-.)> and hence T7-J. — ■" — -^ T T7-5^ 35-J-6^ + 3 3 = ^-t-^ AC, -* 1 — e ' ad; and, It it is in I tions for or, substituting the value of A in terms of w, tan iv = Ctan hv \ ~z — r-—-. - \ 1 -}- 9e The last of equations (43) gives Hence we derive rco^lv = qco,H.E=^-^. {l+AC'^co&'-^v (76) (77) The equation for v in a hyperbolic orbit is of precisely the same form as (7G), the accents being omitted, and the value of A being computed from 4 = ^/^tan'> (78) For the radius-vector in a hyperbolic orbit, we find, by means of the I last of equations (63), (79) (1— ^C")cos4v When t — T is given and r and v are required, we first assume r', we H.B= 1, and enter Table VI. with the argument M= _ q,((-r)T/,'o(i+96) ^i ' B ' 80 TirEORETICAI. ASTROXOMY. in whieli log ('^, - = 9.9()()12T71, and take out the corrospoiuling value of 'w. Tliea wo derive A I'rum the equation A 5ri ^'K 3 1 tan -Aw, 1 -r 9e in the ca^e of the ellipse^ and tVonx (78) in the case of a hyperbolic orbit. \V'ith the resulting vahie of A, we find from Table X. the oorre,-J]ionding value of log J5. and then, using this in the expi'ession ft)r M, Ave repeat the operation. The second result for ^1 ^vill 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 (.' fronx the table, and then v and r by means of the equations (76) and (77) or (79). When the true anomaly is given, and the time t — T is required, we first compute r from tan^ h), T -— H-e in the case of the ellipse, or from tan'' hv, in the c-asc of the hyperbola. Then, with the value of r au argu- ment, we enter the second part of Table X. and take out an a^jproxi- mate value of ^1, and, with this as argument, we find logjB and log ( . The equation T A will show whether the ap})roximate value of A used in finding log Cis sufKciently exact, and, hence, Avhether the latter requires uny correction. Next, to find ?/,', we have ^ , tanjc I i I 9(5 tan^t.^-^.^.^^-^; and, with ^o as argument, we derive M from Table VI. Finally, Wf have MBq^ i •- T- f;i/i^j(l + 9^) (80) by means of svhich the time from tlie ]}erihelion may be accurately determined. POSITION TX SPACK. 81 ,-alue rbolio t. the esslon 11 not i\ptioii ve the of the :^uircd, J argu- |)proxi- 1 loir C. iMuVing Ires uny illy, Avt (80) kiratcly 30. We have thus far treated of the motion of the heavenly bodies, relative to the sun, without considering the positions of tlieir orbits insj)are; and the elements whieh we have employed are the eccen- tvicity JU'-d semi-transverse axis of the orbit, and the mean anomaly at a given epoeh, or, what is e(|uivalent, the time of passing tho |M'rilielion. These are the elements whieh determine the positif)n of the body in its orbit at any given time. It remains now to tix 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 orliit in reference to a known plane must be given; and the Icmcnts which determine this jmsition are the longitude of the periln iion, the longitude of the ascending node, and the inclination of the ])lanc 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 I)y means of tlie tbrmuhc already investigated. The hiiaitudr of the a,^c('iid!nf/ node, or longitude of the })oint tlirougli which the body passes from the south to the north side of the ecliptic, which we will denote by Q,, 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 /hie nf The angle which tlic plane of the orbit makes with the i)lane of the eclii)tic, Avhich we will denote by /, is called tla; indlnatlon of the orbit. It will readily be seen that, if we suppose the ])lane of the orbit to revolve about the lino of nodes, when the angle / exceeds 1Sii>/lhi(lc of /he /irrllirlioii, which is denoted hy ~, fixes the ]>()sition of the orhit in its own plane, and is, in the ease; of direct, Miolioii, the siMii of the hiiifiitiuh' of tlie ascendiii};' node and the an^i'iilar distance, iiieasni'ed in the direction of th(! motion, of tlie perihelion from this node. It is, thc'reH>re, tin! an;j;nlar distance of the perihelion from a point in the ori)it whose ant;iilar distance hack from the ascend i 11!^ node is c(pial to tlu; lonj;itiide of this node; or it may lie measured on the ecliptic (Vom the vernal e(juino.\ to the asceiidiiiu,' node, then on the pliiiie of the orhit from the node to the place of the perihelion. ill the case of retro|z;rade motion, tlw lonifitndes of the successive points in the orbit, in the direction of the motion, (h'crease, and the point in the orhit from which these lon^'itiid(>s in the orhil arc K I mejisiired is taken at an anti;ular distance from the aseendiii}; node c((iial to the loni^itiidc of that node, hut taken, from the node, in tin.' same direction as the motion. Hence, in this <'ase, tll(^ loiiLrittide of tilt? pci'ihclion is e(|iial to the lont;itnde of the ascentliiiii; node dimi- nished Itv the aiii!;iilar distance of the |)erilieIion from this node. It may, perhaps seem desirai)li' that the distinction-, ilincf and r(lr()(/ni one set of formula' would he sulhcient, while in the common form two sets are in part retpiired. IIow<'ver, the custom of astronoiiu'rs d loi)i!,itude of the perihelion is called tli(^ iiiraii /oiif/ifiidc, an ex- an pression w hich can occur onlv iii tlie case o th )f ellipt le oroits. In the eas(! of retro<>rade motion the lonosition of the body at the time f. wu have jf — ; r sin V, (lie line ( l:ikeii at If we (I) \)v\l]nse(jiH^n!tIy, 1)11 in it-' C08« - COB A/«>S (I Q, , sin V cox i — eos h *in(l Q,). sin H nn i sin /j. (H] I hie ^ we H From tlnse we «lerivc tnn I / — ft ) = ± tan n cos /, ta* 'v = =t tan /' niii (I - ^ ), (m ^y\w\\ serve to deterioiiM! / luid h, wli' u ^, i/f aun4 i are given, hinco 84 THEORETICAIi ASTROXOMY. coah is always positive, it follows that I — SI smd u must lie in the same (luadrant Avhcn / is less than 90°; but if / is greater than 90°, or the motion is i-etroj^rade, / — ^ and 3(i0° — u will l)elon<>; to the same (piadrant. Ilenee the aml)i<>;uity whieh the determination of /— ^ l)y means of its tangent involves, is wholly avoided. If we use the distinetion of retrograde motion, and eonsider i always less than 90°, / — SI and — u will lie in the same <|uadrant. 32. By multiplying the first of the equations (81) by sin?*, and the second by cos u, and combining the results, considering only the upper sign, we derive or cos I) sin (u — / + $^ ) =: 2 sin ?t cos « sin^ U, cos b sin (n — ^ -j- SI) '-^ sin 2u sin'' ^'. In a similar manner, we find cos b cos (u — l-{- SI) =^ cos' it -\- sin' 11 cos i, which may be written cos b cos (u — / -f- $2 ) = ,J d -j- cos 2u) +10- — cos 2it) cos i, or cos 6 cos (n — I -{- 52 ) := A (1 + cos i) + 2 (1 — cos i) cos 2u; and henee cos b cos (u — I -j- SI) = cos' ;it + sin' U cos 2u. If we divide this equation by the value of cos 6 sin (« — I -{- SI) alreadv found, wo shall have , , , ^N tan',U'sin2» tan (« — / + Sl)~ ?— rr % , • ir- 1 + tan' Jit cos 2)4 fs.r, The angle u — /+ SI is called the rahii-tion to the (w/lpfic; and tli cxj)ression for it may be arranged in a series whiili converges rapi«ii\ when / is small, as in the case of the ]ilanet-. In order to effeei this development, let us first take the e which is tlie general form of the deveh32)nu'nt of tlie above expression for tan//. Tlie assumed expression for tan cos I ~\- li cos 2" cos O . 4 cos /? sin ). :-= }• cos 6 sin / + ^^^ t'os 2" sin © , ^ sin li = r sin b -\- H sin 2". (88) If we multiply the first of these equations by cos Q, and the second by sin©, and add the products; then multiply the first by sin ©, and the second by cos©, and subtract the first }>roduct from tin se(!ond, we get J cos ,? cos {>. — © ) = r cos h cos (I — © ) ]- R cos 2", J cos ,3' sin (A — © ) -; r cos i sin ( / - © ), J sin ,3 = r sin b -\- li sin 2". (89 1 It will be (»]»served that this transformation is equivalent to the sup- position that the axis of x, in each of the co-ordinate systems, is Ji' we sii tilde is Si, . tioiis (88) b I'v means ot If it be dt'clination, values of fi ticn, denotii now ( POSITION IN" .SI'AC.'K. 87 directed to :i point whose lon(>itii(le is ©, or tliat the system liris Ixun revolved idiout the axis of - to a lu'w position for which the axis of iihscissas makes the an^^'h- Q with that of tlie jjrimitive system. We may, therefore, in general, in order to effect snch a transformation in systems of e(piation,s thns derived, simply diminish the longitnde.- l)y the given angle. The ('(piations (89) will (h'termine /, ,'i, and J when r, h, and I have heen derived from the elements of the orbit, the qnantities A', 0,and 2' k'ing furnished by the solar tables; or, when J, ,5', and / are given, those equiitions determine /, h, and r. The latitude - of the sun never exceeds ±: 0".9, and, therelbre, it may in most eases l)e neg- locted, so that cos 2' =: 1 and sin 2' - 0, and the last equations become J cos /? cos (A — O ) = r cos h cos (/—©) + li, J cos /J sin (A — O ) = V cos b sin (/ — Q ), J sin /3 = /• sin b. (90) If we suppose the axis of x to be directed to a point whose longi- tude is SI, or to the a.scending node of the planet or comet, the e([ua- tions (88) become J COS I'i cos (A - -SI) J COS /? sin (A - -SI) J sin /J r COS u + A* cos - cos (© — SI), ± r sin a cos i + li cos 1' sin ( © —SI), C^l) r sin (/ sin / -f- A* sin -', by means of which ,9 and A may be found directly from SI , i, >', »iid i'. If it be required to determine the geocentric right ascension aned when the motion is direct, and the lower sign when it is retrograde. Let us now put cos Q - sinrt sin A, ^ cos / sin £1 '-■ sin n cos A, sin Q, n sin /> sin l>, ± cos / cos Q, =^^ sin h cos Ji, (04) ill which sin a and sin 6 are positive, and the expressions for the co- ordinates become a; = r sin a sin (A -f- n), y =zr sin h sin (/>' -\- u), (JM) z =r sin i sin ». The auxiliary fpiantities (t, h, A, and />, it will be observed, are functions of Q, and /, and, in computing an ephenieris, are constant ?n long as these elements ai'e regarded as constant. They arc called die i-niiKUinfx for the ecliptic. To deternunc them, avc have, from ecpuitions (94), cot A^=: -\- tan Q, cos /, cos Q, sin ^1 ' sui a cot B = ±: cot Q cos i, ."in ^ ain Ji' sin b the upper sign being used when the motion is direct, and the lower sign when it is I'ctrograde. The auxiliaries sin a and sin b are always positive, and, therefore, sin.l and cos Q, sin /^ and sin SI, resj)ectively, must have the same i^igiis. The quadrants in which A and B are situated, are thus deter- mined. From the equations (94) we easily find cos a =- sin t sin ^, cos b == — sin (' cos £1. (96) If we add to the heliocentric co-ordinates of the body the co-ordi- nates of the sun referred to the eai'th, for which the ecpiations have already been given, we shall have X -f- A' =^ J cos (J cos f., y 4- y --- -1 cos ,3 sin A, z -\- Z = J sin fi, (97) 90 TII KO It KT K A I , A ST IK )NOM Y. ^vlli(•ll siillicc to (Ictcnii'mL' /, ^-i, mikI J. The values of a mid o mav be derived iVom tlicsc by lucan.-^ dl" tlic ('»|iiati(iii,s (!)2). .'};"). W'c shall now (Icrivc (lie roriiiida' for (Ictcnniiiinji' a and H diriTtly. I'^or (his purpn.-c, let .c,;/,:. Ix' tli'' heliocentric co-onliiiatcs of the body referred to the ecpiator, the jMisitive axis of ,(• beinjf directed to the vernal e(|iiino.\. To pass from the system of co- ordinates referred to the ecliptic to those rel'erred to tlu; ('(piator as the fundamental |)lane, we must revolve the systi'm nejiatively aroinid the axis of ./•, so that the axes of :■ and // in the new system make the angle s with those; of the primitive system, £ being the obli(piity of the eellptie. In tliisi easi', we have x" = X, ij' = y cos e — 2 sin s, z" = y sin s + - f O'' -• •Substituting for x, //, and - their value.s from equations (93), aud omitting the aeeenis, we get X = r cos n cos ^ =p /• sin n cos / sin J^ , y=:^r cos II sin £1 con s -\- r s'm n (± cos/ cos Q, coss — sin i sine), g = r cos i( sin £1 sin £ + '' ^'" " ( ^ ^'^^'^ ' t'0'"> Q> >^'i^ - + i^'ii i cos s). (08) These arc the expressions for the heliocentric co-ordinates of tlie ])lanct or comet referred to the ecpiator. To reduce them to a con- venient I'orm I'or numerical calculation, let us put cos SI = ^in « sinvl, zp COS !■ sin J^ ;-= sin a cos^, sin Q cos e == sin b sin B, dh cos i cos SI cos e — sin / sin e =:= sin b cos Ji, sin Si sin s =:= sin c sin C, rt cos i cos SI f"'" ^ + ^i» >■ cos £ =- sin c cos C; and the expressions for the co-ordinates reduce to a; = r sin a sin (A -\- v), y = r sin b sin {B j- n), z = r sin c sin ( C -\- n). (99) (100) The auxiliary quantities, a, b, c. A, B, and C, are constant so long as SI tiiid / remain unchanged, and are called consUmtn for the cquoiof. It will be observed tliat the equations involving a and A, regaid- ing the motion as direct, correspond to the relations between tin parts of a quadrantal triangle of which the sides are i and a, tlu I'OSITIOX ]N SI'Al'K 01 aiijrlt iiH'Iudcd hotwccn tlieso sides lu'injf that wliicli wo dcsiji'iiatc l>y .1, and tlu! aii;i;lc opposite tiic side o hciiio- !)()"- J^. In the case; lit' li and />, tiic relations are those of liie |)arts of'a spherical triant;le s, II lu'inn' the an^le iiichnled which the sides are /;, /, and !*(»' Iiv /' and //, an( iSd 9, tl e anji'le (tpposite tlio su te th de h. Finther HI liie case of c and (', the relations are those of the parts of a 1 tl rie of wliK'li the sides are c. i, aii» U, tl le anir \v ('\ )Cllll fijilicneal triang tliiit included hy the sides / and c, and 1(S()° Q, that included Ky the sides / and s. We have, therefore, the followiii"; aiKlitioiial (■(|iiatiuns cosrt --ir; sin / sill Q,, cos 6 = — cos J2 sin / cos e — cos / sin e, cos c = — cos Q, sin / sin j -j^ cos i c (101) OS In the case of retrofirade motion, we must substitute in tlieso isd- / in place of / The <;'eometrical sifj;iiification of the auxiliary constants for the ('(Hiator is thus made apparent. The anh this line and the co-ordinate axes, iniikcwith a plane passing through this line and perpendicular to the line of nodes. Ill order to facilitate the computation of the eonstants for the equator, let us introduce another auxiliary quantity E^, such that sin i =: e^ sin E^, ±: cos i cos ^2=^0 cos £„, e^ hoing always positive. We shall, thereibre, have tan E„ tan i cos 9>' iSiiK'o both Ty and sin/ are positive, the angle T?,, cannot exceed 180°; and the algebraic sign of tan A', will show whether this angle is to 1)0 taken in the first or second (piadrant. The first two of equations (99) give ami the first gives cot ^4 = -+- ' uu Q, cos ii sm a = cos Q, siu^' -.% ^, %»^^V^( IMAGE EVALUATrON TEST TARGET (MT-S) 1.0 I.I lii|^8 |2.5 •^ ni |2.2 1^ i: III t Ui 12.0 L25 III 1.4 I 1.6 y] /: ^;^ >^ Photographic Sdences Corporation 33 WIST MAIN STRiiT WHSTIR.N.Y. I4SS0 (716) 873-4503 «^ 92 TIIKOKKTlCVL AHTIIONOMY. From tlu! fourth of (fiuations {{)[i), introiluciiig c^ and A'„ we get sin h cos B=^ e„ cos E^ cos c — r, hIii E^, m\ t =- e^ cog (ii'^ -f c ). sill b siu iy — iiiii ft cos t ; But therefore coti? HIU ft We have, also, gin 6 In a siniihir manner, we find COS (E„ + t) co8 i cos ( Ef^ -f- «) tan ft cos A'u cos c cose sin ft ccic sin ii cot C'= and cos t sin (Eg 4- •) tan ft cox Aj ' sin e * sin ft sin c 8mc = sin C The auxiliaries sino, sin 6, and sin a are always positive, and, then- fore, sin -.4 and cos ft, sin B and sin ft, and also sin (' and sin JJ, must have the same sijjns, which will determine the (|uadiiint in which each of the anjjles A, Ii, and ('is situated. If we multiply the last of ec|uations (911) hy the third, and tlu fifth of these cfjuations hy the fourth, and sui)ti'act the first prodiut from the last, we get, hy rctlnction, But sin b sin c sin {C — JS) = — sin t sin ft. sin a cos ^1 = =P cos t siit ft ; and hcnec we derive sin 6 sin r sin ( C — B) sin a cosvl = ± tan I, which serves to cheek the accuracy of the nnmrrieal computation of the constants, since the value of tan / ')l)taine8ITIO.\ IN' SI'AfE. « = r pin It ,«in ' A' -f r), y = rniii //sill ( If -\- r), s =z r .xiiir sir (C -\- v), (102) a transforinntion wliirh is pcrlmps unnccrMsnry, hut wliicli i.s oon- viiii'iit wlii'ii a M-rii's of phircs is t cuinjiiitcd. It will Ik' oltsorvwl that thi' j'oniiuhi' tnr comimtiiin tlic constants II. i, «', A, li, and (\ in the (Uso of direct motion, arc convcrtj-il into tlio-c for the case in which the distinction of retroj^nuU; motion i.s atlopted, hy simply usinj; 180° — / instead of /. 36. When the helif the hody have Imhmi foiiiul, relcrred to the e<|iiator as the fniidamental phme, if we a(Ul to tlu'M' the jjeocentric co-ordinates of tiic sun referred to the same fiiiiilaiiicntal ph»n<', the sum will he the geocentric co-ordinates of the ImmIv eferred also to the e<|iiat latitude of the sun, Y= It m\ O eose, Z = H sin O sin s — i'tnn r, ill wliicii R represents the radius-vector of the earth, © the sun's ](iii;ritual astrononiicjd cplicincrides, such as the lirrlhirr Axtrnnomittchfn Jahrhuch, the Xdulii'dl Alinamic, and ihc .'.meyican Epheuurls and Nauticxd Al- Til Fy»RETICAL ASiTRONOMV. mnnnc, rontain, for cnch year for whicli they arc puMi'sluKl, (be o(|iiat«>rial co-onliiiatcs of tho .smi, rcfrrrcil both to the mean o(|uin(tx and ('(juator of the l>o^iniiingof thoyt-ar, and to the apparent equinox i)i' the date, taking into aceonnt the latitude of the .sun. .'57. Tn the cjise of an oiliptic orhit, we may determine the co- ordinates (lireetly from the eeecntrie anomaly in the following maimer : — Tho equations (102) give, aeeenting the letters a, b, and c, x = r cos V sin n' sin A' -f r sin v sin «' cos A', y = r cos V sin b' sin If -\- r sin v sin h' cos If, « = r cos r sin c' sin C" -f r sin v sin c' cos C". Now, since r cos y — o coa E —- ae, and r sin v ~ a cos ^ sin E, we shall have a; = a sin a' sin A' cos E — ac sin n' sin A' ■■{- a cos f> sin «' cos^' sin E, y = « sin b' sin /^ cos E — ac sin // sin Ji' -\- a cos y sin 6' cos If sin A', 8 = a sin c' sin t" cos E — ae sin c' sin C" -f a cos y sin c' cos C sin £". Let us now put a cos f sin «' cosyl' =^ ^, cos //„ o sin a' sin A' = /, sin />„ — «<* sin a' sin vl' = — ei.^ sin L, = w, ; a cos y sin 6' cos If = kj cos />,, rt sin // sin If := -1,. sin L,, — ac sin 6' sin /i* = — ^ ^^ sin L, = Vj-, a cos f sin c' cos C" = A, cos X„ o sin c' sin ( " = ^, sin />„ — ac sin c' sin C" = — e^, sin L, = v, ; in which sin a', sin 6', and sin c' have the same values as in equations (102), the accents l)eing ad which we shall investij^te in tin.' following chapter, it will always he ad- vis:il»ic to <'ompute the co-ordinates hy means of the radius-vector and true anomaly, since both of these quantities will he reipiired in limliiig the differential coefficients. .'JH. In the ease of a hyiKrhoIic orbit, the co-ordinates may be com- puted directly from F, since we have and, eoiisequcntly, r cos i> ^=a(e — pec F), r sin V — ■ a tan ■! tun F; T — (le sin a' sin A' — n sec Fsin a' sin A' + « tan + tan Fsin n' cos A', y -^ uc win h' sin if — o sec Fm\ b' sin ii' + a tan 4. tan i'sin b' cos if', z — ae sin c' sin C — a sec i'sin c' sin C" + « tan 4- tan i'sin c' cos C". Let us now put Then we shall have acsinrt' sin A' = X„ — a sin a' sin A' =^ fi,^, a tan + sin a' cos A' ^= v, ; nesin// sin if = ^j, — « sin b' sin if = /ij, a tan 4. sin b' cos K ^^Vj\ aosinc' sin C" = A„ — a sin c! sin C = iit, a tan 4 sin c' cos C = v,. X = A, 4- /i, sec F -f ^'x tan F, y = kj-\- /jty sec F -\- Vj tan F, 8 = ^, -|- /i, sec F -\-v^ tan i'. (106) In a similar manner we may derive expressions for the co-ordinates, in the case of a hyjKsrbolic orbit, when tlie auxiliary quantity a is 1 used instead of F. 39. If we denote by :r', Q', and V the elements wliich determine the position of the orbit in space when referred to the equator as the 86 TlIEOHETICAr. ASTIIOXOMY fiiinliunciital plaiu', and by w.) the luiffiihir distniicc Ix'twccn tlic iisc('ii(liii■" J ('" — -). cos }^i' cos J ( ft ' ^- % ) --■ cos ! ft cos .1 ( / + e), ein .W' sin ' ( ft' — mJ - sin ;\ft sin !(» — £), ain i*' cos i ( ft' ~ w„) = cos ^ ft sin ! { / + t). (109) The (puulrant in which i(ft' + w„) or J (ft — <«„) is situated, must Ik; so taken that sin J/' and cos J/' shall l)e positive; and the agriM-incnt of the values of the latter two (piantities, computed hy means of the value of .J/' derived from tan J/', will serve to check the accuracy of the numerir.il cahnilation. For the case in which the motion is repirdcd as rctr<»gra(le, wc iiKist use 180° — i instead of / in these eipiations, and we have, also. --ft + ft' 0' We may thus find the elements ;:', ft', and !', in refei*cnce to the equiUor, from the (ilemcnts referred to the ediptic ; and usinjr the elements so found instead of r, ft, and /, and usini; also the places of the sun referred to the ecpiator, we may derive the heliocentric iuid ••('ocentric j)laces with respect to the equator hy means of the iorniuia' already given for the ecliptic as the fundamental plane. If the [Kisition of the orbit with respect to the ((juato' is given, and its position in reference to the ecliptic is refpiinHl, it is oidy luressary to interchange ft and ft', as well as / antl 180° - /', s remaining unchanged, in these ecpuitions. These fornmlie may alsd he used to determine the position of the orbit in reterence to any plane in space; but the longitude ft nuist then be measured from the place «)f the descending node of this plane on the ecliptic. The value of ft, therefore, which must be used in the solution of the Wjuations is, in this case, eipial to the longitude of the ascending ntnle of the orbit on the ecliptic diminished by the longitude of the descending node of the new |>lane of reference on the ecliptic. The qnaiitities ft', /', and to^ will have the same signification in reference 7 98 TIIKOIIKTK Ah ASTItONOMY. tn tilis plniic (Iiiit tlifv have in n'fcn'iicc to tlu* <'f|iiat<)r, with this dis- tinction, liowcvcr, that SI' i'^ measured from the (hx-entlin^ no(U' of this new |thine of relereiiee on the eeliptie; and e will in this case denot«' the inclination of the eeliptie t(» this |)lanc. U). We have now derived all the forninlip which can be reqnired in the ciise of nndistnrlx'd motion, for the eoni|intation of the helio- centric or jiciM'cntric place of a heavenly ImmIv, referred either to the eclipti( or e«jnator, <»r to any other known plane, when the elements of its orhit arc known ; and the formnliu which have been derived are applicable to every variety of conic; section, thus including all possible Ibrms of undisturbed orbits consistent v,ith the law of inii- versid gravitation. The circle i.s an ellipse of which the eccentricity is zero, anil, conse(|nently, 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 asteroi«l planets between Mars and Jupiter it is sometiuies thought advisable, in order to tacilitate the identili<'ation of comparison stars for a few days succeeding the diseovcry, toconipjite circular elements, and from these ar, ephemeris, The elenjcnts which determine the form of the orbit remain con- stant so Ic»ng as the systi^m of elements is regarded as unchanged; but those which determine the position of the orbit in sjnice, z, Q,, and /, vary from one j-poeh to another on aivount of the change of the relative p(jsition of the planes to which they are referred. Tlnii: the inclination of the orbit will vary slowly, on actc :nt of the change of the position of the «'cHptie 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 nmle will vary, both on account of this change of the position of the plane of the ecliptic, anutation of the geot^jntric places, the longi- tudes of the sun must be referred to the same e(piinox, so that the resulting geocentric longitudes or right ascensions Avill also be re- ferred to that ecjuinox. It will apj)ear, therefore, that, on acf;oiuit of these changes in the values of r, ft, and /, the auxiliaries sin a, sinb, 8\nc, A, B, and C, introduced into the formula; for the co- ordinates, will not be constants in the computation of the places for a series of dates, unleys the elements are referred constantly, in the calculation, to a fixed equin.x and ecliptic. It is customary, there- POSITION IN SPAfE. 00 fi>n', (o rt'»liu'(' tlic elements to the eeliptie iiiul mean ef|nino.v of the lieifiniiin^ of the ynir for whieli the ephemeriH is required, and then to compiite the phiees of the planet or eoniet referrension and declination, the mean iililiipiity of the ecliptic for the date of the tixed e(piinox adopted, in the computation of the auxiliary constants and of the eo-or«linates <»f the sun. 'I'lie pli-'-es thus foinid may he reduced to tlie true (•<|uinox of the date l»y the well-Unown formnhe tor precession and mitatiou. Thus, for the rc(hu-tion of the rij;ht ascension aiul decliua- tjiin from tlie mean equiuox and eipiator of the he^innin^ of the year to the apparent or true eipiinox and cipiator of any dat«', usually the date to which the co-ordinates of the body belong, we luive Att =/ -|- (f sin ( (i + a) tan »), A') —: (J cos ( G -f- o), (110) tlir which the (piautities/, 7, and G arc derived from the data given cillicr in the solar and lunar tables, or in astrouonucal epheuu'rides, >iii(li as have already been menti(Mied. The problem of rcy the intersection of the plane of the orhit and of the planes of the two ecliptics with the celestial vault, we get sin rj cos (SI — 0) ^^ — cos i' sin i -\- sin i' cos i cos Aw, from which we easily derive sin d" — t) -- sin >j cos (SI — <') + 2 sin i' cos i sin' ^Aw. (112) We have, further, sin Aw sin i' = sin ij sin (Q — 0), or . ^ . s\n(Sl—0) Sm Aw ■-=: 8U1 Tj . 'T, Hin I We have, also, from the same triangle, which gives sin Aw cos i' = — cos ( Ji — 0) sin (SI' — 0) + sin (SI ~ 0) cos (Si' — 0) COS)?, (li:0 sin (SI' — Sl)--= — sin Aw cosi' — 2 sin {SI — 0) cos (ft' — 0) sin' Jij, or 8in(ft' — ft) = — sin )? sin (ft — 0) coti' ~ 2 sin (ft — <^)co3(ft'— <^) siuMij. Finally, wo have n' — 7:= ft'— ft + Aw. (114) Since r^ is very small, those equations give, if we ap])ly also the pre- cession in longitude so as to reduce the longitudes to the mean equinox of the date /', Aw _ !>\n(fl--0) sm I Aw' i' = i -fj COS ( ft — 0) -|- 1 sin 2 1, Si'= Si +(it'-t)^^-71 Bm(Sl-0)coii'-}tsm2(Sl-0), (115) ^' = :: + (<'-0-f + '?8in(ft-tf)tan^i'-|l'siu2(ft- iiitrrval t 17'>0".21(/- IToO;, 8u tliat \w liiivc, tinnlly, ~. 351 ^ .'.d' 10" 4- :»»".7iK< - li.VM .V'.'il (t'-- 1). Wlicii the clcinciits r, 52, "H'l ' Imvo Im-oii thus rcdiUH-d (Vom the (M-liptif atui mean <'i|iiiiiteri-ei| to tlie apparent eipiinox of the . 'I'hen, in the eiise of the Jeterniinalion ot' the ri^ht nHcension and (h'clinatioti, usiuj^r the appat'ent oMi(|uity of the eeliptie in the eompiitation of the eo-onlinates, wi- direetly (»l»taiii the phiee of th(> hiKly referred to the apparent e(piino\. lint, in eoni- pntinn a series ol" plae»'>, the ehan^^es whieli tliiis take phiee in the dements themselves from date to date induce correspond in;; chan^^es in the auxiliary ipiantities n, l>^ r, A, II, and (\ so that these are no lonf^'or to ho c(insi(h'red as constants, hut lus eontinnally elian^rin;; tin'ir vahies l>y small ditlcrences. The ditfen-ntial formnlie for the com- putation o*" these chancres, which are easily derived from the eipiation> (!)!>), \vill i)c ;;iven in the next chapter; hut they are perhaps unncccs- sury, since it is generally most convenicni, in the esiscs which occur, to compute tlu' auxiliaries for the extreme dates for which the ephemeris in re(piired, and to interpolate! tlieir values for internu. 103 of niMrviilioii ; f»r, In caso wv know tlic daily or hourly inolioii of the IkmIv ill i'i|;)it a.M't'iisimi dimI dfcliiiatiuii, \V(* inity coiuputc \\u iiiiiiii>ii tliii'iii^ the interval wiiit-li is n-(|uirfsiM'vation. We inav also include tlic alx-rnition dircrtiv in the cphcincris hv ii-iiij: the time / WH'.lHj in compiitiii^ the jfcocciitric places lor the liiiic /, or l»y siihtractin^i t'roin the place tree I'roiii altcrration, coiu- |iiiicd lor the time f, the motion in 7.7H tin nnmlM'r of seconds in which lij^ht traverses the iiKaii distance of the earth from the .siin. It is customary, hr>wevcr, to compnte the t[ili' meris fn>c from iilM'rralion and to siihtract the (Imr itj n/x rratiou, 4'.}1 .~HJ, from the time of ohservation wlwii <'omparin^ ol»servation« "iili an ephemcris, ;u rdiiii.r to the first method ahov*' mentimi'i,. Tj. phic.'s of the fill! used in compntin<; its co-ordinates must also he five from aherra- tidii; 1:1 if the lonj;itndes derived from the sf^lar tahlcs inclntle nlM'rratioii, tlu; proper correction must he appl'cd, in order to obtain till' trill- lon}j;itnde rc(|uired. II. KxAMlM,i:s. — We will now collect together, in the proper oiilrr for nnmcrical calculation, some of the principal formuhe which liave hceii derived, and illustrate them hy numerical examples, coin- im'iiciiijj with the luse of an elliptic orhit. Let it he reijuircd to find the geocentric riffht asc-ension and decliiiati(»n of the planet luiri/iininc 0, for mean midnight at Washinjrton, for the date iSIJo Fehruary 'J4, the elements of the orhit heing as follows: — Epoch ■~=^ \W4 Jan. 1.0 (rreenwich mean time. i»/== r •_>!»' 40". 21 [ hclii)tic and Mean ft .^^20(i 42 4(t .1:5 ■ ,, ! ^. ,^,., ., '"' I h^ouinox, lo()4.0. i= 4 'Mi .)() ..-)1 j ' V>^ 11 1') 51 .02 logrt = o.:wMi.'ny log/i = 2.!)07«088 H = U2«".o.574.5 When a series of places i.s to be computed, the first thing to he tloue is to compute the auxiliary constants u.sed hi the expression.s for the co-ordinates, and although but a single pl.uo is refjuired in the prohKin propo.sed, yet we will proceed in this manner, in order to 104 TirEomrncA i. ahtroxomy. «'xliil)it tho application of tlu> rornmlu". Since the elements r, Ji, and / are referred to the ecliptics and mean e(|uinox of 18(>4.(>, we will first redu(!e them to the ecliptic and mean eo.(). For this reduction wo have t 18()4.(), and (' iHOo.O, which give dt — rw»" ', r)()".2:{9, --.- :{r)2° ">!' 41", r, = 0".4882. Suhstituting those values in the equations (115), we ol)t;iin /' — / r.:. A/ ^ — 0".40, A Ji = - -I- -)\\".(y\ , ATT ^. -f 50".'23 ) and hence the elements whi<'h determine the position of the orbit in reference to the ecliptic of IHOo.O are T= 44° 21' 2.r.;j2, ft 206° 4;r 3;5".74, 4°36'50".ll. For tile siime insiant we derive, from the American Ejilicuu'i'is ttnd Noiiticdl AliiKDHW, the value of the mean obliquity of the ecliptic, which is e = 215° 27' 24".03. The auxiliary constants for the equator are then found by means of tho formulae cot A --- cotli- cot C = sina — tan p, cos /, „ tan I tan Eo - _ , cos SI cos i cos (Eq -\- e) tan ft cos Eg cos e cost sin(A^-j-0 tan ft cos E^ sin e cos ft . , sin ft cos c -. , 8mfc= . — , sui .4 sin Ji sni c sin ft sin e sin C The angle E,, is always less than 1S()°, and the ((uadra:it in which if is to be taken, is indicated directly by the algebraic sign of tan /'7„. The values of sin a, sin />, and sin c are always positive, and, therefore, the angles A, li, and ('must be so taken, with resj)eet to the (puulrant in which each is situated, that sin A and cos ft, sin Ji and sin ft , and also sin ('and sin ft, shall have the same signs. From these we derive A = 2{>({° ;?9' r)".07, 1^ = 205 5;-) 27 .14, C=- 212 32 17 .74, h)g:-inrt=r. 9.1H»!)7ir)(), log sin b =^^ 9.})7482o4, log siuc^: 9.5222192. Finally, the calculation of these constants is prove0".r)2, log )• = 0.4282854. Since 7: — ^=^ 197° 37' 49".58, Ave have n = v-\--—Q, = ;]2()° 41' 40".10. The heliocentric co-ordinates in reference to the equator as the fun- damental plane are then derived from the e<[uations x = r am a sin (A -f n), y ^=r sin /> sin ( B -f- 't^» 2 = r sin c sin ( C + «)» which give, for Eiwynome, X = - 2.(5(51 1 270, 1/ = + 0.3250277, z =: + 0.01 1 048(5. The Aiiwricnn Nautical Almanac gives, for the equatorial co-ordi- nates of the sun for 1865 February 24.5 mean time at AN'ashingtoii, referred to the mean ecjuinox and equator of the beginning of the year, A'==-f 0.9094557, r= — 0.3509298, Z= — 0.1561751. Finally, the geocentric right a.scension, declination, and distance are given by the e(iuations y-\-Y 2 + Z. z + Z , z + y^ tana = ' , -,,, tan o = --—..sni o —--,,, cos a, J=— ,, .1- + A y + J x-\-A sui ') the fii'st form of the ctiuation for tan J being used when sin a is greater than cos a. The value of J must always be positive; and d cannot exceed ± 90°, the minus sign indicating south declination. Thus, we obtain NUMERICAL EXAMPLES. 107 isr 8' 29".2{), .J = — 4° 42' 21 "..")«;, log J =.^ 0.24r)00o4. To reduce a niul o to the true o<1° «' 4(J".71, 5=-~4° 42' 28".73, log J . - 0.24.")0n54. Wlifii only a single place is required, it is a little more expeditious t(i ctiiupute /• from r = nil — e vo9 E), ami then r — E from sin \(v — E ) -- V" sin \- -^ E) ^ 9.9160318, I while the second member of this equation gives log sin \ (V + E) = 9.9165316. In the calculation of a single place, it is also very little shorter to |i(iin|Miti' tii*st the heliocentric longitude and latitude by means of the (i|iiatinn.s (82), then the geixjcntric latitude and longitude by means lot" !8il) or (90), and finally convert those into right ascension and Idirjination by means of (92). When a large number of places are Ix' computed, it is often advantageous to compute the heliocentric If-- 108 TIIKORETKAI. ASTRONOMY. co-ordiiintoH directly I'roin the eccentric anomaly by means of the eiic is, in all respects, similar to that in which the e({iiator is taken as the fundamental plane, and does not re(|iiire any further illustration. 'I'lie determination of the fjreoccntric or heliocentric place in the cases of parabolic and hyperholic motion differs from the process indicated in the preceding example oidy in the calculation of »• and r. To illustrate the case of parabolic motion, let < — T' 75..'{G4 days; log 7 O.OGoOlHfi; and let it be reijuired to find r and i'. First, wc compute m from C »i 7- in which log C^=--- 9.9601277, and the result is Iogj»=:0.0125r,48. Then we find 31 from which gives log3/=1.8S97187. From this value of log M wo derive, by moans of Table VI., V =. 79° 55' 57".26. Finally, )* is found from (I r which gives eos'.'.v' log r== 0.1961120. For the case of hyperbolic motion, let there ha given t — T- 65.41236 days; ^ = 37° 35' 0".0, or h.gc = 0.1010188; and logd — 0.6020600, to find /• and v. First, we compute iV^from in which log^ = 9.6377843, and we obtain log xV-. 8.7859356; iV^= 0.06108514. The value of F must now be found from the equation iV = eA tan F — log tun (45° + ^ F). wlicrcni H = NUMKn'»:AT. KXAMJ'LHS. 10!) If we assumo F- 30°, u jnore appi'oxiniatc value may ho tlfrivcd from tan F. N ■{ lo^r tan (»()'' whi.-li ^'ives F, --- 28° 40' 23", an.l licncc .V, - 0.072G78. TIkmi we (ompiitc the coiTection to Ix; apjtiied to this value of F, hy means of till' o<|iuition LF, (N-N,)coi^'F, s, ?.U—v()hF,) whfTi'in .s ^ 2062fi4".8; and the result is e^F, -= 4.f)0J>7 ( iV ~ X, ) « =. — 3° 3' 43".0. Hence, for a seeoud approximation to the value of F, wc have F, = 25" 30' 40".0. The corresponding value of iVis A', = 0.0G176o3, and hence aF, = r).l!M)(iV^— N,)!> = — 12' 9".4. The tliird approximation, therefore, give.*; F, =^^25° 24' 30".6, and, repeating the operation, we get /'=-25°24'27".74. \vlii( li requires no further correction. To find r, wc have r = a I „ — 1 I , \ cos F J which gives log r = 0.2008544. Then, V is derived from I and we find tan ^v = cot 54 tan }^F, V = 07° 3' 0".0. M'hcn several places are required, it is convenient to compute v and /• by means of the equations 1/ r sm TiV = V^nie -\- 1) V^cos F l^aie — 1) sin \F, Vr cos Av = ^—^'-1 — cos hF. 110 TIIEORETICAIi ASTRONOMY. For tho {jivon values of o and c \\v. have log 1 ' a{c + 1) =^ 0.4782049, logl o{i' — 1) --^ 0.01(X)821I, and hcncf we derivo V = (57° 2' r){)".{>2, log}- = 0.2008545. It remains yet to illustrate the calculation of r and r for elliiJtio and hyperbolic orbits in which the eccentricity differs hut little t'ruu] unity. First, in the case of elliptic motion, let t -- T~ 68.25 day*; c — 0.9075212; and log q -^ 9.7008134. We compute M from M={t-T)^y^^~, wherein log CJ,= 9.9601277, which gives log 3/^=2.1404550. With this as argument wc get, from Table YI., F-= 101° 38' 3".74, and then with this value of Fas argument wc find, from Table IX., A ^ 1 540".08, B = 9".506, C -^ 0".002. 1 e Then wc have log i == ^og r--— = 8.217680, and from the equation V = F+ ^(lOOi) + 5(100/)'+ C(lOOi)', we get V = V+ 42' 22".28 -f 25".90 + 0".28 = 102° 20' 52".20. The value of r is then found from q(l-\-e) jt — — — t 1 -\- e cos v' namely, log J- = 0.1014051. We may also determine r and v by means of Table X. Thus, we first compute M from M= ^oStzU . v'7'o( l + 9e) 3 B Assuming ^ = 1, we get log M^= 2.13757, and, entering Table YI. with this as argument, we find ?<;= 101° 25'. Then we compute A from NUMERICAL EXAMPLES. Ill which f^ivos A ~ 0.024985. With tliis vahio of A as argument, we finni])uted the phu-es of a planet or eoiint for a series of dates e(|uidistant, we may readily interpolate tl e plaees for intennediate dates by the usual forinuhe for interpolatioii. Tliu interval between the dates for which the direct computation is made should also be small enough to permit us to nof^leet the ett'eet of tlie fourth ditferences in the process of interpolation. This, how(!ver, i.s not absolutely necessary, provided that a very extended series of places is to be computed, so that the higher orders of diflerences 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 + nto, and the corresponding value of the co-ordinate, or of tiie function, for which the interpolation is to be made, by /'(a -|" »«')• If we have coiujiuted the values of the function for the dates, or arguments, , 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 J\a) =/Ui) I. Diff. -B+C II. Dift: III. Diff. 2B QC f{a + 2u,) ^/( « )-\-2A-]-4B-\-8C ^'^'^^'^ '^ If we symbolize, generally, the difference /(a + mo) — f(a + (n — 1) o) by /' ())). (117) Hence, to interpolate the value of the function corresponding to a (late 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- iiK'tical mean of the second differences which are found on the same iiorizontal line as the two given values of the function. By extending the analytical process here indicated so as to include tlic fourth and fifth differences, the additional term to be added to qiuition (117) is found to be + rh (^ (f (« + (»* + 1) ">) +/" (« + nio))), ami the correction corresponding to this being applied, only sixth ditlbrences will be neglected. It is customary in the case of the comets \yhich 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 w, equal to four days, for the intermediate dates, we obtain a series of places at intervals of two days; and, finally, inter- 8 114 Til K(>RETICAL ASTRONOMY. polatiii;:; for tlio dates iiitcnncdiato to tli(>se, wp dorivo the places at intervals of one day. AViien a scries of places has been i-oinputed, the use of dill'ereiices will serve us a check upon the accuracy of tliu calculution, and will serve; to detect at once the place which is not correct, when any discrepancy is apparent. The jfrcatest discordanct; will Itc shown in the dillerences on the same horizontal line as thu erroneous value of the function; and the discovdance will he jfreatir and jfrcater 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 nuide of those quantities in whi<'h such an error is likely to occur. The ephenieridcs of the planets, to be used for the comparison of observations, are usually computed for a period of a few wirks before and after the time of opjutsition to the sun ; and the time of the opposition may be found in advance of tlie calculation of the entire ephcmeris. Thus, we find first the date for which the mean longitude of the ])lanet is ccjual to the longitude of the sun increased by 180°; then we compute the etpiation of the centre at this time by means of the e([uation (o.'>), using, in most cases, only the first term of the development, or V — J/='2e8in3/, e being expressed in sec'ds. Next, regarding this value as con- stant, we find the date for which L + equation of the centre is equal to the longitude of the sim 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 (/ — Sl) = tan ii cos i. Let these longitudes be denoted by I and I', the times to which they correspond by t and f, and the longitudes of the sun for the same times by © and O ' ; then for the time t^, for which the heliocentric longitudes of the planet and the earth are the same, we have or t,==t + t,=t'-\- Z— 180° O CO'- F O)~C/'-0 ■180°— O' it'-t), (118) the first of these equations being used when I — 180^ O is less ■ ^f ^^'^ P"t TIMK OK OPPOSIXrON. 115 tlinii /' ISO" - O'. If the tim(> /„ tliftors ronsiilcmbly from t or /', it may 1m' ii('<'('ssMrv, in order to obtain an acciinitc result, to repeat tlic hitter i)art of the calculation, usiM<; /„ for /, and taixiiijj: f at a siiinll interval from this, and so that the true time of opposition shall fall het\v(>en / and /'. The lonj^itndes of the pliiuet and of the sun must 1)(> measured from the sanii" eipiinox. When the eeeentricity is considerahle, it will facilitate the enleula- tidii to use two terms of eipiation (o.'')) in tinlicution, in reference to the homogeneity of th«> different t(>rms. If the ares are expressed hy an abstract nundier, or by the length of arc expressed in parts of the radius taken as the unit, to express thera in seconds we must multiply hy the nund)er 20()2()4.8 ; hut if the arcs are expressed in seconds, each term of the e(juation must contain only one concrete factor, the other concrete factors, if there he any, being reduced to abstract numbers by dividing each by « the number of seconds in an arc equal to the radius. 43. It is unnecessary to illustrate further the numerical application of the various formuhe which have been derived, since by reference to the formula) themselves the course of procedure is ob us. 1 1 may be remarked, liowever, 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 lojfurithms of the sum or difference of two numbers when the loga- rithms of these nnmbei"s are given, the calculation is abbreviated, and is often even more accurately performetl than by the aid of the auxiliary angles. The logarithm of the sum of two numbers may be found by means of the tables of common logarithms. Thus, we have If we put log(a + 6)-loga(l + *-)=log6(l+^). log tan X = 2 (log 6 — log a), no We sliiill Imvc or TIIKOIIKTUAL ASTRONOMY. log (a -\- b) = log a — 2 log coh r, log (n -\- b) —-■■ log b — 2 log win r. The first form is used wlion cosr is greater than sin. 7*, and the scef)ntl form when cnsr is less than sin.r. It should also l»e ol»s('rve errors of these; tahles shall have the least intluence, the value derived from the first e(|Uation is t<» he pre- ferred when cos(^. — ©) is greater than sin (^ — O), and that derived from tlu! seeojul equation when cos(^ — ©) is less than sin (A — ©). The value of J, if the greatest uecm'acy possible is reipiired, should bo derived from J cos ,9 when ^5 is less than 45°, and from J sin ,9 when ,i is greater than 4')°. In the application of munhers to equations (109), when the values of the second members have been computed, we first, by division, find tanjift' ! (o„) and tan ^ (ft' — u)„); then, if sin A (ft' + w„) is greater than cos J (ft' -f <«„), we find cos J/' from the first equation; but if sin \ (ft' + <«„) is less than cos J (ft ' + w,,), we find cos ^/' from the second eciuution. The same princi])lo is ai)plied in finding am It' by means of the third and fourth ecpiations. Finally, from sin \i' and cos Jt' wo get tan U', and hence t'. The check obtiiincd by the agreement of the values of sin Ji' and cos J/', with those computed from the value of i' derived from tan Jt'^ does not absolutely prove the calculation. This proof, however, may Ix; obtained by means of the eciuation sin «■' sin ft' = ain i sin ft, or by sin i' sin w„ = sin e sin ft . In all cases, care should be taken in determining the quadrant in which the angles sought arc 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 iunctious involved. DIFFEUENTIAL FOUMLL.E. 117 ClIAPTKU II. I>VF.«TIOATI()N OF TIIK PFFFKKENTIAL rORMULiK WHlrll EXPIIK'W TMK IIKI.ATIOM IIKTWKKN Till'. (iKOrKNTllK' UH IIKI.IOIKN lUlC ri.A( t>< (>F A IIKAVh'.NLY llODV AXK THE VAIIIATION (IK THE ELEMENTS OK ITH OUlllT. •It. In inaiiy calculiitioiirt n^latiiip; to the motion of a liciivcnly IxmIv, it becomes neeessjirv to (letcrmiiic tlie variations which small increments applied to tiie values of the elements of its orbit will pro- duce in its jfcocentric or heliocentric place. The torm, however, in which the j)rol)lem most frequently presents itself is t'lat in which :i|)pro.\iinate elements are to l)e correctcil by means of tiie (lilfercnccs b('twe( n the [)laces derived from computation and those derivetl from ul)servation. In this cxse it is rc(jnired to tind '\(i vnriatious of the clcnients such that they will cause the dilferences between cahnilation and oi)servation to vanish ; and, since there are six elements, it follows that - \ separate ecpiations, involvinj^ the variations of the elements a.s the unknown (piantities, must be formed. F^ach longitude or right asct.'usion, and eacli latitude or declination, derived from observation, will furnish one e»|uation ; and hence at least three (.'omplete 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 e(piations of condi- tion which are obtained must be reduced to six final e(iuations, from which, by elimination, the. corrections to be applied to the elements may be determined. If we suppose the corrections which must be ajipliod 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 whi(;h involve angular measure being ox|>rossed in parts of the radius as unity, the relations sought may be determined by differentiating the various formuhe which determitio the position of the body. Thus, if we represent by d any co-ordi- nate of the place of the body computed from the assumed elements of the orbit, we shall have, "i the case of an elliptic orbit. 118 THEORETICAL ASTRONOMY. il/(, being the moan anomaly at the epodi T. Let d' denote the vahie of this co-ordinate as derived dircctlj' or indirectly from observation; then, if we represent the variations of the elements by Ci.ir, aJJ, a/, &c., and if we suppose these variations to be so small that their squares and higher powers may be neglected, we shall have do + rfjr^^+ do dti. A.«. (1) The differential coefficients -,— , ,_-, &c. must now be derived from d- dSi 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 diiferential coefficients of the geocentric right ascension and dcqlination with respect to the elements of the orbit, these elements being referred to the ecliptic as the fun- damental plune. Ijet x, y, z be the heliocentric co-ordinates of the body ill reference to the equator, and we have or Hence we obtain =f(x, y, z), do = -.- dx + -,— dy -\- -,— dz. dx ' dy ^ dz dO^ do dx do dx' dr: dy dy do dz d-K dz diT (2) and similarly for tiiG differential coefficients of d with respect to tiie other elements. We must, therefore, find the partial differential co- efficients of d with respect to .r, //, and z, and then the partial differen- tial coefficients of these co-ordinates with respect to *hc elements. In the case of the right ascension we put ^ = a, and in the case of the declination we put d = d. 45. If we differentiate the equations a; -|- JT = J cos ^ cos o, y -{- Y= J cos '5 sin o, s + Z = J sin «J, regarding X, Y, and Z as constant, we find di '^.'o find the a; = r cos u y=:r cog u 3 = r cos u which give, dx dSi ---, and -,'- remain unchanged; and we have, also, dx -rr ^= — r sinitcosa, di dr dr dr di J dz ■ — rsinttcoso, -t- • r sin u cos c. (9) It is advisable, in order to avoid tlie use of two sets of formula?, in part, to regard the motion as direct and the inclination as susceptible of any value from 0° to 180°. If the elements which arc given are for retrograde motion, we take the supplement of i instead of /; and if we designate the longitude of the perihelion, when the motion is considered as being retrograde, by (j:), wo shall have If we introduce, as one of the elements of the orbit, the distance of the perihelion from the ascending node, we have and, hence, du = rfv + t?w, dx dx ^r A , \ -y- = -J- = X cot {A 4- u), dm dv dz_ du> dz dv dy dy d(o dv zcotiC-\- u). y cot(J5 + «), (10) 122 THEORETICAL ASTRONOMY. The values of --> -r'-, 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 givciu by (4), we may derive the differential coefficients of the geo- centric vight ascension and declination with respect to the elements SI, i, and - or co, and also with respect to r and v, by writing suc- cessively a and o in place of d, and ft, /, &c., in place of ~ in the equation (2). The quantities /• and r, however, are functions of the remaining elements and — r- have been d(p dtp dM^ dM^ d.'J. d/x found, the partial differential coefficients of the hel>'X!entric co-oi'di- nates with respect to the elements ,,, , / 2 . , \ . ■ ^y =: ^ dM 4- 1 [- tan f cos v I sm v dv. r' ycosv / ^ If we diflferentuate the equation r = a(l — ecosE), (12) we shall have dr= -da -{- ae sin JEJ dE — a cos ^ cos E d

nntl cos^ = t— r > wc shall have 1 -\-e cos V 1 + c cos u ac sin ^ sin v — a cos 99 cos E = ae cos v" sin' v a cos v" f cos c + c) 1 -\- e cos V 1 + e cos V which reduces to ae sir. E sin r — o cos ^ cos £ =^- — a cos ^c cos v Hence, the expression for dr becomes dr= - da -\- a tan ^ sin v dM — a cos f cos v (?^. a Further, we have T being the epoch for M'hich the mean anomaly is Mq, and kVl + m (14) M = ai Differentiating these expressions, we get dM= dM, + {t — T) dfi, da i ^1^ , a '' f^' and substituting these values in the expressions for dr and dv, we have, finally, dr = a tan ^ sin v dM^ -j- 1 a tan ^ sin v(t — T) — — \dii — a cos tp cos V dtp, (15) , a' cos V ,,,,«' cos ^ ,. mx 7 , / 2 , ^ \ . J dv = 7-^ dMa + , (^ — T) f?M + -f tan <» cos v sm v rf^'. r* " r^ \cos^ / From these equations for dr and dv we obtain the following values of the partial differential coefficients : — dr df dr dFL = — a cos If cos V, =: a tan v sin v, ,cos^ rfv a'cos^ dTL ^a ' tan p COS r isinr, (16) ^=atan^sin. («-r)-|r 206264.8, ^ = ^«-n a/t op. dfi r DIFFERENTIAL FORMULJE. 126 It will be observed that in the last term of the exi>res.sion for , we juivo 8ui)])ose(l n to be expressod in scco/ids of are, and hence the factor 20«j2()4.8 is introduced in order to render the equation homo- geneous. 47. The formula) already derived are .sufficient to find the varia- tions of the right ascension and declination corresponding to the variations of the elements in the case of the elliptic orbit of a planet; l)iit in the case of ellipses of great eccentricity, and also in the cases of parabolic and hyperbolic motion, these formuhe for the differential coetHcients require some modification, which we now proceed to develop. First, then, in the case of parabolic motion, sin ^ =^ 1, and instead of J/j, and n we shall introduce the elements T and q, the differential coeflicients relating to ;t, ^, and i iiaining unchanged from their form as already derived. U we differentiate the equation ^^tzp. _ g! (tan {V + \ tan* {v), regarding T, 5, and v as variable, we shall have -7= = 3 TT— da + ■'i'?^ sec* iv, or, since 7*^ = (f sec* \v^ MT ~ 7^— f^'Z + 2 "i dv. IV2 Q' 2,^ Multiplying through by — -. and reducing, we get dv = — (17) Instead of q, we may use log 5, and the equation will, therefore, oecome 2rU„ log?. (18) in which ?,q is the modulus of the system of logarithms. 126 TIIRORETICAL ASTRONOMY. If WO take the logarithms of both members of the e(iiiation and (lifforentiate, wo find cos' ., V dr = - dq + r tan \v dv. Introducing into this equation the value of dv from (17), we get 1 U (t—T) tan ^ I' \ , k V 2q tan \v dr dT. (19) Now, since — ' ■■ = q (tan ^v + itan'' Iv), and q = r cos^ ^v, wc have V 2q 1 2k (t — T) tan ^v _1 rW'lq (1 4- tan' \v — 3 sin' Av — sin' Av tan' \v) 1 cos y Wc also have k V'lq , kV 2q cos' U' tan iv ^sin v Therefore, equation (19) reduces to ^sinv dr = cos v «fl ,r_z- ft i. If we introduce d log (/ instead of dq, this equation becomes k sin v (20) , OCOSV ,, A-SMIV ^ dr = - -r a log O T-znr dT. ^0 ^^ V2q (21) From the equations (17), (18), (20), and (21), we derive dv _ _kV'2q df ~ /'~' rfy __U:(t — T) dq ~ '•V27 dr ^-si df ~ ~ V" dr dq = COS V, Jir_ rf log ~ qco^v _ ', m dv _ _ 3A(«--jr) v^. dlogq~ 2A,r' and then we have, for the differential coefficients of x with respect to T and q or log q, DIFFERENTIAL FOIlMULiE. 127 dx dx dr dx dv dx dT~ dr'dT^ dc'df d,i~ dx dr dx dv dr dq dv ' dq' dx dx dr ^ dx dlogq dr ' dlogq dr dv 'dlogq' and similarly for the ditrorential cot'tHcionts of^ // and z with respect to these elements. The ex|)ressioiis for the partial (liffereiitial eo- otHcieiits of X, y, and 2, res[)eetively, with resi)e<'t to r and v are the ■same as already found in the ease of elliptie motion. AV(t shall thns ohtnin the equations whieh express the relation between the variations of the geoeentrie ])Iaees of a eoniet and tlu; variation of the parabolic elements of its orbit, and which may be employed cither to correct the approxiniate elements by means of e<[uations of condition fur- nished by comparison of the computed j)lace with the observed place, or to determine the change in the geocentric I'ight ascension and (Iwliiiatiou corresponding to given increments assigned to the ele- ments. 4(S. We may also, in the case of an elliptic orbit, introduce T, q, and e instead of the elements (p, M^, and [t. If we differentiate the expression 5 = a (1 — e), we shall have da We have, also, -dq -{- -de, q q in which Tis the time of perihelion passage, and dM= — ifcV^r+m a- 1 dT— pv^l + m ar'i{t— T) da, Plenee we derive kVl -f- m a~ i kvl + mfrs (« — T) de. (t—T)dq Suhstituting this value of d3I in equation (12), replacing sin^ by e, and reducing, we get dv = = -^J^lPSJL±^dT--i^-^^^kt-T)dq qr _^^ jtV,-a+.n) „_r)_(g + i)„„,)_J_,,. (23) 128 TIIKOnETICAL ASTUOXOMY. In a similar manner, hv snhstituting the values of da and (DI in equation (14), and reducing, we find dr =-- -^- — e emv dT kl^l-\-m(t— T) + I />(** — cos vj— Ul/pil +m) a ^-_|_e8in.)r/7 T) esin v\ 1 )l — e^ de. (24) These e(jnations, (23) and (24), will furnish the expressions for the ^. , ,,„. ^. , «... dv dv dv dr dr , dr , . , partial (litierential coeihcients 777,. -:-> ^-» TTi,''-^'' Jvntl -:-» wluch arc '■ dT dq de dT dq de required in finding the differential coefficients of the heliocentric co- ordinates with respect to the elements T, q, anti e, these quantities being substituted for iJ/„, n, and f, respectively, in the equations (11). 49. When the orbit is a hyperbola, wc introduce, in place of J!/„, fi, and (f, the elements T, <], and a^. If we diflerentiate the equation we shall have iVo = e tan F — log, tan (45° + i^F), dN„ [cosF ^j dF cosF ^ -|- tan i^ de, which is easily transformed into ,-_ r dF , , „ tan4 ,, dK = f^ + tan F- — -- d-^, " a cosii' ' COS+ ' or dF sin F a ,„ a tan^< r tan F " r cos ^z Let us now take the logarithms of both members of the equation tan -\F= tan ^v tan l^*, and differentiate, and we shall have dv = sin t) smv dF sin F sin •4- dF Introducing into this equation the value of — . — ^ already found, we get , asm'y ,,-. lasmv r tan i< " \ r taniV sm COS ■^. sin 4 Ih- DIFFERENTIAL FORMULA. J 29 But, slncn r sin v = a tan i// tan F, and p ^- a tan^ \//, this reduces to If wc differentiate the equation \ cos F I (25) wo get dr = - rfa + ae tan' F-~. -^ -\ ^^ • rt+. a sniF coaF cos + Substituting in this equation the value of ~^~iriy we obtain , r , . a^e tan F , -^ dr = - rfa + ■ ((taM» + 2^). (33) dv Substituting for -.— its value from (31), and reducing, we get dr de g h{i—T) . , , , , , V2q (34) dv dv The equations (31) and (34) fuinish the values of -j- and -j- to be used in forming the expressions for the variation of the place of the body when the parabolic eccentricity is changed to the value 1 + de. When the eccentricity to which the increment is assigned differs but dv little from unity, we may compute the value of -- directly from (XC equation (30). A still closer approximation would be obtained by dv using an additional term of (29) in finding the expression for j- ; but a more convenient formula ma}' be derived, of which the numerical application is facilitated by the use of Table IX. Thus, if we difter- cntiate the eqrucion v^V+A (1000 + B {lOOiy + C(lOOi)', regarding the coefficients A, B, and C as constant, and introducing the value of i in terms of c, we have de dV de ' 2Q0A 4005 ,,„.., 600 C ,,.„.,, r, (loot) — -pf-j^^, (looiy, s(l-fe)- Hl + e) s{l-\-er in which s — 206264.8, the values of A, B, and C, as derived from (IV the table, being expressed in seconds. To find -7-, we have k{ t—T)V\ + e 29! = tan^F+itan='^F, M'hicli gives, by differentiation, k(t-T) de dV 2qi Vl + e cos*'.F' and if we introduce the expression for the value of M us ed a.s the argument in finding V by means of Table VI., the result is Hence we dv _Mc(. rfe~750 by means When t that the t liie express be derived first of the regarded ai If we tak( differentijit To find th( sufficie'it a which give The equati gives and hence Snbstitutiii dv DIFFERENTIAL FORMULA-, Mcos*yV 133 Hence we have dV de 7o(l + e)- dv 3/ cos* A F 200^ 4005 ,,„^., 600C ,,^^.„ ,^^, -=■--- / — -prT~vi 7-, , — ^^7.(1000 -p-, .,(1000', (35) de 75(1 + 6) s(l + .'0 s(l4-e)^^ ^ 8(1 -f e)' /> v / by means of which the \ahie of — is readil" found. de When the eccentricity differs so much from that of the jiarabola that the terms of the last equation are not sufficiently convergent, liie expression for — , which will furnish the required accuracy, may be derived from the equations (75)i and (76)i. If we differentiate the first of these equations with respect to e, since B may evidently be regarded as constant, we get dw ,. k(t — T) cos^Aiw ,„„. ^ (36) de lO l/2qi i?i/,>^(l + 9e) If we take the logarithms of both members of equation (76)i, and differentiate, we get dv dC dw i?in y C sinu' 4de (l + r)(l + 9e) (37) To find the differential coefficient of C wi;h respect to e, it will be sufficie.'it CO take _ 1 4 J wiiich gives The equation gives dA=- and hence wo obtain dC_ G~ -^=lC^dA. . 5(1 -e)^ ,, ^^(iT9e)*^''^^*' (l4-9e)' ^ tan ^i« COS' J, to ^;] + 9tT7 **^" \wdeAr\ • dto. sm w Substituting this value in equation (.'^7), we get dv de (l+de) sin V tan' ^w + 4 sin V C sin V dw ~8hi7 = (1 + ^A) cos^w, l] = (1 + p) cos* ^iv = Ccos* iw. 3/ cos' Xw 75 tan itv ' C^sinv- k(t — T)\/ P 4 0\ln- c)- NUMERICAL EXAMPLES. 135 If we substitute this value in equation (39), and put C" (1 + e) = 2, wc get de 9 kVp (.t-T)-: 8 tan J,v 2(1 + 9e) r' '" ^' (l + e){l + 9e)' and when e =- 1, this becomes identical with equation (31). (40) 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 18G5 Feb- ruary 24.5 mean time at Washington, the differential coolHcients of the right ascension and declination nf the planet Eartjnomc @ with respect to the elements of its orbit, using tiie data and results given in Art. 41. Thus we have a = 181° 8' 29".29, .5 = — 4° 42' 21".56, log J = 0.2450054, logr = 0.4'28285, v = 129° 3' 50".5, % = 326° 41' 40".l, A = '.^96° 39 5".0, B = 205° 55' 27".l, C= 21 2° 32' 17".7, log sin a =. 9.999716, log sin 6 =^ 9.974825, log sin c = 9.522219, log X = 0.425066„, log y = 9.51 1920, log s = 8.077315, £ = 23° 27' 24".0, t—T= 420.714018. First, by means of the equations (4), we compute the following values : — da log cos '-r> and -v., those of cos a, cosi, finil cos c may gcncmliy be obtained with sufficient accuracy from sin a, sin 6, and sine. Their algebraic signs, however, must be strictly attended to. The quantities sin a, sin 6, and sin c are always positive; and the algebraic signs of cos a, cos 6, and cose are indicated at once by the equations (lOl),, 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 ^- and -j-> we have, according to equa- tion (2), da , da dx , . da dii 4- cos 5 -- ^ C0S5-T- = C08- , , a:: ax an which give d8 dn dS dx . a — i— dx dn dy dr: d8 dy dS dz (41) .da . da cos «^— = COS 0-7- rfrr dv + 1.42345, dy dtc d^_dd dn dv + dz dn' = — 0.48900. In the case of Q, i, and r, we write these quantities successively in pljice of re in the equations (41), and hence we derive cos I cos^ da 'da da di , da COS r— = dr = — 0.03845, ^ — 0.27641, 0.08020, dd da dd di dS dr = — 0.09533, = — 0.78993, = + 0.04873. Next, from (16), we compute the following values :- log ^ = 0.179155, d

-7-, and -ry '' dr dv dr dv in connection with the numerical values last found, according to the NUMERICAL EXAMPLES. 137 equations wliich result from the analytical substitution of the expres- - dx dy dz „ . .- /-,\ .^. • i ^r sioiis for -7-. -r-> -j-t okc., in equation (2), writing successively ^, M^, and fJt in place of z. Thus, we have .da, , da dr , .dadv cos - — = cos -— • \- cos —■ J— d^ dr dtp dv d(p dS dS dr da dv d and -r- are de<-erMined as av dr dv dr already exemplified. If we introduce the elements T, q, and c, we shall have .da ^ da dr , . da dv cos5^-j, = cos^-^--.^y + cos^^-.^ dS^ dT dd_ dr dT'^ dv dv^ dt' (43) into account. and similarly for the differential coefficients with respect to q and e. H the mass beins nu>ip:rical examples. 139 ^, , « 1 , . , , ^ (^f (ft' (^>' f''' (ft" 1 (Jf The mode oi calculating the values oi -r™, -r^, -,-' :r' T' ^u*^' t" " ar (IT dq aq cle de (lepeiids on the natux'c of the orbit. In the case of passhijij from one system of parabolic elements to anotiier system of parabolic elements, the coefficients of ac vanish. To ilhistrate the calculation of -rph' "Tm' &c. hi the case of parabolic dT dT '■ motinii, let us resume the values t — 7^= 75.364 days, and log«^ = l).l)()5048G, from which wc have found logr ^ 0.1961120, v = 79° 55' 57".26. Then, by means of the equations (22), we find dr log~ = 8.095802„. log~=7.976397„, dr '^dq dv log ^^=:. 9.242547, log ^ = 0.064602. ^dq If, instead of dq, we introduce d log q, we shall have log dr d log q 9.569812, log , f^ -= 0.391867, . ° d log q " From these, by means of (4.3), we obtain the dilferential coefficients of a and d with respect to Tand q or logf/. The same values are also used when the variation of the parabolic eccentricity is taken into account. But in this case we compute also j- from equation (31) and ^ from (33) or (34), which give, for v = 79° 55' 57".3, de log|==8.147367„, log ^-==9.726869. de In the case of very eccentric orbits, the values of -r^, -7^, &c. are found from dT dT dT'~ ih_ dq'' kV P dT V]} esmv. (44) qr" dq q "^ qyp dr r , r'esinv dv dq q p ' dq the mass being neglected. 140 THEORETICAL ASTRONOMY. To illustrate the application of these formula?, let us resume the values, <—y- 68.25 days, c ^-- 0.9675212, and log*/ =^ 9.7668134, from wliieh we have found (Art. 41) V = 102" 20' 52".20, Hence we derive and log|^-7.943137„, log ^ = 0.186517., log r = 0.1614052. log J) = 0.0607328, log dT' dr 8.180711 , log 3^ = 0.186517, ^ dq If we wish to obtain the diiferential coefficients of v and r with respect to log 5 instead of 5, we have dv dlogq q dv ^ " dq' dr q dr d\ogq~\' dq in which }.q is the modulus of the system of logarithms. Then we compute the value of -7- by means of the equation (i (35), (39), or (40). The correct value as derived from (39) is dv de 0.24289. i The values derived from (35), omitting the last term, from (40) and from (30), are, respectively, — 0.24440, — 0.24291, and — 0.23531. The close agreement of the value derived from (40) with the correct value is accidental, and arises from the particular value of v, which is here such as to make the assumptions, according to which equation (40) is derived from (39), almost exact. dr Finally, the value of y may be found by means of (32), which gives ^ = + 0.70855. de When, in addition to the differential coefficients which depend on the elements T, q, and e, those which depend on the position of the orbit in space have been found, the expressions for the variation of the geocentric right ascension and declination become NUMERICAL EXAMPLES. 141 COS ') Att = COS'> -- ATT 4- COS - — A SI + COS ') -.- Al -f COS ')" - _, A T dr. d^ di dl .da da 4- cos O -r- AO + COS '> T- Ae da ^ de If we introduce log q instead of q, the terms containing q become respectively cos5-ri — -aIoko and ,, aIosjo. It should be ' -^ dlogq *' -' rf log 7 observed that if at, aQ,, and a/ are expressed in seconds, in order that these equations may be homogeneous, the terms containing a 7', dq, and Ae must be multiplied by 206264.8; but if at, aJJ, and Ai are expressed in parts of the radius as unity, the resulting values of cos Att and a5 must be multiplied by 206264.8 in order to express tlicin in i^'econds of arc. The most general application of the equations for cos d Aa and a5 in terms of the variations of the elements is for the cases in which the values of cos 8 Aa and of A«5 are already known by comparison of the computed place of the body with the observed place, and in which it is I'equired to find the values of A?r, aSJ, aj, etc., which, being applied to the elements, will make the computed and the observed places agree. When the variations of all the elements of the orbit are taken into account, at least six equations thus derived are necessary, and, if more than six equations are employed, they must first be reduced to six final equations, from which, by elimina- tion, the values of the unknown quantities a;:, aS^, &c. may be found. In all such cases, the values of Aa and a5, as derived from the comparison of the computed with the observed place, are ex- pressed in seconds of arc; and if the elements involved are expressed in seconds of are, the coefficients of the several terms of the equations must be abstract numbers. But if some of the elements are not expressed in seconds, as in the case of T, q, and e, the equations formed must be rendered homogeneous. For this purpose Ave nud- tiply the coefficients of the variations of those elements which are not expressed in seconds of arc by 206264.8. Further, it is gene- rally inconvenient to express the variations aT, A7, and ac in parts of the units of T, q, and e, respectively ; and, to avoid this incon- venience, we may express these variations in terms of certain parts of the actual units. Thus, in the case of T, we may adopt as the unit of A 2' the «th part of a mean solar day, and the coefficients of the terms of the equations for cos d Att and a5 which involve aT 142 THEORETICAL ASTRONOMY. must evidently bo divided by n. In the same manner, it appears that if we adopt as the unit of A7 the unit of the »«th (r '•iniiil place of its value expressed in j)arts of tin? unit of 7, wc must divide its eoetticient by 10'", and similarly in the case of Ac, so that the equations beeomc cos O Ao = COS " — At: 4- COS o -— A Q + cos o —^ Al + - cos « -, ,„ A 1 ih (IQ, ' (/t ' n clT 8 .fla , « ,(/<* + i-.r COS rt ~ Ar/ + - — ; cos o —- Ac, A5 = — A?r -^ A O 4 Ai + - . — A r -^ Afl (40) in which s = 206264.8. When lofj q is introduced in place of q, the coetfieients of A lojjj q arc multiplied by the same factor as in the eas^e of Aq, the unit of a \ogq being the unit of the mth decimal place of the logarithms. The equations are thus rendered homogeneous, and also convenient for the numerical solution in finding the values of the unknown quantities att, aJ2, a?, aT, &c. When aT, a«/, and Ae have been found by moans of the equations thus formed, tlie AT Aft coirections to be applied to the corresponding elements are n IT and Ae 10^ In the same manner, we may adopt as the unknown quantity, instead of the actual variation of any one of the elements of the orbit, n times that variation, in which case its coefficient iu the equations must be divided by w. The value of Aa, derived by taking the difference between the computed and the observed place, is affected by the uncertainty necessarily incident to the determination of a by observation. The unavoidable error of observation being supposed the same in the case of a as in the case of d, when expressed in parts of the same unit, it is evident that an error of a given magnitude will produce a greater apparent error in a than in o, since in the case of a it is measured on a small circle, of which the radius is cos d ; and hence, in order that the difference between computation and observation in a and d may have the same influence in the determination of the corrections to be applied to the elements, we introduce cos Aa instead of Aa. The same principle is applied in the case of the longitude and of all corresponding spherical co-ordinates. DIFFEUENTIAL FORMULAE. 143 62. The formiilie nlrondy given will determine also tlie variations of the geocentric longitude and latitude corresponding to small in- oroiuents assigned to the elements of the orbit of a heavenly body. Ill this case we put e -- 0, and compute the values of A, 11, siurt, and sin/>» by means of the equations (J>4)i. We have also ('- 0, .*iii (• - sin /, and, in place of a and o, respectively, we write / and /5. But when the elements are referred to the same fundamental plane as the geocentric phuies of the body, the formulie which depend on the position of the plane of the orbit may be put in a form which is more convenient for numerical api)lic!ation. If we ditferentiate the etpiations we obtain x' =^r cos it cos Q, — r sin u sin Q, cos i, •if z=r cos « sin ft + »• sin u cos ft cos i, / = r sin tt sin i, X dx'^-dr r rfsiu u COS ft -f" coau sin ft cos O^^^ — r(cos n sin ft + sin u cos ft cos i) dSl -\-r sin u sin ft sin i di, v' dy' = — dr — r (sin ?t sin ft — cos u cos ft cos du ■\- j*(co3 u cos ft — sin n sin ft cos i) dQ, — r sin « cos ft sin i di, (46) z' dz' = - dr -{- r cos u sin i du + r sin u cos i di, in which x', y', z' are the heliocentric co-ordinates of the body in reference to the ecliptic, the positive axis of x being directed to the vernal equinox. Let us now suppose the place of the body to be referred to a system of co-ordinates in which the ecliptic remains as the plane of xxj, but in which the positive axi3 of x is directed to the point whose longitude is ft ; then we shall have dx = d^ cos ft + d^ sin ft , rfy = — t/^' sin ft + d^ cos ft , dz = (?3', and the preceding equations give dx^=-dr — r sinu du — r ainu cosi dft, r dy = ^ dr -{- r cos u coai du -{- r coau d^l — r "in u sin i di, (47) life dz = -dr -\-r coau sin idu -\-r sin u cos i di. r 144 TIIEORETK'AT. ASTItONOMY. This traiisfornintion, it will he «)1)s(M'V('i1, is ('f|uiv{ilcnt to (liminisliini^ the lonjiitiidort in the oqimtious (4G) by the angle Q tiirough which tlie axis of x has boon moved. Let X„ Y,y %, denote the heliocentric co-ordinates of the earth referred to the sanuj system of co-ordinates, and we liavo X -\- X, = J cos li cos (i — SI), y+ r,= Jcos/9 8in(A— Q), 2 -f Z, = J sin /9, in whicli ?. is the geocentric longitude and /9 the geocentric latitude. In ditferentiating these equations so as to find the relation between the variations of the heliocentric co-ordinates and the geocentric lon- gitude and latitude, we must regard ft as constant, since it indicates here the position of the axis of x in reference to the vernal equinox, and this position is supposed to be fixed. Therefore, we shall have r?.f = cos/5 cosU — Sl)3 = . sin,? cos (A — ft) dx- dy, sin/JsinfA— ft) , cos/5 , -. dy+ . dz. These equations give .dX cos /9 -,— = ax sin (A — ft) COS/? dk cos (A— ft) dy cos /? , = 0, dz dlS dx dl dy P. dz sin/?cos(A — ft) J ' sin/? sin (A — ft) C08/5 > (48) If we introduce the distance to between the ascending node and the place of the perihelion as one of the elements of the orbit, we have dii. = dv -\- d(a, and the equations (47) give dx x -r = _ = cos U, dr r dx dx dy y . — , — = - = sin M cos I, dr r dy dy dz dr dz dz :_ = sm« sm i; r , -= , == — rsmn, - ,"-= / =-r cos w cost, -j— =-y-=rcos«sini; dv duj dv dto dv aw DIFFERENTIAL FORMULAE. (Ix . . (hi rfft 'da j'-' =0 dy ... ■ ,. - = — r Sinn sin », di da dz di =--:0; 145 (49) - r sm n cos i. If wc introduce r, the longitude of the perihelion, wo have du = dv -\- dir — da, and hcnro the expressions for the partial dift'orontial eoofficionts of the heliocentric eo-ortUnatcs with respec^t to z and a become dx dx r sin u, 3^L di: dy r cos u cos t, 2r sin u sin' ^i, ' -— 2r cos u sin' J «, dn dz da — , — = r cos M sin i ; •^ (50) = — r cos K sin i. When the direct inclination exceeds 90° and the motion is rej];arded a.s being retrograde, we find, by making the necessary distinctions in regard to the algebrai(! signs in the general equations, dx di r = 0, di r sin tt sin i. dz di — r sin M cost; (51) and the expressions for -y^, -j-, -r^T' 'i~' ^^' ^^^ derived directly from (49) by writing 180° — i in place of i. If we introduce the longitude of the perihelion, we have, in this case, and hence dx — - = r sin u, dx du === rfv — di: + da> dn dy ,— - = r cos « cos I, — ,— = — r cos v, sm i ; ^'^ ^'^ (52) r cos w sin i. dz dn dz But^ to prevent confusion and the necessity of using so many for- miihe, it is best to regard i as admitting any value from 0° to 180°, and to transform the elements which are given with the distinction of retrograde motion into those of the general ease by taking 180° — i iii«itead of i, and 2a — 7t instead of -, the other elements remaining the same in both cases. 53, The equations already derived enable us to form those for the differential coefficients of X and j3 with respect to r, v. a , i, and w or ~, ])y writing successively k and /9 in place of d, and a, i, &c. in 10 14G THEORETICAL ASTRONOMY. pl.'U'O of TT ill eqiiiation (2). The expressions for the differential coefii- t'ients of r and c, with respect to the elements which determine the form of the orl)it and the position o^ the body in its orbit, being independent of the position of the plane of the orbit, arc the same as those already given; and hence, according to (42) and (43). we may derive the value ■> of the partial ditterential coeiHcients of I and ^5 with respect to ihese elements. The numerical application, however, is facilitated by the introduction of certain auxiliary quantities. Thus, if wc substitute the values given by (48) and (49) in the equations dv .dX ^ dk dx , .dk cos p -J- =: cos p -7- • -, - 4- cos p-y dv dx dv dy and put dv d,3 dx dx d,3 dv dy di d,3 dv ^ rfz rf2 dv' COS i cos {X — Q) =^ Ag sin A, sin (A — SI) — cos. H^a i ■■ n sm — sin (A — Q) cos i -- n cos N, in which Aq and n are always positive, they become (53) dX dX cos /?—-=: cos /? -p dv dv d3 ■ An sin (A -I - u), 1 ■ — (sin /3 cos (A — £1) siu n -\- n cos u sin (iV -}- ,3) ). Let us also put and wc have n i^'m (N -\- ft) ---^^ Ba !iin B, sin fi cos (A — J^ ) -— B,. cos B, cos/3 dX dv dfi COS ,3 -- ^.-- - Ao sm (^1 + u), dv- 1=3^-^^-^--)- (54) The expressions foK co? ft-j- and -,'- give, by means of the sppie auxiliary (piantities, .dk cos p -,- dr --- cos (xl -f «), -j- = V cos {B 4- ^l). dr J m In the same manner, if we put DIFFERENTIAL FORMULJS. 147 we obtain cos (X — SI) = Co sin C, cos i sin (A — ^) =^ C\ cos C; cos i ^= Z>„ sin J), sin (A — J^) sin i=^ D^ cos X); cos /3 -^ _ = ^ Co sin ( C + «), ^ ' - = — ^ J„ sin ,5 cos {A + it) ; dX r . . cos /? —J— = — -T sin i sin u cos (A — ^ ), — — - = - n sin « sin (D + /?). di r ° If we substitute the expressions (55) and (56) in the equations „ di. . dX dr , . dk dv cos /? -.— -— cos li -y~ • -T — \- cos fl -, ,— , d(p dr d

cos v 1 (59) /• sni V, wc get cos ii~=.l- Ag sin ( J. + JF + ii), lu a similar manner, if wc put dr . „ . — 1 nf = <7 sm (r = — a tan «> sm v, dAL '' (60) dv „ a' cos «> = /i sin ir.-= — I a tan ?> sin v(e — T) — ,J 206264.8 j, dr ~ dji dv , rr «* COS ^ ., ™. ,. .. — fi eos JET = (t — T), dft r (61) 148 we obtain THEORETICAL ASTRONOMY. COS /? j^jT =^ I Ao sin (^ + G + iO, ^ = ^-B„8in(5+G + «); cos/3 -^— = -J ^0 sin (^ + ir+ i{)i -^ = ^-J5oSin(£+£r+u). (62) The quadrants in which the auxiliary angles must be taken are determined by the condition that Ag, Bq, Cj,,/, g, and h are always positive. 54. If the elements T, q, and e are introduced in place of ilf,,, //, and (f, we must put g sin C = — h sin 1^= — and the equations become dk de' dr df' dr ff cos G^^r h cos jff = r dv dv^ dq' (63) d,3 de cos /9 ^ = ^ Jo sin (A-{- F-\- iC), B^^m^B-\-F-\-xC)', (64) cos l^^i=~i A sin ( J + G + it). f^,= ^B,,sm(B+G + u); cos /9 -7- =: — ^0 sin (J. + iT^- w), -^=.-^J5oSin(jB + ir+«). In the numerical application of these formulre, the values of the second members of the equations (63) are found as already exem- plified fur 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 ditfcrential coefficients of X and j3 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 supposetl to be invariable, the terms involving e vanish. Further, in the case of parabolic elements, we have which give fj COS G = r^^, ^sinv r tan Iv dv dl" tan G = — tan ^v. Hence there results G = 180° — ^v, and „ and J) n\ust '■ found from the last two of equations (57), using the given value liroctly; and then we sliall have r. dX r . . . ,. ^ ^ cos /? -y. =^ -7 sm t sm u cos (/ — JJ ), - , r == T Da sm It sm {D + ,3). d% J " (66j 55. Examples. — The equations thus derived for the diflercniial coefficients of X and /5 with respect to the elements of the orbit, rpferred to the ecliptic as the fundamental plane, are applicable when any other plane is taken as the fundamental plane, if we consider ) and /9 as having the sam<' signification in reference to the new plane that they have in reference to the ecliptic, the longitudes, however, being measured from the place of the descending node of this plane on the ecliptic. To illustrate their numerical application, let it be miuire + 1.1300 Ail/„ +. 507.25 A//, A„ cos i — cos SI sin to^, , smw , , smwj , dm = -r .-° cos SI dSl '. — . cos I di. " sm Si sm t (70) The equations (68), (69), and (70) give the partial differential co- efficients of SI J h and Wq with respect to SI ' and i', and if wc 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 8uiw„ ^ ^, smw . ., Aw. = -,- -^» cos ft A J2 . ".- cos % hi', ^ i^in SI ^ , sin «>. ,, sm SI sm % ^i = sin Wj sin i' A ft ' -(- cos w^ Ai', Aw =:= Aw' — Aw.. (71) If we apply these formula} to the case of Eurynome, the I'esult is Aw„ = — 4.420Aft' + 6.665 Ai', Aft == — 3.488a ft' + 6.686aj:', Ai = — 0.179a ft' — 0.843Ai'; DIFFERENTIAL FORMULA. 153 and if we assign the values Aft' = - 14".12, Ai' -^ — 8".86, aw' = - 6".64, we get Aw„ = + 3".36, A ft = — 10".0, Ai = + 10".0, Aw ^ — 10".0, and, hence, the elements which determine the position of the orbit in reibrence to the ecliptic. The elements to', ft', and i' wvAy also be changed into those for which the ecliptic is the fundamental plane, by means of e(iuations wliich may be derived from (109), by interchanging ft and ft' and 180° — i' and i. 5G. If we refer the geocentric places of vhe body to a plane whose inclination to the plane of the ecliptic is i, and tlie longitude of whose ascending node on the ecliptic is ft, — which is equivalent to taking the plane of the oi*bit corresponding to the unchanged elements as the fundamental plane, — the equations are 4ill further simplified. Let x', y', z' be the heliocentric co-ordinates of the body referred to a system of co-ordinates for which the plane of ihe unchanged orbit is the plane of xy, the positive axis of x being directed to the as- cending node of this plane on the ecliptic; and let x, y, z be the heliocentric co-ordinates referred to a system in which the plane of xy is the plane of the ecliptic, the positive axis of x being directed to the point whose longitude is ft . Then we shall have dx' == dx, dtf = dy cos i -\- dz sin ", dz' = — dy sin i -(- dz cos i. Substituting for dx, dy, and fh their values given by the equations (47), we get dx' =: — di' — r sin u du r r sinu cosi c?ft, .?/ dy' = '— dr -f- *' cos ii du -f r cos ti cos i Jft , dz' = - dr r r cos M sin i c?ft -\- r sin u di. It will be observed that we have, so long as the elements remain unchanged, r cos u, i/ = r aiii u, z' = 0, 154 THEORETICAL ASTRONOMY. and henco, omitting the accents, so tlmt x, y, z w\\\ refer to the plane of the unchanged orbit as the plane of xi/, the preceding equatioiia give dx = cos it dr — r sin u du — r sin ri cos i dSl, dij = sin It dr -\- r cos u du -f- f cos a cos i dSl, dz = — r cos u sin i dSl + r sin w di. The value of w is subject to two distinct changes, the one arising from the variation of the position of the orbit in its own plane, anil 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 cclij)tic, we shall desig- nate by j siu^', z -\- Z -^ J sin rj, in which X, Y, Zare the geocentric co-ordinates of the sun referred to the same system of co-ordinates as x, y, and z. These equations give, by ditt'orentiation, dx = cos Tj cosO (iJ — J sin ij cos drj — J cos rj sin dO, dy == cos Tj sin dJ — J sin ij sin drj -{- J cos )j cos dO, dz =s\nr) dJ -}- ii cos rj drj ; and hence we obtain cos Tj do z= — drj = — sin — u — II). If we expi'ess r and v in terms of the elements T, ? COS = COH (5 COS (X — (J ), co.s r^ h'\uO z^ n cos ( ^V — i), sin )j = n sin ( ^ — i), (83) from wliich rj and d may be readily found. If wc also put SI), n' sin N' =^ cos i, n' cos iV' = sin i sin (A we shall have cot iV = tan t sin (A — Q,), cos iV' ^ ,. ^ . (84) (85) If ;- is snail, it may be found from the equation sin i cos (A — JJ) sin/- COS); (86) The quadrants in which the angles sought must be taken, are easily determined by the relations of the quantities involved j 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 gel sin (45° — Iri) sin (45° - ^{0 + y)) = cos (45° + ^ (>l — a)) sin (45° — A (/? + i)\ sin (45° — li) cos (45° — ^ (fl + ;-)) = sin (45° + 4 (A — SJ)) sin (45° - ^ (/? ~ i)\ cos (45° — Aij) sin (45° — ^^{0 — y)) = (87) cos (45° + U^ — S^)) cos (45° — Uji + r)). cos (45° — irj) cos (45° — -^{0 — y)) ^ sin (45° + ^ (A — J^ )) cos (45° - ^ (/5 - 0), from w'hich we may derive rj, d, and y. When the problem is to determine the corrections to be applied to the elements of the orbit of a heavenly body, in order to satisfy given observed places, it is necessary to find the expressions for cos;y Ad and ayj in terms of cos/9 £^?< and a/9. If we diiferentiate the first and second of equations (80), regarding SI and i (which here determine the position of the fundamental plane adopted) as con- stant, eliminate the terms containing dr^ from the resulting equations, and reduce by means of the relations of the parts of the spherical triangle, we get NUMEHICAI< EXAMPLE. 159 COS yj do = cos y cos /? dX -{■ sin y dii. Dillcrciitiatiiig the last of tHumtions (80), nml reducing, wo find dfj = — siii Y co.s I'i dX -\- cos y d,3. The equations thus derived give the vuhies of the differential co- (flicients of d and r^ with respect to X and ,9; and if the ditferencea U and A;9 arc small, we shall have cos rj aOz= cos y cos /? aA -f- sin y ^j3, A)j = — sin / cos /S aA -f cos y A/?. (88) The value of y reciuirod in the application of nuuihers to these c(|U!iti()ns may generally be derived with sufficient accuracy from (86), the algebraic sign of cos^ being indicated by the second of equations (81); and the values of rj and d required in the calculation of the diifercntial coefficients of these quantities with respect to the elements of the orbit, need not be determined with extreme accuracy. 58. Example. — Since the si)hcrical co-ordinates which are fur- uisliod directly by observation arc the right ascension and declina- tion, the formulro will be most frequently required in the form for tiiuling ;y and d from a and u. For this purpose, it is only necessary to write a and d in place of A and fi, respectively, and also SI', i', w', '/', and w' in place of SI, i, j --,--- ^-^- -f 2.0978, eoiV // do d

^ r=. _ 4 sin ri sin {0 — u' — F), d

j Afl = + 8".24, Ai; z^ — 6".96. With tlic same values of aw', a^', &c., we have already found cos S Aa = + 5".47, A<5 ^ — 9".29, which, by means of the equations (88), writing a and 8 in place of ). and ,?, give COS1jAo tau

is also a function of f, the coefficients of Ap are changed ; and if we denote by cos o I y- I and I ,— I the values of the partial differential coefficients when the element fjt is used in con- nection with tp, wo shall have, for the case under consideration, .do. cos o 1— d

■' means of the expressions which may be derived by substituting in the equations (15) the value of d/z given by (90), and then we may compute directly the values of cos 5-^—, cos 8 -t-. , -t-> and -n — -• * "^ dip d logp dip d log;) In place of 31^, it is often convenient to introduce L^, the mean longitude for the epoch; and since we have dL^ = dM^ -\-dx=^ dM„ + d'o-\- dSl, and, when ;f is used, dLg = dMf, -\- dx-\- {l— cos i) r? ft . Instead of the elements ft and i which indicate the position of the plane of the orbit, we may use b = sin % sin ft, c = sini cos ft, and the expressions for the relations between the differentials of h and e and those of i and ft are easily derived. The cosines of the angles which the line of apsides or any other line in the orbit makes with the three co-ordinate axes, may also be taken as elements of the DIFFERENTIAL FORMULAE. 163 orbit in the formation of the equations for tlie variation of the geo- centric phice. 60. The equations (48), by writing I and h in pUice of X and /9, respectively, will give tiie values of the differential coefficients of the heliocentric longitude and latitude with respect to x, y, and z. Combining these with the ex})ressions for the differential coefficients of the heliocentric co-ordinates with respect to the elements of tiie orbit, we obtain the values of cos b \l and a6 in terms of the varia- tions of the elements. The equations for dx, dy, and dz in terms of du, dSl, and di, may also be used to determine the corrections to be applied to the co-or- dinates in order to reduce them from the ecliptic and mean equinox of one epoch to those of another, or to the apparent equinox of the date. In this case, we have dv ==(?;: — dSl. When the auxiliary constants A, B, a, h, etc, are introduced, to find the variations of these arising from the variations assigned to the elements, we have, from the equations (99),, cot A = — tan Q, cos r, cot iS = cot Q cos i — sin i cosec Q, tan e, cot C = cot SI cos / -f sin i cosec $^ cot e, in which i may have any value from 0° to 180°. If we differentiate these, regarding all the quantities involved as variable, and reduce by means of the values of sin a, sin 6, and sin c, we get dA = dB = + cosi ,^ sinA . ^ . . ,, — j—dSl ;t,. ;r ^'^ o6 sni i at, 8m*a cose sin'^ b sin B sma (cos i cos £ — sin i sin e cos JJ ) fZj^ r r^ • • f . . . J. , smism Si , • . (cos Si sm t cos £ A- cos i sm e) di A . -,-; — - as. sme dC= —.-—r- (cos i sin e 4- sm i cos e cos SI) dQ sinC, _ , . . . ST., sin i" sin Si , "r „;„ „ ■ (cos SJ sni I sm £ — cos t cos £) di -\ .—^ ds • smc sm'c and these, by means of (lOl),, reduce to 164 THEORETICAL ASTRONOMY. dA = - v-V,— dQ — sin A cot a di, sill' a ,„ cose cose . , coso , uB = — . : , — a Si — sill B cot di ~\ — -^rr de, sill' 6 sin' 6 , ^ sin s cos b dU=^ (91) dSl — sin C cot c di + -r-V- ds. Let us now differentiate the equations (lOl)i, using only the upper sign, and the result is da=: — sin i sin A dQ, 4- cos ^ di, db =^ — sin i sin B dSl -\- cos B di -\- cos c cosec b ds, de = — sin / sin C dQ, -\- cos C di — cos b cosec c ds. If we multiply the first of these equations by cot a, the second hy cot 6, and the third by cote, and denote by X^ the modulus of the system of logaritlinis, we get d log sin a = — /io sin / cot a sin Ad^ -{- >-q cot a cos A di, d log sin b = — ^q sin / cot b sin B dQ, + ^o cot6 eos Bdi -\- A^ ^-.^j^ — ds, cZ log sinc= — Ajsint cote sin CdQ, -|- -'•o cote cos CcZi — \ sin' b cos b cos c sin' c de. The equations (91) and (92) furnish the differential coefficients of A, B, C, log sill «, &e. with respect to SI, i, and e; and if the varia- tions assigned to Q,, i, and £ are so small that their squares may l)e neglected, the same equations, writing aA, aQ,, a/, &c. instead of the differentials, give the variations of the auxiliary constants. In the case of ecpiations (92), if the variations of Q,, /, and c are ex- pressed in seconds, each term of tlie second member must be divided by 20G2G4.8, and if the variations of log sin a, log sin 6, and log sine are required in units of the mt\i decimal place of the logarithms, each term of the second member must also be divided bv 10'". If we differentiate the equations (81)i, and reduce by means of tlio same equations, we easily find cos b dl =^ cc i sec b da + cos b dfl — sin b cos (I — SI) di, db = sin i cos (l — SI) du + sin (Z — SI) di, (93) which determine the relations between the variations of the elements of the orl)it and those of the heliocentrif; longitude and latitude. Jjy differentiating the equations (88),, neglecting the latitude of DIFFERENTIAL FORMULAE. 165 tlie sun, and considering A, /?, J, and © as variables, we derive, after reduction, cos /3 cU = _ cos (A — O ) f^O , d^. E (94) sin/Jsin(;. — 0)f/0, which determine the variation of tlie geocentric latitude and longitude arising from an increment assigned to tlie longitude of the sun. It apjioars, therefore, that an error in the longitude of the sun will produce the greatest error in tlie computed geocentric longitude of a lieavenly body when the body is in opposition. 166 THEORETICAL ASTRONOMY. CHAPTER III. and in declin Thus, we hiv INVESTIGATION OP FORMULA FOR COMPrTIXO THE ORBIT OF A TOMET MOVING IN A PARABOLA, AND FOR CORRECTINO AIU'KOXIMATE ELEMENTS BY THE VARIATION OF THE GEOCENTRIC DISTANCE. 61. TiiK 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 similat'ly for tlie determination of the elements in the ease of an orbit wholly unknown; and since there are six dements, neglecting the mass, — whicli nnist always be done in the first approximation, the perturbations not being considered, — three complete observations will furnish the six equations necessary for finding tlicse unknown quantities. Hence, the data required for the determination of the orbit of a lieavenly body are three complete observations, namely, thr> e observed longi- tudes and the corresponding latitudes, or any other spherical co- ordinates which completely determine three places of the Ixxly as seen from the earth. Since these observations are given as made at some point or at diiferent points on the earth's surface, it becomes necessary in the first place to apply the corrections for parallax. In the case of a body whose orbit is wholly unknown, it is impossible to apply the coiTcction for parallax directly to the place of the body; but an equivalent correction may be a])plied to the places of the earth, acctyding to the formuhe which will be given in the next chapt«.r. However, in the first determination of approximate elc- nient^i of the orbit of a comet, it will be sufficient to neglect entirely tlic coiTcction for parallax. The uncertainty of the observed places of these bodies is so much greater than in the case of Avell-definod objects like the planets, and tlic intervals between the observations whitili will be generally employed in the first determination of tho orbit will be so small, that an attempt to represent the observed places with extreme accuracy Avill be superfluous. Wheii aJ3proximate elements have been derived, wc may rind the distances of the comet from the earth corresponding to the three observed places, and lience determine the parallax in right ascension in which a i of the comet of observati( observation, equatorial n sun. In order t place by met; tion must a time of ob» but if J is n( in the first a The transi into latitude which may I and writing and also which will Since cos/9 tho same si taken. G2. As so have been ej to the same DETERMINATION OF AN ORBIT. 167 and in declination for each observation by means of the usual formulse. Thus, we have ■Kp COS '=: A [n-"] = rr" sin («" - u), (2) [,•',•"] =. r'r" sin («" - «'), If wc designate by x, y, z, x', y', z', x", y", z" the heliocentric co- ordinates of the body at the times t, t', and t", wo shall have x' = r sin a sin (A + u), x' =z r' sin a sin (A -f- «')> x" = r" sin a sin (^1 -|- it"), in which a and A are auxiliary constants which are functions of the elements il and /, and these elements may refer to any fundamental plane whatever. If we multiply ^ 3 first of these equations by sin {u" — ?/'), the second by — sia {u" — u), and the third by sin (it' — u), and add the products, we find, after rediction, a comet, u X X sin {u" — xi') , sin (it" «) + -^ sin (it' ■ T it) = 0, which, by introducing the values of [/*;•'], [rr"], r.nd [r' r"], becomes [rV'] X — Irr"] x' + [r/] x" = 0. [rV] we get -V] ,."T ' If WC put nx — xf + n"x" = In precisely the same manner, we find " -\rr"^ (3) (4) nt/-i/' + «'y' = 0, nz — z'^ n"z" = 0. (5) DETERMINATION OF AN ORBIT. 169 Since the coefficients in tliesc equations arc indepcnrlcnt of the posi- tions of the co-ordinate planes, except that the origin is at the centre of the sun, it is evident tliat the three equations are i(U'nti<'al, and express simply the condition that the plane of the orbit passes through the centre of the sun ; and the last two might have been derived from the first by writing successively (/ and z in place of .r. Let I, )J, X" be the three observed longitudes, /9, /9', [i" the corre- sponding latitudes, and J, J', J" the distances of the body from the eiu'th; and let J cos /? = p, A' cos ;/ = p', A" cos ,5" = p'\ which are called curtate distances. Then we shall have x = p cos A — B cos O , y =^ p sin X — E sin O , z =^p tan i3, x' —- p' cos A' — R cos ©', y'=/,'siuA'— iJ'sinO', z' = p tan [i', x" = p" COS ?." — Ji" cos Q", y"^p"mi)." — R"smQ", 2" -= p" tan <5", in wliich 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 com])utation 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 formuhe w'ould 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 := ji (jo cos A — i? cos O ) — ip' cos A' — R' cos ©') + n"(/>"cosA" — ii"cosO"), = « (/) sin A — i? sin O ) — (/>' sin A' — R' sin ©') + n"(/f>"sinA" — irsinO"), = np tan ^ — p' tan ,5' + w'V" tan ,S". (6) 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 imknown quantities in the case of an orbit wholly unknown, namely, n, n", p, p', and p" ; and if the values of n and n" are first found, they will be sufficient to determine p, p', and p". 170 TIIEOTIETICAI. ASTEONOMY. The determination, however, of n and n" to a snflfioient degree of acniracv, bv means of the intervils of time between the ol)servation8, requires that // should bo approximately known, and henee, in general, it will become necessary to derive first the values of n, >/', and f*' ; after which those of ft and p" may be found from ecpiations (6) by elimination. But since the number of equations will then exceed the nundier of unknown quantities, we may combine them in such a manner as will diminisli, in the greatest degree possible, tlio effect of the errors of the observations. In special cases in which the conditions of the problem are sueli 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 equim \- to the place of the sun at the time <', and again in the second, fp"! the equinox to the second place of the body, we must diminish tl longitudes in these equations by the angle through which the axis of x hais been moved, and we shall have = nip cos(>l — Q')—Rco3(Q'— Q)) - (r' cos(k'— ©') — E') + 7i"(p" cos(;."— 0') - Ji" cos(0"- ©')), = n(psm(X — O')-{-Emi(Q'—Q))—r'sm(k'—Q') + h" ((>" sin (A" - O ' ) - li" sin ( O " - O ')), (7) = 11 (f> sin (X' — X)-\-lism(Q— )! )) — li' sin ( © ' — A') — n" (/." sin (A" — /') — R" sin (©" — k')\ = np tan ,3 — p' tan ,5' + n"p" tan /5". If we multiply the second of these equations by tan ^9', and the fourth by — sin(// — ©'), and add the products, we get = 7iVaan/5'sin(;."— ©') — tan,'/' sin(/l'- ©')) — ?i"i2" sin (©"—©') tan ,/+?)/> (tan /3' sin (A — ©') — tan ,3 sin (/'—©')) + nR sin (©'—©) 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' by [i?-R'], and we shall have iRR''\ = RR' sin(©'— ©), and similarly IRR"] = RR" sin ( © " — © ), iR'R''-^ ^ R'R" sin (©"-©')• OUBIT OF A HEAVENLY BODY. 171 Then, if we put N: lli'Ii"] ,11-, > we obtain [7;A"'] N" = [/»'A"] .» /,'"i ' UiJi"] (9) i?"siu(0"-0') = i2 8in(0'- O) ..r, iV" Stil)stitnting this in the equation (8), and clivitling by the cocffioicnt of //'', the result is ;" tan fi' m\ (X — Q') — tan ,? sin (/.' — O ') u" tan ,5" sin {X' _ © ' ) — tan ,i' sin (A" — ©') X\ AsinfO'— 0)tani9' "^\«" iV")tan/i"gin(A' ©') — tan/5'8in(r— ©')' Let ns also put , ,, _ jtan /i' sin (A — O ' ) — tan + (y^-|^)^"^- (11) We may transform the values of M' and 31" so as to be bettt^r 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 ©', and we shall have sin (A' — ©') tan w' = tan /5'. (12) Let j3^, i%" be the latitudes of the points of this circle corresponding to the longitudes k and /", and we have, also, tan /?o = sin (A — ©') tan ^v', tan /V = sin (A" — ©') tan w'. (13) Substituting these values for tan/3', sin(^ — ©') and sin (A" — ©') in the expressions for 31' and 31", and reducing, they become 31' sin (/?„ — /9) cos /5" cos ,V' sin ((3" — fi^') cos/?(,cos/3 M" = tan w' sm (©' - ©) -^, — ^. (14) 172 TIIEOUKTICAL ASTUON'OMY. n When tlio value of ", has boon fouiul, ^(Hiufion (11) will ^ivc the relation hotwot'ii p and />" in terms of kiiown quantiticH. It is evi- dent, however, from etjuations (14), that when the apparent [>ath of the eomet is in a piano passing through the second plaee of the sun, since, in this case, 1^ ~- 1% and ["t" —- ft,^" , we shall have M'- d and M" ~ oo. In this ease, therefore, and also when i% — /9 and [i" — /9|," are very nearly 0, we nuist have recourse to some other ccpiation which may be derived from the equations (7), and which docs not involve this indetcrniination. It will he observed, also, that if, at the time of the middle obser- vation, the comet is in opposition or conjunction with the sun, the values of M' and M" as given by e(|uation (14) will be indeter- minate in form, but that the original equations (10) will give tlio values of these quantities provided that the ap})arent ])ath of tlio comet is uot in a great circle passing tlu'ough the tjccond place of the sun. These values are M'=-- sinU-Q') sin (/'—©')' M"=- sin (Q^—Q) Hence it appeal's that whenever the aj>parent path of the body is nearly in a plane passing through the place of the sun at the time of the middle observation, the errors of observation will have great influence in vitiating the resulting values of M' and M" ; and to obviate the difficulties thus encountered, we obtain from the third of equations (7) the following value of //' : — Bin(A' — ;i) + -^i?sin(0-A') siaCA" — A') i^ii'sinCO' — /)+i2"sin(0" — A')^^^^ sin (/" — A') We may also eliminate p between the first and fourth of equa- tions (7). If we muliiply the first by tan ^9', and the second by — cos (A' — ©'), and add the products, we obtain = ft'V (tan /3' ('OS (A" — ©') — tan /?" cos (A' — ©')) — n"i2"tan/5'cos(©"— ©')+«/>(tan/5'cos(A— ©') — tan/5cos(A'— 0')) — nE tan [n' cos (©'—©) + ii' tan /S', from which we derive OIUUT OF A IIKAN'KNr.Y liODV 173 7." tnn ,y 009 {X O ' ) — tnn ,i^'08 (A' —_0') tan ,J" cow ( k' —O') — tanji' cort'i k" ~Q ') (HI) 7i"' tan fi' coh (0"— © ') + -^i7 It tan ,'/ eoa ( © '— O ) — J, A" tan ,5' tan/i"co8U'— ©') — tan,'j' coa(-i"— "©') * Let u.s now (U'note by I' the int'Ilnation to the cclijitio of a p;r('nt circlt' passin}^ tliroiij^h the second place of the comet and that point of tlic ecli[)ti(^ wluKse longitude is ©' — 00°, which will therefore be the longitude of its ascending node, and we shall iiave COS (A'— ©')tan7': tan ,y \ (17) and, if we designate by /9, and /5„ the latitudes of tlie points of this circle corresponding to the longitudes ). and /", we shall also have tan p, — cos (A — ©') tan I', tan fl„ ~ cos (A" — ©') tan /'. Introducing these values into equation (IG), it reduces to „__ 11 sin(,J, — /J) cos /5" COS/?,, '' ~ *" «" ' sin"(/V'^^* J ' "cos7"eos /37 (18) tan /' cos /5" COS/?,, n sin(/y ,'/ /5„) R" cos CO"— ©') + -7-Rcos(0'— ©) n (19) AM n I 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 si'cund place of the sun diminished by i)0°. In this case we may use C(|iiation (11) provided that the path of the comet is not nearly in the ecliptic. AVhen the comet, at the time of the second observation, is in quadrature with the sun, equation (19) becomes indetenninatc in form, and we must have recourse to the original equation (16), wliich does not necessarily fail in this case. AVIien both equations (11) and (IG) are simultaneously nearly in- ck'tcrniinate, so as to be greatly affected by errors of observation, the relation between p and />" 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 inap})lic;ablc, 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 THf:ORETICAL ASTRONOMY. (M. The fornuilif iilroady derived are suffioiont to determine tlio reL'iion Ix-tween •>" and (> wlien tlie values of n and n" are known, and it remains, tlieretbre, to derive the expressions lor these ijuan- tities. If we put k m ~t) ^ t", k (t" — t') = r, (20) k (t" - 1) =- r', and express the values of x, y, z, x", y", z" in terms of a;', t/', z' by expansion into series, wo liave a; 1= a; ax' r" , 1 f/V r' ,"S 1 f/V T '»~' J'a (H ' h ^1.2' dV ' k' 1.2.3' df h^ r . 1 f/V r' ^' + -;/.-T + T 1 d!>x' T» (/« yfc ' 1.2 dt' L^ ' 1.2.3 di^ /r* 4- &c., (21) and simihir expressions for ?/, i/", 3, and z". We sliall. however, take the phuie of the orbit as the fundamental plane, in whieh case s, ;', and z" vanish. The fundamental equations for the motion of a heaveidy body relative to the sun are, if v.e neglect its mass in comparison with that of the sun, dV Pa/ df' + r" "" ' d'y' , ^V df ' r"' 0. If we differentiate the first of these equations; we get (Px' _ 3/tV dy_ P dx' df' """ '7*' ' dt r" ' ~dT Diilerentiating ngain, we find rfV __ / 1 12k' I dr^ \ » 3^^ d.V \ , , 6^ df ~ \ r" "" " ?»' \ "(It f ■+" -/♦ ■ dt' } "^ ^ r'* 6/fe» di-;^ d3[ dt ' dt' d'l/ Writing y instead of x, we shall have the expressions for ■--: and --—• Substituting these values of the differential coefficients in eqiia- tions (21), and the eorres})onding expressions for y and y", nnd putting -^ 1 ^11 r-"i ORBIT OF A HEAVENLY BODY. -'" dr' 175 • r'* ^ kr* ' dt ,"3 t"« (//•' .2 ^ ^/ (22) h" -- 1 _ . ^' 4- 1 i!- 'K ^ '" k Hr'''^^kV'*'~dt ' wc obtain X --— ax , dx' ui' = a'V -i- b" dx' 'fit' y = ay'~b dij di' f = ay-i-b"^. From these equations we easily derive , , , x'di/ — y'dJ t/x —x'y =h — •'—^ , , , ii.x'dy' — y'dx' ^-^ —dr~' y"x' (23) fx — x"y = {ab" + a"b) ///A ^'<^y' — y'doi^ dt Tiic first members of these equations are double the areas of tlie triaiitflos formed by tlie radii -vcctores and the chords of the orbit ootween tlie places of the comet or planet. Thus, yx-x'y^irr'l y"x' - x"y' ^Ir'r^'l fx~a^'y^{rr"-\, (24) and x'dy' — y'dx' is double the area described by the radius-vector x'dv' n'dv' during the element of time di, and, consequently, - — - — '' ia double the areal velocity. Therefore we shall have, neglecting the mass of the body, x!d]f — ifdx' M V^ki/p, in which p is the semi-parameter of the orbit. The equations (23), tlicrofore, become ['•*•'] ^ hk ^p, [rV] = b"k y/p, [r7-"] = {ab" -{- a"b) k y'p- Substituting for a, 6, a", b" their values from (22), we find, since r' - - -I- t" 176 TlIKOllKTICA Ti A.STIIONOMY. ['•'•"] =r'v-(i-^;;!J+ r' dr' /c7' ' dt ' ' r'Hr — r") dr' /cr'* dt ...). (25) r ■' f"i r 'j''i From tlioso o(| nations tlio values of h =; r- -77.1 'Hi^^ '»'" '^- T^'^n^ 'n^^v [?•/• J Irr ] be (h'l'ivc'd ; and tin; results arc (2Gj ?i = which values arc exact to the third powers of the time, inclusive. In tiic case of the orbit of the earth, the term of the third o'M . bein is reduced t<> „ superior order, and, therefore, it may be uc<;lectcd, so that in tlii- casc wc shall have, to the same degree of aj)proximation as in (2()), ^27) [rV] From the e(iuatious (20) or from (25), since -7, ~- f — ,--,- we find 1 \ / yj L''''J n ^ - / + 1 r' -i- r'" dr' kr'* ' 'dt ■■■)■ (28) dr' Since this equation involves /•' and .-. it is evident that the value of -,,. in the case of an orbit wholly unknown, can be deteni. only by successive approximations. In the first approximation "1 the elements of the orbit of a heaveidy body, the intervals bctwcin the oi)servations will usually be small, and the series of terms of i2>* will converge J'apidly, so tliat we may take rt _ T ORBIT OF A HEAVENLY BODY. 177 and similarly N N" T Houce tho equiition (11) reduces to P" = ^„IM'P. (29) It will be ol)ser%-ed, further, that if the intervals botAvcen the observa- tions are equal, the tern of the second order in equation (28) vanishes, and the su])positinn that —, = —, is correct to terms of the third order. It will be advantageous, therefore, to select observa- tions whose intervals approach nearest to equality. But if tho observations available do not admit of the selection of those which give nearly ecjual intervals, and these intervals are necessarily very uno(iual, it will be more accurate to assume n n" N_ N'" and compute the values of N and N" by means of equations (9), since, according to (27) and (28), if r' docs not differ nuich from R\ till' error of this assumption will only involve terms of the third order, ev<'n when the values of r and r" differ very much. Whene\er the values of p and o" can be found when tiiat of their rati'* is given, we may at once derive the corresponding values of r anil r", as will be subsequently ex])Iained. The values of /• and /•" may also be expressed in terms of t' by meuns of scries, and wi- have dr' •iJ -2 from which we derive /' neglecting terms of the third order. Therefore r' ■ k (r" - r") 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 ih n "77 n rr"sin(v" (0 rr' sm{v' — v) ' (34) iV and that of ^-y, from (32); and the value of 31 derived by means of these from (33) will not retpiire any further ci^rrection. 65. When the apparent path of the binly ii^ ^*»K*h that the value of 3r, as derived from tbe first of etiualions (10^ is either indeter- minate or greatly aftVeti-c by errors of observHtii«*i. the equations (15) and (16) must be employj'u. The ]a^^t temfc f4" t'lese equation, may be changed to a form which is more convtmienl in the approximations to the value of the ratio of jf/' to ft. Let Y, Y'j Y" be the ordinate^ ai the sun when the Jixis of ORBIT OF A HEAVENLY BODY. 179 ab-cissas is directed to that point in the ecliptic whose longitude is /,', and wc have Y ^E sin(0 —A'), Y' ^B' ainO' —X'), Y"^E"Bm(,Q" — n. 'Sow, in the last terra of equation (15), it will be sufficient to put n n ~>7 N"' and, introducing Y, Y', 1 ", it becomes ( -^r Y-lrY'+ F"]cosec(A"-A'). 1 (35) It now remains to find the value of -77- From the second of equa- ?/. lions (26) we find, to terras of the second order inclusive, We have, also, and hence n N" Therefore, the expression (35) becoraes But, according to equations (5), NY—Y'+N"Y"^0, and the foregoing expression reduces to . 1 1!lrr'4-r"^/ 1 1 \ J?'sin(Q^-;/) "*■ ^^ t" ^ "•" ^ \ r'» E" j sin (A" - X') ' since Y' — B' sin(0' — A'). Hence the equation (15) becomes „_ n sin (X' - ;,) , rr^ /I 1 \ jg^ sin (A^-QQ '^ ~ '^ n" ■ sin (A" - A') 5 "?'" ^^"^ "T" "^ ^ \ 73 ii'3 / ^ sin~(A" - A') " ^- "^ 180 If wo put TIIEORETICAI. ASTRONOMY. ,r 11 sin (A' — k) 1," rz- ^.^l_.:L_.i;,(.'+,'0!^^f^'-O') R'll 1 11 sin (A' — A) /^ \7^ jK'O' we have (37) Lot us now oonsidor tlio equation (IG), and lot us dosignato by A', X', X" the ahs(!issas of th(! earth, the axis of abscissas l)oinfj; direotod to tliat point of the ecliptie tor which the lojigitude is ©', then X ==Rcoii(Q-Q'), X' = M', X"=R"cos{Q"-Q'). It will be .sufficient, in the last term of (IG), to put n "77 n N 1 . and for -,-, its value in terms of N" as already found. Then, since NX-X'-^N"X"=0, R'tanfi' this term reduces to _ . -' (>' , ," ( 1 _ ± \ — _- 8 7" ^ ^ "^ ^ \ 73 ie'3 ; tan /'/' COS (A'— ©') - tan fi' cos (A"— ©') ' and if wo put Af> — JL t«" P' cos (^ — Q ' ) -^tiui /? co3(A'— Q') ■*'» ~ n"' ' till) /J" cos (A' - O') — tiui ii' cos (A" — O')' tan ft' (38) 22' ' "' * 71 T"^"^ Hr" ii''/tan/3'co8(;i-0')— tan/'ico8(;i'— O') p' the equation (16) becomes (39) M=='-^Af'F'. In the numerical application of these formulaj, if the elemeiits are not api)roxinnitely known, we first assume n T when the intervals are nearly equal, and ORBIT QF A HEAVENLY BODY. 181 n N as given by (32), wliou tlic iiitorvals are very iinef(iial, and ncj^lccfc IIk^ llictors /'' and F'. TIk; values of ,o and |r>" wliicli are thus ol)- tained, enable us to find an ai)j)roxiniate value of /•', and with this a more exaet value of-.-, may be found, and also the value of F ov F'. Whenever equation (11) is not materially atfeeted by errors of observation, it will furnish the value of i)[ with more aeeuraey than the e<[uations (37) and (39), sinee the negleetcid terms will not be so great as in the ease of thess e(juations. In general, therefore, it is to he prefeiTcd, and, in the ease in whi(!h it fails, the very eireumstance tliiit the geocentrie j>atli of the body is nearly in a great eirele, makes tlie values of i<' and F' diller but little from unity, sinee, in order tl.'it the apjiarcnt path of the boily may be ncjarly in a great eirele, '/•' must differ very little from W. 66. AVheu the value of 3£ has been found, we may proeeed to determine, by means of other relations between (t and ft", the values of the (|uantities themselves. The eo-ordinates of the first place of the earth referred to the third, are 2/, = i«;"sinO" — /isinO. ^ ^ If we represent by g the chord of the (uirth'.-i orbit between the places corresponding to the first and third observations, and l)y (r the longi- tude of tlie first place of the earth 'as seen from the third, we shall have x, = g cos G, yi=^ 9 sin G, and, consequently, R" cos CO" — O) — i? = .7 cos ( G - O), .... ii"sin(0"-0) =^8in(6!-0). ^ ^ If "]/ represents the angle at the earth between the sun and comet at the first observation, and if we designate by w the inclination to the ecliptic of a plane passing through \\\v, places of the earth, sun, and comet or iilanet for the first observation, the longitude of the ascending nodt of this plane on the eclij)tic will be O, and we siiall have, in accordance with e(]uations (81),, cos 4 n=: COS /9 cos (A — © ), sin 4 cos IV = cos /? sin (A — O), sin ■i sin w =; sin ,5, 182 from wliich TII EORETICAL ASTIIONOM Y. tan IV = tau4 = _tan i? sTnCA--©)' tana — O) cos IV (42) Since 008/9 is always jmsitivo, cos 4- and cos(,'> — ©) must have the same sif^n; and, further, -v^ cannot exceed 180°. In the same manner, if «/' and ■\l/" rei)resent analogous quantities i'or the time of the third observation, we obtain tan w" = tan/S" sin (A" - ©")' tan (k" — 0") Wc also have tanV' = cos?o' COS 4" == cos /5" cos (r — O"). r' =J^-{-E'— 2JE cos ^, (43) which may be transformed into r' = (p sec 13 — li cos ^y + iiJ^ siu» 4 ; and in a similar manner we find r"' = (p" sec /5" — li" cos +")» + li'" sin= 4". (44) (45) Let n designate the chord of the orbit of the body between the first and third places, and we have But X' - {^" - ^y + (y" - yy + iz" - zy « = /> cos A — J? cos O, y ^= P sin A — i2 sin ©, g z= /) tan /S, and, since ()" = Mo, x"= Mp cos k"—R" cos ©", 2/" = iW"/> sin A" — i?" sin ©", s" = il//>tan/5" from which we derive, introducing g and (r, x" — x = Mp cos A" — /) cos A — ■ _r/ cos G, y" — y = 3//) sin A" — |r> sin A — rj sin (?, 2" — 2 = Mp tan/5"— p tan/?. Let us now put Tlien we have ORBIT OP A HKAVKNLY MODY. Mf> cos X" — /; cos ^ v: /)Ji C'OS X COS ff, Ml> sill >■" — p sill k ^. fill cos C sin i/, Jl//* tan fi" — p tan ,3 = /^/j sin C. af' — a; =^ /)/i cos C cos /T" — j/ cos G, i/' — y -- fih cos ; sin H — y sin fr, 2" — 2 -^ /'/i sin JT. 183 (40) Squaring tlicse values, and adding, we get, by reduction, x' = ^»/i' — 2ff ph cos ? cos ( (? — if) -|- y'^ ; and if we put wc have cos C cos ( G — //^ = cos 9", x' =:; (/>/t — g cos ^)' + (jr* sin' , A_co.^ M hR cos 4- = c, jf cos ^ — b"R" cos 4." := c", ph — g cos ^J* = (/, R sin 4 = -B, i?"sinV'=-5", ft", (51) and the equations (44), (45), and (49) become (52) i? 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. Thev will serve to determine r and /•" in all eases in which the unknown quantity d eau be determined. If /> is known. 184 TIIKOIJKTICAIi ASTKONOMY. (/ hccomos known directly; l)nt in tlu; case of an unknown orhit, tlicsc c'(|natioii.s arc a|)|tli(al)lc only when // or >l may l)c determined directly or indirectly I'ntm tlie data Inrnished l)y observation. G7. Since (lie e(|iiation.s (512) involve (w(t radii-vectores /• and /•" and tli<' chord K joining; their extremities, it is evident that an addi- tional ei|nation involving- these and known <|Uantities will enahle uh to derive d, it" not • /' rr sin ;V (f — v) = ._!. - (w -t- w). 2/y/' Introducing this value into equation (55), we find mt" ~ t) ^ m^ :^ n\ Replacing m and n by tlieir values expressed in terms of ?•, r", and X, this becomes Qkit" -t) = (r + r" + x)t =p (r + r" - x)J, (56) the upper nifa being used when v" — v is less than 180°. This equation 'xprcises the I'elation between the time of describing any parabolii! arj 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 IMAGE EVALUATION TEST TARGET (MT-3) 1.0 I.I IS ■" 1^ ■" Ui |2.2 12.0 t Ifi IL25 i 1.4 141 1.6 - =Binr. f57) and, since x, r, and r" arc jwsitive, sin;-' must always be positive. The value of ;•' must, therefore, be within the limits 0" and 180°. From the last equation we obtain (r + r"r - x» and substituting for k' its value given by this becomes x» = (r + r"y — 4rr" cos' { {v" — v), =>«' cos' Y = 4rr" cos» '.(i/' — v) Therefore, we have {r^r") fi\i COS and also tan / = 2Vrr"coa\{v" — v) (58) (59) Hence it appears that when v" — v is less than 180°, y' belongs to the first quadrant, and that when v" — v is greater than 180°, cos;*' is negative, and y' belongs to the second quadrant. If we introduce /-' into the expressions for vi^ and n^, they become which give m»=(r-f r")(l +sin/), n' = (r + r"Ml— sinA m*=(r-\- r") (cos ^ + sin .y)», n'=ir-\- r") (± cos hj' ^ sin i, ')»; and, since y' is greater than 90° when v" — v exceeds 180°, the equation (56) becones 6/ (r + r")* = (cos \r' + sin {y'f — (cos \y' — sin yf. PARABOLIC ORBIT. From this ccjuation we got 6t' 187 or (r + r")^ 6/ 6 cos' A/ sin \y' + 2 sin' \y', = 6 sin \/ — 4 sin' 1/ ; (r 4- r")l and this, again, may bo transformed into 2l(,. + r")^^ \ ]/2 / \ 1/2 / Let US now put or autl wc have 3r' V^2(r + r")^ sin V 8ma; = — --, V 2 sin y = V2 sin a;, = 3 sin a; — 4 sin' a; =^ sin 3a;. (60) (61) (62) When v" — V is less than 180°, y' must be less than 90°, and 'lenco, in this case, sin x cannot exceed the value J, or x must be within the limits 0° and 30°. When v" — v is greater than 180°, tlio angle y' is within the limits 90° and 180°, and corresponding to these limits, the values of sin .c are, respectively, \ and \\ 2> Hence, in the case that v" — v exceeds 180°, it follows that x must be within the limits .30° und 45°. The equation : sin 3a; V2{i' + r")' is satisfied by the values 3.r and 180° -- 3.r*; but when the first gives .r less than 15°, there can be but one solution, the value 180° — 3.i' being in this case excluded by the condition that 3.c cannot ex<'eed 135°. When x is greater than 15°, the required condition will be Batisficd by 3a; or by 180° — 3a;, and there will be two solutiv.ns, eorresponding respectively to the cases in which v" — v is less than 180'', and in which v" — v is greater than 180°. Consequently, wlion it is not known whether the heliocentric motion during the intervals" — < is greater or less than 180°, and we find 3x grcjiter than 45°, the same data will be ssitisfied by these two diH'crent solutions. In practice, however, it is readily known which of the 188 THEORETICAL ASTKONOMY. two solutions must bo adoptod, sinco, wlioii tlio interval t" — t in not very larj^c, the lielitMrntrio motion camiot oxctnid l.SO°, nnlt-ss tlu' ])crili(.'lion distanre i.s very snuill ; and tlu; known circnmstancoti will gcni'i'iklly show whet her sueh an asMimption is a(hnissil)le. Wu shall now put 2r^ aud wc obtain We have, also, and hence Therefore (r -I- r") » 8U1 • -f- r") (66) From equation (64) it appears that ;y must be within the limits and J) ^. We may, therelbre, construct a table which, with r^ as the arj^ument, will give the corresponding value of //, since, witli a given value of jy, 3a; may be derived from equati(»n (<)4), and tlicii the value of ft from ((55). Tabic XT. gives the values of // corro- eponding to values of jy from 0.0 to 0.9. 69. In determining an orbit wholly unknown, it will be necessary to make some assiiniption in regard to the approximate distance <»!" the comet from the sun. *n this ease the interval t" — I will gent- rally Im) small, and, conseq.\ently, x will be small compared with r and /'". As a first assumption we may take /• =- 1, or r -|- ;•" — 2, aud n = 1, and then find x from the formula X = tV2- PA II A no I, ir OR HIT. 189 With this value of x we eonipute il, r, nnd r" by means of the o(|Ua(i(ms (o2). iliiviiij; thus found approximate vahies of r and r", we compute jy l)y nu'ans of («l;J), and with this value we enter Tal)lo XI. and tak(; out the eorrespondinj; vulue of /i. A seeond value fur X is then found from (()(>), witli which we recompute rand /•", a'>d jirocced as iM'fore, until tiie values of these ((uantities remain un- (■li;in<)) will be exactly satisfial when the true value of y is u.-4ed, it follows that /(y) = o, and hence, when :^y is very small, so that we may neglect terms of tile .second order, we shall have y<^ — y„ == ^ Ay = vl (y„ — y). Lot lis now denote three .successive approximate values of log (r + »'") yo — yo = «. tlien we shall have yo" - yo' »'. o = i4 fy, — y), o' = yl(y;-y). Eliminating A from these (Kjuatious, we get y (a' — o) = a'y^ — ay^, from which y=y.-a'---=y„ a' o — a (67) 190 THKOItETKAI, ASTIIOXOMY. Unless the nswiimcd vnliirs nrc (•on.Bi|iro\ini!ito nearest to the truth, derive ;/ with still greater aeeumcy. In tiie nunu-rical a|»|»li«"jUion of this eijuation, a and a' may Ik? expressed in units o" the last «ieciiiial place of the lopirithms employed. The solul "on of ecpiation (•'it}), to find t" t when x is known, i.s readily etlw ted by means of Table VIII. Thus we have 3r' V^2(r-\-r")i and, wlicn y' is less than 90°, if wc put =: gin ;}r, we get or -r fin 3j! iV= . ,, sui;' r'=:\V2Xmir'^r-\-r")u \Vh('n ;-' excecd.s 90°, we put N' = sin 3a;, t' =. I \/~2 N' (r + r")h (68) and we have (69) in wliieh log J] 2 9.()733937. With the argument y' we tako from Table VJII. the corresponding value of A" or N', and liy means of these equations r' k (t" — t) is at once derived. The inverse problem, in which r' is known and x is recpiirod, nuiy also be solved by means of the sjuue table. Thus, we may lor a liivt approximation put x = ^ l/2. and witli this value of x compute K = v/2'iVl/r-f>' PAR Vnoiwf ORBIT. 191 or by 3 in wliicli loe— ,- V 2 _ 3 T'tiinf 0.326G()()3. Then wc rooompiito tf, r, and »•", niid |)n»c«'<'.\inia- liims aiT liU'ilifated hy means of ('(jnation ((17). It will l>e obscrvttl that d is ('oinputotl from an/*, in ord(?r that d may be negative. The eiiuution (47) shows tli.;t when x is greater than y, we have g cos tp < yh, and hence d must in this case be positive. But when x is less than .7, either the positive or the negative value of d will answer to the given value of f. and the sign to bo adopted must be determined ironi the physical conditions of the problem. If W(! sup|K)so the chords r/ and x to be proportional to the linear voloeities of the earth and comet at the nnddle observation, we have, the occcntricity of the earth's orbit being neglected, =w?. whieh shows that x is greater than r/, and that d is positive, so long « r' is less than 2. The comets are rarely visible at a distance from .lie earth which much exceeds the distance of the earth from the sun, and a comet whose nvdius-vcctor is 2 must be nearly in opj-.osition 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 sicnorally bo 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 TIIKOIIKTICAL ASTIIOXOMY. 70. WluMi the vnlvK's of /• and /•" have Ik'cm fiiinlly (lotormiiicd, as jui-t cxplaiiiocl, the exact value ot" tl may Ix; computed, uiul then we have P = d -f- ff C08 y» p"=Mp, from whicli to find ft and f»". According to the tujuatioiirt (90),, wc have r 009 b C09 (/ — 0) ::^- /> C08 ( > — O) - r c().s6 sin (/ — O ) - /' sin (^ — 0)i }• 8in b =^ (> tan ,i. (70) R, and al 80 r" coHi" co9(r- O") = p" co8{r— O") - R', r" cos //' sin {I" - O") - //' sin (A" - Q"), r"Hin6" --//'tan,3r", (71. (72) in wliich / and /" are the lieliocontric longitudcfl and b, h" the corie- sponding hcliwentric hititudcs of the comet. From these c(jnati((iis we find /•, /•", /, /", b, and b" -, and the vahies of r and r" thuH found, should agree with the final values already ol)tained. When I" is less than /. the motion of the comet is retrognule, or, rather, when the motion is such that the heliocentric longitude is diminishl ig instead of increasing. From the tHpiations (82),, we have tan i sin (/ — Ji) = tan b. tan i sin (/"— ft ) = tan 6", (73) which may lie written ■±. tan J (sin (/ — a;)co8(a; — ft ) "|- ^^in (^* — JJ ^ cos(/ — a;)) = tan6, ± tan / (sin ( t'— x) cos (x — ft ) -|- sin (« — ft ) cos {f— x)) — tan b". Multiplying the first of these equations by sin(^" — .r), and the second by — sin (/ — x), and adding the products, we get ■±. tan i sin (x — ft ) sin {H' — /) = tan b sin (/" — x) — tan b" sin (l — x); and in a similar manner wc find ± tan i cos (« — ft ) sin (f ' — = tan b" cos {I — a;) — tan 6 cos {I" — x). Now, since x is entirely arbitrary, we may put it equal to /, and we have l.U: ORHIT. . 103 tnii h, tun 6"— Uinb vtmil" ~ I) (74; niuir-t) 1 tan J sill it — Q) = ± tan / cos (/ — Q) =-- ± the lower sijxii Ix-in^ ust'tl when it i.s doiiirt'd to introduce tlu; distinc- tion ol" retrograde motion. Tlie torinnlie will lu' better adapted to io^aritliniie ealeidation it" wo put X - J(/"+ /), whence I"- x ^(1" - I) and / -.r \[l -^ /"); ant! we obtain . . ,,,,„,, ^v ^i» ^tt" '[• b) ^ • ' 2 e«)M ft cos ft eo.x \(i — / ) ^ . ^. /•» . IX ^1 «"• »ft ft) tan* cosl.', (/ + O — Jj) = ± o — , ,„ . . , „, jx- ^' ' ' 2 cos ft cos ft sui \ ( r — /) Tliese ocpiations* may also he derived directly i'rom (I'.l) hy addition and sui)tracti()ii. Tims we have ± tan /(sin (/"— ft ) + j*in (/ — ft )) =^ tan ft" + tun ft, ± tun/ (sin (r— ft) — 8ia(/— ft)) = tan ft"— tan ft; and, fince nn (/"— ft) + si" (/ - ft ) -- 2 sin A (r+ / - 2ft ) cos \ (l"~ I), HinCr- ft ) — 8in(/ - ft) ^-= 2 cos .J (/"+/- 2ft ) sin \ (l"-~ I), these become * • • /!/;" I /^ r.\ ^ Utan ft" -f tan ft) tant8m(i(/' + 0-ft) = ± ,.„j, ,(/"_/)- • tan / cos (J (/" + /)— ft) =--± ^( tin) ft"- tan ft) "singer— ' (76) which may be readily transformed into (75). However, since h and It'' will be found by means of their tanji-ent.s in the numerical appli- cation of ecpiations (71) and (72), if addition and sulitraetion loo;a- ritlunx are used, the e([uations last derived will l)e more convenient than in the form (75). As soon as ft and i have been computed from the prccodini? c(|ua- tions, we have, for the determination of the arguments of the latituile n and Ji", tan u = ±: Now wc have tana-ft)^ tann"=d:^^-'"-«J. (77) cos I COS I U = V -{- w, in which w — ;: — ft in the case of direct motion, and w — ft — tc 13 194 THEOIIKTKAL AHTI{()N(»MY. ■wlicn tilt' distinction of rctrognuU' motion i«n(lo|»tcd; und wo i-hall Imvo u" — u -rz v" — r, and, consofinpntly, x« = r' 4- »•'" — 2r»-" cos Cu"— u), K» = (»•" — r oos («" — u)Y -V »•' sin' ( It" — «). W (78) (7!») The vmIuc (•(' X derived from this erjiintion should ngrce witli tliat already found from (00). We have, further. or r = q sec' i(it — w), 1 ir ^ 1 — 7» COS .1 (M — 0») =: , _, 1/7 Vr i^' — q see' ] (?t" — w), -— ,-^- cos A (n" — itt) = — -—. By addition and subtraction, we get, from these ecjuations, - - (cos ] in" — w) -f- cos \ (n — w)) = -^ + - - .-, Vq Vr vr -'.- (cos A ( u" — w) — cos A (i< — w)) = -7^ >«, from which we easily derive 2 11 -- cos ^ (J, («" -f «) — w) cos .j («"— tt) = -7- + -7=-, Bnt - -- sin A (.'. (?t" + «) — m) sin j (u" — «) =*= —-7- 7=. Vq ' ' Vr Vr" J. _ _i _ _i / ijY _ « rr \ ■ V7^V?~ t/r?'\^ r '^^'?/' (80) and if wo put tan (45° + 0')-^^, since \ — will not differ much from 1, lT + Vf = 2 sec 2^', I>AI(AIU»L1C OltUIT. TluTfforo, the 0(|Unti<>iiH (80) IxH^nnio Yq Hill I in — It) I' »T -^ C08 J (i(l( + ") - «") „ .7 V Vq ens | (it — h) v rr 195 (81) from which the vahics of y and to may 1«! fuuiul. Then we nhall have, for the lougitiKh; of tho jHTihcliou wlicii the motion is dirctt, and jr = ft — w, when / mircstrictod exceeds 90° and the distinction of retrograde motion is ailopted. It remains now to find T, the time of perihelion pussaf^e. We Imve = u — «o, v"=:n"-w. With the resulting; vahios of v and v" we may find, by means of TiiMe VI., the correspond iiij; vahies of M (which must Ikj distin- puislicd from the symbol M already used to denote the ratio of the curtate distances), and if these values are designated by J/ and J/", ;v(! shall have or T=^t~ M m ., M" I ) ni ill which HI -- -" > and log C„ -- 9.9601277. When v is negative, the corresponding value of J/ is negative. The agi'cement between the two values of 2* 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 whi(!h we derive the two values of N and compute the corresponding values of T from the equation 2 - T= t — oT -AV' sin V, 2 in which log5, = 1.6883273. When v is greater than 90°, we de- lUU Til i:« »UKTI( AL ASTUOXOM V rive the vali* 's of iV from tlir tultle, and <'oiu|>iitc the coiTt'spoiulin^' vulucrt of T from 71. The ol(.'ments q ami 7' may Ix' lU'rived dingily from tlio value- «)f r, I'", and x, as jlcrivcd from tlu! iMjuations (o'J), witliont first finding the position of the plane of {\\v. orliit an«l tlii> position of the orl>it in its own plane. Tims, the e(|nations (SO), re|)lacing u and )(" by their values r I m ar.d v |- w", become 2 11 — = co« \ (v" + vj COS ] (v" — 1>) = — = -f -7-r-. Addinjj; toj^ether the squares of those, ari.l roducinp, wo get 1 \'r^,--^,^oB},iy"-v) or 9 7 = sin'' ^ (v" — ry jT"sin».J(i»" — r) r" + r — 2 V/»V' cos ;1 {v" — v) Combining this equation with (59), the result is __ rr" sin' j jv" — v) 9— ,._!_/'_ X cot r" and henec, since x = (r + r") sin ^', rr" q = — sin' \ {v" — v) cot {/. We have, further, from (78), x» = (r" — )•)' + 4rr" sin' A (v" — v), from whii'h, putting r" — 7* sm V = , vc derive cos V = sni A (v — v). X (82) (83^ (84) (85) Therefore, the equation (83) becomes PARABOIJC OUIUT. q = ^(t -f r") con' }/ pos'w, 197 l)V moans oC which tj is (Icrivcd din'ctly y means of the formula (84), so that comi/ is |iit>iti\M'. When y' cannot l>ro(Ui('t.s and rodiiciug, wo ca.sily Hiul Vr co« \v ~Vq 1 V7' sin Jr, adtl- Hl'iico we have gin i ((.'" — r) sin \ v c os \ i v" — v) 1 Vq Vr V'/ 1 . , cotUi'" — lO Vq ' Vr 1 V^,."sin^(v" — v)' (92) Vq cos iv = — 7=, Vr which may be used to compute q, v, and v" when v" — r is known. When J (r" — (•) and i(r" • r), and lience v" and i-, have Ikh'u determined, the time of perihelion passage must be found, as ah'cady explained, by means of Tabic VJ. or Table VIII. It is evident, therefore, that in the determination of an orbit, as soon as the numerical values of r, r", and x have been derived irom the e(iuations (o2), instead of "ompleting the caletdatiim of the ele- ments of the orbit, we may h.^vl q and 7', and then, by means of these, the values of >•' and v' may be computed directly. When ti>is has been ett'ected, the values of n and n" may be found from (3), or that of -T, fnmi (34). I'hen we compute (> by means of the first of ecjnations (70), and the corrected value of M from (33), or, in the special cases already examined, from the eciuations (37) and (39). In this way, by successive approximations, the determination of pi.ra- bolic elements from given data may be carried to the limit of aecin-acy which is consistent with the assumption of jiarabolic motion. In the ease, however, of the ecpiations (37) and (39), the neglected terms mpy be of the second order, and, consequently, for the final results it will be necessary, in order to attain the greatest possible nccuraiy, to derive p M- from (15) and (16). When the final value of 3/ has been found, the determination of the elements is completed by means of the formuhe already given. PAHABOLIC ORBIT. 199 72. KxAMri.R. — To illnstriilc tlio application of the formula' for tlio calciilatioM of the paracolic eleinciits of the orl)it of a ('(tinot l)y a niiinoiical oxami)lo, let us take the following ob.« 20'..-) 18 G 11 54.7 1(5 G 35 11 .G 19* 14™ 4M)2 19 25 2.84 19 41 4.54 + 34° G' 27".4, 36 3G 52 .8, + 39 41 20 .9. Tiii'se places are referre(r to the a])paront equinox of the date and arc already corrected for parallax and aberration by njoaris of approximate values of the jfcocentrie distances of the ct)niet. Hut if approximate values of these distances are not already known, the corrections for parallax and aberration may be neglected in the first (letcrniination of the apjjroximate elements of the unknown ori)it of a comet. If we convert the observed right ascensions and declinu- tioiis into the corresponding longitudes and latitudes by means of O(|uations (1), and reduce the times of observation to the meridian of \\'ashington, wo get WnshiiiRtoii M. T. 18(i4 Jan. 10 7* 24" 3* 13 G 38 37 16 7 1 54 297° 53' 7".6 302 57 51 .3 310 31 52 .3 -f 55° 4()' 58".4, 57 39 .35 .9, + 59 38 18 .7. Next, we n.'duco these places by applying the corrections for pre- cession and nutjition to the mean ecpiinox of 18G4.0, and reduce the times of observation to decimals of a dav, and we have t =10.30837, r = 13.27(582, r.^. 16.29299, X =. 297° 52' 51 ".1, r r_: 302 57 34 .4, X" =^ 310 31 35 .0, ,? =. + 55° 46' 58".4, ir = 57 39 35 .9, fi"=. + 5d 38 18 .7. For the same times we find, from the American Xaiitical Almanac, O -=290° 6'27".4, O' =. 293 7 .57 .1, 0" = 29G 12 15 .7, log R = 9.9927G3, logie' =9.992s:i(), log /r = 9.99291(5, which are referred to the mean equinox of 1864.0. It will gene- rally be sufficient, in a first ai)proxiniation, to use logarithms of five decimals; but, in order to exhibit the caleuhition in a more complete form, \i _' shall retuin six places of decimals. Since the intervals are very nearly o(iual, we may assume !; \''"' 200 Then we have M TIIEORETICAI. ASTRONOMY. n ?l T 'N"' and /" — <; tanks' sin (X — Q')^— tan/5 sin (^-'^-^G^ t' — t ' tan, J" sin (A' — ©') — tan,*' sin (A" — 0')' ff mi (G—Q)=: R" .sin (©" — O), (J cos ( Ct' — 0) --'-- R" co.s( 0" —Q) — R; h cos : coaiH—k") = M— cos (a"— A), /i cos : sin (H—).") ^ sin (a" — ;.), A sin C = J/ tan ,5" — tan ,? ; from which to find M, G, f/, II, ^, and h. Thus we obtain log iV=. 9.821)827, G == 22° 58' 1".7, log5r=^!).019 — b"R" cos i" = c", we obtain A, B, B", &c. It will generally be sufficiently exact to find sin ^ and sin ^l/" from cos ^ and cos\//"; but if more accurate values of '^ and i//" are required, they may be obtained by means of the ecpiations ^42) and (43). Thus we derive log A =^ 9.006485, log B = 9.91 2052, log B' = 9.933366, log 6 = 9.438524, log 6" = 9.562387, c == — 9.125067, c" = — 0.150562. h cos ,5" , „ NUMEP.ICAL EXAMPLE. 201 Then wc have T' = k(t"-t), 2t' 2/ X = Vr-\- jil^y V(l+')" + 5^ from which to find, by successive trials, the vahics of r, r", and x, that of // being found from Table XI. with the argument r^. First, we assume log X = log t'v2 = 9.163132, and with this we obtain log r = 9.913895, log r" r= 9.938040, log (r + r") = 0.227165. This value of log(r + r") gives rj = 0.094, and from Table XI. we find log/i - 0.000160. Hence we derive log X = 9.200220, log r = 9.912097, log r"= 9.935187, log(7- + r") = 0.224825. Repeating the operation, using the last value of log(r + r"), we get log X =: 9.201396, log r == 9.912083, log r" = 9.935117, log(r + r") = 0.2;^783. The correct value of log(j* + r") may now be found by moans of the equation (67). Thus, we have, in units of the sixth decimal place of the logarithms, a = 224825 — 227165 = — 2340, a' = 224783 — 224825 = — 42, and the correction to tlie last value of log{r -\- r") becomes a' a' — a 0.8. Therefore, log(j- + r") = 0.224782, and, recomputing rj, n, x, r, and r", we get, finally, log X = 9.201419, logr = 9.912083, log r" = 9.935116, log(r + r")-= 0.224782. The agreement of the last value of log(r + '"") with the preceding one shows that the results arc correct. Further, it appears from the 202 THEORETICAL ASTRONOMY. values of )• !uiorihc'lion and was rt'ccdin^ IVoin the sun. iJy nn-ans of tho vahies of r and /■" wi- nnj^ht oonipnto ai)|)r(».\i- niati' vahics of /■' and . from the equations (.'JO) and (•»!), and then ((I ^ f. a more approximate vahie of „ from (28), tliat of ^,^ being found from (;{2). IJiit, sinee r' differs but little from Ji', the ditt'erenoo between „ and ,rr,; is vcrv small, so that it is not neeessarv to eon- H J\ sider the second term of the seeond member of the etpiation {'•i'-)); and, since the intervals are very nearly e(iual, the error of the as- sunjption n T is of the third order. It should be observed, however, that an error in the value of J/ affects //, ^, li, and luMice also A, h, h", c, and r;", and the resnilint; value of fi may be affected l)y an error which con- siderably exceeds that of ^^. It is advantap'ous, therefore, to select observations which furnish intervals as nearly ecjnal as possil)le in order that the error of J/ may be small, otherwise it may become necessary to correct J/^and to repeat the calculation of f, r", and x. We may also compute the perihelion distance and the time of I'cri- helion ])assafjje from /•, /•", and J£ by means of the e(piations (8(J), (89), and (01) in connection with Tables VI. and VIII. Then r' and c' may be computed directly, and the eomp'"te expression for M may be employed. In the first determination of the elements, and espe< lally when the corrections for parallax and aberration have been neglected, it is un- necessary to attem[)t to arrive at the limit of accuracy attainable, since, when approximate elements have been found, the observations mav hv more convenientlv reduced, and those which ini-lude a lonyior interval may be used in a more complete calculation. Hence, as soon an r, r", and x have been found, the curtate tlistances are next deter- mined, and then the elements of tlie orbit. To find ft and //', we have rf = + 0.122395, the positive sign being used since x is greater than g, and the forinulte h ' give log /> 1=9.480952, log//' =-9.310779. NIMKRICAr- KXAMPLE. 203 From tlicso values of (> and //', it appt-ars ihat the comet was very luar the earth at tite time of the observations. The heliooeiitrie j)hurs are then fonnil l)y means of the equations (71) ami (72). Thus we obtain / r..: 106° 40' oO".r), 6 =-- + ;}:}° r 10".0, logr = 9.0120^2, r^lVl ;U I) .}>, Z»"-^ + 28 05 5.8, Iogr"=U.!»;5511(i. Till' agreement of these values of /• and r" with those previously liiund, cheeks the accuracy of the calculation. Fiu'ther, since the litliocentric l(»nu;itu(lcs are increasing, the motion is dirvof. The longitude of the ascending node and the inclination of the orbit may now be found by means of the ecpiations (74), (75), or (7U); and we get Si == 304'^ 43' 11".5, i = 04° 31' 21".7. The values of w and n" are given by the formula; tan u tan(/— Si) cos I tan It" tan(/"— Q,) COS! (' and / — Si being in the same quadrant in the case of direct motion. Thus we obtain u = 142"^ 52' 12".4, u" = 153° 18' 49".4. Then the equation x» = (}•" — r cos ( h" - n)y + r' sin' («" — u) log x = 9.201423, gives and the agreement of this value of x with that previously found, proves the calculation of Q,, i, u, and u". I'Vom the e({uations * r- wo get tan (45° -}- = 37<'38'43".l. 204 THEORETICAL ASTRONOMY. and v = u — m = 2V 12'6".l, Then we obtain log m = 9.Jt()01277 — 3 log ry -^ 0.129061, and, corresponding to the values of v and r", Table VI. gives log M= 1.267163, log M" = 1.4241o2. Therefore, for the time of perihelion pa.ssage, wo have T^t ~ '^'^^ =-< — 13.74364, m and T^t"- M" m :/"— 19.72836. 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 p If we find T by means of Table VIII., we have log iV^^ 0.021616, logiV"=r. 0.018210, and the equation 9 2 T=t — -^Nri sin v = t" — -^ N"r"^ sin v", in which h>g.^,=-- 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. ff = 60°23'17".8) ^ ,. . i-» on« ^o 11 - I t-cliptic and Mean log 7 = 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 f) and f}" have been found. If, by means NUMERFCAI- r^XAMPLES. 205 of tli('«o elomonts, wc compute' n and »", and correct the value of ^f, the elements which will then he ohtained will a|)j)roxiniate nearer the tnu! values; and each successive correction will ftu'nish more ai'carate results. When the adopted value ol' .1/ is exact, the result- iiijf elements must by calculation rej)roduce this value, and since the (diiiputed values of ?., )'\ ,9, and (i" will he the same as the observed values, the «'omputed values of /' and [i' nujst bo such that when suhstituted in the e(piation for J/, the same result will be obtained a- when the observed values of // and (i' an; used. liut, according til the e(piations (l.'J) and (14), the value of .l/(U'pends only on the inclination to the ecliptic of a great circle |)assinir 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- onliiiates of any point of the groat circle joining these places of the sun and conu't will, in connection with those of the extrcMue j)laces, give the same value of J/, and when the exact value of M has been iwd in deriving the elements, the computed values of // and ^•i' must jrive the same value for \c' as that which is obtained from observa- tion. But if we represent by ^' the angle at the earth between the sun and comet at the time i\ the values of ^' derived by observation and by computation from the elements will differ, uidess the middle jjiace is exactly represented. In general, this ditferenee will be small, aud since ic' is constant, the equations cos ij = cos (5' cos (A' sin 4' cosn»' ■= cos S sin (A' - sin V sin w' = sin {i\ ©'), (93) give, by differentiation, (94) cos ^ d)! = cos n'' sec ,S' d^' , d-i' = sin v/ cos (A' — ©') d\\ From these we got cos [-i' dk' _ tan U' — OO d^ ~ sin/ ' which expresses the ratio of the residual errors in longitude and liititiide, for the middle place, when the correct value of M has been u>e(l. Whenever these conditions are satisfied, the elements will be correct on the liypothcsis of parabolic motion, and the magnitude of the final residual::, in the middle place will depend on the deviation of the actual orbit of the comet from the parabolic form. Further, 20G TII RORKTIf A L ASTRONOMY. wlit-'ii c'lf'inontH li!iv(- Ikm'ii (Icrivcd from a value of 3/ which has iidt been finally convctod, if we compute /' and (i' by means of those elements, and then fnn ,5' tan ,^ tan IV = .— -~_-y^,, 8m(A' — O'j tlie comparison of this value of tan«'' with that given by obs(M'va- tion will show whether anv further correction of M \n neeessarv, and if the difference is not greater than what may be due to iniavoidalilc errors of ctileulation, we may regard M its (jxact. To compare the elements obtained in the case of the example given with the middle place, we find w'r^32°3r 13"..5, Then from the erpiations It 148° 11' 19".8, log/-' = 9.9228:]6. we derive tan (r — JJ ) = cos i tan u', tan b' = tan i sin il' — J^), r = 109° 46' 48".3, b' = 28° 24' 56".0. By means of these and the values of 0' and R', we obtain >l' = 302° 57' 41".l, /5' = 57° 39' 37".0 ; and, comparing these results with the observed values of X' and ,i', the residuals for tlie middle place are found to be Com p. — Obs. cos ;/ aA' = + 3".(}, A/3 = + l".l. The ratio of these remaining errors, after making due allowanro hx unavoichible errors of calculation, shows that the adopted value of M is not exact, since the error of the longitude should be less tlnin that of the latitude. The value of w' given by observation is log tan ?t)' = 0.966314, and that given by the computed values of )J and /?' is log tan xo' = 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- Xr.MKRICAL EXAMPIii^. 207 rection. Since the diflbrcnee is siduII, wo may dorivc tlio oorrcrt viiliit' of M l)y u.siiig tlio same ussutued value of -,,. and, instead of tlie value of tan w' derived from ol)Hervation, a value difrerinj^ as imicli from this in a contrary direction as the computed value differs. Tims, in the )>resent example, the computed value of lo^ tan w' is 0." cos (A" A) — Ii COB /), (J CM A' sin ( G — A) ■----.-- It" cos />" sin ( .1" A ), 0)(y ) (/ sin A' ^ Ti" sin //' Ji sin 7^ t'roMi which //, A', and (f may l)c found. It' we dcsii^nati! hy .r„ i/„ z, the co-ordiimtcs of tlu> first place of tlic conu't referred to the third place of the earth, we shall have x,= J cos ') cos a -{- y cos A' cos G, y, = J cos " HJn o -{- (/ cos A' sin G, z, = J sin o -\- y sin A'. Lot us now put und we get a-, ^ h' cos :' cos //', y, = h' cos C' sin W, 2, = h' sin C', /i' cos C' cos(H' — O) = J cos 'T cos (» — ^) + i7 cos K, h' cos :' sin (//' - G) = J cos -1 sin (a — G), h' sin C = J sin >> -{- g sin A", (97) from which to determine H', !^', and h'. If we represent by ' = c, R cos ■^ -- i, ii" cos +" = 6", (103) and we shall have (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 J" by means of the approximate elements known, and find r" from the last of these equations, the value of r having been already found from the assumed value of J. Then x is obtained from the equation 2r'__ Vr + r" Ih fi being found by moans of Table XI., and a second approximation to the value of J" from J" = c±v/x'^— C" (105) The approximate elements will give J" near enough to show whether the upi)er or lower sign must be used. With the value of J" thus found we recompute r" and X as before, and in a similar manner find a still closer approximation to the correct value of J". A few trials will generally give the correct result. When J" has thus been determined, the heliocentric places are found by means of the formulee r cos b cos (I — A) = J cos S cos (a — A) — R cos D, r cos b sin {I — A) = J cos S sin (» — A), r sin 6 = J sin 5 — R sin Z>; (106) VAIUATION OF TIIK (iECM'KNTUIC DIHTAXCE. 211 /'coHrcoHfr- A") /•"n.s/>".Hin(r-vl") /•" sin //' J" cos A" cos ( a" - A") — n" cos /)", J" COH <)" m\ ( a" A"), (107) J" sin -5" - i{" sin />", in wliicli /), h", /, /" arc tlic heliocentric .splicriciil co-ordinates rc- lirrcd to the equator as the t'nndaniental phiiie. The vahie-* of r and >•" tJnind IVoin these ecpmtions must agree with those obtained I'roni The eh'inents of the orbit may now l)e determined by means of the (■(|iiations (7")), (77), and (HI), in connection with Tabh-s VI. and VI I J., as abT:idy explained. The elements thns derived will be re- il'ircd to the eijii:itor, or to a plane passinjr through the centre of the sun and parallel to the earth's equator, an\', in which a' and d' denote the differences between eom[»utation and olworvation. Next we assume a second value of J, which we repre- sent by J -j- oJ, and compute the corresponding system of elcmenb*. 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 — uJ, and denote the ditfer- cnces between computation and observation by a and d; then we shall have «=/(J-<5J), a'=/(J), a"=/(J + ,5J), and similarly for d, d', and d". If thtso three numbyrs are exactly represented by the expression X m + n-^-^ + ijjf' in which J + .r is the general v "'ue of the argument, since the values of u, a', and a" will be such th t 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. imicli from J, or within the limits x = — SJ and x = -f- SJ, the as- sunu'd vahios J — dJ, J, and J -}- (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 7/i, v, and o, we liave m — ?i -|- = a, whence m n m ^^ a , K«"-«). m -\- n -\- o = A(«"+fl) Now, in order that the middle place may be exactly represented in right ascension, we must have ''(*!,)'+"(,;j)+"'=0' from whicli wc find X - 2o ^" l/?j,* — 4mo) = p, or x—p,U = 0. In the same manner, Jie condition that the middle place shall I)e exactly represented in declination, gives In order that the orbit shall exactly represent the middle place, l)otli conditions must be satisfied simultaneously; but it will rarely happen tliat this can i)e efiected, and the correct value of x must be found from those obtained by the sej)arate conditions. The arithmetical mean of the two values of x will not make the sum of the squares of the residuals a mininuun, and, therefore, give the most probable value, uidess thu \anation of cos 5' Aa', for a given increment as- signed to J, is the same as that of aS'. But if we denote the value of X for which the error in a' is reduced to zero by x', and that for Avhich Ao' = 0, by x", the most probable value of x will be X- n^x' 4- n'Kv" (108) in which n ^=^- h{(i" — a) and n' = l{(l" — d). It should be observed that, in order that the ditferences in right ascension and declination shall have equal influence in determining the value of x, the values of a, a', and ", respectively. Thus we obtain I = 159° 43' 14".2, b = + 10° 50' 14".0, logr = 0.323447, r^l44 17 47 .8, 6" = + 35 14 28 .7, logr" = 0.052347. The agreement of these results for r and r" with those already obtained, proves the accuracy of the calculation. 8incc the helio- centric longitudes are dimiuishing, the motion is retrograde. Then from (74) we get and from Q, = 165° 17' 30".3, tan(7— Ji) tan ?i =^ . — -, cost tan u i = 63°6'32".5; tan(l"-Q) COSi we obtain «--12° 10' 12".6, u" = 40° 18' 51".2, the values of — ic and I — SI being in the same quadrant when the motion is retrograde. The equation (79) gives log x = 0.090630, which agrees with the value already found. Tiie formulse (81) give v" = u" iu = 129° 6' 46".3, and hence we have y = u — w == _ 116° 56' 33'.7, from which we get T== 1858 Sept. 29.4274. From these elements we find log r' = 0.212844, v' =. - 107° 7' 34".0, and from log 5 = 9.760326, (^ = — 88° 47' 55".l, u 21° 59' 12".3, we get tan {I' — R) =; — cos i tan u', tan 6' = — tan i sin {P — Q), r = 154° 56' 33".4, b' = + 19° 30' 22".l. NUMERICAL EXAMPLE. 217 By nioxins of these and the values of O' and R', wc obtain -I' =. 137° 39' 13".3, /5' = + 12° 54' 45".3, ami comparing tlieso results with observation, we have, for the error of die middle i)lace, C.-O. cos;j'a/.' = — 27".2, A/}' = — 23" .1. From the relative positions of the sun, earth, and eomot at the time /" it is easily seen that, in order to diminish these residuals, the gco('entri(! distance must be increased, and therefore we assume, for a .second value of J, log J = 0.398500, from which we derive ^^'=153° 44'57".6, log 6'=-^ 9.912587, log J" = 0.311054, :' = + 7° 24'26".l, lege = 0.472115, log /•" = 0.054824, Then we find the heliocentric places / =: 159° 40' 3.3".8, b = + 10° 50' 8".6, r=144 17 12 .1, h" = + 35 8 37 .8, and from these. log A' = 0.4H802(), log r = 0.324207, log X = 0.089922. logr =0.324207, log ,•" = 0.054825, S^ = 165° 15' 41".l, « z= 12 10 30 .8, w = 128 54 44 .4, T= 1858 Sept. 29.8245, v' = — 106° 55' 43".8, /'= 154 53 32 .3, /= 137 39 39 .7, ■i = 63° 2'49".2, «" = 40 13 26 .0, log q = 9.763620, log/ = 0.214116, m'= 21° 59' 0".6, 6' = + 19 29 31 .9, /5' = +12 55 2 .9. Therefore, for the second assumed value of J, we have C.-O. cos/yAA'^. — 1",5, A;S' = — 6".l. Since these residuals are very small, it will not be necessary to make a third assumption in regard to J, but we may at once v the eoofheicnt of a' in that equation and taking the sum of all the equationti thus formed as the final equation from which to find ,v, the observations being supposed equally good. 220 TUEORETICAL ASTKOSOMY. CHAPTER IV. DETEUMIXATION, FROM THRKK COMI'I-ETK OIlSEUVATloNS, OF THE ELEMENTS OF THE Olilirr OF A HEAVENLY HODY, INIXIDINO THE ECCKNTUICITY OK FoUM VV THE ( OXIC SECTION. 77. TnK f'ornuilio which have thus far boon dorivcd for tlio dotcr- miiiation of the olemeuts of the orhit of a hoavonly body by means of observed plaeos, do not suffice, in the form in which they liave been given, to determine an orbit entirely unknown, except in tlie partlcidar case of parabolic motion, for whicii one of the elements becomes known. In the {general case, it is necessary to derive at least one of the curtate distances without makinjj any assumi)tion as to the form of the orbit, after which the others may be found. I>iit, preliminary to a complete investigation of the elements of an un- known orbit by means of three complete observations of the body, it is necessary to provide for the corrections due to jjarallax and aber- ration, so that they may be ai)plied in as advantageous a manner as possible. When the elements are entirely unknown, we cannot correct tiic observed places directly for parallax and aberration, since both of these corrections require a knowledge of the distance of the body from the earth. But in the case of the aberration we may either correct the time of observation for the time in which the light from the body reaches the earth, or we may consider the observed place corrected for the actual aberration due to the eond)ined motion of the earth and of light as the true place at the instant when the light left the planet or comet, but as seen from the place which the earth occu- pies at the time of the observation. When the distance is unknown, the latter method nuist evidently be adopted, according to which we apply to the observed apparent longitude and latitude the actual aberration of the fixed stars, and regard this place as corresponding to the time of observation corrected for the time of aberration, to be efTected when the distances shall have been found, but using for the place of the earth that corresponding to the time of observation. It will appear, therefore, that only that part of the calculation of the DKTKIOIINATIOX OF AN OIUUT. 221 (■IciiK'iits wliicli involves the titiics of oUscrvatioii will have to In- rc- |Kat('tl alter the eorresi»omliiifj: (li>taiiees of the hody from tlie earth liiive been found. First, then, by nieans of the a[»i)arent obliijuity of the eeliptie, the observed apparent rij^ht ascension and declination iiitist !)(' converted into apparent longitude and latitUv.e. Let /^ and ,t. respectively, denote the observed ajyparent loiif^itiicie and latitude; iiiid let 0,i be the true longitude of the sun, 2',, its latitude, and A', it.> distance from the earth, corresponding to the time of observation. Then, if / and ,^ denote the longitude and latitude of the planet or coinct corrected for the actual aberration of the fixed stars, we shall have /. -/>.„ = + 20".445 cos (i — 0„) sec ,? + 0"..^4:{ cos (;. - 281°) sec y tiie radius of the earth at the place of observation, ex[)ressed in i)arts of the etpiatorial radius as unity. Then ^„ is the right ascension and *'» fo> Zo = i>o *^in "o >*>» l>o' in which;r„-=8".5711G. J^'t J^ be the distance of the phmet or eomet from the true place of the observer, and J, its distance from the j)oint in the ecliptic to which the observation is to be reduced. Then will the co-ordinates of the place of observation, referred to this point in the ecliptic, be x, = (J, — J,) cos /? cos^, y, =:(J, — J„)coS(?sin-'., z, =:(J, — Jjsiu/5, the axis of x being directed to the vernal erpiinox. Let us now designate by O the longitude of the sun as seen from the point of reference in the eeli})tic, and by li its distance from this point. Theu will the heliocentric co-ordinates of this point bo Z= — A'cosO, r= — A'siuQ, Z = 0. The heliocentric co-ordinates of the centre of the earth are Xo = — Iio COS -0 cos ©0, Yo--=^ — Ii„ cos 2„ sin O 0, Zq = — lif, sin -,. But the heliocentric co-ordinates of the true place of observation will be X + x„ r+y„ Z-{-z„ or and, consequently, we shall have Zq + »0' i? cos O — (J, — Jj) cos /? cos A = jRg cos -„ cos ©o — Po sin tt^ cos b^ cos/j, a sin © — (J, — Jq) cos ,3 sin A = /^^ cos -„ sin ©, — />, sin -„ cos b„ sin /„, — (^/ — -Jo) sbi ,3 — Bo sin 2; — p^ sin -„ sin b^. If we suppose the axis of x to be directed to the point whose longi- tude is ©Q, these become DETERMINATION OF AN ORllIT. 223 E cos ( O ~ 0„) — ( J, 1„) cos ,5 cos (A — 0,) = Ji„ cos I'o — r„ ^i" "o t'"« ^ cos (/„ — 0„), li^m (0 — 0„) — (J, — Jo) co8,J sin (A — 0„) =^ (2) — /'„ sill T„ COS Ao sin (■/„ — 0o), — ( J, — Jj sin ,3 = /;„ sin i; — /v sin t:,, sin />„, from which 72 and may bo (letcrniined. Let us now put (J, — J„) COS /?=:/) ; (3) then, since rr^, 1\, and — 0o arc small, those equations may be rwliiccd to Ji = Dcos(X — 0„) — 7r„ />„ cos b^ cos (/„ — ©„) + 11^, EiQ — 0o) -- JJ sin ( A — J — 7r„ /;„ cos 6„ sin (/„ — 0„), — Z> tan ,J — Tu ;r,^ sin />„ + ii'„ 2;. Hence we shall have, if rr„ and -'„ arc expressed in seconds of arc, 5^sin6o-^ 206264.8 " R =i?o + -Dcos(>l — 0„) — -^' -aPoCOsbaCOsda—Qo. 206264.8 (4) = Oo4- 206264.8 D sin (X — 0„) — -, p„ cos b„ sin (/„ — QJ from which we may derive the values of and R which are to be u>'cd 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 ]>lane 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 ohsorvcd, but at a time different from ^„, to be determined by the interval which is required for the light to pass over the distance J, — Jy. Consequently we ought to add to the time of observation the quantity (J, — J„) 497'.78 = 497'.78 D sec jS, (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 k derived from equations (1) and the longitude © 224 THKOKKTirAI- ASTIIONDMY. derived from (4) slxtidd he rediieed l)v applying the eorreetioii lor mitntioi) to tlie mean ecpiinox of the date, an;h the jjlace of observation on the earth's surface in- tersects the plane of the ecliptic, are derived from the e»piations (I). Then the places of the stm and of the planet or comet are redii(0(l to the ecliptic; and mean e(piinox of a fixed date, and the results tliii.s obtained, toirether with the times of observation, furnish the data lor the determination of the elements of the orbit. When the distance of the body corres|)onding to cacli of the observations shall have been determined, the times of observation may bo corrected for the time of aberration. This correction is necessary, since the adopted places of the body arc the true 2)l!U'cs for the instant when the light was emitted, corresponding respectively to the times of observation diminished by the time of aberration, hut as seen from the places of the earth at the actual times of observation, respectively. When (9 - 0, the erpiations (4) cannot be applied, and when tiie 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 magnitutlc, it will be advisable, if more suitable observations are not available, to neglect the correction for parallax and derive thy elements, using the uncorrected [)laccs. The (li.sl:u>ces of the body from the earth which may then be derived, will (:nai>.<; us to Jipi)ly the correction for parallax directly to the observed plsijos of the body. When the approximate distan'.'cs 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 tliese places are adopted, it will be necessary, for the most accurate solution possible, to retain the latitude of the snn in the formula) which may be required. But since some of these formulae acquire greater simplicity when the sun's latitude is not introduced, if, in this case, we reduce the geocentric places to the PKTKUMINATION OF AN OlMUT. 225 piiiiit ill wliicli a |H-rp('i)(]irtiliii' let i'all from tlic ctMitir of tlio earth t(i tlic plaiK' of tli(> ecliptic cuts that phuio, tiio loiijritiide of the suii will reniain uuchaiij;eros.sed in seconds. This correction having been applied to tlie geocentric hititude, the latitude of the sun becomes . 1=0. The correction to be applied to the time of observation (already (liiiiinished by the time of aberration) due to the distance J, - J„ will be absolutely insensible, its maximum value not exceeding 0'.0()2. It should be remarked also that before apj)lying the equa- tion (()), the latitude -'„ should be reduced to the fixed ecliptic which it is desired to adopt for the definition of the elements which deter- inino the position of the plane of the orbit. 78. When these })relimiiiary corrections have been applied to the (lata, we are prepared to proceed with the calculation of the elements of the orbit, the necessary formula) for which we shall now investi- gate. For this purpose, let us resume the equations (G).^ ; and, if wc; multiply the first of these equations by tan /9 sin/" — tan ,9" sin/, the second by tan/9" cos/ — tan/9 cos/", and the third by sin(^ — /"), and add the products, we shall have (\=-)iR (tan ,5" sin (A — O) — tan ,S sin (A" — ©)) - r (tan ,5 sin ( /" — A') — tan ,3' sin (A" — A) -\- tan /5" sin (A' — A)) ~Ii' (tan ;5" sin (A — ©') — tan ,3 sin (A" — ©')) -}- n"R" (tan ,i" sin (A — ©") — tan ,3 sin (A" — ©")). (7) It should be observed that when the correction for parallax is applied 15 226 TIIEOR ETIO A L A ST ItONOM Y. to the rt\acc (if the sun, />' is the projection, on the plane of the celiptie, of the distunee of the body from tlie point of reference to whiclj the observation lias been reduced. Let us now desijunate l)y K the lon', or of J', when the values of n and n" have !)een determined, since a^ and A'' are derived from and //', when the corrcetio.is for parallax arc applied to the places of the sun, being as already noticed in the case of //. 71). If we multiply the first of equations (6).j by sin ©" tan^i", the second by — cos O" tan^?", and the third by a'niU" — 0")j ''"*^^ add the products, we get 0-:^ H/'(tan;j'''sin(0''-A)---tan,?sin(0"— A"))— Hieva'.M}"sin(0"--0) -// (tan ii" sin ( ©"— A' ^-tan ,3' „in (©"—A" ))^~U' tan [i" sin (.©"—©'), (13> which may be written 0=^»/'(tanr3sin(A"— ©")—tan,5"sin (A—©"))— »ii!tan,j" sin (©"—©) + />'(tan,J"sin(A'— ©") — tan ,:'osin (A"-^ ©")) — ^'(tan,j'— tan,J„; sia(A" ~ ©") -j- A" tan,?" sin C©"— ©'). Introducing In<-o tliis the v.dues of tan ^5, tan^i", and tan,?u in terras of / and A", and reducing, the result is — ?(/> sin (A"— A ) sin ( © "— K) — uR sin (©"-©) sin (A"— A') -^ />'sin a"— I' ) sin ( ©"— K) - i>'a, sec,j' sin (A"- ©") + A' sin (©" — © 'j sin (A" — A^;. Tlioreibre we obtain ''~n\ sin (A"" — A) + sin'CA" — sin(A"— O") A) sin (©"—A')/ sin( A"— ii) A^in (©"—©')— »7^sin (Q"~Q) sinCA" — A)sin(u" — A') " ' n But, by means of the equations (9)3, we derive R' Hin(©" — ©') — nR sin(©" —Q) = (N—n)R sin (©"— ©), 228 THL.)RETICAL ASTRONOMY. and the preceding equation reduces to _///sin(>l"-;/) fl„ sees' P — — znrrvr rr ~r sin (/"—©") ■X) sin (A"—;.) sin(0 — O") \ "-K)j 4-1 I ^^\ ^^si"^ Q"— ) sin (/" — K) '^\ nj~ sin (/'' — A) sin CO" — A') " (14) To ol)tain an expression for p" in terms of />', if we multiply the first of equations (6)3 by sin O tan /3, the second by — cos © tan ,'i, and the third by sin {?< — O), and add the products, we shall have 0=nV'(tan/3sin(r—O)— tan /S" sin (A—©))— H"i2" tan /3sin(O"—0) — o'(tan/Ssin (/'—©)— tan/5' sin (A— Q))+/i' tan, S sin (©'—©). (lo) Introducing the values of tan/9, tan/9', and tan/5" in tex-ms of ^ and /, and reducing precisely as in the case of the formula already fouD'j. for /), we obtain ^ ~«"\sin(A"— r sin (A — © ) X) sin(/."— A)'sin(© — A') o„ sec ii ) / N" \ A^"sin(©"— © )sin(A — A^X ~^\ n" J sinU" — A)sin(© (16) K) Let us now put, for brevity. 6 = is? sin (© — A) R'sm(O'-K) or , i?"sin(©"-A) . d = , / : sec /5' sin (/"—;.)' h = /?i?"sin(©"-0) aoSinCA"-;.) M, sin (A'' -//) R" sni (;." - ©") I J ' m: ' sin [)." — I) sinU'— A) M,= sin (;." — /) /j.sin(;."— AO -/ d ) Asin(-l — ©) h ,,„ /( sin(-i - -A-) ~ « (17) and the equations (11), (14), and (16) become p' sec /5' = c + ?i6 -f n"d, P ^- m: n + i>/. + M. (-f)' '(-?')■ (18) If n and n" are known, these equations w'll, in most j^'ASCS sufficient to determine (>, ft', and (> be DETERMINATION OF AN ORBIT. 229 80. It will be apparent, from a consideration of" the equations which have been derived for p, (»', and (>", that under certain circum- stances they arc inapplicable in the form in which they have been given, and that in seme cases they become indeterminate. When the groat circle passing chrough the first and third observed places of the body passes also tiirough the second place, we have a„ = 0, and equation (11) reduces to n' R"m\{Q" — K) + nRsm{Q — K) = R' siniQ' ~ K). If the ratio of n" to n is known, this equation will determine the quantities themselves, and from these the radius-vector 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= ©', or K^^ 180° + ©', and hence or 7i"R" sin (Q" — Q') — uR sin (Q' - H^ _ R s in (Q^-Q) n "ie"sin(0" — O'}' o) = o, from which it appears that the solution of the problem is in this case impossible. If the first and third observed places coincide, we have ^ =^ ?." and ^ - ^", and each term of equation (7) reduces to zero, so that the pv '/1cm becomes absolutely indeterminate. Consequently, if the i!ta rt)' nearly such as tc render the solution impossible, according t u:( ( onditions of these two cases of indetermination, the elements 'v!'irh may be derived will he greatly affected by errors of observa- tion. X<' however, / is equal to ?/' and /3" differs from fi, it will be possible to derive f/, and hence f> and ft" ; but t le formula; Avliich luive been given requii some modification in tl is particular case. Thus, when k /' , we i. K=X" = ^, 7 = 90°, and ;9,^-=90' and hence «„, as determined by equation (iO), becomes -. Still, in tliis case it is not indeterminate, since, by recurring to the original equation (9), the coefficient of f>', which is — «„ sec/9', gives o„ = sin /3' cot 7 — cos /3' sin (/' — K), au'l when X = ^", it becomes simply cfg = — cos ,5' sin (/' — K). (19) 230 THEORETICAL ASTRONOMY. Wlicnover, therefore, the differeace /" — X is very small compared with the motion in latitude, a„ should be computed by means of the equation (19) or by means of the expression Avhich is obtained directly from the coeflicient of p' in equation (7). When / =- /" -=: A', the values of .¥„ J//', M.„ and M," ca:-iot be found by means of the equations (17); but if we use the original form of the expressions for p and p" in terms of p', as given by equations (18) and (15), without introducing the auxiliary angles, we shall have _(>' tan,S'; : n tau (J siu ( -G^') — tan,rsin(/'— 0") 0") N n Hence tan z'j" sin (;. — ©'') "^ \ ' 11 ) tan /? sin ( A" — ©'') — tan /"J" sin (/. ^n /5^in (/' — ©) — tan ,5' sin (A — ©) tan ,i sinT^/'^^o7-~taii^i'' i*i^n (A^^^oT / . _ iV; V irtaD/Ssin(Q^'-G) "'" \ 'n" j tan /5 sin U" — © ) — tan /j" sin (;. 1/ — ^"Zi.'"iA" ~ Q' O — ta n;?''sin(/^— Q") 0")' -O) M, tan /J sin (/" — 0") — tan ,5" sin (/ ,f tan /? sin (^' — ©) — tan ,S' sin (k ©")' -©)• iK iW = tan /5 sin (A" — ©) — tan ,'i" sin (A R tan /5" sin ( 0" — 0) tan /? sin (■ ' — ©") — tan ,f" sin (A — ©")' i^'^tan /5 sin ( ©" — ©) lanT^ sin (A"-"- ©j — tan /5" sin (A — ©) ' (20) arc the expressions for 3/,, 31/', 31.^, and 31.," which must be used when / - -/" or when X is veiy nearly equal to /"; and then p and //' will be obtained from equations (18). These expressions will also be used when A" — / =^ 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 ])lace of the sun, the last two of the equations (18) become indeterminate, and other foi .luke must be derived. If wc multiply the second of equa- tions (7)3 by tan/5" and the fourth by — sin (A" — ©'), and add the products, then multiply the second of these equations by tan ,5 and the fourth by — sin (A — ©'), and add, and finally reduce by means of the relation wc get NE sin (©' - 0) = N"R" sin (©" - ©'), detj:umixation of ax orbit. 231 (>'_ tan ,?" sin (// — Q ') — tan ,5' sin ( )!' — © ') n ' tun ,y' sin {?. — ©') — t"an /J sin ( A"~— Q ') /r tan;/' sin (O" Q') sin ( / — ©') — tan /J sin (A" - ©')' „^P^ tan ;/ sin ( A — ©') - tan /? sin (/■' — ©') ^' n" ■ tan ;5" sin (I — ©0 — tan /J sin (/" - - ©') ^" ^ "^ \ n," N" } tan (J" sin (). — ©') — tun'^f siu('<"— ©')' Tlicsc equations ai'e convenient for determining ft and f/' from p' ; Ijiit they boeome indetcriniiv:^ when the great circle passing tlu'ougli the extreme phices of the body also passes tlirough the second place of the sun. Therefore tliey will generally be inapplicable for the cases in which the equations (18) fail. If we eliminate f/' from the first and second of the equations (6)3 wc get = )ip sin (/." — k)—nR sin (/." — ©) — p' sin (/." — A') + R' sin (A" — ©') - H"Ii" sin (A" — ©"), from which we derive // sin (A"-/') ''■^.T-sinll'^-A) ^22) uE sin (/" — ©) - - .R' sin (A" — Q') + n"R" sin (>." — © ") "^ /i sin (A" — A) Eliminating ^o between the same ecpiations, the resu^~ is ^=,7' f/ sin(A' — A) sin (A" — A) 7iR sin (A ©) — i?' sin (A - ©') + n"R" sin (A (23) ©") n" sin (A" — A) These formula} will enable us to determine f> and ,0" frron ,n' in the special cases in which the equations (18) and (21) are inapplicable; but, since they do not involve the third of equations (6)„ they are not so well adapted to a complete solution of the jtroblem as the lOrmuUe previously given whenever tliese may be applied. If we eliminate successively p" and () between the first and fourth of the equations (7).j, we get _ p' tan ;5" cos (A' ~ ©') — tan ,3' cos (A" — ©') n tan /j" cos (A — ©') — tan fi cos (A" I ^}^^^ niicosC©'— ©) — ii'- ©') vi"/i" cos (©"-©') n tan /5" cos (A — © ') — tan ,3 cos (A" — ©') „ p' tan if cos (A — ©') — tan ,9 cos (A'— © ') (24) n" tan , J" cos ( A — © ') — tan ,5 cos ( A"— © ' ) tan /5 „ie cos ( © ' — © ) — i?' + «"i2" cos (©"—©') n" tan ;?" cos (A — ©') — tan /? cos (A" — ©') 232 TIIEORETU AL ASTRONOMY. Mliic'li may also be used to dctorniine /> and ft" when the equationa (18) and (21) cannot be applied. When the motion in latitude is greater than in longitude, the.se equations are to be preferred in!«tead of (22) and (23.) 81. It would appear at first, without examining the quantities in- volved in the formula for (>', that the equations (26)., will enable us to find 11 and n" by successive approximations, assuming first that n = ,-> T n r" and from the resulting value of p' determining /•', and then carrying the approximation to the values of n and n" one step farther, so as to include terms of the second order with reference to the intervals of time between the observations. But if we consider the equation (10), Ave observe that a„ is a very small quantity depending on the difference ^' — /9„, and therefore on the deviation of the observed path of the body from the arc of a great circle, and, as this a^jpears in the denominator of terms containing n and n" in the equation (11), it becomes necessary to determine to what degree cI approxi- mation these quantities must be known in order that the resulting value of (»' may not be greatly in error. To determine the relation of Oy to the intervals of time between the observations, we Jiave, from the coefficient of p' in equation (7), Oo sec ;i' = tan ,3 sin (A" — k') — tan /S' sin (A" — A) -[- tau/S" sin (A' — /). We may put tan ,3 = tan ;/ — At" -\- Br'"' — ...., tan ,5" = tan /5' -\- At -\- Bt' -\- . . . . , and hence we have flo sec /5' = (sin (/." — A') — sin (A" —A) + sin (A' — A)) tan /5' -f (r sin {X'—?.) — r" sin (A"— A')) ^+(r^ sin (A'— A)+r"^ sin {a"— a')) B-\-. ., which is easily transformed into cfo sec /5' = 4 sin i (A' — A) sin ^ (A" — A') sin A (A"— A) tan ,5' (2o) -f- (r sin {k'—k)~T" sin (A"— A')M+(r" sin a'—).)+T"-' sin (A"— A'))J5-f. ... If we suppose the intervals to be small, we may also put sinV(A" — A) = ^(A" — A), and sin (A" — k)-= A" — A, sin (A' _ A) = A' — A. DETERMINATION OF AN ORBIT. Furtlicr, we may put k = A' - ylV + B'r'" — . .., A" = A' 4- yl'r + 5V + 233 Sulistituting tlicse values in the equation (25), neglecting terms of tlie fourth order with respect to r, and reducing, we get a, ^ tt't" (4.4" tan ,5' + A'B — AB') ccs ;5'. It appears, therefore, that a^ is at least of the third order Avitii reforonce to the intervals of time between the observations, and that an error of the second order in the assumed values of n and n" may produce an error of the order zero in the value of {>' as derived from 0(Hi!ition (11) even under the most favorable circumstances. Hence, in general, we cannot adopt the values ?i = -r> II -T) T omitting terms of the second order, without affecting the resulting value of f/ to such an extent that it cannot be regarded even as an apj)ro.\imation to the true value ; and terms of at least the second order must be included in the first assumed values of a and n". The equation (28)3 gives (26) dr omitting the term multiplied by ,-. which term is of the third order with respect to the times ; and hence in this value of — , only terms .'at least the fourth order are neglected. Again, from the equations (26)3 we derive, since r' = r + r", n + n" = 1 + -^ji, % ,'3' (27) in which only terras of the fourth order have been neglected. Now the first of equations (18) may be written : /,'sec/5' = (?i + n") 6 + 1+^ (28) in which, if we introduce the values of — and n + w" as given by (26) and (27), only terms of the fourth order with respect to the 234 THEORETICAL ASTRONOMY. times will be noffjec^ted, and consofjuently the rcsultiii}^ value of r/ will 1)0 att'cctcd with only an error of the ^eeond order when '" (n + n" — 1), we may adopt, for a first approximation to the value of />', P = (;]0) and 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 ori)it, the in- tci'vals should be as nearly equal as possible, since the nearer tliov approach to equality the nearer the trutli will be the first assumed values of i* and Q, thus facilitating the successive approximations; and when «„ is a very small quantity, the equality of the intervals is of the greatest importance. From the equations (29) we get 71 -'■- 1+P\ ^2?'='/' n" = nP; and introducing P and Q in (28), there results ^Pd + P' (31^ ,'sec.S' = (l + ^)^ (32) This equation involves both f>' and r' as unknown quantities, but by means of another equation between these quantities f/ may ho eliminated, thus giving a single equation from which r' may be found, after which p' may also be determined. Dtn'KRMIXATIOX OF AX ORBIT. 235 82. Let ^' reprc'si'iit thu aiiijlo at the earth between the sun and ])lan(>t or co'net at the sceunJ ub.servation, and we shall have, from the eqiiations (OS),, tan ,'/ tan It' --• '.— ;-, Hin(/.'— O')' , tauU'— ©') tan 4 = ; , COSU' COS 4' — COS (i COS (// — ©'), (33) l)v means* of which wc may determine ^', whicli cannot exceed 180°. Since COS,*' is always positive, cosa^' and cos(/'— ^O') must have the same sij»'n. Wc also have r'^ = J'^ + i2'» — 2 J'i?' cos 4', ^ which may be put in the form r'^ = (/>' sec ,5' — E cos ijy + E- siu'^ 4', from which we ffet // sec /5' = E cos ^' ±iV r" — E' sin^ 4'. (3t) Substituting for p' sqc^3' its value given by equation (32), we have . {i-^§,y'^^-c^Ecos^'±V7^=nr^i^. For brevity, let us put '■" ~ '1 + P' Cg C = Kq, and we shall have k„ — 4 = E cos 4' =h 1/7^^— ie'^sin'H'. (35) (36) AVhen 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 (lata furnished by observation. We shall now represent by 2' the angle at the planet between the sun and earth at the time of the second observation, and we shall have E sin V sin z' (37) 236 THEORETICAL ASTRONOMY. Subsstitutiii}; this value of /•', in the precoding equation, there results and if we put (/'o ~ K cos 4.') siu z' ^ y^" sin 4' cos z' = A . 3-7, ij„ sin r = J{' m\ V, 5;„ cos C =: /•„ — W COS 4', ,o/rsin»V' (38) (39) JH„ the condition being imposed that wip shall always be positive, we have, finally, sin {z =h ») = (Ho sin*/. (40) In order that m^ may be ])ositive, the quadrant in which ^ is taken must be such that -y^ shall have the same sign as /„, sinca sin \^' is always positive. From equation (37) it appears that sin z' must always be positive, or3'<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 A'^ Therefore, /_sm iz' + 40 _ R' sin iz' -\- ^') sin 4.' sin z' , i2'sin(2'+4') sm2 cos /5', (41) and, since p' is always positive, it follows that sin (2' + 1^') must be positive, or that z' cannot exceed 180° — '^'. 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 /5' = 0, i|/' = 180° when the body is in opposition, and '^' =- 0° when it is in conjunction. Consequently, it becomes impos- sible to determine r' by means of the angle 2'; but in this case the equation (36) gives k I ps R' + r', when the body is in opposition, the lower sign being excluded by the condition that the value of the first member of the equation must be positive, and for ^' = 0, I I '0 '^0 ^ R' the upper sign being used when the sun is between the earth and the piyrKioirxATioN of an ounrr. 237 pliinot, and the lower sij^n when the i)l!iiiet is between the eartli and tlic snn. It is hardly necessary to remsirk that tlie case of an obser- vation at tlie superior eonjnnetion when /9' = 0, is i)hysieully impos- ,und from these eijuations by trial ; and then we shall have p' = ,•' - W when the body is in opposition, and ,,' = R' — r' when it is in inferior conjunction with the sum. For the case in which the great circle passing through the extreme observed places of the body passes also through tiie middle place, wliicii gives o^ -- 0, let us divide equation (32) through by c, and we have \ ^ 2/'» / 1 + P />' sec /5' The equations (17) give ^>_ J?siu(0 —K ) ~c'~ A"sinCO'— isT)' and if we put we shall have 'L + p'l c ' c d c t-'ni i2"sin(0"j ie'"sin(0'- -K) 1+P since c = co 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 ?n„ sin* z' ■ sm z cos ; = cos z sm 2.'J8 TIIF/HJr.TK A r, A.vrnoNOM V. and, 1)V squaring; iiixl ri'ilucinjrj this becomes »/( 'siii''2' 2)ii„ cos X sin* z' -\- m\^ z' — sin''' C — 0. WluMi ^ is witliin the limits — 90° and -f 90°, cos^ will ho ])ositi\v, and, «/„ hcinji' always |)()sitivo, it appears from tlie aljfehraie si^ns i,[' the terms of the eijuation, aeeordini;- to the theory of (>(|nation-;, th;it in this ease; there cannot he more than tl)nr real rc»ots, of which thnv will l)(! positive and (»ne neu,ative. When ^ exceeds the limits - \HP and I 90°, cos ^ will ho ne!j;ative, and hence, in this ease als(», there cannot he more than fonr real roots, of which ono will he [xtsitivo and thre(! nef:;ativc. Further, since sin" ;J^ is real and ])ositive, tlurc must he at least two real roots — one [)ositiv(! and the other nej^ative — whether cos ^ he ne}j;ative or i)ositive. We may also remark that, in finding the roots of the e(piation ( 10), it will only ho necessary to solve the cfjuation sin (z' — C) = ■»»„ sin* /, (4:!) since the lower sign in (40) follows directly from this hy s;d)stittiti- 180° — z' in j)lace of :;'; and hence the roots derived from tiiis w comi)rise all the real roots belonging to the general form of the equation. The observed places of the heavenly body oidy give the diroctina in s|)ace of right lines passing through the places of the earth iiiul the corresponding places of the body, and any three points, one in each of these lines, which are situated in a plane passing through tlic centre of the sun, and which are at such distances as to fullil the condition that the aroal veUx'ity shall be constant, according to tlio relation expressed by the equation (30),, must satisfy the analytic;!! conditions of the problem. It is evident that the three places of tlie 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 Further, it follows from the equation (37) that this root must be s' = 180° — 4' ; and such Avould be strictly the ease if, instead of the assumed values of P and Q, their exact values for tlie orbit of the earth were adopted, and if tiio observations were referred directly to the centre of the earth, in the correction for parallax, neglecting also the perturbations in the motion of the earth. DKTKRMINATION OF AN ORIJIT. In the cn.se of the enrth, 239 n = N = /?'W" sin (©"-©) l{R"m\(Q"'-Q)' anil iht' coinplctt} vulucs ol' P ami Q beeomo /.'/."siii(0' - ©) K{-Ui y J{Ii"mi(Q"~Q) ''-^)' and !«iiK'e the appi'oxiiniitc values T Q =- rr" (litfcr l)ut little from these, as will ai)p(>ar from the equations (27)^, tlKi'f will be one root of o(juation (43) whieh gives z' nearly eijual to 1S0° — a]/'. This root, however, cannot satisly the physical con- (litiniis of the problem, wliich will rc(pilre that the rays of light in coming from the planet or comet to the earth shall proceed from points whieh are at a considerable distance from the eye of the ol)scrver. Further, the negative values of sin ;:' arc (excluded by the nature of the problem, since r' must be positive, or z' < 180° ; and of the three positive roots which may result from equation (43), that being excluded which gives s' very nearly equal to 180° — ^', there will remain two, of which one will be excluded if it gives z' greater tiian 180° — 1|/', and the remaining one Avill 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° — -y^', in which case both will sausfy the physical conditions of the problem, and hence the observations will be satisfied by two wholly different systems of elements. 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 sanie 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 TIIKOIU'/riCAT. ASTRONO>fY, the approximate viiluos of P and Q are computed directly, by moam of ap])ro.\inuite elemeiu.s already known, from the equations rr' )i\n{u'—n) ♦ />•" sin («"—«')' n _ 9 '3 / rr'»in(u'—u) + r'r" si n(u"— 7t') \ V-^'- \ rr" sfn (.«"- «) V' (-i 44) it may occur that eos ^ is negative, and the positive root will actually belontr to the orbit of the eomct. The condition that one value of s' shall be very nearly e([ual to 180° — i^', requires that the adopted values of 7^ 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 Avill correspond to the orbit of the earth. 84. We may find an expression for the limiting values of ?/!,, and ^, within which ecpiation (43) has four real I'oots, and beyond which there are only two, one positi\e and one negative. This chaniie iu tlie number of real roots Avill take place A\hen there are two ccpial roots, and, consequently, if we proceed under the supposition that equation (43) has two c(pial roots, and find the values of >/)(, and ^ which Avill accord with this supposition, wo may determine the limits recpiired. Differentiating equation (43) with respect to z', we get cos (z' — X) --- 4>«, sin V cos z' ; and, in the case of equal roots, the value of must also satisfy the original equation as derived from tliis sin (z' — :) ^oSinV. To find the values of vIq and ^ which will fulfil this condition, if we eliminate m^ between these equations, we have sin z* cos {z' — = 4 cos z' siu (2' — i'), from which we easily find sin (2/ — C) = « sin C. (46) This gives the value of ^ in terms of z' for which equation (43) has DETERMINATION OF AN ORBIT. 241 equal roots, and at whicli it cea.ses to have four real roots. To find the corresponding expression for ?<*„, we have m„ sin (z' — C) cos (/ — C) sin *z' 4sinV cof z' in which we must use the value of ^ given by the preceding equation. Now, f'nce sin (2s' — i^) must be within the limits — 1 and H 1, the limiting values of sini^ will be + ? iuid — §, or ^ must be within the limits + 30° 52'.2 and — 86° 52'.2, or 143° 7'.8 and 21G° 52'.2. If (^ is not contained within these limits, the equation camiot have equal roots, whatever may be the value of m^, and hence there can only be two real roots, of which one will be positive and one negative. If for a given value of ^ ■ vc compute z' from equation (45), and call this z^\ or sin (22/ — ») = 3* sin C, we may find the limits of iiic values of 77)^, within which equation (43) has four real roots. The equation for zj will be satisfied by the values 2V-:, 180° -(22,-:); and hence there will be two values of ?»„, which we will denote by Hii iMid JHj, for which, with a givei; value of !^, equation (43) will have equal roots. Thus we shall have '"i = sin (2,,' — O sin*2g' ' and, putting in this equation 180° — (2:;/ — c^) instead of 2z^/ — i^, or 90° — [zq — ^) in place of zj, cos z^' ll follows, therefore, that for any given value - .' ^, if »(„ is not within the limits assigned by the values of »(, and m.,, equation (43) will oidy have two real roots, one positive and one negative, of which the latter is excluded by the nature of the problem, and the tbnnor may belong to tlie orbit of the earth. But if P arid (^ ditler so nuich from their values in the case of tlie orbit of the earth that z' is not very nearly ecpial to 180° — 4''> ^''"^ positive root, when ^ exceeds the limits + 36° 52'.2 and - 36° o2'.2, may actually satisfy the eonditioris of the problem, and belong to the orbit of the body observed. 16 242 THEORETICAL ASTBONOMY. AVlion C is within the limits 143° 7'.8 and 216° 52'.2, there ^vill be four real roots, one positive and three negative, if m^ is within the limits «i, and m.^ ; but, if riiQ surjrasscs these limits, there will be only two real roots. Table XII. contains for values of C from — 36° 52'.2 to + 36° 52'.2 the values of my and r,i.,, and also the values of the four real roots corresponding respectively to ?/ij and wij- In every case in which equation (43) has three positive roots and one negative root, the value of m^ must be Avithin the limits indicated by m, and m^, and the values of z' will be within the limits indicated by the quantities corresponding to mj and nij for each root, which Ave designate respectively by 2/, z./, z^', and z/. The table will show, from the given values of hIq and 180° — 4'', whether the probleia admits of two distinct solutions, since, excluding the value of z', which is nearly equal to 180° — -x//'? ^^^ corresponds to the orbit of the earth, and also that which exceeds 180°, it will appear at once whether one or both of the remaining two values of z' will satisfy the condition that z' shall be less than 180° — 1]/'. The table will also indicate an aj)proxiraate 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, Ave have />' = 0, and henco from (36) Ave derive k Jo Substituting this value for k^ in the general equation (32), Ave have and, since p' must be positive, the algebraic sign of the numerical A'alue of /„ Avill indicate Avhethcr r' is greater or less than li'. It is easily seen, froni the forniulre for /„, b, d, &c., that in the actual application of these formula;, the intervals between the obserA'ations not being very large, I^ Avill be positive Avhen /9' — /9„ and sin (©'— K) haA'c contrary signs, and negatiA'C Avhen /9' — /9q has the same sign us sin (O' — K). Hence, Avhen O' — A' is less than 180°, /•' must be less than R' if /9' — /9„ is positive, but greater than R' if /9' — /9^, is negative. When O' — .jfiT exceeds 180°, r' Avill be greater than /i' if /9' — /?(, is positive, and less than R' if /9' — /9q is negative. We may, therefore, by means of a celestial globe, determine by inspection Avhether the distance of a comet from the sui* is greater or less than DETER51INATION OF AN OUBIT. 243 tliat of the earth from the sun. Thus, if we pass a great circle tlirou(rh the two extreme observed places of the comet, ;•' must bo greater than W when the place of the comet for the middle observa- tion is on the same side of this great circle as t'le point of the ecliptic which corresponds to the place of the sun. But when the middle place and the point of the ecliptic corresponding to the place of the sun are on opposite sides of the great circle passing through the first and third places of the comet, r' must be less than R'. 85. From the values of p' and /•' derived from the assumed values t" P = — and Q = Tr'', >vo may evidently derive more approximate values of these quantities, and thus, by a repetition of the calcula- tion, make a still closer approximation to the true value of p'. To derive other expressions for P and Q which are exact, i)rovided that / and p' are accurately known, let us denote by s" the ratio of the sector of the orbit included by r and r' to the triangle included by the same radii-vectores and the chord joining the first and second places ; by s' the same ratio with respect to r and r", and by « this ratio with respect to r' and >•". These ratios s, s', s" must neces- sarily be greater than 1, since every jjart of the orbit is concave toward the sun. According to the equation {30)„ we have for the ureas of the sectors, neglecting the mass of the ho'^y, 2^" V'P> i^'v% ■z' I p> and therefore we obtain s"[rr'-]=T"^p, s'[rr"-]=T'^/p, s[,V'] = T|/>. (^46) Then, since we shall have and, consequently, [rV] [»t"] r s n = -, • -, T S n" = t" s' T S P = t" s T S- Substituting for 8, s', and s" their values from (46), we have (47) (48) o_o,,.[^v'] + K]-K'] III' (49) 244 THEORETICAL ASTRONOMY. The auf^iilar distance between tlic perihelion and node being denoted by 0), the polar equation of the conic section gives P P 1 -f e cos (u — w), 7 = 1 + e cos (u' — Hi), ^ = 1 + e cos (it" — w). (50) If we multiply the first of these equations by sin [u'' — w'), the second by — sin [u" — u), and the third by sin («' — u), add the products and reduce, we get P P sin (it" — u') — ■'-, sill (m" — «) -f ^ sin («' — u) = sin {u" — it') P and, since sin («" — ?f) + sin (it' — u) ; sin (u" — «') = 2 sin \ («" — it') cos J (it" — it'), sin (ii" — i() — sin ( u' — it) = 2 sin A (it" — it') cos h (it" + it' — 2u), the second member reduce^: to 4 sin ^ (it" — it') sin I (it" — it) sin A (it' — it). Therefore, we shall have Arr'r' sin -^ (it" — i'') sin \ (it" — it) sin ^ (it' — i',) P r'r" sin (it" — «') — rr" sin (it"— it) + r/ sin (it' — it)* If we multiply both numerator and denominator of this expression by 2rr'r" cos ;\ (it" — it' ) cos ^ ( "" — it) cos \ (it' — it), it becomes, introducing [/v'], [^■'■"], and [r'r"], _ Jr'/'Mn-'2, M 1 P ~ [r'r"] + [r/] — [i-r"] " 2rr'r" cos A (it"— it') cos A (it"— it) cos ^ (it'— it)' Substituting this value of jj in equation (49), it reduces to TT ^ "~ W ■ r,." nns I r^i." — ss" rr" cos ^ (it" — it') cos \ (it" — it) cos \ (it' — it)' (51) 86. If we compare the equations (47) with t he formula (28)3, we derive g"_1 .t'-t'" , (t»4-t'^') rf/ •" "^ * yfcr'* ■ rf< (52) DETERMIIiATION OF AN ORBIT. 245 Consequently, in the first approximation, we may take = 1. If the intervals of the times arc not very unequal, this assumption will differ from the truth only in terms of the third order vith respect to the time, and in terms of the fourth order if the intervals are equal, as has already been shown. Hence, we adopt for the first ai)proximation, q = rr", the values of r and 7" being computed from the uncorrected times of observation, which may be denoted by i^, fj, and tj'. With the values of P and Q thus found, we compute »•', and from this p', p, and [)", by means of the formula) already derived. The heliocentric places for the first and third obser'^ations may now be found from the formuke (71)3 and (72)3, and then the angle n" — u between the radii- vectores /• and r" may be obtained in various ways, precisely as the distance between two points on the celestial sphere is obtained from the spherical co-ordinates of these points. When u" — it has been found, we have iir sin {xc' — ?t') = -— sin («" — «), n"r" (53) sm {u — u) = —J- sm \u — u) from which •%' W and u' — u may be computed. From these results the ratios s and s" may be computed, an ^ then new and more aj)|)ro>-imate values of P and Q. The value oi' u" — u, found by tiidng the sum of u" — u' and u' — ti as derived from (53), should aifrce with that used in the second members of these equations, within the limits of the erroi's which may be attributed to the logarithmic tables. The most advantageous method of obtaining the angles between the radii-vectores is to find the position of the plane of the orbit directly from /, I", b, and b", and tlien compute u, u', and a" directly from SI and i, according to the first of equations (82),. It will bo expedient also to compute r', Z' ai i b' from />', //, and ,9', and the agreement of the value of /■', t» ns found, with that already obtained from equation (37), will check the accuracy of part of the numerical 246 THEORETICAL ASTRONOMY. calculation. Further, since the three places of the body nuist be in a plane passing through the centre of tlie sun, whether F and Q arc exact or only approximate, wa nuist also have tan h' = tan i sin (/' — Q,), and the value of i^ derived from this equation must agree with that computed directly from ()', or at least the difference should not cxcood what may be due to the unavoidable errors of logarithmic calcula- tion. We may now compute n and n" directly from the equations n = r'r" sin (ft" — «') rr" sin {u" — %i)^ „ rr sni(?<'— lO ,... rr sm {u — u) but when the values of u, d', and u" are those which result from the assumed values of P and Q, the resulting values of n and n" \vill only satisfy the condition that the plane of the orbit passes through the centre of the sun. If substituted in the equations (29), they will only reproduce the assumed values of P and Q, from which they have been derived, and hence they cannot be used to correct them. If, therefore, the numerical calculation be correct, the values of n and n" obtained from (54) must agree with those derived from equa- tions (31), within the limits of accuracy admitted by the logarithiuii' tables. The differences u" — u' and u' — u will usually be small, and hence a small error in either of these quantities may considerably aft'ect the resulting values of n and n". In order to determine whether the error of calculation i,s within the limits to be expected from the logarithmic tables used, if we take the logarithms of both members of the equations (54) and differentiate, supposing only n, n' ti and v! to vary, we get rfloge?^ = — cot(w" — li!) dii' , d lege n" = + cot (?t' — u) du'. Multiplying these by 0.434294, the modulus of the common system of logarithms, and expressing du' in seconds of arc, we find, in units of the seventh decimal place of common logarithms, dlogn = — 21.055 cot («" — v!)du', d log n" = + 21.055 cot (V — u) du'. If we substitute in these the differences between logn and log?i" as found from the equations (64), and the values already obtained by DETERMINATION OF AN ORBIT. 247 means of (31), the two resulting values of (hi' should agree, and the magnitude of du' itself will show whether t)ic error of calculation exceeds the unavoidable errors due to the limited extent of the logarithmic tables. When the agreement of the two results for n and n" is in accordance with these conditions, and no error has been made in computing 71 and n" from P and Q l)y means of the equa- tions (31), the accuracy of the entire cideulation, both of the (jiuiu- tities which depend on the assumed values of P and Q, and of th<>^e which are obtained independently from the data burnished by observa- tion, is completely proved. 87. Since the values of n and n" derived from equations (o4) cannot be used to cori'ect the assumed values of P and Q, from wliich r, r', «, u', &c. Jiave been computed, it is evidently necessary to comj)ute the values for a second approximation by njcms of the series given by the equations (26),, or by means of the ratios « and s". The expressions for n and n" arranged in a series with respect to the time involve the differential coefficients of r' w ith respect to t, and, since these are necessarily unknown, and cannot be coiiveniently determined, it is plain that if the ratios ■>>• and s" can be readily found from r, r' , v" , u, u', u" , and r, r' , z", so as to involve the relation between the times of observation and the places in tiie orbit, the_,' may be used to obtain new values of P and Q by means of equations (48) and (51), to be used in a second approximation. Let us now resume the equation M=E-^-e^mE, or h{t—T) 3 and also for the third place k{f—T) Subtracting, we get a* 3 a^ = £ — c sin E, ^E" — esmE". E—2esm^ {E" — E) cos A (E" + E). (55; This equation contains three unknown quantities, a, e, and the dif- ference E" — E. We can, however, by means of expressions in- volving r, r", u, and m", eliminate a and e. Thus, since p-—a{l — e^), we have T'Vp =- a?V^l — e' {E" —E—2esm^ {E" — E) eos J- (£'■ -f- E)). (56) 248 THEORETICAL ASTIIONOMY. From the equations l/r sin }^v = VaCTfe) sin IE, VV' sin {v" --= Va{\-\e) sin \E'\ Vr cos Iv = 1/0(1— e) cos ^E, Vr" cos ^i>" = Va{\ — e) cos Ai,'", since y" — v = u" — u, we easily derive VW' sin i (it" — tt) = aVV^' sin | (i;" — E), (57) and also a cos I (£" —E) — ae cos ^ (-E" +-£)=: l/^ cos ^ (tt" — it), or e cos I (£" + £) = cos I (£" - £) - ^ '"''''''"' ^'^'*'~''^ . (08) Substituting this value of e cos^{E" -{- E) in equation (56), we get z'V'p r= aVl^^' {E" — E— sin (E" — E)) + 2al/r^=~e^ sin -i (E" ~ E) cos i («" — u) V^', and substituting, in the last term of this, for aVl — e^, its value from (57), the result is t' I'p = aVl — e' {E" —E — sin (E" — E)) + rr" sin («" — u). (59) From (57) we obtain "^V ^ ""i> sin4(^"-£) ' or / rr"sin(it'^ — tt) \'' 1 0=1 1 — e' = \2Viy'cosK^t"-«)^ i^sin-U (£"—£)' Therefore, the equation (59) becomes 1 E" — E—sin {E"—E)j [rr"] r'Vp 'p suiniE"—E) \ 2l/rr" cos i («" — «)/ (60) Let x' be the chord of the orbit between the first and third places, and we shall have x'^ := (r + r")^ — 4rr" cos^ -K"" — «). Now, since the chord x' can never exceed r + r", we may put x' = (r + r")sinr', (61) and fi'om this, in combination with the preceding equation, we derive 2]/»V' cos ^ («" — u) = (r + r") cos /. (62) DETERMINATION OF AN ORBIT. 249 Substituting this value, and [7t"] = -7 k _p, in equation (60), it reduces to E" — E-mi{E" — E') f;;74 + .7-=1. (63) sin'-i(^"— -^') (r + r")" cosV' ^' ' a The elements a and e are thus eliminated, but the resulting equation involves still the unknown quantities W — A' and ,s'. It is neces- sary, therefore, to dci'ive an additional equation involving the same unknown qn.antities in order that E" — E maybe eliminated, and that thus the ratio h', which is the quantity sought, may be found. From the equations r=^a — ac cos E, r" = a — ae cos E", we get r" + r = 2a — 2ae cos A {E" + E) cos A {E" — E). Substituting in this the value of c co^\{E" -\- E) from (58), we have r"-\-r = 2a sin» \{E" — E) + 2V^' cos A («" — u) cos J (£" — E), and substituting for sin^(JE^"— E) its value from (57), there results But, since ^ 2r" we have r4-r"--. ^ 8'-' (r-fr")'co8V from which we derive 8in'J(J5;"— ^) = 2prr" cos'^ ^ (w" — tt) 1 2/^/ 1 \« ■«'=' \2l/r^'cosK«"— «))' 7 + 0- + »•") cos / (1 - 2 sin' \ {E" - E)\ 1 ^'» sin'' y 7? (;r + r")'cosV cosr' / ' (64) which is the additional equation required, involving E" — -S/and s' as unknown quantities. Let us now put svn^\{E"—E) 3 -pil E"—E — sm{E" — Ey m =: ,."^»noa9v" (r + )•")» cosV (05) sin' .J/ cos;* 250 TIIEOUKTICAL ASTRONOMY. and the equations (63) and (G4) bcoomo m! '1,1 1/ 's"'^h'~'^' (66) When tlie vahie of y' Is known, the first of these equations will enable us to determine s', and henec the value of x*', or )i'n\^\{E" — E), from the last e(juatiou. The calculation of y' may be facilitated by the introduction of an additional auxiliary quantity. Thus, let tan/ ^'-4 (67) and from (62) we find 2»/ rr" cos / = cos ;J (n" — u) \'-r~-jr = 2 cos }, (u" — ti) cos'/ tan ■/, or We have, also, which gives cos / = sin 2x' cos ^ («" — «). x'" = (r + r"y — Arr" cos^" \ (u" — «), x'^ = (r — r")" + 4n-" sin" -} («" — u). (68) Multiplying this equation by cos" J (ft" — u) and the preceding one by sin" | {u" — v), and adding, we get x" ^ (r + r")' sin- ^ (u" — ii) + (r — r"y cos' ' («" — u). (69) From (67) we get cos''/ : sin^/: and, therefore, r + r' ,"» r — r" cos 2;?' = — j — jf, r -\- r so that equation (69) may be written . ^ „.- = sin» / = sin' ^ («" — «) + cos' 2/' cos' A (w" — u). We may, therefore, put sin / cos (t' = sin ^ («" — t()> sin / sin G' = cos | («" — it) cos 2/, cos >»' = cos -^ {u" — u) sin 2/, (70) DETEIlMrNATIOK OF AN OKHIT. 251 from which y' may he derived by means of its tiuigent, .so that sin y' shall he positive. The auxiliary angle G' will l)e of suhscquent ii.so ill (li'tcrmining the elements of the orbit from the linul hypothesis for Paml q. 88. We shall now consider the auxiliary quantity y' introduced into the first of ecpiatious (G(i). For brevity, let us put and we shall have This gives, by differentiation, g^\{E"-E), , sin^.7 ^ ~ 2(/ — sin^' ^y o i J 4 sin' 7^7 -T- — 3 cot ff dg — ,y •f~f-, y ^U — SHI 2g or dl By' cot g — 4^ cosec g. The last of equations (65) giv^es x' = a'nv l(/, and hence ^^2 cosec j7. Therefore wc have dy' 6y' cos g — Sy" __ 3 (1 — 2x'') y' — 4?/'» dx' sur g 2a;' (!—«') It is evident that we may expand y' into a series arranged iu refer- ence to the ascending powers of x', so that we shall have y' = a + ,3.r' 4- yx" + oV^ + sx"- + Zx'^ + &c. Differentiating, we get dxf dx' ^ = /9 + 2yx' + Sdx" + i^x" + 5:x'* + &c., dy' and substituting for j-j the value already obtained, there results 2iSx + (4r — 2/3) x" + (6'5 — 4y) x" + (8^ — 6 r?5» h — 2" „ — I 9 I 3 U -2 1 I 1 !t I ;t u (I v! I /6 I A'p ( 7 1 ^ If Ave multiply through by y, and put we obtain ft'___'( ya I f)'.' ,.'3 1 13 84 ^4 I ft 08 8 ^4 •» — 3 5* T^ |51;. •'^ T^ 0737S* T^ ial 'J3 IB* I ^ 3 8 J 7 8 (I I 8 ,,.'6 _l ,«rp .^Oy_ 5 _!_.,' __,.'. (72) (78) Combining this with the seeond of equations (66), the result is m' If we put wc shall have 'fl^y'+^-^—i+Z + f'. ,' = m 1+/ + t"" (74) i^o y— «'^_,' But from the first of equations (66) we get m and therefore we have y -=,'»(/-!); »'! r»' (76) As soon as ^' is known, this equation will give the corresponding value of s' . Since f ' is a quantity of the fourth order in reference to the diftbr- ence \ {E" — E), we may evidently, tor a first approximation to the value of yj', take m and with this find s' from (75), and the corresponding value of x' from the last of equations (66). With this value of x' we find the corresponding value of f ', and recompute tj', s', and x' ; and, if the DKTKRMINATION OF AN OUniT. 253 valiit' of c' (Icrivi'd from the la(?t value of x' ditlers from that already used, the operation must ho ropcatnL It will he ohservcd that the iSeries (72) for ?' converges with j^reat rapidity, and that for E" — E^^dA° the term containing .r'" amounts to only one unit of the seventh decimal place in the value of ^'. Tahle XIV. gives the values of c' corresponding to values of .i-' from 0.0 to O.-.i, or from E"~E=0 to E" — E V.V1° GO'.G. Should u wise occur in which E" — E exceeds this limit, the expression ,_ 8in-^ \ jE" - E) y — E"—E — mWE"—E) may then be computed accurately hy means of the logarithmic tables ordinarily in use. An approximate value of x' may be easily ibund with which y' may be com])uted from this equation, and then ^' from (7''5). With the value of ? ' thus found, r/ may be comi)uted from (74), and thus a more approximate value of x' is immediately obtained. The ecpiation (7o) is of the third degree, and has, therefore, three loots. Since .s' is always positive, and cannot be less than 1, it Mlv.ws from this equation that tj' is always a positive (quantity. The equation may be written thus : r;V h' 0, ami 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 log.s'^ corresponding to values of tj' from 3y'=0 to jy'=^0.6. Wliou r/ exceeds the value O.G, the value of s' must be found directly from the equation (75). 89. We are now enabled to determine whether the orbit is an ellij)sc, parabola, or hyperbola. In the ellipse x = sin"'^ \ {E" — E) is positive. In the parabola the eccentric anomaly is zero, and hence a'--=0. In the hyperbola the angle which we call the eccentric anomaly, in the case of elliptic motion, becomes imaginary, and lionce, since fi\n\{E" — ^) will be imaginary, ;«' nuist be ncgativ^e. It follows, therefore, that if the value of x' derived from the equa- tion , m' ., is positive, the orbit is an ellipse ; if equal to zero, the orbit is a parabola ; and if negative, it is a hyperbola. 254 theore:tical astronomy. For the case of ])arabo]ic motion wc liave x' = 0, and the second of equations (6(3) gives „ m' ^'^If- 176) If we eliminate s' by raoaiu* of both equations, since, in this case, y' = i we get Substituting in this tl' .' vahies of m, and I given by (65), we obtain '»_' wliich gives (r + r")i Qt' (^ + r")^ 3 sin 1/ cos •/ -f 4 sin" A/, 6 sin hy' CDs'* Ay' + 2 sin' A/, or OM-r") This may evidently be written — ; =-^ (sin 1/ + cos Jr')* + (sin \/ — cos A/)' ^-' 6r , rTT^Ti ='-" '^^ + siii/)i :f (1 — smr')^' {r-\-r )a the upper sign being used when y' is less than 90°, and the lower sign when it exceeds 90°. Multiplying through by (." + r")^-, and replacing {r -'- /•") sin y by '/., we ol)tain 6r' n^ (r + r" + x)^ qr (r + r" - x)!, which is identical with the equation (06)3 for the special case of parabolic motion. Since x' is negative in the case of hyperbolic motion, the vai u; nf ?' determined by the series (72) will hv diflerent 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 tablo, it ^vill be necessary, in the case of the hyjx'rbola also, to comj)Utc llie value of ?' directly, using additional terms of the series, or wc may modify the expression for y' in terms of ii"' and £! so as to lie apjtlicablo. If we compare equations (44), and (56)i, we get tan hE--- V~-l tan hF; DETERMINATIOX OF AN ORBIT. 255 and lionce, from (58),, a -j- 1 Wc have, also, by comparing (Go), with (41)i, since a is negative in the liypcrbola, ff' + 1 cos-E which gives Now, since 2ff ' ■' 1 /- sin E = — 7c — V — 1. ill which c is the base of Naperian logarithms, we have E l/— ~i =: log„ (cos E + l/^' sin E), which reduces to or £= r — 1 loofe"". By means of these relations between E and A.r ^y -f- Ei.f +...., and if a.?' and a^ are very small, we may neglect terms of the second order. Further, since the employment of x and y will reproduce the same values, we have /(^,y) = o, f'ud hence, since a.x' == a;„ — x and a?/ == ?/« — Vi x^^A (xo — x)-\-B{]i^~ y). 17 258 THEORETICAL ASTRONOMY. In a similar manner, we obtain 2/0' — 2/0 = ^' (•*'o — ^) + -S' (yo — 2/). Let us now denote the values resulting from the first assumption for P and Q by P, and (^j, those resulting from P„ (^^ by Pj, ^^o, and from Pj, ^2 ^y A> ^3J ^"<^) further, let P,-P,=a', Then, by means of the equations for xj — Xq and y^' — 7/0, we shall have a ^A(P-x) + B(Q-y), a :A(P,-x) + B{Q,- AiP,-x) + B(Q^- h =A'(P-x) + B'(Q-y), ■y), b'^--^'^A'{P-x) + B\Q,~y), ■y), h"^A'{P-x) + B(Q-y). If we eliminate A, B, A', and B' from these equations, tLj results are P(a'b" — a"b') + P, (a"b — ah") + P, (aV — a'b) x = y = from which we get („'6" — a"b') + ia"b — ab") + (ab' — a'b) ' Qia'b" -a"b' ) + Q, (a"b — ab") + ^, («&' - a'6) ( «'6" — a"6') + (cd'u — ab") + (a6' — a'b) ' .r = P fa" + a') (n'b" — a"b') + a" (a"b — ah' ) y=Q: " {a'b"— a"b') -f (a"6 — a6") + {ab' — a'b)' (b" + b') (a'b" — a"b') 4- b" (a"b - a6") (82) {a'b"— a"b') + (a"6 — ab") + (a6'— a'6)" In the numerical application of these formukc it will be more con- venient to use, instead of the numbers /■•, P,, P^, Q, §„ etc., the loga- ritlnns of these cjuantities, so that a = log Pj — log P, b -- log C^,— - log Q, and similarly for a', b', a", b", — which may also be expressed in units of the last decimal place of the logarithms employed, — and we shall thus obtain the values of log a: and log?y. With these values of log.c and log^ for log P and log Q respectively, we proceed with the final calculation of r, r', r", and m, ?t', u". When the eccentricity ic small and the intervals of time between the observations are not very great, it will not be necessary to employ the equations (82); but if the eccentricity is considerable, and if, in addition to this, the intervals are large, they will be required. It may also occur that the values of P and Q derived from the Ifwt hypothesis as corrected by means of these formula:, will differ so DETERMIXATIOX OF AN ORBIT. 259 much from the vahies fouiul for x and ij, on account of the neglected terms of the second order, that it will be necessary to recompute these (jiiantities, using these last values of F and Q in connection with the three preceding ones in the numerical solution of the equations (82). 91. It remains now to complete the determinstion of the elements of the orbit from these final values of P and Q, As soon as Q,, i, and u, u', u" have been found, the remaining elements may lie de- rived by means of/", r', and u' — u, and alsu from >•', ?•", and u" — u' ; or, which is better, we will obtain them from the extreme places, and, if the approximation to P and Q is com})lete, the results thus found will agree with those resulting from the combination of the middle place with either extreme. We must, therefore, determine s' and x' from r, r", and u" — u, by moans of the formulte already derived, and then, from the second of equations (46), we have y = ('''-'-"''"f"— >)', (83) from which to obtain p. If we compute .s and s" also, we shall have / sr'r" sin («" — «') \ ^ / /'rr' sin (u' — u)\^ P=\- r )=^\ V )' and the mean of the two values of ^j obtained from this expression should agree with that found from (83), thus checking tlio calcula- tion and showing the degree of accuracy to w'hicli the approximation to P and Q has been carried. The last of equations (6o) gives sh\\(E"—E)=\/x', (84) from which E" — E may be computed. Then, from equation (57), since c = sin if, we have COS^P rr (85) ■ sin -](-£;"— ^') for the crdculatiou of a cos f. But }> = «(1 — r) = a cos" r r r ¥" (87) by means of which e and u) may be found. Since cos 2x' 2Vri j> we have • + r"' si°2/---qr-,7r' 2;; ^ + ^-2=:- '■ '"^ Vrr"ii\n2-/: J) j> 2j^cot2;^' --% r r and from equations (70), , sin ^ (ii" — u) tan G' cot 2/ cosy sin 2/ : cosy- cos i (tt" — v) Therefore the formulae (87) reduce to e sin (w — ^ iu" + ^()) = rT=r, t^" ^'' e cos (w — ^ (it" + ''0) P_ cos / V rr' P (88) sec i (u" — w;, cos /'' Vrr" from which also e and <« may be derived. Then sin

; and if we compute r, r', 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. Accoi'ding to the equation (45)„ we have tan IE = tan (45° — \', (90) tan \e" = tan (45° — \ 2fi4 TIIEORETICAI. ASTRONOMY. or from a = zi mi" rr" cos" \ («" ~ u) (x" — z')' which is derived directly from (89), observing that the elliptic senii- truii.sverse axis becomes negative in the case of the hyperbola. As soon as to has been found, wc derive from it, «', and u" the corresponding values of v, v', and v", and then comi)ute the values of F, F', and F" by means of the formula (57), ; after which, by means of the equation (09),, the corresi)onding values of A', N' , and N" will be obtained. Finally, the time of perihelion passage will be given by T=^i 3 3 3 N^t'-~N' = f-~N" wherein log;,„Z; -- 7.8733G575. The cases of hyperbolic orbits are rare, and in most ol iliose which do occur the eccentricity will not differ much from that of the para- bola, so that the most accurate determination of T will be effected by means of Tables IX. and X. as already illustrated. 93. ExAMPiiE. — To illustrate the application of the principal for- mula; which have been derived in this chapter, let us take the follow- ing observations of Earynomc @ : Ann Arbor M. T. (w'a 1863 Sept. 14 15* 53"* 37'.2 1» 0" 44'.91 21 9 46 18.0 57 3.57 28 8 49 29.2 52 18.90 + 9° 53' 30".8, 9 13 5 .5, + 8 22 8 .7. The apparent obliquity of the ecliptic for these dates was, respect- ively, 23^^ 27' 20".75, 23° 27' 20".71, and 23° 27' 20".65 ; and, by means of those, converting the observed right ascensions and declina- tions into apparent longitudes and latitudes, we get — Ann Arbor M. T. Longitude. Latitude. 1863 Sept. 14 15* 53'" 37'.2 170 47; 2,1" m + 3° 8'43".19, 21 9 46 18.0 16 41 36 .20 2 52 27 .4(3, 28 8 49 29.2 15 16 56 .35 + 2 32 42 .98. For the same dates we obtain from the American Nautical Almanac the following places of the sun : NUMERICAL EXAMPLE. True Longitude, Latitude. loRi^o- 172° ^'42".l — 0.07 0.0022140, 178 37 17 .2 + 0.77 0.001 :wr)7, 185 2f5 r,4 .8 + 0.H7 0.0005174. Since the elements are supposed to be wholly uiikuown, tlie pliioes of tlie planet nuist l)c correeted for the aberration of the fixed stars as given by ecpiations (1). Thus we find for the correetions to be applied to the longitudes, respeetively, - 18".48, and for the latitudes, + 0".47, — 19".49, + 0".30, 20".8, + 0".14. When these corrections arc applied, we obtain the true places of the planet for the instants when the light was emitted, but as seen from the places of the earth at the instants of observation. Next, each place of the sun must be reduced from the centre of tlio earth to the point in which a line drawn from the planet through the place of the observer cuts the plane of the ecliptic. For this purpose we liave, for Ann Arbor, / = 42° 5'.4, log /)„ = 9.99935; and the mean time of observation being converted into sidereal time gives, for the three observations. 0, =- 3* 29"* 1', ".48, log tan / = 9.3884640, /9o = 2° 52' o!)"^ \l log «o =: 0.801 J5583„, log b ^ 2.54oG342„, log c ^ 2.2:}280oO^, logrfr= 1.2437914, log/ = 1.3587437,., log/i -= 3.9247G91. The forniuhe ,. sin a" -A') , .i?"sin(A"-Q") M, = -. — rrr, ;t +J M, sin (A" — A) sin ( y — A) sin (A" — A) hmii(X"—K) d / d RsinU — Q) m: = hmi{k—K) give log il/,"== 9.0690383, logiJ/;'==0.7306025„. log i¥, = 9.8940712, log iV,=j: 1.9404111, The quantities thus far obtained remain unchanged in the suc- cessive approximations to the values of P and Q. For the first hypothesis, from P=-, T b + Pd 1 -I- P' Q = "", "'0 — "o "> Tjo sin C = i?' 3in V, ijo cos C = /jfl — P' cos v. ^o=-iCo§, /„ ««n FT!" ij,ie"sin^4 268 THEORETICAL ASTKOXOMY. we obtain lOgT =r : 9.0782249, ]ogr"=:=.9.0()45575, logP=- 9.98()882(], log Q ^ 8.1427824, log^„ = : 2.22985(i7„, log /'a -.0.0704470, log^o - .0.07i(i091, loglJo :r.:. 0.;W2()'.)25, C = : 8° 24' 49".74, log?Ho=1-2449l;>(J. The (quadrant in whicli !^ must bo situated is determined by tJie eoii- dition that r^^ shali liave the same sijj;n as /„. The \'ahu> of z' must now be found by trial from tlie C(|Uation sin iz' — ?) =r 7Ho sin* z\ 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 tAvo of tliese roots give z' greater tlian 180^ —■4'') and are excluded )w the condition thi'.t z' must be less than 180°— ^'. 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 ecpuition by a few trials to be z' ^ 9° r 22".96. The root which corresponds to the orbit of the earth is 18° 20' 41", and differs very little from 180° — t^'. Next, from / =: J?\sin4' sin z' ' R'm\(z'-\-^') sms cos/S', n i^p(i + ,'?.). .."^-=«p, Ave derive logr'=.0..']025r,72, log n==^- 9.70(11229, log/, =:=. 0.0254823, hgp' ^0.0123991, logJi"=^9.()92*555, log/>"^ 0.0028859. The values of the curtate distances having thus been found, the heliocentric places for tlie three observations are now computed from NrJrERICAL EXAMPLE. 269 r cos/) cos (/ — O) r cos/> ^in (J — ©) rnmb ?-'co.s6'cos(r— 0') = /ocos(A — O) — E, = P sin (^~ Q), = /) tan /5 ; :/)'cos(/'— ©') — -R', r' COS // sin (7' - - ©') :--. // sin (/' - © '), r sin l> -^ // tan fi' ; 1-" t-os b" COS (Z" - - ©") = f>" cos (A" — ©") - r" COS 6" sin (/" - ©") = />" sin (/' - ©"), B", r" sin h" ■ p"tvin(i", wiuoli give / ::. 5° 14' 39".a3, /' = : 7 45 11 .28, r'-^lO 21 34 .57, log tan b --= 8.4615572, log tan 6' :r^ 8.41075.55, logtan6" = 8.:i497i)1l, logr =r 0.:{040094, log r' == 0.;5025»>73, logr" ^0.3011010. The agreement of the value of logr' thus ol)tained with tliat already tbund, is a proof of part of the calculation. Then, from tantsin(.Ur'-t-0— S^) tan h" 4- tan b iTcos^U" — 0' tan b" — tan b tau V - wo get ^ , / 111 , i\ ^ •» mil u — tunc/ cos I cos^ cos I SI = 207° 2' 38".16, i = 4° 27' 23".84, u . 158° 8' 25".78, «' = 160° 39' 18".13, «" -- 163° 16' 4".42. The equation tau b' ■-= tan i sin (I' — SI) ^■ivos log tan/>' 8.4107514, which diifers 0.0000041 from the value already found directly from //. This ditlerence, however, amounts to only 0".05 in the value of the heliooeutric latitude, and is due to errors of calculation. If we com{)ute w and n" from the ecpiatious n = r'r"mi(u"-u') rr"Hin(,i«''— <0' n" = rr' sin{((' — n) rr" sin (u" — it)' the results should agree with the values of these (piantities ]n'eviously fomputed directly from P and Q. Using the values of «, w', and 1'" just found, we obtain log « =9.7061158, log ri":= 9.6924683, 270 THEORETICAL ASTRONOMY. which (liifor in the last decinial i)laf'e.s from the values used in findiii"- [) and ft". According to the equations d log H =: — 21 .055 cot ( it"— ?t') du', d log n" = 4- 21.055 cot («' — ?<) da', the differences of log » and log/*" being expressed in units of the seventh decimal place, the correction to «/ ne(;essaiy to make the two values of log/* 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 com])lete, although near enough for the first approximation. It should l)c observed, however, that a great circle passing tlirough the extreme observed places of the i)lanet passes very nearly through the third place of the sun, and hence the values of p and f>" as determined by means of the last two of e(inations (18) are somewhat uncertain. In this case it would be advisable to com- pute o and fi", as soon as ft' has been found, by means of the equa- tions (22) and (23). Thus, from these equations we obtain log i" = 0.0254918, log/." = 0.00288 74, and hence I = 5° 14 40". 05, log tan h - = 8.401 5019, log r = 0.3041042, I" = 10 21 34 .li), log tan b" - = 8.3497919, log) ,11 0.30] 1UI7, 9,^ = 20 r 2' 32".97, i = 4° 27' 25". 13, U- = 158^ ' 8' 31" .47, u' = 160° 39' 23" 31, «" = 163 ° 16' 9". 22. The value of log tan b' derived from /' and these values of Q> and i, is 8.4107555, agreeing exactly with that derived from ft' directly. The values of n and n" given by these last results for u, u' and n'', are log // = 9.7001144, log ii" == 9.0924040 ; and this proof will be complete if we apply the correction du'^=—0".lS to the value of u', so that we have h" ~ «' = 2° 36' 46".09, n' — u = 2° 30' 51".66. The results which have thus been obtained enable us to proceeil to a second approximation to the correct valuers of P and Q, and we may also corn.'ct the times of observation for the time of aberration by means of the formulae t^t^—Cp sec ,?, f = f,; ~ Cp' sec f/, t" = t," - Cp" sec jf', \v\u'f*»n log C-= 7.760523, cx^eantA m partt< of a day. Thus we get 4 ^-- 257,67467, if ---. ^.#074^ <" = 27 1 .38044, NUMERICAL EXAMPLE. 271 and hence Wr = 9.0782331, log r' .= 9.8724848, loL'r"== 9.0045692. Thou, to find tlie ratios denoted by s and s", wc have m sin ycofiG = sin -S {u" — u'), sin y sin G = cos A {u" — u') cos 2/, cos /- = cos tI («" — v!) sin 2/ ; tan;^"..-^^, sin •/' cos G" --- sin h (u' — u), sin r" sin G" = cos l iu' — u) cos 2/', cos /' = COS A (,n' — u) *^hi 2/f" ; T* . sin' .')' •^ cos y ' sin^V: •^ ~ cos>" ' (r'+/-")'cosV' ,"2 m" = (,._|-,.')3C0SV"' from wliich we obtain X =■ 44° 57' 6".00, y== 1 IM 85 .90, log?»i = 6.84H2ll4, logj = 6.1^8185, X" --= 44° 56' 57".50, /' = 1 15 40 .69, log Hi" =. 6.8168548, log/' = 6.0834230. From the.se, by means of the equations m m using Tables XI 11. an«l XIV., we compute s and s". First, In the ca^o of s, we assume m : 0.0002675, and, with this a« the argument, Tablf X III. trive.s log s'- ^ 0.0002581. Hence we obtain j;' == 0.0OfX>92, ami, with this as the argument, Table XIV. gives f 0.00lote form here givoii, since tliese ratios may then l)e fomid by a simpler process, as will appear in the secj^uel. Then, from r rr" .." ' we find ^ ~" 6'«" ' r?' cos h (u" — u'fcos \ [vT— u) cos ^ (u' — n)' log P= 9.9863451, log Q = 8.1431841, wit!\ which the second approxiination may be completed. AVc now compute (',„ /•„, /,„ z', etc. })recisely as in the first api)roximation; but we shall prefer, for the reason already stated, the values of /> and //' computed by means of the equations (22) and (23) instead of those obtained from the last two of the formukc (18). The results thus derived are as follows: — log r, r=. 2.2298499„, log k, =-- 0.0714280, log [„ ^_ 0.07 19540, log r/„ ^ 0.3332233, C :^ 8° 24' 12".48, logw„= 1.2447277, / = 9° 0' 30".84, log 7-' =. 0.3032587, log/*' = 0.0137G21, log )i ^-. 9.70(!1153, log )>"--=.-. 9.r)924(i04, \ogi> =. 0.02G9143, log//' = 0.0041748, I = 5° 1.5' 57".26, log tan b = 8.4622524, log r = 0.3048308, ^' = 7 46 2.76, log tan 6' =8.4114276, log /•' = 0.3032.W, I" = 10 22 .91, log tan b" = 8.3504332, log r" = 0.3017481, Q, = 207° 0' 0".72, i = 4° 28' 35".20, u = 158° 12' 19".54, u' = 160° 42' 4o".82, u" = 163° 19' 7".14. The agreement of the two values of log?'' is complete, and the value of log tan b' computetl from tan b' tan I sin d' — Sl)> is log tan 6'== 8.4114279. agreeinir with the result derived directly from /}'. The values of n and n" obtained from the equations (54i are log a -.=: 9.7061156, log n" = 9.6924^..:^ which agree with the valiu^* already used in computing « and //', and the proof jf the calculation is ••omplcto. We hmv^, tliinvtbre, u"— u' = 2° 36' 21".32, «'— u = 2° 30' 26".a?\ n"— « = 5° G' 47".60. From these values of it" — u' and u' — **> we obtain log s = O.«>001284, log a" = 0.0001193, NUMEEICAL EXAMPLE. 273 aud, recomputing P and ^, we get log P :=. 9.98G3452, log § = 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 tan/ = -^-, sin y' cos Q' = sii. h (tt" — u), sin y' sin G' = cos \ (u" — u) cos 2/', cos / = cos J Qii" — u) sin 2/', m' = (r -\- r" f cos' y .3„" sin' A/ cos J'' ' 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 formulce m +/ + * — s'» —J ' and Tables XIII. and XIV., we obtain Then from we get log 8'' = 0.0009908, log .7;' = 6.54941 16. -( s'rr" sin (u" — u) logi> = 0.3691818. The values of logp given by -( s/r" sin (u" — u') y_l 8"rr'mUu'—n) Y are 0.3691824 aud 0.3691814, the mean of which agrees with the result obtained from lo" — u, and the differences between the separate results are so small that the approximation to P and Q is sufficieut. The equations a cos y = 003^9 = sin^ (u" — u) sin^ {E"—E) _P a cos ^' Vrr", •>j4i 271 TIIEORETIOAL A.STnOXOMY. give 1 iE" — E) .-- 1° 4' 42".!»03, lofx (n cos .--. (100) ioB-=f v:. For the values lo^,' T = 9.0782331, log t' ^^ 9.3724848, log r" = 9.0045092, log /•' =-^ 0.3032-387, tiic^e formulsD give log 8 = 0.0001 277, log s' :-=: 0.00049o3, log s" = 0.0001 199, whicli diller but little from the correct values 0.0(M)1281, 0.0001954, and 0.0001193 previously obtained. Since secV=-l + 6 .si^^y + &c., the second of equations (G5) gives Substituting this value in the first of equations (GG), we get If we neglect terms of the fourth order with respect to the time, it will be sufficien:. lu 'bis equation to put y' --- f, according to (71), and hi'iice we have and, since .s' — 1 is of the second order with respect to r', we have, to terms of the fourth order, 8'»(8'-l)=l0g,s'. IMAGE EVALUATION TEST TARGET (MT-3) II I.I 11.25 Photographic Sdoices Corporation 23 WIST MAIN STREIT WEBSTfR N.Y. I4SM (716) S73-4503 rtV iV •1>' s> '^ %^ 4^> I/a i 280 Therefore, THEORETICAL ASTRONOMY. ^-s^-t'oj—:^ (101) whicli, when the intervals are small, may be used to find s' from r and r". In the same manner, we obtain t"' log s = t^, -(7q.-7Tya. log s" = ti, (^"Ty.- (102) For logarithmic calculation, when addition and subtnu'tion loga- rithms are not used, it is more convenient to introduce the auxiliary angles ^, x't *^"*^ X"> ^Y w^***"'* of which these formula! become , ., T'''cos*y , , ., t'^cos'/ , ,, ., t"»cosV' /<««% log 8 = iX, -~ ^ ,, \ log «' = iX, — -^ ^ , log «" = ^^, ^ ^ , (103) in which log ^A„-= 0.7627230. For the ^M'st approximation those e(juations will l)e sufticlent, even when the intervals are considerable, to determine the values of « and h" required in correcting P and Q. The values of r, r', r", and /•" above given, in connection with logr=:: 0.3048308, log r" = 0.3017481, give logs ==0.0001284, log «' = 0.0004951, logs" = 0.0001193. These results for log« and log«" are correct, and that for logs' differs only 3 in the seventh decimal place from the correct value. ORBIT FROM FOUR OBSERVATIONS. 281 CHAPTER V. DETERMINATION OF THE OUHIT OF A HKAVKNI.Y BODY FROM FOUR OBflF.KVATIONS, OP WHICH THE SECOND AND THIRD MIST BE COMI'I^FIE. 95. The formultc given in the preceding chapter are not sufficient to (li'terniine the elements of the orbit of a heavenly hody when its apparent path is in the plane of the ecli|)tic. In this case, however, the position of the plane of the orbit being known, only four elc- nioiits remain to be determined, ai,d four observed longitudes will furnisii the necessary equations. There is no instance of an orbit whose inclinr.tion is zero; but, although no such case may wicur, it may happen that the inclination is very small, and that the elements derived from three observations will on this acc'ount be uncertain, and esjjeeially so, if the observations are not very exa<'t. The dilli- culty thus encountered may be remedied by using for the tlata in tiic determination of the elements one or more additional ol)scrvations, and neglecting those latitudes which are regarded as most uncertain. The formula;, however, are most convenient, and lead most exj)e- ditiously to a knowledge of the elements of an orbit wholly unknown, when they are made to depend on four observations, the si-coiul and third of which must be complete ; but of the extreme observations only the longitudes arc absolutely required. The preliminary reductions to be applied to the data are derived precisely as explaincil in the preceding chapter, preparatory to a de- torniination of the elements of the ori)it from three observations. Let /, r, t", i'" be the times of observation, r, /•', r", r'" the radii- vcctores of the body, «, u', u" , u'" the correspomling arguments of the latitude, R, li', R", R'" the distances of the earth from the sun, and O, O', 0", ©'" the longitudes of the sun corresponding to these times. Let us also put and [//"] = r>r"' 8in («'" - «'), [,•"/"] = r"r"' sin (u"' — m"), n' \r"r"'^ n (1) 282 TUKOIJKTICAIi iV)?T«OXOMV. Then, lu-cording In the ('(luatioiis ('))„ we i^hall have ny->'-f/."V" 0, iV Lot /, /', /", //" bo the ohsorvod lonirididcs, ,'i, ,V, ,i", ,i"' tho oh- served latitudes eorres|)oiidiii" cos r — /r cos o"), — ?t (/> i^in -I — Ji sill O) — (// »iin /.' — 7»" sin O' ) + «"(>/' sin r — i2" sin 0"), O = u'(//eos/l' — i^'cosO') — 0/'co8/"- A'"eusO") (3) + „"'(//"eosr'-/?'"cosO"'), = n' (// sin A' — it" sin ©') — (f>" sin /•" - If" sin ©") -f »'"(//" siiU"'-7r sin 0'"). If we niultijdy the first of those eqnations by sin ^, and the second by —cos/, and add the j)rodnets, we get = nJi sin (X - Q) — {p' sin (;.' — ;.) + /?' sin (A — ©')) 4- n" {p" sin ( / ' — ;. ) + n" sin (.;. — © ")) ; (4^ and in a similar niannei, from the third and fonrth equations, wo find = «'(//sin(r' — A')--/rsin(>l"'— ©')) (5) - (//' sin (A'"- ;") — li" sin (/"- ©")) - n"'Ji"' sin (A'"- 0'" ). Whenever the values of n, n', ii'', and »'" are known, or may he dotennined in functions of the time so as to satisfy the conditions of motion in a conic section, these e(|uations become distinct or inde- pendent of each other ; and, since only two uidcnown quantities // OKIUT FKOM VUVn ui;sKI!VATI()NS. 283 ami /'" arc involved in tlioni, tlicv will cnal)li' us tu dctcrniinc tlit'sc cuitiil*' distances. Lit us now put cos ,j' sin (A' — k) =A, (•(.s,J"i.in(A"'— r):^. C, all'' the iM'cccding oijuations give cos ,j" sin (■/"-;.) =n, COS -J j-in i A (6) ,,,,' sfc ,j' — /;«"//' sec ,J" - - nli sin {k—Q) — Ji' sin (/>. — ©') + /."A"'sin(/i. - />//,-' scc,i'- f//' 8cc,J"=- n'Ji' sin (/'"- ©') - /^ sin (/.'"- ©") O"), <7) + //"7r.sin(;."'-o"'). If we assume for n and n" tlioir values in the ease of the orliit of tlic earth, which is {'(luivalcnt to ncglcctin;^ terms of the sccctnd order in llic c(|iiaiions (-0).,, the .sccon«l mcnduT of tlw first of these e(jua- tiiiii< reduces rigorously to zero ; and in the; same manner it esm he slutwn that when sindlar terms of the second order in the corre- t.l)nnding expressions for n' and n" are neglected, the second mendier of the last cijuation reduces to zero. Jlence the secer of each of these etiuatious will generally differ from zero hy a t|uantity wliicli is of at least the second order with respect to the intervals of time hetween the ohservations. The eoeflicients of (i' and />" are of the lirst order, and it is easily seen that if we eliminate //' froui those eijuations, the resulting e(|uation for // is such that an error of tile secdud ()rder in the values of /i and ii" may produce an error of tlu' order zero in the result for »/, so that it will not he even an appr >.\iniation to the eorreet value; and the same is true in the case of //'. It in necessary, therefore, to retain terms of the second order in the fii-st assumed values for /), n', n", and n'" ; and, .since the terms of the second order involve /•' and /■", we thus introduce two a, (17) wo shall have, since To' -= r + r"', ■P" = -.,, ( 1 ~ i — /7i — 1» When the intervals are equal, we have (18) P'— — P"— _ and these expressions may be used, in the case of an unknown orbit, for the first approximation to the values of these quantities. The equations (13) and (17) give n = n"P'', n"' = n'P"; and, introducing these values, the equations (12) become (19) ^^ d^i 286 TIIF.OHKTKAI- AHTKOXOMY. * = 1 + />' ( 1 + '^' ) ' /''•'•" + ^"''' + ^') - «'. *" = 1 + P" ( 1 + ^C ) C/'"-^' + i^"''" + c") - a". Let us now put P'd'-\-e' 1 + / f " n J' — / » i+1 ill — J f and \vc shall have Wc have, further, from equations (10), (20) (21) (22) (28) If wo substitute these values of ?•'•' and r"^ in equatims (22), the two resultiuff equations will contain only two unknown (juautiiies .<•' ;iiul x", when P', P", Q', and Q" are known, and hence iliey Mill l)e suiKcient to solve the problem. IJut if we elVe<>t the eliminatinn <3 of Xq" for x" wo derive a new value of x' from the second of those equations, which we denote by .1'/. Then, recomputing x" and x', we obtain a third approximate value of the latter quantity, which may be designated by x./ ; aid, if wo put •''1 •''0 — ^0' a;,' = a/, 288 TlIKOIlCTICAIi AHTHOXOMl , W(? sliall Imvo, norordiiij; to tlic tHjimtion (67),, the ncccR'>n:y clianj^cs being made in the nutation, «' = x/- (28) The vahie of x' thns obtained will give, by means of the first of e(juations (27), a new vahie of a*", and the .snbstitution of this in the last of these eijuations will show whether the correct result has Im-cu foun//'— /e'^siu'V instead of by putting Q' and Q" equal to zero. 97. It should be observed that when ^' =^ ^ or X'" = ?-", the equa- tions (22) arc inapplicable, but that the original equations (7) give, in this case, either p" or // directly in terms of n and n" or of it,' and n'" and the data furnished by observation. If we divide the first of equations (22) by /«.', we have h! ':={^+m-^^H The equations (21) give I h' and from (11) we get a! e _ K~ K~ Then, if we put 1 1 +P" '' 1 + P' ' /? cos 4' , K sin (X — 0') H h' B iJ"sin(>l- O") i?8iu(/l — 0) c: ^/I' + F (29) ORBIT FROM FOITR OIWERVATION8. 289 its Viiluo may Im- fiumd f'nuu the rcsiiltH foi* , and ., durivcd l)y meann of tht'j't' cHjiiatiniiis, and \\v. shall have \\\\v\\ X' ~- X, wc have h' = oc, and this formula becomes the value of -, l>eiuj; fjiven hy the first of equations (20) This C(|Uiition and the second of e(|iiations (2U) are sufTieient to determine 3-' and .!•" in tin- s|H>cial ejisc' under eonsidi'ration. The see(»nd of ecjuatious (22) may be treate•'", h, and it'". Thus, we have [r'r"] = rV" sin («"-«), and, from the equations (1) and (3)j, 19 290 THKOllKTK A 1. AHTIIONOMY. n / M =-^,-t>-V'], n Therefore, [rr"j -^llr'r"], [f"n -■ 'i Ir'r"], n r sin ( u' — u) = — r" sin («" — «'), n rsni („" _ „) ^l / Bin („" _ u'), n n (32) r'" sin (u'"- u") = 4; / sin (u" - «'), n r'" sin (u'" - u') =-- 1, »•" sin (u" - «')• From tl».» fir.st un'>l .second of these equations, by addition and ^^ub- trac'tion, we get rsin ((h' - «) + i (n" - «')) = '^+-?^' sin i (n" - «'), r cos ((it' — tO + \ (u" — «')) = -=^^' cos J (n" - «'), ('33) from whieli we may find r, u' — «, and n ^= u' — [u' — m). In a simihir manner, from the third and fourth of equations (32), we obtain r"' sin ((«'" — It") + { (u" - u')) = --+,— sin A («" - it'), IV r'" co8((n"' - It") + { («" - «')) = ^,7— cos^ (it" - tt'), (34) / ,," ,,"' from whi(!h to find r'" and it'". When the approximate values of v', r', ?•", )•'", and tt, «', tt", have been found, by means of the preceding equations, from the assumed vahies of P', P", Q', and Q", the second approximation to the elements may be commenced. But, in the ease of an unknown orbit, it will be exiKnlient to derive, first, approximate va.ues of /' and r", using P' — _- P" = T^ and then recomijute P' and P" by means of the equations (14) and OniUT FIIOM FOrn OUHFUVATroNS. 2fll (|H), licfori' fitxiiii); ii' ami ii". The Utiiis of tln' wcond (»nl«'r will tliii.H l)c coiiiplctcly taken into account in the iii'^t approxiniatiun. W. If the tiiiH'.M of ohscrvation have not U'on «'orrc<'t«M| for the time of aiH'i-ration, as in the case of an urhit wholly unknown, this currection may he applietl iH'fore the secon' and (»" arc alrcatly jjiven, there remain only // and //" to lie tiiund. If wo eliminate />' from the first two of (H^'iations (.'I), tlu* result is '-^f „n"m\(X"—k') ('.]r» n sin ( A — i) nit 8in(A' — G) — /e' sin i^'—Q, |- /t"A'"M n fA' — ")_ -r ' - ' ^j^i^jii (^k' ~ k) ' ' uhI, hy eliminating ,o" from the last two of these ofpiations, we also u)>tuin (36) ,n' Bin (/';;— _A') " n"'m\(i"'~X") n' Ji' »\u {k" - Q ') ~ Ji" urn C/l"— CT) + n'" Ii' " ain (k" - Q^'^ n"'miU"'-X") hy means of which p and //" may he found. The combination of the lirsl and second of equations (.']) gives P~^ cos (A'— A) n"p' n cos(r— ;) (37) + nl{ cos (X — Q)~Rcos(i — ©') -f n" R" cos {k — ©") n and from the third and fourth we get V;^co8(r'-r)-'|-^:'co3(r n (38) + n' Ii' cos (A'"— 0') —R" cos (X"'~ Q" ) -\-n"' R'" coii(X"'—0"') n" Further, instead of these, any of the varioiis formuhe which have been given for finding the ratio of two eurta^^ distances, may be employed; but, if the latitudes /9, ^9', &e. are very small, the values of f) and p'" which depend on the differences of the observetl longi- tudes of the body must be preferred. 292 TIIKOUKTKAr- AHTRONOMY. The values of // and p'" may also lie in place of J cos ^9. 'i'iijs process all'ords a verificali<»n of the inimerical caI(;ulation, nainelv, the Vidnes of X and X'" thus found .should a^rei; with those furnished l)y observation, and the a^re(!ment of the computed latitudes ^9 and [•i'" with those ohserved, in d ti . the data furnish th(.' extreme latitudes fi and ^9'", t<» compute /> and //" as .soon as (>' and //' have iiicii found, and then find /, /"", h, and />'" directly front tliese by means of the formula' (71),. Tlu; values of Q, and / may thus hi\ obtained from the extreme places, or, tlu; heliocentri exactly with the result.s obtained directly from /»' an' and jt" by niean.s of (Tl).,, nuist agree with those derived from .r' and .r". The corrections to be applied to the times of observation on account OKIMT FROM FOUR OBSKRVATIONS. 293 of :il«'rrntion may now l)o foiiiid. Tims, if /„, /„', /„", and /,/" are the uiioorrecttstl thncs of observation, the (rorre'.. Q - i L » (41) «7' ■ rr" cos i ( 11" — 'u!) eos ^ ( 11" — v) cos A («' ■ - « )' In a similar manner, if we desij;nate by m'" i\u> ratio (»f the sector formed i)y the radii-vectores r" and r'" to tin; trianj^le formed by the same radii-vectores and the chord joining their extremities, we tlnd P" ^_ — ■* ' lit Q" TT J>* (42; * .W" rV" C08 \ ill!" - «") (!08 ^ («'" - n') cos i («" - n') The formuhc for finding the value of s'" arc obtjiined from those for « by writing x"'y X"'y ^'""> *^'*'- '" p'^^*'^' "* Jf' ?'» ^'> ^'^'"'^ '^'•'' ""^ing }•', /•'", \i!" — »," instead of /•', /•", and \i" — n\ respeejtively. By means of the results obtained from th(! first approximation to tile values of /", P" ^ Q', and (j>", we may, from ('(piations (11) and ( 12), deriv(! new and more nearly accuriite values <»f these; (|uantities, and, by ri']K!ating the culeidation, the approxiniations to the exact values mu • be earrieil to any extent whic^h niay be desirable. When tlu-e(! approximate values of P' and (/, and of P" and C/', have been derived, the next approximation will l)e fiu'ilitated by the use of the formula' (H2)^, as already explained. \Vh<.!n the values of 7^', P", Q', and Q" have been derive(l with sullicieiit aeeuracy, we proceed from these to find the elements of the orbit. After S2, /, r, /•', r", r'", «, u', n", and n'" have been found, l.ie remaining elements may be derived from any two radii-vectores 294 THEORETICAL ASTRONOMY. and the corrospoiuling arf^mnents of the hititude. It will be most aeciirate, however, to derive the olcnients from r, r'", u, and it'". If the vahies of F', P", (}', and Q" liave been obtained with groat aeeuraey, tlie res-.ilts lierived from any two places will agree with tho!<(! obtained from the extreme places. In the first j)lace, from sin^j cos Gj = sin ^ («" sin Y^ sin 6rj cos r^ cos^fw" cos A (n" u), u) cos 2/o, - u) sin 2xo, (43) we find Yq and Gq. Then we have r,=^k(f t), W'o = ' (,. + /")S cosVo »»» COS)'/ (44) '?o Jo» 4"./o "1"" ^0 from which, by means of Tables XIII. and XIV., to find s^^ and x^. We have, further, "'"sin («'"—«)' P >oZ^ ')■• 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 npi)roxinia- tion to P'y P", Q', and Q" has been carried. The elements are now to be computed from the extreme places jn-ecisely as explained in the preceding chapter, using r'" in the ])lace 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 fornnihe 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. Irreenwich M. T. 1863 Sept. 20.0 Dec. 9.0 1864 Feb. 2.0 April 30.0 14° 30' 35".6 9 54 17 .0 28 41 34 .1 74 29 58 .9 + 9°23'49".7, 2 53 41 .8, 9 6 2 .8, + 19 35 41 .5. NUMERICAL EXAMPLE. 295 Those norniiils give the geocentric; phiee-s of the planet referred to the mean equinox ami equator of 18(!4.(), and free from aberration. l'\)r the mean obliquity of the eeliptic of 18()4.0, the American Nautical Almanac gives e = 23° 27' 24".49, and, by means of this, converting the okserved right ascensions and declinations, as given by the normal places, into longitudes and lati- tudes, we get (irccinvich M. T. 1803 Sept. 20.0 Dec. 9.0 1804 Fob. 2.0 April 30.0 ?. 16° 59' 9".42 10 14 17 .57 29 53 21 .99 75 23 46 .90 + 2° 56' 44".58, — 1 15 48 .82, 2 29 57 .38, — 3 4 44 .49. Those 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 eeliptic of 1864.0 are H- 0".60, +0".53, + 0".36, + 0".19. For the reduction of the geocentric latitudes of the planet to the jwiiit in which a perpendicular let fall from the centre of the earth to the plane of the ecliptic cuts that plane, the equation {^^)^ gives the corrections — 0".57, -0".38, — 0'M8, and — 0".O7 to bt; a|)plied to these latitudes respectively, the logarithms of the approximate dis- tances of the planet from the earth being 0.02618, 0.13355, 0.29033, 0.44990. Thus we obtain t = 0.0, /' = 80.0, r =.135.0, r = 223.0, ; =16° 59' 9' A' =10 14 17 k" = 29 5.3 21 A'" = 75 23 46 .42, .57, .99, .90, <5 = = -|-2°56'44".0], - — 1 15 49 .20, = — 2 29 57 .5t), = — 3 4 44 M ; and, for the same times, the true places of the sun referred to the mean equinox of 1864.0 are O =177° 0'58".6, 0' = 256 58 35 .9, O" =312 57 49 .8, 0'" = 40 21 26 .8, log ft =0.0015899. log R' = 9.9932638, log li" = 9.9937748, log R"' = 0.0035149, -' y 206 THEORETICAIi ASTRONOMY. From the ecjuations _ tan/5' sin(/l'— ©')' _ tan (5" sin (/. we obtain tanitf ■■ tan w" : ^ , tan (/'-©') tan 4 = ; , cosij; tan (/" - O") TTJ O") tan 4" = v"08 tV V --^ 113° 15' 20'M0, V'= 76 56 17 .75, log (/r cos 4') =9.58nfi777„, log(/ir siuV) =9.9564624, log (/r cos V j = 9.a47«.S48, log(ir8in+") = 9.9823904. The quadrant in wliich -vl/' mu.st be taken, is indicated by the condi- tion that eot«\|/' and co.s(/' — G') must have the same .sign. The same condition exist.s in the ca.se of ■»//". Then, the formuhe A — • cos /5' sin (X' — A), C= cos /i"sin(r '—/"), B A~"' C R sin (>l — ©') h', a! = Ji' cos 4' + ^=rcos;5"sin(A" — A), Z) = cosrj'sin(A"'— A'), a" = i?"cosV'- d' c' =/t'iJ"co8 4" + c" = h"R co^A,' — R sin ( A— 0) A ' R"^m{r—Q") C R'mxU — Q") ~~A ' /e's i(r'-o') d": i?"'sin r'— ©'") give the following results: — log^=:^9.0699254„, log B = 9.3484939, log /i' = 0.2785685,, loga' = 0.8834880„, logr' =0.9012910„, log d' = 0.4650841, log C = 9.8528803, log Z> = 9.9577271, log r = 0.1048468, logrt" = 9.9752915,,, logo" =9.7267348.,, logcZ" = 9.9096469„. We are now pi'cpared to make the first hypotheses in regard to the values of P', Q', P", and (/'. 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 NUMEPiCAL EXAMPLB. 297 P'= — P" _ _!_ tlipn approximaio values of /•' and r" are readily obtained by means of the cy putting Q' = and (^" — 0, namely, «' = <+/V-A"- 1 -rr after which the values of x' and x" will be obtiiined 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- ficrvations will generally be small, and hence the value of x' derived from the assumption of (^' =- and (^""--Owill be sufficiently ap- proximate to facilitate the solution of equations (27). As soon as the approximate values of r' and r" have thus been found, those of P' and P" must be recomputed from the expressions With the results thus derived for P' and P", and with the values of §' and Q" already obtained, the first approximation to the elements must be completed. When the elements are already approximately known, tiie first assumed values of P', P", Q', and Q" should be computed by means of tlicse elements. Thus, from n = rV'sin(v" — v') rr" 8in(v"- n" rr' sin (v' — v) n' r"r"'^m{v"'~v") m\{v"'—v'y rr" ^h\{v" - rV'sinrv" -v') rr ^ ~i'r"''sm{v"'-v')' we find n, n', n", and n'". The approximate elements of Eunjnome give V =322° 55' 9".3, logr =0..308327, v' =353 19 26 .3, log/ =0.294225^ v" = 14 45 8.5, logr" =0.296088, v"'= 47 23 32.8, log/" = 0.317278, \il 298 TIIEOKETICAL ASTRONOMY. and heiiec we obtain Tlien, from we get logn ^ 9.rM,']052, log h' ~ 9.825408, •* — ~iri n n'" n log P' =^ 9.84G216, log P":^ 9.807763, log n" ---^- 9.806836, logn'" =^9.633171. (2' =(„ + »" -!),.'», ^' = (n'+ n'"- 1) r"», log(^ ^9.840771, log (/'=. 9.882480. The values of these quantities may also be computed by means of tiie equations (41) and (42). Next, from we find P'd' + c! I +'P'"' W'-l- c" l + i ^•d — 1 j^'pn ^ ~\+ P'' h" [ill ' J 11 !>"• log..' =.0.041 344„, log r„" ---= 9.807665„, log/' = 0.047658,, log/" = 9.889385. Then we have ■r' + q ' r; ^(1+^;) •^"^' x" + a" a; = r(i+:q ^ tan 2' = / = i?' sin V x' ' ii' sin +' tan 2 ' = ,y-^-, X „ i2"sin4" sin/ cos 2' sin 2" cos 2" from which to find r' and r". In the first place, from x' ^-. l//»— i2"sinH', we obtain the approximate value log a;' = 0.242737. Then the first of the preceding equations gives loga;" = 0.237687. - NUMERICAL EXAMPLE. 299 From thi.s we get 2" = 29° 3' 11".7, log (•" ^ 0.296092 ; ami then the equation for x' gives log x'^ 0.2427(58. Heuce we have ^ = 27° 20' 59".6, log r' = 0.204249 ; and, rcpouting the operation, using these rcsult.s for x' and r', we get log .1;"=-- 0.237678, log .c' = 0.242757. TliP correct value of log.)' may now be found hv inean.s of equation (28). Thus, in units of the sixth decimal place, we have fl„ = 242768 — 242737=^ + 31, a,' = 2427o7 - 242768 ^ — 11, and for the correction to he applied to the last value of log .r', in units of the sixth decinuil place, A log x' = a„ — a, - + 3. Therefore, the corrected value is log a:' = 0.242760, and iVoni this we derive log .r" = 0.237681. Those results satisfy the equations for x' and x", and give 2' = 27° 21' 1".2, 2" = 29 3 12 .9, logr' =0.294242, log r" = 0.296087. To find the curtate distances for the first and second observations, the formulte are R sin (z' + 4') ,„, ^ = ; ; cos if C08,T, .■: = Kli't!£+j:},^,r, am 2 su) z which give log p' = 0.133474, ■ log p" = 0.289918. Then, by means of the equations 300 THEORETICAL ASTRONOMY. r'coHh'vG'iiC-Q') ==t>'cos(A'-Q')-R', r' cos b' sin (f — ©') = />' sin ( A' — ©'), »•' gin h' = // tan ,5', r" cos 6" cos {/" - O") ^ //' cos (X" — ©") — /?", r" cos 6" sin (/" - ©") = f>" sin (>■" - ©"), r"8in6" =^/'tan,j", wo find the following heliocentric places : r = 37° 3')' 2()".4, log tan // == 8.182H(il„, log r' = 0.2fl4l>4:i r =. 58 58 15 .3, log tan b" = 8.634209„, log r" = 0.2it«i(l^7. The agreement of these values of log /•' and log >•" with tho.se obtaimd directly from x' and x" is a partial proof of the numerical calcula- tion. From the equations tan i sin (A d" -f /') — R ) = ^ (tan b" -\- tan 6') sec S (T — T), tan i cos (A {I" + /') — Q) == a Uii" ^" — tan6') cosecA {F' — /'», tan(/'— ft') cos t tan It' ■we obtain „ tan (r- ft) tan II = SI = 20(5° 42' 24".0, n' -= 190 55 6 .G cost i ^ 4° 36' 47".2, «"=212 20 53 .5. Then, from -"=1^(^ + 1)' ft n n"P\ : n'P", we get log n" = 9.806832, logn' =9.825408, log ft =9.653048, logft"' = 9.633171, and the equations r sin ((ft' - ft) + ^ (ft" - «')) = I-i-"— sin i («" - «'), r cos ((«' — ft) + i (ft" — «')) = — cos ^ (ft" — ft'), r" + n'r' r'" sin ((ft'" — ft") + ^ (ft" - ft')) = r'" cos ((ft'" - ft") + ^i (m" - «')) = /' — n'r' n" sin A («" — ft'), cos^(jt — «;, NUMERICAL EXAMPLK. 301 give logr ^.0.308:570, lo}r/-"'.^0.:51727:{, V = 100'' :{0' 'u'fy, u'" = 244 5U 32 .o. Next, by moans of the fonmilii; tan b tan b' tun (I — SI) = cos / tan u, tan (/'" — ft ) =^ cos / tan u'", P cos(A — O ) = r cos 6 cos (/ — ©) + Ji, f> mi (A — O ) =r con b sin (/ — O), /> tan /? = r .sin i ; tanisinC/ — ft), tan t. sin (T— ft), cos (r — ©'") sin(r'-0"') f'" tan ,J r"'cos6"'cos(r- >•'" cos 6'" sin (/'" - 0"') + R" ■ ©'"), '" sin b' we obtain / = 7° 10' ol".8, b = + 1 32 14 .4, X = 16 59 y .0, /9 = + 2 o6 40 .1, log/> = 0.02.5707, r = 91° 37' 40".0,. b'" = - 4 10 47 .4, /" = 75 23 46 .9, fi"' = — 3 4 43 .4, log//" =: 0.449258. The value of ^'" thu.s obtained agrees exactly with that given by observation, but A differs ()".4 from the observed vahie. This ditler- onco (Iocs not cxeeeil what may be attributed to the unavoidable errors of calculation with logarithms of six decimal places. The (littorences between the comi)uted and the observed values of ,9 and fi" show that the position of the plane of the orbit, as deterinincd l)v 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 tlie case of an orbit entirely unknown, to complete the correction for aberration according to the cijuations (40). The calculation of the quantities which are independent of P', P", (/, and Q", and which are therefore the same in the successive livpotheses, should be performed as accuratelv as [)ossible. The c' value of ~, required in finding x" from x', may be computed directly from f ~ h'^ h'' d' the values of yi »"ciug found by means of the equations (29) ; 302 TIIEOHKTICA I. AHTIIONOMY. uiul a similar iiu'tluul may I »e adopted in the carto of *• FiirtliiT, in t\w ('omputation (tf x' and .r", it may in some (u^c's l)c advisalilo to employ one or l)otli of the efjiiations (22) for t\w. Miial trial. 'I'liii-, in tl)C i)r('sent case, .r" is fomid from the first of ('(jnations (27) liv means of the difference of two larj^er nundwrs, aiivl an error in tiie last decimal place of the logarithm of eitiier of these ninnhers ulUcts in a f'reater tan/' = 4'-, n sin y cos (r =: sin A («" — n'), sin y" cos G" = sin S (n' — u), sin y sin G = cos \ {u" — u' ) cos 2/, sin /' sin G" = cos \ (u' — u) cos 2/", cos J' =coSo(tt" — n') sin 2/, cos/' =cosA(n' — u)s'm'lx", sin/" cos G"':=sini(«"'- -u"), sin/" sin G"'^cosy(u"'- u") cos 2/" cos/" =^cos\{u"'- ■u") sin 2/"; m t' cos';f v''cos''/ "' = ,-c..V"' 7tt"' = t""co8«/" "r"»cosV"" J sin" {y cos ^ ' sinM/' ^ ~ cos/" r= sin^J/" cos/"' m ,'" = m'" ^ ~2+J + f' ^ ~'-\-j"+r 1+/"+^"" X x'" = 8"" '' ' in connection with Tables XIII. and XIV. we find s, s", and h'". The results are Iogr=n0.n7r)0441, ^=10 42 ")') .U, logm . -«.1H<5217, logj --^ 7.y4«()!»7, log* -^0.0(Wr,248, NUMERICAL EXAMPLE. log r"-_ 0.1880714, /'.--- 44° 32' 1".4, r"-^ 15 i;{ 45 .0, log wi" 8.51 G727, log/' -= 8.2«5()()i;{, log a" =^0.0174621, 303 log r'"-^- 0.1800(541, /.'" 45° 41' 55" .2, r" 1(5 22 48 .5, log m'" - : 8.5}»05!»(5, log/"--. . 8.325:5(55, log «"'=^ : 0.02040(53. Tlicn, l)y means of tlie forinuhc r J< P' Of t" TT • M,i" rr" cos \ («" — u') cos .J («" — «) cos J (u' — n)' P" = t"'" « TT ^' "" - «*.'" ' ?r"' cos A («" r"» ri' we obtain logP' ^9.8402100, log P"=. 9.8077(515, u") cos ^ («'" — «') cos ^ («" — u')' log (^ == 9.840753G, log (}' = 9.8824728, with which the next approximation may be completed. We now recompute cj, oJ',f',f"y x', x", &c. precisely as already illustrated; and the results are L)gc;=. 0.5413485,., log/' = 0.047(>(>14„, logs/ ==0.2427528, g' = 27° 21' 2".71, log r' =0.2942369, log ^' =0.1334635, log»i =9.6530445, log h' =9.8254092, Then we obtain logc„" = 9.8076649,., log/" =9.8893851, log a/' =0.2376752, 2" = 29° 3' 14"..., logr" =0.2960826, log^" =0.2899124, log n" = 9.8068345, log7t'" = 9.6331707. V = 37° 35' 27".88, r'=58 58 16 .48, log tan b' = 8.1828572„, logtani"=8.6342073„. log/ =0.2942369, logr" =0.2960827. These results for log J*' and logr" agree with those obtained directly from s' and z", thus checking the calculation of '^' and i]/" and of the heliocentric places. Next, we derive ft = 206° 42' 25".89, n' = 190 55 6 .27, i = 4° 36' 47".20, «" = 212 20 52 .96, 304 TIIKOFIKTICAI, ASTUONOMV. niul from u" -— n', r', r'\ », n", n\ and u"\ we ohtnin logr =^ o.:1()m;{7;{4, « . i(i()° ;{()' :):)".4"), log r"'r^O.:n 72(174, n"'.-244 ')!> :M m. For the purpose of provinjj the accuracv of the minioriciil rostilt>i, wc compiiU! also, tis in the first approximation, /= 7° KJ' ^A"^A, 6=+ 1 ;{2 14 .07, • A = IG oj) 9 .:{8, /9 = + 2 m 39 .04, lojr/>=:().()2.")(}!)(;0. /'"= 01° 37' 41".20, 6'"=— 4 10 47 .:{({, r'= 7.") 23 4(> .99, /9"'=:— 3 4 43 .33, log//" - 0.44925,39. The values of /. and I'" thus found differ, respectively, only 0".0| and 0".09 from tiiose given l)y the normal places, and hence the accuracy of the entire calculation, hoth of the (piantities which arc independent of /", P", Q', and Q'\ and of those which depend (in the successive hypotheses, i.s completely proved. Thi.s condition, however, nmst alway.s be satisfied whatever may be the twsunieJ values of P', P", q\ and (}» . From /•, r', », «', &c., we derive log « =- 0.0080254, log «" = 0.0174637, log s'" = 0.0204076, and hence the corrected values of P', P", Q', and Q" become logP': logP": 9.8462110, : 9.8077622, log (/ = 9.8407524, 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 To == k it'" ^ 0, tan xo = \ — ' sin y-j cos Ga = sin ^ (v!" — u), sin Yo sin ^a = cos ^ («'" — \C) cos 2xq, cos Y^ = cos ^ (tt'" — u) sin 2/^, «Jn = (r + r'")' cosVo' Wo _ sin" \Ya_ -'" cosYo ' ^0 — "7T Jv NUMEUICAL KXAMl'LE. 30o wc get r.r 4-2' 14' :{()". 17, logtiiii ^'„ M.o.v.M !»:.:?__, l(.^,'»;i„ !t."]7!MC.'»;, loj,'.i-„ M.iHl(»M:!!»7. I'lii' ((iriniilii givw /.„n"'»'inrH"'— u)\» )' lojry, o.;{7l'.Mni; and if we comimtc t\\v. sniue (jiiuiitity i)y iiU'IIIIm of* / xr'r"^\i\ ( >("— ,i' ) \' I V'/Y'«in ( »' - » ) \'' / .■<"' r" ,'" An ( .t"'-»/' > v' ?'^( -- r ) \ r" )^^\ r"' )' till' sopanito results aro, rospoctivcly, (>..'J712'"5!>7, ( .•')7r2 1 IH, and 0.'"{7r24l4. The clitKTeiices hetweeii these results nre ^ery siimll, uii*l arise hoth from the iiiiav(>i".li in the values of ii' a and a'" - n" will pn hice an entire ueeordunce of the particular results. J''rom the equation.s acosv> sin .J(h"'— It) sin ((£'"- E) V'rr QOitp a cos v> we ol)taui (£'"-£) = 17° 35' 42".l 2, logfrt cos 9') ^ 0.37!)G883, The formulic logcosic^r^O.OinorjlH. i'An{u> .-^,(«"'-f-«)) P cos Yo 1 '■>•' rr::^^ tan G, 0) e cos(w — tJ (n"' + n)) = -— -- — sec .J («"' — u) ffive oj = 197° 38' 8".48, V 11° lo'52".2: cos yo V j*r"' log e = log sin

9, log^ = 2.9678027. For the eccentric anomalies:' Ave liave tan IE ^ tan I ^u -— w) tan (45° — ^^), hill IE' =.:tanU«' — w) tan (4-5° — I^?), tan 5 E" -.-: tan ] (u" — w) tan (4o° — l,p), tan Ie'" =^ tan l («'"— w) tan (4o° — l-p), from which the results are E =329° 11'46".01, £'=:354 29 11 .84, E" =-^2° 5'33".63, E'" = 39 34 34 .65. The value of }[E"' — E) thus derived differs only 0".03 from tliat obtained directly from Xq. For the moan anomalies, we have M =rE~c sin E, M'^E' — esinE', M" =E"-e sin E", M"'^E"'-esinE"', which give M = 334° 55' 39".32, M" ^ - 9° 44' 52".82 M' = 355 33 42 .97, M'" - - 32 26 44 .74, Finally, if 3/„ denotes the mean anomaly for the ejioch T= 1864 Jan. 1.0 nicaii time at Greenwich, from M^r=M-!i{t — T) = M' — ii{i! - = iV" - // ( V -T) = M'" - II (r • T) T), we obtain the four values il/o = 1° 29' 39".40 39 .49 39 .40 39 .40, the agreement of which completely proves the entire calculation of the elements from the data. Collecting together the several results, we have the following elements : NUMERICAL EXAMPLE. 307 Epoch =^ 1864 Jan. 1.0 Groonwicli mean time. JI/= 1° 29' 3i»".42 r :^ 44 20 3? .37 SI = 20(5 42 25 .89 I pA'lijjtic and Mean E(iuinox 18G4.0. i=. 4 3« 47 .20 j •" = 0.2000826, u" =^ 212° 20' 33".(;8, r== 58 58 10 .50, h"--=— 2 27 50 .00, /'= 29 53 21 .99, /?■' = — 2 29 57 .02, log |o"=. 0.2899122. Honco, the residuals for the second and third places of the planet a«! — Comp. — Obs'. aa' := — 0".22, A,j' = + 1".53, aA" = .00, A,J" =^ — .00 ; and the elements very nearly represent the four normal places. Since the interval between tlie extreme places is 223 days, these elements imi?t represent, within the limits of the errors of observation, tlio entire series of olx^fcrvations on which the normals are based. It may be observed, also, that the successive approximations, in the case of intervals which are very large, do not converge with the same degree of rapidity as Avhcn the intervals are small, and that in such cases the numorifal calculation is very much abbreviated by the determination, in the tirst instance, of the assumed values of P' , P", Q', and Q" by means of approxinuite elements already known. For the first determination of an unknown orbit, the intervals will gene- rally be so small that the first assumed values of these quantities, as determined by the equations P P — ^»i \^ a ,"3 / » V — 2 ■ • > Mill not differ much from the correct values, and two or three liv|)otlieses, or even less, will be sufficient. But when the intervals are large, and especially if the eccentricity is also considerable, several hypotheses may be required, the last of which will be facilitated by using the equations (82)^. The application of the formula; for the determination of an orbit from four obser\ations, is not confined to orbits whose inclination to the ecliptic is very small, corresponding to the cases in which the method of finding the elements by means of three observations fails, 4 i 310 THEORETICAL ASTHOXOMY. or at loa.st becomes very uncertain. On tlie contrary, tlie.se formula; apply cipiall} well in the case of orbits of any inclination whatever, and .since the labor of com[)Utin}^ an orbit from four observations does not much ex(;eed that required when only three; observed places are used, while the results must evidently be more approximate, it will be cxpei'ient, in very many cases, to use the formulai given in this cha[)tcr both for the first approximation to an unknown orbit and for the subsequent determination from more complete data. CIRCULAR ORBIT. 311 CHAPTER YI. ISVmTIOATIOS OP VARIOUS FORMl'L.K FOIl TI[K COURECTIOX OK THE APPUOXIMATE ELEMENTS OF THE ORUIT OF A UEAVEXLY UODV. 103. In the case of the discovery of a planet, it is often couvc- niont, before sufficient data have been obtained for the determination of elliptic elements, to compute a system of circular elements, an cphemeris computed from these being sufficient to follow the planet for a brief period, and to identity the comparison stars used in dif- ferential observations. For this purpose, only two observed places are required, there beinu; l)ut four elements to be determined, namely, J2, i, (I, JUid, for any instant, the longitude in the orbit. As soon as a has been found, the geocentric distances of the planet for tiic instants of observation may be obtained by means of the formuhc A —R cos i 4- v/a' — R' siii'^ 4, J" = ii"co3V'+V'a» R"'mi'^", (1) the values of -v// and ij/" being computed from the ecpiations (42),, and (43)3. For convenient logarithmic calculation, we may first find z and z" from Rsin^ . „ ^"sinV smz a sm z since the formula) will generally be required for cases such that these angles may be olv-ained with sufficient accuracy by means of their sines. Then we have i? sin (2 + 4) 5 p = :j:rz ^^^ '^> f = T—r, cos ,i , (3) sm z sni z from which to find p and p". These having been found, we have jTJsinfA — O) tan(/— O) = sin 6 =^ ft tan (3 (-1) for the determination of I and b, and similarly for /" and b". The if-! ;n2 THEOUETlCAIi ASTUONOMY. inclination of tlic orbit and the lonf^itude of tlic ascending node are then lound by means of tlie forniulic (7o);„ and the ar}j;ninents of tiie latitndc by means of (77);i. Since u'' — n is the distance on the celes- tial sphere between two points of which the heliocentric spherical co-ordinates are /, 6, and /", 6", we have, also, the equations sin («" — tO sin B = cos b" sin (l" — I), sin (u" — u) cos B = cos b sin b" — sin b cos b" cos (I" — /)> cos («" — It) =: sin b sin b" + cos b cos b" cos (l" — /), for the determination of »" — n, tiie angle opposite the side 90° — />" of the spherical triangle being denoted by />. The solution of the; and (>" by means of the equations (2) and (3), and the corresponding heliocentric places by means of (4). If the inclination is small, u" — (t will (litter very little from /" — /, Therefore, in the first approximation, when the heliocentric longitudes have been found, the corresponding value of t" — t may be obtained from equation (5), writing /" — / in place of n" — u. If this comes out less than the actual interval between the times of observation, we infer that the assumed value of a is too small ; but if it comes out greater, the assumed value of « is too large. The value to be used in a repetition of the calculation may be computed from the expression log a = i (log it" -t) + log k - log («" - «)), the dif!erenee u" — n being expressed in seconds of arc. With this we recompute p, ft", f, and /", and find also b, b", ft, i, u, and u". Then, if the value of a computed from the last result for m"— u differs from the last assumed value, a further repetition of the calcu- CIRCULAR ORIIIT. 813 latioii 1)oronios necossaiy. But whoa three snocossivo a])i)roximate values of (> readily inter- polated aeeordinjj; to the proeess already illustrated for similar eases. As soon as the vahie of a has heen obtained whieli eoni|)letely gatisfics e(|uation (o), this result and the eorrespondin»; values of SI, i, and the ar<>;uinent of the latitude for u fixed epoch, complete the system of eireidar elements whieh will exactly satisfy the two observed places. li' we (Ujiiote by ii^ the art _ d dJ " />' (/'J _ d" dJ"""]/' cos '5' Ao' I) ^ JJ" ' ^. d , , d" ,,, ^0= j^^J + ^.,^J". (7) (8) 111 tlio «ime manner, computing the plaeos for various dates, for which observed places are given. In* means of each of tlie three systems of elements, the equations for the correction of J and J", as deter- mined by eaeli of the additional observations employed, may be I'onned. 105. For the purpose of illustrating the application of this method, let us suppose that three observed jtlaces are given, referred to the eclijttie as the fundamental plane, and that the corrections for })aralla.\', aberration, precession, and nutation liave all been duly applied. By means of the approximate element^j already known, we compute tiic values of J and J" for the extreme places, and from tliese the helio- centric })laces arc obtained by means of the equations (71).j and (72).,, writing Jcos/9 and J" cos^j'" in place of ft and p". The values of Ji, /, u, and u" will be obtained by means of the formuhe (70)., 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 J + D and J" we compute in precisely the same manner a second system of ele- ments. Next, we assign to J" an increment /)", and from J and J" -~ D" a third system of elements is derived. Let the geocentric longitude and latitude for the date of the nuddle observation com- puted from the first system of elements be deignated, respectively, by // and /9/ ; from the second system of elements, by X^' ^^'i^ t%' > and from the third system, by A^' and fi/. Then from (9) we compute a, a", d, and d", and by means of these and the valuea of D and D" we form the equations (;.;_a;)oos;V, d -;V-/5/. (V-V)eosii of the convctions to be ni)pli('(l to the first assumed values of J niid J", l)y means of the ditU'renees b'.'twccii observation and computation. The observed hmiiitUiU.' and hititiidc beinj^ (U'lioted by // and ,'i', resj)ectively, we shall have cos;?' A/.'=;(/ (11) -//)cos;j', for findin*; the values of the seeond members of the erjuatifuis (10), and then bv elimination we obtain the values of the eorreetions iJ and aJ" to be ai)j)lied to the assumed values of the distance:!. Finally, we compute a fourth system of elements corresponding to the geocentric distances J — a J and J" + aJ" either directly from these values, or by interj)olatiou from the three .systems of elements already obtained ; and, if the first assumption is not considerably in error, these elements Avill exactly represent the middle jilaee. It shoidd be observed, however, that if the second system of elenuiits re])resents the middle place better than the first system, /.„' and {ij should be u.sed instead of // and ,9/ in the equations (11), and, in this case, the final .system of elements must be computed with the distances J -f i^ + a J and d" + aJ". Simihirly, if the middle 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 olo- mcnts still exhibits residuals, on account of the neglected terms of the second order, a repetition of the calculation of the corrections , aJ and aJ", using these residuals for the values of the seeond mend)ers of the ecjuations (10), will furni.sh the values of the dis- tances for the extreme places with all the precision desired. The increments I) and D" to bo assigned successively to the first assumed values of J and J" may, without difliculty, be so taken that the true elements shall dificr but little from one of the three systems computed ; and in all the formuhe it will be convenient to use, in- stead of the geocentric distances themselves, the logarithms of these distances, and to expre.ss the variations of these quantities in units of the last decimal place of the logarithms. These formula) will generally be applied for the correction of VAIUATIOX OK TUT) GKOCKNTItlC DISTANi^KH. ar a|)l)ro\imate olcineuts liy iiicaiis of M'vcral observed places, wliidi limy hv either siiifxle ohservatioiis or nonual places, eacli derived from K'Vt ral ohservations, and the two phu cs selected for the eoiupiitation (if the eh'inents from J and J'' should not only l)e the most accurate j)()ssil)le, l)iit they shonld also he such that the resulting elements are not too much all'eeted by small errors in these {geocentric |)laces. Tliey shotdd moreover be as distant from eacli other as possible, the other c()nsiderations not being overlooked. When the three systems of elements have been computed, each of the remaining oltscrved places will furnish two equations of condition, according to eipiations (10), for the determination of tlie corrections to be applied to the nssumed values of the geocentric distances; and, since the nund)er of equations will thus exceed the number of unknown (piantities, the entire group must be c(}nd)ined according to the method of least siiiiares. Thus, we multiply each e '^"^^ ^'^^ position of the plane of the orb! I: as determined from these by means of the equations (76), will be referred to the equator as the funda- mental plane. The formation of the equations of condition for the corrections aJ and aJ" to be applied to the assumed values of the distances will then be effected precisely a.s in the case of }. and ,-1, the 018 TIIROUKTICAI- ASTHONOMY lU'cossarv clmnjijert hv\u^ iimdc in tho iiotiition. In a similar manner, tli(! calculation may be elU'ctcd for any otiu'i* I'undanicntal phuu; wliicli may l»c a(l<)i)tc(l. It rilioiild !)(' observed, furtlicr, that when the ecliptic is taken as the f'nndamental plane, the j;eocentrie latitndes shonid he convctcfj l»y means of the e(ination ((J)^, in order that the latitudes of the sun shall vanish, otherwise, for strict acenraey, the heliocentric places must be deternuned from J and J" in accordance with the eipiatioiis (89),. lOG. The partial differential cocHicients of the two spherical co- ordinates with respect to J and J" may be computed directly by means of dill'ercntial formuhe; but, excc[>t for special cases, tliu numerical calculation is less expeditious than in the case of tiie indi- rect method, while the liability of error is much greater. If wc adopt the [)lane of the orbit as determined by the approximate values of J and J" as the fundamental plane, and introduce ;f as one of tlio elements of the orbit, as in the ecpiations (72)^, the variation of tlio geocentric longitude 9 measured in this plane, neglecting terms of the second order, depends on only four elements; and in this ease tlio differential formulie may be applied with fiieility. Thus, if we ex- press r and v in terms of the elements dMg . dr dn dJ "^ dji ' dJ' dMg dv dfjL dJ ' '^ (Ifx ■ dJ' dv dJ "^ di\L dMn . dv dfi dJ '^'dJl'dJ' In like manner, wc have dr" dr" dJ dtp d

tt' and — — rv — - are rtJ dJ dJ «J known, the equations necessary for finding the differential coefficients of the elements ;f, ^, Ji^,, and fi with respect to J are thus provided. In the case under consideration, when an increment is assigned to J, VAIUATIO.V OF TWO OEOCENTItlC DISTANf'KS. ;J19 tlio value of J" rctiminiiij; uncliiuij^cil, /•" and v" f X '"'*^ •'•'^ cliaiigi'*!, and iioiiuc dr" = 0, (/J = 0. To tiiid . . and , . — , Irom tiie equations J cos ij cos = X -\- X, J cos yj sin :=y -\- Y, in wliicli )y is the goooentrlc latitude in reference to the plane of the orbit coiiipiited from J and J" as the fundamental plane, autl A', Y the gooeentric co-ordinates of the sun referred to the same plane, wo sot dx = cos Tj cos dJ, rfy = C0S1J sin COS rj sin dJ =^ sin it (/*• -}- r cos u d{v -{■ x)' Eliminating, successively, d{o + x) and f?/-, we get dr dJ =: cos tj COS (fl — t<)> ,-,-^~ = - cos 1J SUl (0 — U). dJ r (12) (13) Therefore, we shall liave (h , dv d

?" sin (0" (14) n'). u' I Since the geocentric latitude tj is affected chiefly by a change of the position of the plane of the orbit, while the variation of the longitude C is independent of SI and l when the squares and products of the variations of the elements are neglected, if we determine the elements which exactly represent the places to which J and J" belong, as well as tiie longitudes for two additional places, or, if we determine those which satisfy the two fundamental places and the longitudes for any number of additional observed places, so that the sum of the squares of their residuals shall be a minimum, the results thus obtained will very nearly sati .y che several latitudes. Let 0' denote iiie geocentric longitude of the body, referred to the plane of the or')it computed from J and J" as the fundamental plane, for the date r' of any one of the observed places to be used ibr cor- recting these assumed distances. Then, to find the partial differential coefficients oi' 6' with res?)ect to J and J", we have COSIJ t/J ,dO' dy , ,d(r dtp 'd^'dS--^''''''^'cW'dS , do' dii r-r. COS r, -r-' ■- -f COS f) j- • , r + ^OS J) , ,J^ , do' dM, dM^ dJ + cos ^ COS 5? , do' dr dii dJ , do' COS f) .— ■ dtp ,dO' _d:i^ dfi ' dJ" (15) ,dO' dy , ,dO' dtp , , do' dM, ''' "^ dx • dJ"- + ''' "> d^ • dr - ^- ^«^ '^ dii: ' "d^ + cos Tj and by means of the results thus derived, we form the equation ,dO^ cos rj' AO' = COS 7j''Jj\aJ-\- COS r/ V .„ A J". , do' ' dr (16) A fourth observed place will furnish, in the same manner, he addi- tional equation required for finding aJ and aJ". If more than two VARIATIOX OF TWO GEOCENTRIC DISTANCES. 321 observations arc used in addition to the fundamental plaee.s on .'liirJ' tlie assumed elements as derived from J and J" are based, the several longitudes will furnish each an equation of condition, and the most probi'blc values of aJ and aJ" will be obtained by eombininji; the entire group of equations of condition according to the method of least squares. 107. In the actual application of these foimuhe to the correction of the approximate elements, after all the preliminary corrections have been api)lied to the data, we select the proper observed places for determining the elements from the corresponding assumed dis- tances J and J", according to the conditions which have alr(!ady been stated, and from these we derive the six elements of the orbit. Since tlie data furnisiied directly by observation are the right ascensions and the declinations of the body, the elements will be derived in reference to the equator a',, the plane to which the inclination and the longitude of the ascending node belong. These elements will exactly represent the two fundamental places, and, if the assumed distances J and J" are not much in error, they will also very nearly satisfy the remaining places. We now adopt as the fundamental plane the plane of thu ap{)roxi- mate orbit thus dotevmincd, and by means o;' the equations (83).^ and (85).,, or by means of (87)2, writing a, (?, Q', and /' in place of /, ^9, SI, and /, ''C'spectivcly, we compute the values of 0, iy, anil y for the dates of the several places to be employed. Then the re^jiduals for each of the observed places are found from the formultc cosi; A6' :r= sin y A'5 -|- cos^ cos '5 Aa, at; =: cos >* A'5 — sin ;* cos '5 Ao, (17) the values of Aa and Ar? for each place being found l>y subtracting from the obser\ed right ascension and declination, respectivel} , the rijrht ascension and declinntio!i eom})uted by means of the elements derived from J and *'. The values of d, 3j, and y being required only for finding c^;^^ A^, A;y, and the differential coelHcients of 6 and ij, with respect to the elements 0'' the orbit, need not be determined with great accuracy. Next, we compute j-: and TT ^^O"' equations (12), and from /ip\ ii 1 £> fl>' dr" dv dv" dr r, , ,. 1 • . (lb)., the values of -, ^— , — , -j-, -— , cvc, by means ot which, dtp d

^^". from which to find a;^' ; and in a similar manner A^, aJ/(„ and sn may l)e obtained. It, from tlie values 01 ~~"/T~ " fnd rj„ — 1 we compute VARIATION OF TWO GP:OC'EXTRIC DISTANCES. 323 i.s equal J ami d(v-^x') ^■/, and apply tli':sc corrootions to the values of v aiul v" fouiul tVoin J and J", we obtain the true anoniidie.s corresponding to the distances J + A J and J'"' -\- aJ". The corrections to be applied to the values of /• and /•" derived from J and J" are given by A}' dr dJ aJ, ov If aJ and aJ" are expressed in seconds of arc, the corresponding values of A/" and Ar" must be divided by 2002()4.8. The corrected results thus obtained should agree with the values of r and /•" com- ])uted H' of the orbit. Thus, if the observed longitude and lati- tude and J ues of ^ and i are given, the three equations (91), will contain oj. three unknown quantities, namely, J, r, and u, and the values of these may be found by elimination. When the observed latitude ,9 is corrected by means of the formula (6)^, the latitudes of the sun disapjiear from these equations, and if we nmltiply the first by sin (O — $^)sin/9, the second (using only the upper sign) by — cos ( O — SI) sin ,3, and the tiiird by — sin {?. — O) cos /5, and add the products, we get sin /9 sin (O — SI) tan u = T—. — 5- . _ cos t sm.j cos (O from which ic may be found. SI) — sin I cos ;5 sin (A — © )' (19) If we multiply the second of those equations by siu;?, and the third by — cos /9 sin (A — SI), and add tiie products, wc find i?sin(0 — SI) r = sin ^l (sin i cot /J sin (A — SI) — cos i) (20) The expression for r in terms of the known quantities may also be found by combining the firs*, and second, or by combining the first and third, of equations (91),. If we put n C09N-- ?i siniV- : sin (5 cos (O cos /? sin (A — -Si), o). the formula for m becomes tan T( = cosiV cos (iV-f i) tan (O — Si). (21) The last of equations (91), shows that sin it and sin/9 must have the same sign, and thus the quadrant in which w must be taken is deter- mined. Putting, also, 7)1 cos M ^^ sin u, - ?Hsiu3f =siuttcot/3sin(A — ft), VARIATION OF THE NODE AND INCLINATION. n2o we have r = cosil/ Rs\i\(Q -- 9,) cos ( i¥ + i) sin w (22) AVhen any other phmc is takoa as the f'unclamental piano, the latitude of the sun (which will then refer to this plane) will be re- tained in the equations (91)i and in the resulting expressions for (t and r. Tiie value of u may also be obtained by first computing v and i// by means of the equ.-i^' ns (42)3, and then, if z denotes the angle at the planet or comet between the earth and sun, the values of u and z, as may be readily seen, will be determined by means of the rela- tions of the parts of a spherical triangle of Avhich the sides are 180° — (2 + ij/), 180° + O — «, and u, the angle opposite to the side n being that which we designate by w, and the side 180° + O — S2 being included by this and the inclination i. Let *S'= 180° — (z + \|/), and, according to Napier's analogies, this spherical triangle gives w o I \ cos i (i — w) tan i (5 + «) = :-.-—. — r ' cos i {% + w) coti($^ — O), wc ^ sinA(i — w) .,.^ -v tan I {S - .) = --r^^^p^ cotUa-Q), from which S and u are readily found. Then we have g = 180° — + — 6; It sin ■4' (23) (24) r = sm2 to find r. If we assume approximate values of Q and /, as given by a system of elements already known, the equations here given enable us to find r, n, r", and «" from k, /9 and /", ^9", corresponding to the dates t and t" of the fundamental places selected, and from these results for two radii-vectores and arguments of the latitude, the remaining elonients may be derived. From these the gfocentric place of the body may be found for the date t' of any intermediate or additional observed place, and the difference between the computed and the observed place will indicate the degree of precision of the assumed values of SI and i. Then we assign to Q the increment oSi, i remaining unchanged, and compute a second system of elements, and from these the geocentric place for the time t'. We also compute a third system from SI and / + di, and by a process entirely analogous to that already indicated in the case of the variation of two geocentric >^u THEOKKTICAL ASTRONO.A/Y. distances, wo obtain the numerical values of the difrerential eooffi- cicuts of /' and ^3' with respect to Ji5 and /. Thus the equations cos /S' aA' = cos [i' - , _ A JJ + cos /5' -y 6,1, (25) for finding the corrections aS^ and At to be ajiplicd to the assumed values of those elements, will be formed; and each additional obser- vation or normal place will furnish two equations of condition for the determination of those corrections. If the observed right ascensions and declinations are used directly instead of the longitudes and latitudes, the elements Q, and i must be referred to the equator as the fundamental plane, and the declina- tions of the sun will appear in the formula) for u and r o])tained from the equations (91),, thus rendering them more complex. Their deri- vation offers no difficulty, being similar in all respects to that of the equations (19) and (20), and since they will be rarely, if ever, re- quired, it is not necessary to give the process here in detail. In general, the equations (23) and (24) will be most convenient for finding /• and u from the geocentric spherical co-ordinates and the elements Q, aud /, since %', i//, xo" , and i^" remain unchanged for the tlircic hypotheses. When the equator is taken as the fundamental plane, 4- is the distance between two points on the celestial sphere for which the geocentric spherical co-ordinates are vi, D and a, «?, those of the sun being denoted by A and D. Hence we shall have sin 4 sin B = cos '5 sin (a — A), sin 4 cos £ = cos D sin d — sin D cos li cos (a — A), cos 4- :::= sin Z) siu (J -|- COS D cos 8 cos (a — A), (26) from which to find t]/ and B, the angle opposite to the side 90"^ — d of the spherical triangle being denoted by B. Let K denote the rigiit ascension of the ascending node on the equator of a great circle passing througli the places of the sun and comet or planet for the time t, and let w^ denote its inclination to the equator; then we shall have sin «?,, cos (A — K) = cos B, sin u'o siu {A — K) = sin B sin J), (27) cos it>. = sin B cos D, from which to find w. and K. In a similar manner, we may com- VARIATION OF THE XOPE AND INCLINATION. 327 putc the values of ;/" — n, ^, and / from the holiocontric sjihorical co-oriHiuite8 /, ft and /", b'\ From the ccjuatioud tan ^iS, + u) tan J (.S', — u) cos j (/' — K'q) 'cos.J(t'+ U'o) sin I (i' — ■!<'o'* sin A ( i' + icj cot\{Sl'~K), cot\{gi'-K), (28) the accents being added to distinguish the elements in reference to the equator from those with respect to the ecliptic, the values of <% and u (in reference to the equator) may be found. Let ,\ denote the angular distance between the place of the sun and that point of the equator for which the right ascension is A", and the ec^uatiou cot Sq = COS ?/'g cot (K — A) (29) gives the value of ,%, the quadrant in which it is situated being deter- mined by the condition that cos s^ and cos (A' — ^i) shall have the same sign. Then wc have *S' = aSq — s„, and li sin -i^ sin 2 (30) r = from which to find r, 109. In both the method of the variation of two geocentric dis- tances and that of the variation of SI and /, instead of using the geocentric spherical co-ordinates given by an intermediate observa- tion, in forming the equations for the corrections to be applied to the assumed quantities, we may use any other two quantities \vhi(;h may be readily found from the data furnished by observation. Thus, if we compute /•' and u' for the date of a third observation directly from each of the three systems of elements, the differences between the successive results will furnish the numerical values of th 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 Tj = n cos a, yo = P sm a, 2j = /) tan S. (32) If we represent the right ascension of the sun by A, and its declina- tion by D, we also have VARIATION OF ONE OKOCEXTRIC DISTANCE. 329 Xr=/2co8Z)cos.l, F=/ecosZ)si!i^l, Z=Iiii\nD. (33) The tundanieiital equations for the uiulisturbed motion of tlie planet or comet, neglecting its mass in comparison with that of the sun, are df "^ V ~ "' dt' "^ 1^' ~ "' but since .r = a-o — X, y = y^—Y, and, neglecting also the mass of the earth, ^a -r p3 — "> R' d'Y lc'Y_ {2 r em" '^» dt' B? d'z k'z __ df +> -"' 3 = Zo — Z, dt' ^ Jte ' these becouie ~di (34) Substituting for aT„, ?/„, and ?;„ their values in terms of a and d, and putting we get k-'p cos a -f- ^ = V, (36) d ^+-fcOSa + ~0, Differentiating the equations (32) with respect to t, we find Tt = cos a dp It dok />sma^, diJa .dp da '-dt-'''"'di+f'''"'di' w-'''''"rt+'''''d-f (37) 330 TIIEOHETICAL A8TH0NOMY. Ditlbrcnliatiiig again with rospoct to t, ami .sub^itituting iu the cfjuu- tions (30) the vahics thus fouiul, the results are ik'r , d',> 2;;^.|)eos.-,,=o. (38) (^ + ;;;;')tan. + 2.e..|.^ + 2.se..tan.;;;: + ..ec.|^ + :.0. If we multiply the first of the.sc equations by sin a, and tlie second by — cos (X, and add the products, we obtain lit f sini 7j cos o — p (Pa (It^ da dt Now, from (35) we get ? sin tt — rj cos a =^ ^M jm j)ii COS D sin (o — A), and the preceding equation becomes dp dt da dt (39) dp The value of ~7 thus found is independent of the differential co- efficients of d with respect to t. To find another value of -A using all three of equations (38), we multiply the first of tliese equations by sin A tan o, the second by — cos A tan d, and the third by — sin (a — ^1). Then, adding the products, since ^ sin A = rj cos A, the result is from which we get dp di = iP cot (a -A)^+ sec^ .3{ 2 '£ + cot ^g-^ ) + ^ coto ^ ^ i\ da . , .. , «/t5 cot (o — A)- cot S sec' 8 — dt dt (40) VARIATION' OF ONE GEfK'KXTRIC DTSTANTE. 331 W hen tlic ecliptic is taken as the fiindainental plane, the last term of the numerator of" the .> is given, that of -j- will be determined ill terms of the data furnished directly by observation and of the dittorential coefKcients of a and o with respect to t from ecpiation (3!)), or from (40), the latter being i)referred when the motion of the body in right ascension is verj' slow. The value of -^ having been found, we may compute the velocities of the body in directions parallel to the co-ordinate axes. Thus, since a-o = .c + X, the equations (37) give yo^y+ y, 2o = 2 + ^, dx dt ~ dp . da = cos a ^- — p sm a -r- - dt dt dX dt' dy_ dt ~ dp , da =:,ma-+pCOSa-- dV di' dz _ dt " = tanl(l-c„0 (! + »«„) K' (41) ?/)„ dciiotiu}^ tlie mass of tlie oartli, and (•^, tlic eccentricity of its orbit. Tile polar equation of the conic .section gives (It rh'. sin V dv 2> ' (It' hct r denote the longitude of the .sun's perigee, and this equation gives !„ (it (It 1 V\-e} If we neglect the .square of the eccentricity of the earth's orbit, we liave simply do ^"l/l + wio dH (It The values of R' dt kVl 4-jnueosin(0 — T). '3) a© dR -r- and -, having been found by means of these . . dX dY formula?, the equations (43) give the required results Ibr ~, -. and dZ -jj, and hence, by means of (42), we obtain the velocities of the comet or planet in directions parallel to the co-ordinate axes. 112. The values of x, y, and z may be derived by means of the equations x= d cos '5 cos a — X, y := J cos d sin o — F, z — d 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, tlie position of the plane of the orbit being known, we may compute r and u directly from the geocentric right ascension and declination by means of the equations (28) and (30). But if we use the values of the heliocentric co-ordinates directly, multiplying the first of equa- tions (93), by cos SI, and the second by sin ft, and adding the pro- ducts, we have VARIATION OF OXK (JEOCKNTIIIC DISTANX'E. 333 (47) »• Hill It =--« cosec (, r c'Ori It = X (Mw SI -\- y »*'» SI, from wlilcli r iiiul it may l)o found, the !ii'f;uinent of the latitiuloJt Ixinu; ri'ft'rrod to tlie phuiu ol' .17/ a.s tlie I'uiuUuneutal plane. The e(iiiatiou r» = a;' -f- y' + z' gives and, since we t^hall have dr X dj; y dy z dz dt }• ' dt r ' dt r ' dt ' (48: dr )''e sin v dv di "^ p di' dv k Vp dt ~ ~r' ' Vp dr ecosv P (49) '--1. r from which to find e and v. Then the distance between the ytcvi- helion and the ascending notle is given by w = ?t The sonii-transverse axis is obtained from p and e by means of the relation P 1 — e^ Finally, from the value of v the eccentric anomaly and thence the moan anomaly may be found, and the latter may then be referred to any epoch by means of the mean motion determined from n. In the case of very eccentric orbits, the perihelion distance will be given by P and the time of perihelion passage may be found from v and c by means of Table IX. or Table X., as already illustrated. The equation (21)i gives, if we substitute for / its value in terms of p, denote by V the linear velocity of tlie planet or comet, and neg- lect the mass, dr'' Let \//|3 denote the angle which the tangent to the orbit at the ex- tremity of the radius-vector makes with the prolongation of this radius-vector, and we shall have 334 THEORETICAL ASTKON< Y. rVcoS'^g- dr 'lit dx , dy , dz ""'dt-^y-dt-^'it' so that the prccct^ng equation gives Hence we derive +^'T equations Vr sin ^^ = kVp, ,. dx , dii , dz (50) from which Vr and i^q may be found. Then, since wc shall have k' a W -V, (51) by means of which a may be determined, and then e may be foiiml by means of this and the value of p. The equations (49) and (50) give e sin (ii — w) and, since -^rsm 4-0 cos ,1.0, V e cos (ii — w) = -,- r sin' ^^ ■ 1, k' a these arc easily transformed into 2ae sin (u — w) = (2a — r) sin 2-^^,, 2«c'.^ cos (n — w) = — ( 2a — r) cos 2-i>(, — }•. If we multiply the first of these equations by — cos (i and the .second by sin w, and add the products; tiien multiply the first by siu« and the second by cos u, and add, M'ti obtain 2ae sin w = — (2('- — r) sin {2'^g + n) — r sin u, ,r.y-. 2oecoswr^ -■(2a — ?•) cos(24/„ -f (t) — rcosw, These equations give the values of (n and e. 113. We have thus derived all the formula) necessary for findins,' the elements of the orl>it of a heavenly body from one geocentric distance, provided that the first and second diffenintial coefficients of a and d with respect to the time are accurately known. It remains, VARIATIOX OF OXE GEOCENTRIC DISTAXCE. 335 thoroforc, to devise the means by whieli these differential cdefiieients niiiy 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 r')erration, and to f trm the normal places which shall represent the series of ol)scrved places. We may now assume tiuit the deviation of i.he spherical co-ordinates conii)utcd by means of the approximate elements from those which woidd be obtained if t!>e true elements were used, may be exactly represented by the formula c.O^A + Bh f Ch\ (53) h denoting the interval between the time at which the deviation is expressed by A and the time for which this ditfercnee is c^d. The f^iacrences between the normal places and those computed w'ith the approximate elements to be corrected, will then sulfice to form ccpia- tions of condition by mf^ans of which the values of the coellii'ients A^ B, and C may be determined The epoch for vln'ch h ---^- may be chosen arbitrarily, but it will j^cncrally be advantajj;cous 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 aseeni-ion will give an equation of condition, according to (53), and the three etpiations thus formed will furnish the numerical values of A, B, and C. These having been deter- mined, the c({uation (53) will give the correction to be a])plied 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 ccjuations, from which, by elimination, the most probable values of A, B, and C'will 1)0 derivaxl. In like manner, the corrections t > be ai)plicd to the coinpufed latitudes may be determined. The.se corrections being npplicd, the ephcmeris thus obtained may bo assumed to rcpr(>sent the apparent path of the body with great precision, and may be cm- ployed as an auxiliary in determining the values of the differential coefficients of a and d with respect to t, liOt f(a) denote the right ascension of the body at the middle epoch or that for which h --- 0, and lct/(« ± nio) denote the value of a I'c any other date separated by tlie interval tuo, in which (o is the interval between the successive dates of the ephemeris. Then, if we put n successively erpial to 1, 2, 3, &c., wc shall have 336 TIIEOKETICAL ASTR0N05IY. Function. I. Diff. II. Diff. III. Diff". IV. Difl'. V. Diff. /'"(«+«>) /(a-2-)P"~!"^/"(«-2«>) /(a — w) J f(C') fir II -\ /(« + ") V„TL /("■+■"' /'"(»+<») /(a + oo;) The series of functions and diifercnees may be extended in the same manner in either direction. If we expand f{a + no)) into a series, the result is i-r I \ I ^^* ■ 1 '^"''* 2 2 11 '^^''* 3 3 1 1 ''** « 4 I f (/< dt' df or, puttnig tor brevity A = -j: ) =/'" (a ), &c., we obtain da _ 1 dt w dt' d*a dt* d'a '(if (/'(«) - J/"'(«) + .'o/'W - rhr^a) + &c.), 1 :.- ir («) - .1/'' («) + 2I 0/"" («) - &c.), (54) dt £=7!r(r(«)-&c.). ^) = -.\ (/""(«) -&c.). VARIATION OF ONE GEOCENTRIC DISTANCE. 337 by means of which the successive differential coefficients of a with respect to / may be determined. The derivation of these coefficients in the case of o 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 diff'erential coefficients of a and u with respect to t will also be expressed in seconds, and must be divided by 206264.8 in order to express them abstractly. ,TT ■, Til 1 n ff"' d'^a (Id (I'o , , >V e may adopt directly the values oi -^. -^i -vr. and -.-j determinea by means of the corrected ephcmcris, or, if the observed places do not include a verv long interval, we may determine only the values of jj, -,.^-, &c. by means of the ephemeris, and then find -,- and y- directly from the normal places or observations. Thus, let a, a', a" be three observed right ascensions corresponding to the times t, V, t", and we shall have da da', (f-o+.A5.'a'-o df d'a' '+'^v'-n+i^ ^a' ' df' d'a' ij d*a d*a'. (t''-n'+kim(-^''-n'-^,Y'j^(f~ty+&c., d^ di* which give I + U.--0 ,sOPa' _a df d^a' d^ tf—t de ,rfV di* &c., (55) K^"-0^^ ^(*"-oS. -&«• df da These equations, being solved numerically, will give the values of .- and we may thus by triple combinations of the observed and df places, using always the same middle place, form equations of con- dition for the determination of the most probable values of these differential ecefficienta by the solution of the equations according to the method of least squares. In a similar manner the values of r; and -, - may be derived. dt dt 114. In applying these formula; to the calculation of an orbit, after the normal places have been derived, an ephemeris should be eonij)utod at intervals of four or eight days, arranging it so that one of the dates shall correspond to that of the middle observation or normal place. This ephemeris should be computed with the utmost 22 338 THEORETICAL ASTRONOMY. care, since it is to be employed as an auxiliary in determining quan- tities on M-liich depends the accuracy of the final results. The com- parison of the ephemeris with the observed places will furnish, by means of equations of the form A + Bh + Ch' = Aa', A' + B'h + J'h' = Ad', h being the interval between the middle date t' and that of the ])lace used, the values of A, B, C, A', &c.; and the con-ections to be applied to the ephemeris M'ill be determined by A + Bntu 4- Cii'"'^ = Aa, • A' + B'noj -\- C'li'io^ = Ad. The unit of h may be ten days, or any other convenient interval, observing, however, that mo in the last equations must be expressed in parts of the same unit. With the ephemeris thus corrected, Ave da (Pa dd d-d , , 1 • 1 mi compute the values oi -j:' Zui' Ti' ^ IF^ ^^ ^^^^^^^y explamed. iJiese differential coefficients should be determined with great care, since it is on their accuracy that the subsequent calculation principally de- pends. We compute, also, the velocities ~, ~~, and -jr ^7 means dQ dR ^^ '^^ ^' of the formulae (43), -j- and — being computed from (46). The quantities thus far derived remain unchanged in the two hypothesis with regard to J. Then we assume an approximate value of J, and compute /J = J cos 5 ; and by means of the ((juation (40) or (39) we compute the value of -^7- It will be observed that if we put the equation (40) in the form p the coefficient ^ remains the same in the two hypotheses. The three , V ^ (Jp equations (38) mav be so combined that the resulting value ot -^ will not contain -^-- This trausformatio : is easily effected, and may be advantageous in special cases for which the value of --t— is very Civ uncertain. The heliocentric spherical co-ordinates will be obtained from the RELATION BETWEEN TWO PLACES IN THE ORBIT. 339 assumed value of J by means of the equations (106)3, ^^^ *^^^ ^^" tangular co-ordinates from X = r cos b cos I, y^=^r cos h sin /, 3 = ?' sin b. The velocities -jr, -■-, and -rr will be given by (4r*), and from these and the co-ordinates x, y, z the elements of the orbit will be riom- puted by means of the equations (32)i, (47), (49), &c. AVith 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 + J, and compared with the observed places. Let the ditt'crcnce between computation and observation for either of the two spherical co-ordinates be denoted by n for the first system of elements, and by n' for the second system. The final correction to be applied to J, in oroer that the observed place may be exactly i-eprcsented, will be determined by ^(«'-») + «-0. (56) Each observed right ascension and each observed declination will thus furnish an equation of condition for the determination of aJ, observing that the residuals in riglit ascension should in each case be multiplied by cos rj. Finally, the elements which correspond to the geocentric distance J + a J will be detcrpiined either directly or by intorpolation, and these must rej)resent the entire series of obterved places. 115. The equations (02)3 enable us to ^...d 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 efpiation 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 value of r and r" corresponding to a given value of a. To derive t!ns expression, let us resume the equations 'MO THEORETICAL ASTRONOMY. f.==E"-E-2e sin i (E" - E) cos -J (E" + -E), .„. r + r" =- 2a — 2ac cos ^ (£" — £) cos ^ (£" + £). For the chord x we have x' == (r + r")* — 4rr" cos' i («" — u), which, by meaii- of (58)4, gives «' = (r + r'O' — 4a'^ (cos' I {E"—E)-2e cos i {E"—E) cos ^ (J?'^+S)+e» cos" 1 (!:''+ J?)) ; and, substituting for r + f" its value given by the last of equations (57), we get x^ = 4a' sin' ■} {E" — E){1— e' cos' ^{E" + £)). (58) Let us now introduce an auxiliary angle h, such that cos h = e cos ^ (j;" + E), the condition being imposed that h shall be less than 180°, and put g = ^{E"-E)', then the equations (57) and (58) become t' — = 2g — 2 sin g cos h, r -|- »•" = 2a (1 — cos g cos li), K = 2a sin g sin h. Further, let us put h — g = d, h-{-g = e, and the last two of equations (59) give r -|- r" + X = 'la sin" Js, r + ?•" — X = 4a sin' ^S. Introducing d and e into the first of equations (59), it becomes — = (e — sin e) — (5 — sin S). a2 (59) (60) (61) The formuljB (60) enable us to determine e and d from r + '>'", ^j and a, and then the time t' =^k (t" — t) may be determined from (61). Since, according to (58)^, v'n'"co8^(n" — u)=a{cosg — cos A) = 2 sin ^e sin ^5, ■'tELATION BETWEEN TWO PLACES IN THE OEBIT. 341 and since sin ^e is necessarily positive, it appears that when n" — u exceeds 180°, the vahxe of sin J5 must be negative, and wlien n" — it = 180°, we have ^ = 0; and thus the quadrant in which d must be taken is determined. It will be observed that the value of ie, as given by the first of equations (60), may be either in tlie first or the second quadrant; but, in the actual application of the formulfe, the ambiguity is easily removed by means of tae known circumstances in regard to the motion of the body during the in- terval t" — t. In the application of the equations (52)3, by means of an approxi- mate value of X we compute d, and thence r md r". Then we com- pute e and o corresponding to the given value of a, and from (61) we derive the value of r— < = r- k If this agrees with the observed interval t" — t, the assumed value of K is correct; but if a difference exists, b}' varying )t we may readily find, by a few trials, the value which \s'll oxactly satisfy the equations. The formulaj (70)3 will then enable us to determine the curtate distances p and p", and from these and the observed spherical co-ordinates the elements of the orbit may be found. As soon as the values of u and u" have been computed, since € — ^ == E" — E, we have, according to equation (85)^, cos sin J- («"— tt) /— 77 which may be used to determine (p when the orbit is very eccentric. To find p and q, we have p = a cos' . Therefore^ if we put '^^-ia-Jj,^^^''-^'' (75) (76) the expression for r^' becomes ro' = '^-^V. (77) Tal)lc XV. gives the value of log N corresponding to values of 8 from £ = to e == 60°. If the chord H is given, and the interval of time t" — t is required, we compute Ar„' by means of (76), and, having found rj from , xl/r + r" '» = ~~27x~' as in the case of parabolic motion, wo have \i should be observed that although equation (76) is derived for the case of a small value of x, yet it is applicable whcuever t!ie differ- ence e — 5 is very small, whatever may be the value n' x. For orbits which differ but little from the parabolic form, it will in all cases be sufficient to use this expression for Ar^'; and for oases in which the difference between e and d is such that the assumption of cos ^£ = cos \d, X -{- x' = 2x, &c., made in deriving equation (70), does il 340 THEOIIKTICAL ASTRONOMY. not nfTord llio roquirod iKicuracy, we may compute both Q niul Q' direct ly, iind then wo have AT, ;=i(i-^)o-+r"-H)i. (78) Tlic vahics of the Hu!tor J I 1 — - I may be tabulated direetly with -J — {IS the vertical argument and j— as the horizontal argument; but for the few oases in which the value of iV given by the equation (7o) is not sufliciently accurate, it will be easy to compute Q and Q' by means of the formuhe (GO) and (08), and then find Ar^' from (78). Further, wlu n there is any doubt as to the accuracy of the result given by (76), for the final trial in finding x from r + r" and r^, by means of the equations (73) and (74), it will be advisable to compute APu' from (78). It apj)ears, therefore, that for nearly all the cases which actually occur the determination of the value of x, corresiionding to given values of a and 31= — . is reduced by means of the equation (72) to the method which is adoi)ted in the case of parabolic orbits. The calculation of the numerical values of r -j- r"+ a and r + r"~K will be most conveniently cffoctcd by the aid of addition and sul)- ti'action logarithms. If the tables of common logarithms are used, we may first compute sin r = — I — r,, r + r and then we liave r + r" + X = 2 (r + r") sin^ (45° + J^/), r + r" — X = 2 (r + r") cos'^ (45° + I/). 118. In the case of hyperbolic motion, the semi-transverse axis is negative, and the values of sin Js and sin ^d given by the equations (60) bocome imaginary, so that it Ls no longer possible to compute the inverval of time from r + r" and x by means of the auxiliary angle i e and 8. Let us, therefore, put sin' ttS — w' 8in'^<5 = n'; then, when a is negative, m and n will be real. Now we have and = sin V— 7?l', ^e l/ — 1 = logs (cos ^e -f- V^ — i sin ^s). RELATION BKTWEEN TWO PLACES IX THE ORBIT. Hence wc derive 347 « = 2 sin V — ,n* = - y — log. (l/ 1 + m' + m), directly with the argument m. The «anic liiltlc, using n as the argument, will give the vniue of Q/. Talkie XA'I. gives the values of Q corresponding to values of m from ri> - to m =-- 0.2. When the values of r -f r", r', and a are given, and the chord '/i is required, we may compute at,/ from (78), r,/ fi-om (77), and finally H from (73). It may be remarked, also, that the formulae for the relation between r', )•-]- ?•", K, and a sufliee to find by trial the value of a wlien r + ''" and X art! given. Hence, in the computation of an orbit from assumed EELATIOX BETWEEN TWO PLACES IN THE ORBIT. 340 values of J and J", the value of x intiy be computed iVoni /•. /•", and u" — II, and then a may be found in the manner hero indieated, If wo substitute in the eqnations (84) the values of sin |£ and sin Id in terms of r -\- r", K', and a, and then substitute the resulting values of Q and Q' in the equation (Go), we obtain 1 + B§0 ^ (('• + r" + x)i ::f (r + r" - x)?) + &c, ^^''^ (JL the lower sign being used when u" — u exceeds 180°. When the eccentricity is very nearly equal to unity, this series converges with (Treat i'a])idity. In the case of hyperbolic motion, the sign of a must t)e oliangod. 119. The formuhe thus derived for the determination of the chord x for the eases of elliptic und hyperbolic orbits, enable us to correct an uppioxiiuate orbit by varying the semi-transverse axis a and the nuiu J/ of two curtate distances. But siui-e the formuhe will gene- riilly be applied for the correction of approximate parabolic elements, or those which ave nearly 2)arabolic, it will be expedient to use - and 3/ as the quantities to be determined, lu the first place, we compute a system of elements from M and /=-; and, for the determination of the auxiliary quantities pre- limiiKiry to the calculation of the values of r, r", and X, the equa- tions (41).j, (00)3, and (51)3 will be employed when the ecliptic is the fiindiuuental plane. But when the equator is taken as the funda- menlal ]>lane, we must first compute y, K, and G by means of the "'Illation-^ '>'*5):i- Then, by a process entirely aiialogous to that by ■>\vieh the equations (47)3 and (00)3 were derived, we obtain h cos : cos (H— a") --= M- cos (a" — a), h 00s : sin ( ^ — a'') =--- sin (a" — a), (86) h sin C = M tan 'J" — tan d, frosn which to find II, ^, and h ; and also cos ^ = cos ? cos K cos (G — H) -\- sin ? sin K, (87) from which to find ". When the equator is the fundamental plane, we have pj= AcQ^ (?, J" cos .5". Froni p, p", and the corresponding geocentric spherical co-ordinates, the radii-vectores and the heliocentric spherical co-ordinates /, /", 6, and b" will be obtained, and thence Q,, i, u, u", and 'the leniaiuing VARIATION OF THE SEMI-TRANSVERSE AXIS. 351 clemonts of tlie 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 sin \ (E" -\-E) = ,. !^'~ '' ... e cos i (E" + E) = cos i (e + <5), from which c and \{E" -\- E) may bo found, and hence, since \{E" — E) ^\{t — d), we derive E and E". The values of q and V may then be found directly from these and quantities already obtained. Thus, the last of equations (43)i gives cos \v cos \E cos {v" cos \E" V~q " V'r ' Vq l//' ^lultiplying the first of these expressions by sin Jy", and the second by — sin |t', adding the products, and reducing, we obtain sin 2 (v" — v) sin -] y cos I {v" — v) cos ^E cos .\ J5"' Vq Tlierefore, we shall have Vr --=sin-.Vy : Vq ' 1 , -7- cos SV ■■ Vq ' COS IE cos ^E ITT" l/r"sin^(tt"— «) (88) Vr tan A (tt"— ti) COS IE 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 Since r^=a — ae cos J?, and (j* = a(l — c), we have C09>E-- 1 — 1 — and hence 1 — t-\ cos»iE=l cos'A£"=l 9 » "1-1 Jl (89) 352 THEORETICAL ASTRONOMY. When the eccentricity is nearly equal to unity, the value of q given by apj>roxiniate elements will be sufficient to compute cos^E and cos^E" by moans of these e(piations, and the results thus derived will be substituted in the equations (88), from which a new value of (J results. If this should differ considerably from that used in com- puting cos^E and cosJJS'", a repetition of the calculation will give the correct result. In tiie case of hyperbolic motion, although E and E" are imagi- nary, we may compute the numerical values of cos^E and coa^E" from the c([uations (89), regarding a as negative, and the results will be used for the corresponding quantities in (88) in the computation of 7 and v for tlie hyperbolic orbit. Next, we compute a second system of elements from 3/and/+ 8f, and a third system from 3f-\-d3Ian(lf,8fsim\ o3/ denoting the arbitrary increments assigned to / and M respectively. The com- parison of these three systems of elements with addit'- nal 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 AiJ/and a/ to be ai)plied to J/ and /respectively. The formation of these equations is effected in precisely the same manner as in the case of the variation of the geoceutiiC distances or of ^ and /, and it does not require any further illustration. The final elements will be ob- tained from J/+ a3/, and/+ a/, either directly or by interpolation. We may remark, further, that it will be convenient to use log Ju 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 exjieditious to make two hypotheses in regard to 31, putting in each case - = 0, and only compute elliptic or hyperbolic elements in the third hypothesis, for which we use il/and f=3f. The first and second systems of elements will thus be parabolic. 120. Instead of il/and - wc may use J and - as the quantities to 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)3, and (103)3, we compute the auxiliary quantities C, B, B", etc., il- quired in the solution of the equations (104)3. We assume, also, an approximate value of J" and com[)ute the corresponding value of /•", 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 with EQUATIONS OF CONDITION'. 353 the assumed value of -. will satisfy the equations (104)3 ^^^*^ (^^) or (61). The corresponding value of J" is given by A"^e±Vx'—C\ When J" has thus been determined, tb.e heliocentric places will be obtained by means of the equations (106)3 ^^^^ (10~)3, and, finally, the corresponding elements of the orbit will be computed. If the ecliptic is taken as the fiuidamental plane, wo put D = 0, vl = O, and write ?. and /9 in place of a and d respectively. If we now compute a second system of elements from J + o J and /= -, and a third system from J and/+ df, the comparihon of the three systems of elements with additional observed places will furnish the equations of condition for the determination of the corrections A J and a/ to be applied to J and - respectively. When the eccentricity is very nearly equal to unity, we may as- sume /=0 for the first and second hypotheses, and only compute elliptic or hyperbolic elements for the third hypothesis. 121. The comparison of the several observed places of a heavenly hotly with opo of the three systems of elements obtained by varying the two quantities selected for correction, or, when the required dif- ferential coefficients are known, with any other system of elements sucli that the squares and products of the corrections may be neg- lected, gives a series of equations of the form mx -f- ny ^=p, m'x -{- n'y = r>', &c., in M-hich x and y denote the final corrections to be applied to the two assumed f[uantities respectively. The combination of these equations which gives the most probable values of the unknown quantities, is effected according to the method of least squares. Thus, we multiply Oiicli equation by the coefficient of x in that ecjuation, and the sum of all the equations thus formed gives the first normal equation. Then we multiply each equation of condition by the coefficient of y in that equation, and the sum of all the j)roducts gives the second normal equation. Let these equations be expressed thus : — [mwi] X -f- [inii] y = Imp"], [inn'] X + [/tji] y ^ [np], 23 \\ 354 THEOKETICAL ASTRONOMY. in which [»i»i]-— m^+m"'+«i"^+ then in the ease of the system of parabolic elements we havc/=0, and the com- parison of the middle place with these and also with the ellnitic or hyperbolic elements will give the value of do O.-Or ill which d^ denotes the geocentric spherical co-ordinate computed from the parabolic elements, and 6.^ that computed from the other system of elements. Further, let tid denote the difference l)etween oomputation and observation for the middle place, and the correction to he applied to /, in order that the computed and the observed values of d may agree, will be given hy do df £,f j-A0 = O. Honcc, 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 tlio corresponding elements will bo those which best represent the observations, assuming the adopted value of 3i to be correct. 123. The first dotcrmination of the approximate elements of the orbit of a comet is most readily eflected by adopting the ecliptic as the fundamental plane. In the subsequent correction of these ele- ments, by varying - and 31 or J, it will often be convenient to use the cfiuator as the fundamental plane, and the first assumption in regard to JI will be made by means of the values of the distances given by the approximate elements already known. But if it bo desired to compute 31 directly from three observed places in reference to the ccjuator, without converting tiie right ascensions and declina- tions into longitudes and latitudes, the re(piisite formulae may be derived by a process entirely analogous to that employed when the curtate distances refer to the ecliptic. The case may occur in which only the right ascension for the middle place is given, so that the corresponding longitude cannot be found. It will then be necessary to adopt the equator as the fundamental plane in determining a system of parabolic elements by means of two complete observations and this incomplete middle place. If we substitute the expressions for the heliocentric co-ordinates in reference to the equator ui the equations (4)3 and (0)3, we shall have = n (p cos o — R cos D cos A) — (/ cos o' - R' cos U cos A') + 7i" y sin a"— R" cos D" cos A"), = « (/) sin a — i? cos D sin A) — (p' sin a'— R' cos D' sin A') (92) + n" (p" sin a"— R" .03 D" sin A"), = )i (p tan d — RsinD) — (p' tan <5'— R' sin D') + n"ip"tanr — R"smD"l in which f>, />', p'' denote the curtate distances with respect to the equator, A, A', A" the right ascensions of the sun, and D, D', B" 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 0=^11 {p sin (a'— a) — i? cos Z) sin {o!—A)) + R' cos D' sin (o'— ^') — n" (/>" sin (a"— a') + R" cos D" sin (a'- ^1")), and hence 3/-= sin (a' — a) n (93) p IV sm(,o — a') nR cos D sin (a'— ^)— i2' cos D' sin {o:—A')+n"R" cos D" sin (a'— ^") pn" sin (tt" — a') VARIATION OF TWO RADII-VECTORE8. 357 This formula, being independent of the declination '" a third system. The comparison of these throe systems o lements with an additional or intermediate observed place will furnish the equations for the determination of the corrections Ar and ^r" to be ap[)lie(l to '/• and r", res})ectively. 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 i)lace8 may be employed in forming the e(pi:i- tions of condition for the determination of Ar and Ay". 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 }•„ and v,/', respectively ; then, by means of the equations (88)i, we obtain R sin U — O) sin(^ — ^) (96) from which / may be found ; and in a similar manner we may find l". If we put we have tan (Z — A) R'shi'i^ — Q), i?sin(A — O) (97) liet S denote the angle at the sun between the earth and the place of the planet or comet projected on the plane of the ecliptic ; then we shall have v'AniATION OF TWO UAnil-VKCTonKH. 359 6": P ■■ : 180° + O - /, sin (7 — A) ' ami tan b p tan /S (98) (99) by means of wliioh the lieliooentrie latitudes b and b" may be found. The calculation of the elements and the correction of /•„ and /•„" are then ellected as in the case of the variation of r and /•". In the case of parabolic motion, the eccentricity bein^ known, wc may take r^ and T as the quantities to be corrected. If we assume approximate values of these elements, r, r', r", and r, r', v" will be given i nulled lately. Then from r, /•', /•" and the observed spherical co-urdinatos of the body Me may compute the valuts of u" — u' and 1/' — i(. In the same manner, by means of the observed places, we compute the angles »/"— u' and u' — n correspond ing to ii + oij and 7', and to r/ and T -\ ol\ d! OF OIISEIIVATIONS. 1 25. Wli KX the olcnicntH of the orbit of a heavenly body an; known to such a decree of approximation that the scpiares and jjroduets of the corrections which should be applied to them may be nejjjlected, by computing the partial differential coefficients of these elements with respect to each of the obs(;rved spherical eo-ordinates, \\v may form, by means of the differences between computation antl observa- tion, the ccjuations for the determination of these corrections. Three complete observations will furnish the six efpiations 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 piven, the number of ecjuatious will exceed the number of unknown (piantities, and the problem will be more than determinate. If the observed places were absolutely exact, the cond)inatiou of the equations of condition in any manner whatever would furnish the values of these corrections, such that each of these ecpiations 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 j)laees, are entirely different, ^^'hen 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 camiot 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 I.KAHT SQITARKS. 361 tlu' resulting values of the tuiknown quantities will he those whieli, \vhil(^ they do not eoinplctciy siitisly the several equations, will allord the hijjjhest dejrree of prohaliility in favor of their neeuraey. It will be of interest also to determine, as far as it may he possihlc, the degree of aeeuraey whieh may hu attributed to the separate results. But, in order to simi)lify the more general problem, in whitih the quantities sought are determined indireetly by observation, it will bo expedient to eonsider first the sim[)Ier case, in whieh a single (piantity is obtained directly by observation. 120. If the accidental errors of observation could be obviated, the (liflerent determinations of a magnitude directly by observation would be iilentieal ; but since this is impossible when an extreme limit of precision is sought, we adopt a victtn or average value to be derived from the separate results obtained. The adopted value may or nuiy 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 7and jtrubalilr value. It is evident, from the very nature of the case, that wc approach here the confines of the unknown, and, before wc pro- ceed further, something additi(tnal 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 arc more probable than large errors, and that errors surpassing a certain limit will not occur. Wc may also assume that in the case of a large number of observations, errors in excess will occur as frequently as errors iu defect, so that, in general, positive and negative residuals of equal absolute value are equally ]>robab]e. It ai)pcars, therefore, that the relative frequency of the occurrence of an accidental error J in the observed value will depend on the magnitude of this error, and may be expressed by e witliin the limits — o and + d will be expressed by i ^/ -»'A* dJ; (14) and in anothv- $^im t^ '>bservations, more or less precise, the pro- 366 THEORirnCAL ASTIJOXOMY. I)!il)ilitv tliat the error of an observation is witliin tlie limits — d' and T O WI Since 11 he ■ h">ii"- lU. (15) -i-hS ~_ j' c-"'^' dJ = ---f e-"'^' d ihJ), — « —liS it appears that the intejjjrals (14) and (15) are equal when hd = h'd'. llc'iice, if we put ^' = 2/(, these integrals will be equal when d=^-2d', and an error of a given magnitude in the first series will have the same ju'obability as an error of half that magnitude in the second series. The second series of observations will therefore be twice as accurate as the first series, and the constant h may be called the iiwuHnrc of prcolx'ion of the observations. The greater the degree of precision of the observations, the greater will be the value of li. The relative accuracy of two scries of observations may also he detc'Muined by a comparison of the errors whic^h are committed with e(|ual facility in each sc.'ries. If we arrange the errors of the several observations in each series in the order of their absolute magnitude without reference to the algebraic si^n, the errors which occu[)y the same position in reference to the extremes in each case will serve to detei'mine the relation sought. We select that, however, which occu- pies the middle place in the series of errors thus arranged, and since the number of errors which exceed this is the same as the number of errors less than this, if we designate the error which occupies the middle place by /■, the probability that an error is within the limits — r and + r will be equal to \. The probability of an error greater than }' being the same as the probability of an error less than /■, the error /• is called the jiohahfe error. The relation between r ami h is easily determined. Thus, we have Mill give the relati or, ptittiug hJ — <, -J dJ 5' /. 'V<.-^L S144311. (16) If we expand c"** into a series of asc^ct*(iing powers of t, multiply by dt, and integrate btjtween the limits » * auJ T, we get X METHOD OF LKAST 8QUAKES. .-'^ ,]f — Ji 1 T"> _!_ I 1 _l_ I „ ' '« - ^ 3 J t- Ti ^ _ 2 ' 1 . 2 . 3 ^ M . 2 . 3 . 4 3()7 -&^'., (17) wliich converges rajiidly when T is small. To find the value of T wliieli corresponds to the value 0.44311 assigned to the integral, we nmipnte 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 Avhen the sum of the terms of the series is 0.44311, we have r==/(r = 0.47694, or 0.47(594 h (18) wliich determines the relation between the probable error and the measure of precision. The })robability that the error of an observation, without regard to sign, does not exceed nr, is expressed by nhr (19) 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 niiniber of observations. Hence, if we assign different values to n, the integral (19) computed for the several assumed values of nhr = 0.47694/i will give the relative number of errors of a given magnitude. Thus, if we put n = ^, we obtain 0.2385 fidiii which it appears that in a series of 1000 observations there ought to be 264 observations in which the error does not exceed A?-. It has been found, in this manner, that in tiie case of an extended series of observations the number of errors of a given magnitude a>;signed by theory agrees very closely with that actually given by the sories of ol>servations; and hence we conclude that the error com- mitted in extending the limits of the summation in the expression (1) to - X and + oc, instead of the finite limits which it is pi-esumod that the actual errors cannot exceed, is very slight, so that the form K-^f S^^rr ii'-^'v 308 tiij:oui:tical astronomy. of thr fiinrtion ' I/tt" Pe" Iii tlio case of a siiijfle observation, if /* denotes the probaljility of tlie error zero, and F' the probability of the error d, we have P' = Pe — hV Hence it appears that if /(^ denotes the nieasnre of precision of the arithmetical mean of m ol)servations, the relation between Ii^ and h, the measure of precision of an observation, is given by h,^ =: «i/i' ; (27) and if r^ is the probable error of the arithmetical mean, and e^ its mean error, avc have, according to the equations (18) and (20), ^•o^ r £ (28) V m These expressions determine the probable and the mean error of the arithmetical mean of a number of observations when these errors in the ease 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 miniber 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 x, 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 niunber of observations, the nearer will these residuals approach the abr^ohite errors. If J, J', J", &c. are the actual errors of the observation.^, and V, v', v", (fcc. those which result from the most probable value ol X, we shall have, denoting the arithmetical mean by .% and the true valne by x^ -f 8, A = v — 8, A'=.v'~ d, J"=.v" — d,&c.; METHOD OP I-EAST SQUAKES. 371 and hence )H£' = [J J] ::^ [iv] + mo' (20) This equation will enable us to detenninc the mean error of an ob- servation when o is given; but, since this is necessarily unknown, sonic assumption in regard to its value nuist be made. If we assinne it to be equal to the mean error of the arithmetical mean, the re- niaiiiing error will be wholly insensible, and hence the equation (29) becomes Therefore, we sliall have and, according to (21), ~\m-l' Afm — l r = 0.6745 (30) (31) These equations give the values of the mean and probable errors of a single observation in terms of th(! 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 ^ m (m — ly r, = 0.6745 (32) When the number of observations is very large, the probable error of :a\ 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 ""-^l^' ami hence we have, according to (23), r = 0.8453 For the mean error of an observation we have e = ,l/j,7r = 1.2533 m' (33) (34) 372 TIIKORKTIfAL AHTHONOMY. Tl' the niiiiil)('r of olwcrvfttious is very jj;roat, the results given liy thew! C(jU!itii)ns will aj^reo with those fj;iven by ('U)) and (-il); but tor any liinit<'(l series of observed values, the results obtained l)y niean.s of the mean error will alford the j:;reatest acH'uraey. 1.'12. The relative aeeuraey of two or more observed values nf a f|Uanli(y may 1h' expressed by means of what are called their ir< It/ldK. If the ol)servations are made under })re(!isely similar eireumstantrs, 80 that there is no reason for preferrinj^ one to the other, they are said to have the same wei«i;ht. The weiji;ht must therefore depend on tiiu measure of preeision of the observations, and hence on their j)robal)le errors. The unit of the weight is entirely arbitrary, sinee only the relative weights are recjuired, and if we denote the weight by p, tlic value of J) indicates the number of observations of e((ual accuracy which must be condnned in order that their arithmetical mean may hav(! the same degree of preeision as the observation whose weight is p. Hence, if the weight of a single observation is 1, the arithmetical mean of in such observations will have the weight m. Ijet the pro- bable error of an observation of the weight unity be denoted by /•, and the probable error of that whose weight is p' by »•'; then, ac- cording to the first of equations (28), we shall have or Vp'' -.p'r'\ For the case of an observation whose weight is p" and whose pro- bable error is r", we have :yV'»: --p'r'\ from ^vhieh it appears that the iveights of tivo observations arc to each other hiverseli/ as the squares of their probable or mean errors, and, aceordinc/ to (18), directly as the squares of their measures of precision. Let us now consider two values of x, Avhieh may be designated by x' and x", the mean errors of these values being, respectively, e' and e"; then, if we put X: : X' ± X" and suppose that both x' and x" have been derived from a large num- ber m of observations (and the same number in each case), so that the residuals v„ v/, v'", &c. in the case of x' and the residuals v„ v/, (,", &c. in the case of x" may be regarded as the actual errors of ubser- MF.THOI) OF LKAfiT HQUARKH. 373 vation, tlu; errors of the vulue of X^ us deterniined from the scvorul observations, will be V ± v„ v' ± !•/, v" ± V,", Ac. Let the mean error of A' be denoted by K; thou wo have m£' = ^(y± v,y = [I'l'] ± 2[i'iv] + [y,i',] ; and since the number of observed values is supposed to be so ^reat that the frecpieney of nej!;ative i)r<>duets vr, is the same as that of the similar positive prodnets, so that [_ni,'] - 0, this ecpiation gives or i'« = £'■' + S"\ Combining X with a third value x'" whose mean error is e'", the moiui error of x' ± x" ± x'" will be found in the same manner to be Kjiial to £''^ -'f e"^+ s'"''; and lience we have, for the algebraic sum of any number of separate values, and, according to the last of equations (21), R ^ Vr' -h r'' + r'" + etc., (35) (3G) R hoing the probable error of the algebraic sum. If the probable errors of the several values are the same, we have r = r' ^:^ r" ^:= &c, and the probable error of the sum of m values wnll be given by R =z )Vm. IIouco the probable error of the arithmetical mean of m observed values will be _R _ ^ m V m which agrees nith the first of equations (28). Let P denutf! thi! weight of the sum X, p' the weight of x', and jj" that of x" ; rhen we shall have y _ r'^ + r'" pI - ^" + ^"' ^'iu ^>. IMAGE EVALUATION TEST TARGET (MT-3) l/.A fA 1.0 I.I 1^128 |2.5 ■iS 12.2 us C« ''!2.0 11:25 i 1.4 1.6 '/y ^>. ^'J> ^ 'V' Hiotographic Sciences Corporation 23 WEST MAI'l SVRECT WEBSTER, NY. USSO (716) 873-4503 \ 4' V> 4^'^. <> ^ ^P?<\ ^^^<5:\^^ '^V i? 374 THEORETICAL ASTRONOMY. PP from wliich we get Since the unit of weight is arbitrnry, we may take f37) P =7« 1 . 1 „ P =/7,'&c.; and lience we have, for the weight of tiie algebraic sum of any nuniljer of values, or, whatever may be the unit of weight adopted, 1 P^ i 4. J_ _L X. 4. pf -r p>, T ^/// -r (39) In the ease of a series of observed values of a quantity, if we designate by r' the probable error of a residual found by comparing the arithmetical me:in with an observed value, by r the probiil)lc error of the observation, by x^ the arithmetical mean, and by n any observed value, the probable error of according to (36), will be n •To + V. ,.' V > m Tq being the probable error of the arithmetical mean. Hence wc derive ' Wl — 1 and if we adopt the value r' ^0.8453 .0 H m the expression for the probable error of an observation becomes r = 0.8453 l/m(»i — 1)' (40) in which [v] denotes the sura of the residuals regarded as positive. and m the number of observations. 133. Let n, n', n", <&c. denote the ol)8erved values of .r, and let/), p'y p"f &c. be their respective weights ; then, according to tiie deti- METHOD OF LEAST SQUARES. 375 iicc wc derive nition of the weight, the v:ilue n may be regarded as the arithmetical mean of p olwervatioiis whose weight is unity, and the same is true in tiie ease of n', n", &v. We thus resolve the given values into }> -\- j>' + p" + . . . . observations of the weight unity, tuid the arith- metiml mean of all these gives, for the most probable value of x, pti + p'n' 4- p"'i" -f &c. [;jh] p-\-p'-\- V" + A^'- [;0 (41) The unit of weight l)eing entirely arbitrary, it is evident that the relation given by this equation is eorreet as well when the (juuptities ;>, y/, p" , ifec. are fractional as when they are whole numbers. The weight of .t*u as determined by (41) is expressed by the sum and the probable error of x^ is given by '•0 = n Vp + J^^f-V (42) when r, denotes the probable error of an oljservation whose weight is unity. The value of r, must be fouhd by mej ns of the observa- tions themselves. Thus, there will be p residuals e.\])ressed by n — a-j, p' residuals expressed by »' — ,r„, and similarly in the ease of n", n'", &c. Hence, according to equation (31), we shall have 0.(5745 JtM. » ?/i — 1 (43) in which m denotes the number of values to be combined, or the number of ((uantitics n, n', 11", &c. For the mean error of j-'y, we have the ecpiations ' Ml — 1 ,^, V'lp^ \[,n-l)lp] If different determinations of the (juuntity x are given, for which the probable errors are r, r', r", + J"i 4- &c. shall be a minimum. Hence it appears that when the observations are C(|iially precise, the most probable values of the unknown tpiantities are those which render the sum of the squares of the residuals a niiniinum, and that, in general, if each error is multiplii!tl 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 r, v', f", dv „ dv" dx dx dx dv , .dv' ^ „dv" , dy dij di, dv ,dv' „dv" . %lz^' dz-^' dz-^-' = 0, &c. If we tliflll'rontiatc the equations &c. ax -\- hy -\- cz -\- du + ^'f + /' + "■ = ^'» ti'x + b'y + cz + d'n + c'w + ft + n' =^ v', a"x -f b''y + c"z + d"u + e"w +ft + n" = v", &c. &c. with reHpoct to x, y, z, &c., successively, we obtain dv" „ . -J- = a , &c. dx dv dx-""' dv' dx-""' dv , -dy-'' dv' ,, ^"1 dij b", &c. &c. &c. (fee. (•18) (49) (60) Introduoinj; these vnhics into the e(iuation>« (48), and substituting for V, v', v", etc. thoir vahies given by (49), we get [aa] X + [nh'] y -f [o^] z -f- [ad'] a -f [ae] w -\- [«/] t -\- [ow] -- 0, [«6]J- + [hh]y + [6c] ^ + [bd] n + [A^'jw -|- [6/]< + [*"] "-- 0, [op] X 4- [/><•] J/ + [cc] z + [cf^ It + [cc] i<; + [r/] t + [c/j] =. 0, , . [ad] X -(- [Af/] y + [cf/] 2 + [dd] n + [f/c] lo + [df] t + [rf/i] -^ 0, [««■].(• + [Ac].'/ + ['•''] 2 + ['^e]M + [efi]w -I- [f:"n< + [ch] =0, (51) iu which [aa] z=aa -\- a'a' -\- a" a" -f- [o6] = a6 -f a'6' + a"b" + [ar] = ac + a'c' -f a"c" + [ii] .-: ft6 + 6'6' + b"b" -f &c. <&c. (52) The equations of condition are thus reduced to the same number as the number of tlie unknown quantities, and the sohition of these will give the values for which the sum of the squares of the residuals will he a minimum. These final equations arc called nornml equations. When the observations are not equally precise, in accordance with the condition that AV -\- h'h'^ + h"h"'^ + &c. shall be a miuiinuin, METHOD OF LEAST SQUARES. 379 cncli ofumtinn of condition ninst l)c nuilti|)li<'(l hy tho mcnsnrc of ]tn'('ision of the ohscrviitioii; or, since tho wci^jht i.s propMrtional to tlio .s(|niire of the niciisun? of precision, each c(|nation of (^onditiim innst he nnihiplied hy the scpiare root of the weijjht of the ohscrva- tioii, and tho several e(iuations of condition, heinj; thns rcdncjid to till' same unit of weight, must he comhineil a.s indieatejl l)y the e(jua- tions (51). \',M). Ft will he ohserved that the formation of the fii*st normal e(|iiation is (jifected hy multiplyinji; eaeli e(|uation of condition i)y the ('((enicient of .c in that e(|Uution and then taking; tlu^ sum of all the cfiuations thus formed. The seeoiid normal ecpiation is olttained ill the same manner by multiplyine derived from one of the others by multiplyinj; it by a (!on- stant, or if one of the e(|uations may be derived by a combination of two or more of the remainin}r e([uations, the number of distinct rela- tions will be less than the nnml)er of unknown quantities, and tlu; proljjem will thus become indeterminate. In this case an unknown fliiantity may be expressed in the form of a linear function of one or more of the other unknown quantities. Thus, if the iiumber of iiulependent ecjuations is one less than tho number of uid 4- [,/•.!] < 4- [fH.l] :: . 0, (59; Tlu'S(! (Mjtialioiis aro synuiu'triciil, iiuA of llio same form as the normal (•(Illations, the cocrticicMts being distingnislied l>y writing tho nnmt-ral 1 within the hracki-ts. Tiio unUnown (|uantity .r is thus eliminated, an- // may be eliminated from the e<|uations (o{>), the resnlting e(|na- tioiis lieing nMuU'red symmetrical in form hv the introtluetion of the numeral 2 within the braekets. Thu.s, we put ^''■^'^-[bhA-]^^'-'^^^^''-^^' [/W.I], [c(U]-[Jj;^j?|[W.l]=:[«/.2], ['•/.I] [brA] IbbA] [/>/.!] =[r/.2]; (60) [rf(/.l] [66.1] [6(M] = [dfl.2l t''^-^]-[6U]t'^^'^^='^'^^'-2^^ [Af'n-^l;^w^:i = w-^i (61) ^''•^^ ~ [60I t^'-^^ =^ '^''•2^' and the equations become * [f/H.l]-[*jJ-j3[;6,,.i] = [rf».2], [/»■!] [66.1 J [6h.1] = [>.2], (63) [cc.2] z 4- [of/.2] u 4- [pe.2] If' 4- [c/.2] < -f [rH.2] = 0, [«^2]z 4- [fW.2]it 4- [de.2]iv + [({t\2]t + [«//i.2] = 0, [fv'.2] 2 4- [(/('.2] n 4- [(V'.2] ?D 4- [f/.2] < 4- [^'".2] = 0, [r/.2] 2 + [(i/.2] u 4- [^/.2] t(; + [jOr.2] ^ 4" [>.'2] - 0. (64) To eliminate z from these equations, we put t<'''.2]-[-^3[crf.2] = [rf». -f 1 " [""J L""J -f [""1 ["''1 „ , [""1 [m] •' ^ [na] ' "^ L""] " '^ [.""] " "^ [""J ' "^ t""] ^ "'" [///;. 1 ] "^ [A/>. 1 ] "^ [hh. 1 ] "^ [/»/.. 1 f '^ [f>t>. I ] ^+[...2J»+t. < 1( i] ' [r,r>] [,r.2] <+ [/"••■•I [jtr-^] 0, 0, 0, 0, 0, 0, (74) tlic s. It will he (thMTVcd, fiirtlHM*, tliftt lM)th in tho nornml (Mjuations and in thoHO whicli result iit'ti r fach HUccc'SMivcj oliiuination, th(^ ('(M'nicicnt.s which appear in a li(iri/iintal lino, with tho exception of the coenicitMit involvintr the iilMiliitc torniH of tho equations of condition, an; found also in the (virrespondinjj vertical liiu'. Tho form of tho notation [/>i.lj, ['"■•1]> iVr. may bo symbolized thus : r 1 [»/'/0 r T [/?r. 0^ 4- D], (75) in wliich a, ^9, y, donoto any three letters, and // any numeral. 'fhe ('(juations (74) are derived for tho ease of six unknown (|uan- tities, which is the number usjuilly to \w determined in the correction of the elements of the orbit of a heavenly body; but there will l)e iKHlitViculty in oxtendinj; tho process indicated to the ease of a <;reater niiiiilu'r of (uiknown (piantities, except that the nund>er of auxiliaries syiiihoiizod generally by (75) increases very rapidly when the number of unknown quantities is increased. 1")7. In the numerical application of the forraulnc, when so many quant Ities are to be computed, it Iwoomes important to be able to clu'ck tho accuracy of the calculation in it.s successive stages. First, then, to prove the calculation of tho coefficients in the normal cpia- tiuu.s, we put o+6-fc4-rf+e +/ =«, a' + 6' + c' + rf' + e' +/' =. s', Ac. 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 TIlKOIliniCAIi AHTKONO.MY. [an] -1 [/».] -I- [rn] -j [r, iiiiiltiplyiii^ i)y cadi of tli)* (;()('f!i('i(>l [ab] + [W>] + [br] -f- [/,,/] 4- [/>,.] + [/./•] - - [H, ["< ] -f {.be] -f ['•'•] -f- [rd] + []. The expressions for the auxiliaries [nH.2], [?j»..3], ifec. arc [nu.2] = [nn.l] - [*J|;JJ [6».l], [nn.3] = [nH.2] - [^^^ [r«.2], [»".4]^[«„.3]-[^;;;|][rf«.3]. [jjH.o] = [nH.4] — ^'J"^--' [cH.4], [n«.6] = [nn.5] - ^'^ [//<.5]. (83) The process here indieated may be rcatlily extended to the erne of a prcater number of unknown quantities, and we have, in general, when // denotes the number of unknown quantities, [yv] = lnn.fi]. 2» (84) 386 THEORETICAL ASTRONOMY. This o([uation affords a coinplotc verification of the entire numerioal oalciiliition involved in the determination of the unknown (juantitios from liie original equations of eondition. Tlius, after the elimination has l)een completed, we substitute the resulting values of x, y, z, (V'c. in the efjuatictus of condition, and derive the corresponding values of tlie residuals r, r', v", r], &c., since only the coefficient q is required. Tlie most probable value of .r is found from (90) by the condition of a mininuim of the squares of the residuals, namely, that [ai-] = 0, [6y] = 0, [c!-] = 0, &c. The j)rocess here indicated for the determination of the weight of the final value of x is general, and applies to the case of any otlier unknown quantity provided that the necessary changes are made ia the notation. Thus, the reciprocal of the weight of y is deterinin(3(l by writing, in the normal equations, — 1 in place of [/>«], and putting [«»], [en], t&'j. equal to zero, and completing the eliminatioii. It is also the co-^fficient of [6i'] in the value of y when the eliminatioii is effected with the symbols [av'], [6i'], &c. retained in the second members of the normal equations. 139. It may be easily shown that when the elimination is effectccl by the method of successive substitution, as already explained, the METHOD OF LEAST SQUARES. 389 oocfrioicnt of the unknown quantity which is miule the last in the oliinination, in the final equation tor its determination, is eijual to the woigiit of the resulting value of that (juantity. Thus, in the eu.se of the eijuations for six unknown quantities, since the reciprocal of the weight of the most probable value of t is the value of t obtained from the normal ecjuations by putting//i = — 1, and an, bn, vn, (te. equal to zero, the equations (63), (67), (69), and (71) show that we have [/«] -= [/H.l] = [/«.2] =. [/«.3] =: [/«.4] == [/«.5] - - 1, and hence, according to (72), for tlie recipi'ocal of the weight of t, i>.= [//-5]. (93) [//••5]; which gives 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. Hen(!e, by repeating the elimination, fiiucessively changing the order of the quantities, so that each of the unknown quantities may have the last place, the weights will be (Ictermiiied 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 ([uantities will be given by means of the auxiliaries used in the elimination. Thus, the reciprocal of the weight of w is ol)tained by putting en = — 1, and the other absolute terms of the normal ecjuations etpial to zero, and finding the corresponding value of «'. This operation gives [e«.4] = -l, [/».4] = 0, Hence the equation (73) becomes '■*' -■ [ce.4] [ee.^ijf.^ and substituting this value of t in the last of equations (70), we get or (94) W: 390 THEORETICAL AHTUOXOMY. / which gives the weight of to in terms of tlie auxiliary quantities required in tlie determination of its in(j*it probable value. Jf the order of elimination is now eompletely reversed, so that x is made the last in the elimination, the weights of x and ti will be determined by the equations _[«a.5]_.. ._ (9») [auA] IbbAl A third elimination, in whieh z and ?6 are the unknown quantities first determined, will give the weights of these determinations. It appeal's, 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 furnisii 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 whieh 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 Avill nave the denominator D := [«a] [ift.l ] lcc.2-] Idd.S^ lee A] [//.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 unkndwn 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 coeflieient of the quantity first determined; then we shall have D^laa-] [66.1] [ce.2] [rfrf.3] [ee.4] [jf^.S] = [aal [66.1], [cc.2]. [r/r/.3], [//.4]. [.e.5] = [«a], [66.1], [cc.2]„ [ee.3], [//.4], [rW.5] = [««]. [bb.ll ldd.2l [ee.il [ffAl [.0.5] - [««], ['■^■1]* l = [aa][i6.1] [cr.2] [.(hIM] leeA'] \_fj\rq = [««][W>.1] [.r.2] [(W.:?][ir.4] [l [«■.:]]„ [./J.4], [i/>.o] = [^6] Kl],. ldd.'>l [ecMl UJ-^l ["«-'^]. and from these equations we obtain P„ = [dilo^ ;).=:[ec.5] : [ee.4], J...4] [frA]_ ■ idd:2] [dd.S] (96) ^0.2], [c^.2] [w.2] ■ [.cc.lX [ii.l], [Aft.l] [««]. 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 fjr y),^ is identical with (94), the expression for p„ introduces the new auxiliary quantity [./(/'. 4]j, and that for 2>e introduces two new luixiliar'"- The exi)ressions for the new auxiliaries [.)(y.4],„ [.i(jr.4]^,, [fr.3]^, '", B' ', 7?'^, and /i^, and wc shall have 0: [/,6.1] (101) 4-Ky:f^" + j::;T;^5"'+^'\ [66.1J ['"''•-J [rW.3] [<'('.4J In a similar manner, we ohtain the following ecinations for the de- tirinination of the eliminating factors necessary lor finding the values of the remaining unknown quantities: 0: [rrf.2] [rrr2] t«'.2]' "^ [f^/.3J ^ "^ ^ ' + C"", [(/..3] " [j^ +[^^.4]^ +[#5]^ — z [m.2] [rf».3] [w.2] "^ [(W.3] 4_ [''"•4] ^.iv , \. />'->} C\ - [rW.3] + [ee.4] ^ + [.^J.5j ^ [ee.4] ^ [1.5] ^ ' _ , _ [.At 5] »r - (103) 394 THEonKTKA L ASTRONOMY. The first of tlicsc; cqiiutioiis will f>i\'o the recipnx-nl of tlio woijjlit of .r, wIk'Ii \v»' )»ut [rui] - 1, and tliu otlu'i" ahsolute tiTiiis ol' tlie iioniiiil ((iiiiitions i-cjual to zero; tliu >sl'coii(1 will give the rcciprociil of the weight of // hy putting [//»] -^ — 1, and the other absohite terms oi" the normal ecjuations e(iiial to zero; and, continuing (he process, finally the last ecpiation will give the reciprocal of the weight of t when we jjiit fn '-- — 1, and [''/'], ['>"], [t'"]» ^^^'- t'M"'i' t" '•''•'"• It remains, therefore, to determine the particular values of [/>".l], [(•/(. 2], ttc, and tiie exi)ressions for the weights will he complete. li' we multiply the first of etpiations (100) by [««], it becomes [6h.1] =^ [a»] A' + [bn]. 104) Multii)lying the second of ecpiations (100) by [an], and the first of (101) by [/>/(], adding the products, and introducing the value of [/>«.!] just found, we get [6c.l] [e)i'\ — [pH.l] which reduces to [^6.1] [Ah.I] -\- [a/i] A" 4- [6«] B" = 0, ini,-]A" + ib„-\.ir + [<•«] = [c«.2]. (105) Multiplying the third of cujuations (KX)) by [«u], the second of (101) by [/>"]) '"'•' ^''^' ^^'^^ **^ (1^'-) kv [''"]> "thling the products, and ro- dueing by means of (104) and (lOo), wc obtain ^ Idn] - Idn.l} + [J^'-J^^ [/>..!] which, by means of the expressions for the auxiliaries, is further re- duced to [«»] A'" + [6«] £'" + [ch] C" + [rf»] =: [rf/i.3]. (106) In a similar manner we find, from the remaining equations of (100), (101), and (102), the following expressions: ri07) [o»]^l" + [in] -6"+ [ch] C' + [rf/i] />"+ [en] = [eH.4], [«»] ^^ + [ft»] iJ> + [tvt] C + [rf«] -O' + [e/i] £" + [/«] =^ [/«-r)] The equations (104), (105), (106), and (107), enable us to find the particular values of [in.l], [e?i.2], etc. requiretl in the expressions for the reciprocals of the weights. Thus, for the weight of x, wc have [au] := — 1, Ibii] = [oi] = Idii] = [ch] = [/n] = ; MKTMOD OF KKAST SQUAHES. and those ('(luatioiis give .305 .1', [_n,.2] A", [J«.3] If,,.-}-] = ~ A\ .r, Fiir tli(! case of the weight of //, we iiiive \b,,-\ - - 1 , \_a„-\ - [c«] r^ [f/«] =-. [e/i] = [/»] =- 0, ami the satne equations give [/>//.!] -- — 1, [fH.2] = — li\ \jh,SS-\ = — E", [>.5] = -i:- Wc have, also, for the weight of 3, [r./i] - -1, [w.3] + [ec.4]+[./)y5]* 1 _ 1 A'^*^ £-[ee.4]+UJ-.5]' 1 1 (108) i>, L./J-5]" The eqiuvtions (103) and (108) will serve to determine separately tlie value of each unknown quantity and also that of its weight, the 306 TIIKORKTKAI. ASTIIOXOMV. aiixiliiiry racfors A', A", II", tVrc. Iiaviim hcon found from the or|iiu- tioiis ( 1(»0), ( KH), and (102). If we n-vcrso the operation and w- (•oniposc tlic (>(|(iutions (71) by nii'ans of tin; t'xprcssions for the un- known (|Uantitit's jrivon hy (!<>.'}), the conditions wliirh iuinicdiatclv follfiw furnish another scries of (Mjuations for the determination of tlu; auxiliary liietors. The e(|uations thus (h'rived will };ive first th' valiici of .1', /,"', ("", D% and A" ; tlH;n, those of A", 11'", (''% Jr-, and ' be i)ut in the form A A' A" (109) in which D is the determinant formed from all the coeflfieients of the ludvuown (piantities in the normal equations, and in which ^1, A', A'', &v. are the partial determinants recpiircd in the elimination. TIiuh, A is the determinant formed from the coefficients of all the unknown quantities except .r, 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", A"', &c. are fornied in a similar manner. Now, since the value of x which results when we put [rnt] ^=: — 1, and the other absolute terms of the normal equations equal to zero, is the reciprocal of the w'eight of the mo!-t probable value of this unknown quantity as given by (109), we have P.= t> (110) In like manner, the expression for the most probable value of y will be MKTIIOD OF I-F.AST Stil.A KF». I) [/'"] ij [en'] — Ac, 397 (111) /), />", 11" , ttc. bo'mj; tli<* partial (Ictcniiiiiaiits foniicd when the co- cllii'k'iits ttt" y lire omitted; and I'or it.i \veij;;ht we have P, I) l}' (112) The fnrtiiulfP for the most prohahle vahie of z and for its \vei<;ht arc entirely aiiahtj^oiis to tiiose for x and y, Mt that the pntcess hcrt; indi- latcd may l>e extended to the ease (»f any nnnd)er of nnknosvn (pian- titics. Jt ap[>ears, therefore, that the \vei;;ht of the most prohaMe value of any nid^nown (puntity is found hy dividin^r the; (complete iltttrminant of all the eoellieients hy the partial determinant formed when wo omit the normal expiation eorrespondinjij particularly to this unknown (piantity, and when we omit also the coefficients of this (lUiuitity in the remaininf; normal ecpiations. The iK'culiar arranj^ement of the coefficients in the normal o = [««] [66] — [o6]', and hence P» [«a] [66] — [ab-y [66] P» = laa] [66] — [«6]' [Ott] Wlien the number of unknown quantities is increased, the expressions for the determinants necessarily become much more' complicated, and lieiice 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 (juantities is rendered uncertain l)y the similarity of the signs and coefficients in the normal equations. a»8 Til K( mi'.TKA I. ASTK( »NOM V. ami ill wliifli tlic ]M'oI)Iciii Iiccoiim's nearly inilctcriiiiiiatc. Sniiicfinit's it will l»c |»o.ssil»lc to ovcrcniiic the «lillifiilty thus ciinnuitrrcd hv a siiilalilc <'haiip' of the cleinciits to ho (Irtcniiiiicd ; hut, ^;»'iici'allv, lid* a coiuplctr uikI sutisliictory .solution, additional data will l»c rciiuind. It often happens, howi!ver, that several of the iinUiiowii ipiaiititics may Ix; aeciirately deteriiiined from the jfiven ecpiatioiis when ilic values of the <»thers are known, hut that the <'ertaiiity of the det< r- miiiation ttf tlu' saiue (piantilies is very (greatly impaired when all the unknown fiuautities are derivecl simultaneously from the saiiio e(|uations. Let us suppose that one of the unknown (juantitien is, from the very nature of the prohlem, not suseeptihle of an aeciiratc determination from the datji employed. The e(juation.s will tlieii Iiresent themselves in a form approaehin^ that in which tlu' minilicr of independi'iit relations is one less than th(.' niimher of unknown quantities, so that it will he neeessary to deterinine the other unkiunvn fpiantities in tcrniH of that whose value is necessarily uncertain. In this case the elimination should be so arraii}j;ed that the ipiaiitity which is regarded as unct.'rtain is that whose value would he lir»t determined. Then, if its eoelUeient in the Hnal ecpiatioii, corro- spondinji; to (72), is very small, a circumstance which indit^atcs at once the exi.steiioo of the uncertainty wlien it is not otherwise sus- pected, the process of elimination should not he eoin}»leted, and the auxiliary fpiantities should he determined only as far as those ro- )»:.\] [r„A] [n,A\ ^o~ [,/,/.:;] [,r.4J />', 3J)}» OVA) and lionoc x~r,+ A't, ^'•=-[,..4]- u = I/, + 7)7, i« =^ w, + E't. (lUj As snoii as / is (Ictcniiiiu'd hy .soino iiuli'iK'ndciit (.'; heeii determined l)y ne^lectinj^ t entirely, it" we denote tlu? mean error of the final ado|>te<[)ondin<5 values of the other variahles will he given by ^; -- (O' + '-I'^i^', s" -- ('-y + fi'^v. c = (O'' + <^"' <^"s'. ■c=(>:/+^'/>v. (e„v+A"i;v. (11^) These in whieli ($,), (e^^), ite, denote the mean errors of .r„, »/,„ tt( f'urniidie shf)W, also, that when one of the variables is nej^leeted, the e(|nations assign oo great a degree of preeision to the results thus olitain(,'d. Wlieii there arc two or more unknown qjiantitios wliieh cannot be (leterniined from the data with sutlieient eertainty, the prolilcm must lie treated in a manner entirely analogous to that here indieated; but, since cases of this kind will rarely, if ever, occur, it is not necessary to pursue the subject further. 144. The weights which arc obtained for the most probable values of the unknown quantities enable us to find the mean anil [)robablc errors of these values. Let e denote the mean error of an observa- tion whose weight is unity; then the mean error of v will be (116) ^P. and, in like manner, the exprcssioua for the mean errors of y, 2, u, &c. will be S=-^> ^=-f- e„= ' &C. (117) Vp, Vp. ^P„ It remains, therefore, to determine t*^ , 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 TIIKOUKTICAL ASTRONOMY. rial)l<'s. rf tlu'sc residuals were tlic actuiil fortuitous errors of oljscr- vatioii, the nicau error of an observation would l)o m heiiijj; tlie number of ecjuations of eondition. Tliis value is evi- dently an approximation to the correct result; but since by supposinir tlie residuals v, r', v", iVrc. to be ti.e actual errors of th(( several ob- served values of the function, we ussif;u too hij^h a de«frei' of pre- cision to the s(!veral results, the true value (tf s must necessarilv iii' greater than that j^iven by this ecpiation. liCt the true values of (lie unknown (piautities be x \- A.r, // |- A//, 2 -|- as, ite., the substituti;iii of which in the several e(|uations of condition wi)uld j;ive tlic residuals J, J', J", itc. ; then wc; shall have (('AX- + //At/ -I- c'AZ f- J] Ay + [cJ] A3 -f- [f?J] Alt 4- ... . (110) If we form i!.;' ii(»rmal ccpiations from (118), it will be observed timt they arc of the same form as the normal ecpuitions formetl from the original e(|uatious of condition, providcil that wo write —J in phui' of V ; and hence, according to (85), wc; have We have also, AX aJ -j- a J -f" " -^ "t~ [aJ] = aJ + a' J' -|- a" J" + and the product of these equations gives [ttJ] A.'C := oa J' + a'alS' + a"tt"J"» + . . . . -f rto'J J' 4- rto"JJ" + The mean value of the terms eontaining JJ', JJ", &c. ia zero, and ;;; t^r-^ COMIUXATION OV OHSK'tVATIONS, JOl 8 errors of obscr- nd tako tho sum flir tlic mean values oi' J-, J'-, J"-, tVc. we must, in cai'Ii case, writo c. Ili'iiiv tlio iiu-an valiio of the [irotliict [(f JJ A.i' will hv 1111(1 this, l)y iiK^aius of the lirst of eiiuations (88), is fcrther nMliicetl to Ill a similar manner, we obtain the value r inr the mean value of each of the products [/>J]a//, [cJJa:, iV:e. Now, the terms aihlcil to [re] in the second memher of the e(|uation (11!>) are necessarily very small, and, althoii<;h their exact value cannot l»c determined, we may without scnsihlc! error adopt the mean values ot" the several terms as lull' determined, so tiuit the eipiation heconies (1201 Therefore, since (121) II heiiifjj the numher of unknown <|Uantities. |JJ] iiir, we shall have ' III — H ^ HI — [I l>v means of w-hieh tlu! niean error of an observation whose wciirht is unilv may be determined. When // 1, this eijuatiitn bi-comes identical with {'.)()). I'or the determination of the probable errors of the linal values of the imknown (luantities, if /■ denotes the probable error of an obser- vation of the weij^ht unity, we have the folluwin{ stpiares is derived, enable us to combine the (lata furnished by observati(»n so as to overcome, in the ^rcalesl il('L;icc p(i.->ililc, the ellect (»f those accidental crr which no rcfine- niciit of tlicory can sueeesslully eliminate. 'I'he problem of the cur- icclion of the a|)proximate elements ot" the orbit of a heavenly body hv means of ji series of observed places, re(piires the :ipplication of nearly all the distinct rcsidts which have been derive(l. The (irst :'.|»|iro.\iinat(' elements of the orbit of the body will be determined h'eiii three or fliur observed places according to the iiicthods which 26 402 THEORETICAL ASTUOXOMY. liavo been ulrciuly explained. In the ease of a planet, if the inelina- tlon is not very small, the method of three geoeentrie ])laees may be employed, but it will, in general, afford greater accuracy and require but little additional labor to ba.sc 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 appai'cnt Avhether 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 [)laces sepai-ated by considerable 'ntervals of time; but it will be preferable to use places Avhich may be regarded as the average of a numbci of observations made on the some day or during a few days before and after the date of the average or nonnnl place. The ephemeris computed from the approximate elements known may be assumed to represent the actual path so closely tl t, for an interval of a few days, the difVerence between computation and observation may be regarded as being constant, or at least as varying proportion- ally to the time. Let n, n', n", <.tc. be the differences between com- putation and observation, in the ease of either s})herical co-ordinate, for the dates t, f', t", &c., respectively; then, if the interval between the extreme observations to l)e comI)ined in the formation of the normal place is not too great, and if we regard the observations as c{[ually j)recise, the normal difference /;„ between com})ulation and ol)scrvation will I»e found by taking the arithmetical mean of the several values of n, and this being a[)plied with the pro|)cr sign to the computed sjiherical co-ordinate for the date ^„, which is the mean of /, /', t", ttc, will give the corresponding normal place. Jiut when different weights p, p', p", &c, are assigned to the observations, the value of »,, must be found from np + n'p' + n"p" + """ i> +/+/' + .... ' and the weight of this value will be equal to the .sum p+p'-]-p" -{- (123) roMHIXATION OF OHSi:UVATK)N'S. 403 The flate of tlie normal place will bo ilotorniined by (124) If the error of the ephonioris ean be considered as nearly constant, it is not necessary to determine f^ with great |)recision, since any date not differing much from the average of all may be adopted with suf- ficient accuracy. It should be olwers'cd further that, in order to obtain the greatest accuracy practical)lc, the sj)herical co-ordinates of the body for the date ^|, should be computed directly from the elements, so that the resulting normal place may be a.s free as possible from the etfcct of neglected differences in the interpolation of the ephemcris. When tliC diffc^rences between the computetl and the observed places to be combined for the formation of a normal place cannot be considered as varying proporti(jnally to the time, we may derive the error of the ephemeris from an e([uation of the form of (03)5, namely, £iO .4 + ^r + CV», the coefficients A, B, and C being found from equations of condition fonaed by means of the several known values of a^ in the case of ouch of the spherical co-ordinates. 14(). 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 now system of elements sliould be computed by means of some one of the methods which have already been given; and these fundamental jdaces being judi- ciiMisly selected, the resulting elements will furnish a jiretty close approximation to the truth, so that the residuals which are Ibund by comparing them with all the directly observed ])laces may be regarded as indicating very nearly the actual errors of those places. We may then pi'occod to investigate the character of the observations more t'tilly. But since the observations will have been made at many dif- ferent places, by different observers, Avith 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 theorv of errors which has been developed in this chapter, it has been assumed that all constant errors have been dulv eliminated, and that the onlv errors which remain are those accidental errors which must ever continue in a greater or less degree undetermined. The greater the number and 404 THEORETICA L AHTHONOMY. 1 ))crfi'cti()ii of the ohservation.s oniployod, the more nearly will those errors be doterinined, iind the more nearly will the law ot" their dis- tribution conform to that which has been assumed as tlie 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 ])eculiar, such as may arise from the idiosyncrasies of the diHereiit observers, from the systematic errors of the adopted star-places in the ease of differential observations, and from a variety of f)tiior 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 jnore or less extended, as rcfjular may be eliminated. AVhen this has been accomplished, the residuals which still retnain will enable us to form an estimate of the degree of aceui'acy wiiich may be attributed to the different series of observations, in order that they may not only be combined in the most advantageous maiiiuT, 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 A\;»uld 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 attemj)t should be made to eliminate all errors which may be accurately or approximately determined. The places of the comparison-stai's used in the observations should be determined with care from the data available, and should be reduced, by means of tlie proper systematic corrections, to some standard system. The rodiie- tion of the mean places of the stars to apparent places should also l)e made by means of uniform constants of reduction. The observations will tlius be uniformly reduced. Then the perturbations arising from the action of the planets should be computed by means of forniulaj whicli will be investigated in the next chapter, and the observed COMBINATION OF OBSKRV'ATIONS. 405 places sliould be freed from these ]>erturhations so as to give the places for Ji system of osculating elements for a given date. 147. The next step in the process will he to compare the i)ro- visional elements with the entire series of observed [daces thus cor- rected; and in the cal(!ulation of the cphemeris it will be advan- tageous to correct the places of the sun given by the tables whenever observations are available for that jnirpose. Then, selecting one or more epochs as the origin, if avc compute the coetttcients ^1, B, C in the equation ^0 ^ A + Br + Cr\ (125) in the ease of each of the spherical co-ordinates, by means of eipia- tiiiDs of condition formed from all the observations, tlie standard ephemeris may be corrected so that it may be regarded as rei)re.'hich gives 2.5, 2.5, respectively. 5.1, 2.5, 3.6, 2.5, 4.1, 2.5, 408 THEORETICAL ASTRONOMY. Ill this manner tho wciirhts of tlie observations in the series made l)y each observer must be determined, usinj; throughout tin; same value of £. Then the diti'erenees between tlie jihiees c()Uij)utod from the provisional elements to be eorrected and the observed plaees eor- re<'ted for the eonstant error of the observer, must be combined ac- cording to the eijuatioiiH (123) and (125), the adopted values of p, y>', />", ttc. l)eing those found from (128). Tims 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 he applied to the elements. The relative weights of these normals will be intlicated by the sums formed by adding together the weights of the observations coud)ined in the formation of each normal, and the unit of weight will depend on the adopted value of e. If it be do- sired to adopt a different unit of weight in the case of the solution of the ecpiations of condition, such, for example, that the weight of an ecpmtion of average precision shall be unity, we may simply divide the weights of the normals by any number -p^ which will satisfy the condition im})osed. The mean error of an observation whose weight is unity will then be given by the value of e being that used iii the determination of the weights p, p', sion i'nr limliii^ tiie liiuitiiii; value of n tiierefore beeoinos Hftf - I C at -r 1 — . / ff »^ 2»l Vr. (i:U) \\\ means of this e<|uatioii wo derive for {fiveii values of m the eor- nsponiliu^ values of /(/*/•— 0.47094/1, and lienee the values of /;. For couveuient aj)|)lieatioii, it will he preferable to use ,- instead (jI' '•, ami if we jmt /i' ^^ U.0744i>yj, the limitinj^ error will he i\'z, and the values of //' eorrespondiug to given values of //i will be as exhibited ill the following table. TABLE. Ill 6 1.732 III 20 ..' HI n' »i n' 2.241 55 2.008 f»0 2.17:i 8 1.8G3 25 2.32(5 00 2.(i38 95 2.791 10 l.OHO 30 2.3!)4 05 2.()()5 100 2.-S07 12 2.0:57 35 2.450 70 2.090 200 3.020 14 2.100 40 2.498 75 2.713 300 3.143 Ifi 2.104 45 2.539 80 2.734 400 3.224 18 2.200 50 2.576 85 2.754 500 3.289 Aecording to thi.s method, we first find the moan error of an obser- vation by means of all the residuals. Then, with the value of m as till' argument, we take from the table the corresponding value of «', and if one of the residuals exceeds the value n's it must be rejectod. Airaiii, finding a new value of s from the remaining m — 1 residuals, and rejieating the operation, it -vill be seen whether another observa- tion slioidd be rejected; and the process may be continued until a limit is reached which does not require the further I'ojeetion of ob- servations. Thus, for example, in the case of 50 observations in wliieh the residuals —11". 5 and +7".8 occur, let the sura of the squares of the residuals be [vv-] = 320.4. Then, according to equation (30), we shall have e = ± 2".56. 412 THKOUKTICAL AKTIU)X()MY. ('orn'spoiidinf; to tlic value m 50, ilu' tiiMe gives n' -'- 2.570, I'lid tlio litiiiliiiif value of tlie error Im-coiucs n t - - (» .n ; and lience the re!-j)i»ii iing places computeil from the provisional elements to be corrected, taken in the sense computiitioii minus observation, give the values of w, n', n", etc. which are tho absolute terms of the equations of condition. By means of tlic^^t' elements we comj)ute also the values of the diif'erential coefficients of each of the spherical co-ordinates with resj)ect to each of the elcment.s to be corrected. These differential coefficients give the values of the coefficients a, h, c, a', b', Ac. in the equations of condition. Tlu' 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- COUIIKCTION OF Till: KLKMENTS. 413 iujci' that the mimcriciil valiU's of tlifso {'oclliciciUs wlioiild he mw- I'lilly »'li«('k('(l l»y (lii'crt calculiitioii, assiiiiiiiii^ variations to tin- I'lc- iiKiits, or l»y means ol" (liircrcnct's wlu'ii tiiis test can 1h' succi'ssliiUy !i|)|»litM|. In assi(rnint; incicnionts to the clcincnts in order to clirck the tiirination oC tiie ecjuations, tliey slioiild not l)i> so lar^(! tliat the iKirleeted terms of tho second order l)ecome sensilih', nor so .small that tlu'V do not afUtrd th(! recinired certainty i)v means of the aurecment of the corresj)ondini; variations of the spherical co-ordinates as uhtained by siil)stitution and hy direct calculation. As soon as the ecjuutions of condition have been thus formed, wo iiiiihiply each of tliem by the scjnare root of its weight as trivcn by till' adopted relative \vei.t!;hts of the nor iial places; and these ecpia- tioMs will thus be reduced to the same weij^ht. In general, the iiiiiiierical values of the coelHcients will be such that it will be con- venient, althoui^h not essential to adopt as the unit of wei;i;ht that wliich is the average of the weights of the normals, so that the iiinnbers by which most of the etpiations will be multiplied will not (litVcr much from unity. The reduction of the equations to a unitt)rm measure of precision having been etfected, it renuiins to combine them ;ii'c(irding to the method of least s(|nar('s in order to derive the most |iroiKil)le values of the unknown (piantities, togc^ther with the relative woiglits of these values. It should bo observed, however, that the mimerieal calculation in the coml)ination and solution of these equa- tions, and especially the reipiired agreement of some of the checks of the calculation, will be facilitated by having the numerical vidues of tilt' several coelfK-ients not very une(iual. If, therefore, the coeilicient '( of any unknown quantity x is in each of the equations numerically much greater o'" much less than in the case of the other unknown (|iiaiitities, wc may adopt as the corresponding unknown quantity to liL' determined, not .r but u.r, v being any entire or fractional number siuh that the new coefficients -, — . &q. shall be made to aijreo in inaL:iiitude with the other coefficients. The unknown (quantity whoso value will then be derived by the solution of the equations will be w, and the corresponding weight will be that of v.r. To find the weight of X from that of vx, we have the equation V. = ^'i^ (132) In the same manner, the coefficient of any other unknown quantity may be changed, and the coefficients of all the unknown quantities may thus be made to agree in magnitude within moderate limits, the i 414 THKOUiyriCA I. ASTIIONOMY. ii(lvaiit;ii);o of wliidi, in tlio minicrical solution of tlio ('(inations, will !)(' ajjparcnt hy a coiisidcralioii of (lie mode of provinu,- the calciihi- ti' is rccpiircd, the correspond inijj decimal. It may be '.cMiarked, (iirllier, that tlic introdnction of v is frenerally rcipiireil only when the eocllicient of one of tiu; nidcnown (inanlities is very larti;e, as frc((;:ently happens in the case of the variation of the mean arrange the calculation in a convenient forn:, applyinjj; also the cli('i'k> which have been indicated. 'I'lie most convenient arranu'cmciit will be to write the loj;'arithin-; of tin; absolute terms /(, /(', ii'\ Ac. inn horizontal line, directly under these the logarithms of the c(i(Hiri(iits It, d', d", iVrc., then the !oy;aritliinH of h, />', //', itc., and so on. 'flicn writini!;, in a correspond inu; form, the values of loj>;", h)ix n', A-c. on a slip of paper, by brin"]i *Vrc. will 1)(! readily formed. A^aiii, writing en aiiolh(!r lip of paper the Io<;arithms of <(, a', a", tt(\, and piacinii this lip snceessively over the lines eoutaininir the eoellicicnts, \m' derive the values [<"']> ["'']> [''*']» ^^'^' '^'''*^' multiplicjition by 6, '", ilcrivctl [/>/'], ['"']> \/>'l\i ^^'''m !'"•' finally [/J] in the case of six un- known ((uanlitics. In ilirminu; llicsc sums, in tlio cases of sums of |iiisitivi' and nciiutivc »inantitios, it is i-onvcnirnt as well as condncivo (ii aicuracy lo write the positive vali,<'s in one vertical column and the ntt;ative values in a separate coli nin, and take the dillirence of the sums of tlu( uumhers in the resp.'ctive columns. The proof of the ']> \_'''^]) ^^'•'- Then, according to tin; eipiations (7iJ) ai\il (77), thcM'alues thus ("omul should ai^reewith those ohtaiuecl i)v takini;- the corresj)ondinj^ sums of the coellieients in the normal oiliialions. '1 lie noruKxl ecpiations heiufji; thus derived, tlu" next step in the [Mdccss is the determination of tlie values of the auxiliary (piautitles ii((('>>;M'y for the formation of the eipiations (7 I). An examination of tlie eipiations (')!), ('»')), tVrc, hy means of which these avixiliarios ;ii'i' ilrtcrinined, will indica C at once a c^onvenient and ';y>icni,i{it' :irr;ni l""I) '""' *''" iciilv under them (he corresponding^ loj^arithins. ]ve.\(, we write iiiiilcr these, eonimeiiciiiji; with ["/'], the values of [/>/> |, [/>cj, [/^f/j, . . r/w], r/;/)l; then, add in tj; the loiijaritlun of the factor , , to the l(it;;irillims of [''/>], ['"'], t'tc, sucecsively, we write; the value of , , \S), the values of ['•'•.I {i |'''/.l], . . ('••'''.1], [<'"•!]' which are to he placed under the cor- ivs[KMidin}j; quantities [ir'], [o(/]. Aw. Next, we suhtraet from these, ix'speeiivoly, the m'oducts [hrA] U>cAl [bnAl 416 TIIFORETICAL ASTRONOMY. iuul thus derive the vjilucs of [cc.2], [cf/.2], . . [<'«.2], [c?i.2], M-hioh are to he written in the next horizontal line and under theia tluir logarithms. Then we introduce, in a similar manner, the coeiHcionts [f/f/], [(/('], . . [d)i], writing [tW] under [cJ.2]; and from eaeh of these in sueeession we subtract the products [adj [rta] lad], [ad} [rtfl] [as], lad] [aa] Ian], thus finding the values of [fW 1], [f^c.l], . . [dn . 1]. From these we subtract the products Ibd.l] [66.1] Ibd.l], Ihd.l] Ihb.l] [6e.l], [trUJ [66.1] [6«.l], respectively, which operation gives the values of [(W.2], [c?c.2], . . . [f?/i.2]. From these results we subtract the products [cc.2] [cf/.2j [cc.2] [ce.2]. [cd2] tcc.2] [cn.2], and derive [f?r/..3], [(:?c.3], . . [f/».3] under which we write the cor- responding logarithms. Then we introduce [cc], [ C'/-!]* ^s.l], and [c«.l]; then subtracting from these the products [6c.l] [66.1] [^^.1], [!u|[Vl]"-[^^^[^"-l]' we obtain the values of [ee.2], [e/.2], [cs.2], and [cn.2]. Again, subtractinsr [cc.2] [ce.2], [ce.2] i^m, [cc.2] lai.2], [ce.2] "- -" [cc.2] '■■•'•"-" ■ • [cc.2] we have the values of [cc.3], [c/.3], [t',s.3], [cu.3]; and finally, sub- tracting from these the products [rfc.3] [(W.3] Ide.S], ^'"'^[rf/.3],..E-^[^-3]. Idd.S] Idd.S] ' we derive the results for [fc.4], [^/.4], [e.s.4], and [c?i.4]; under which the corresponding logaritluns are to be written. If there are six unknown quantities to be determined, we mast further write in a horizontal line the values of Ijj], f fj], "nd ] fn\, COREECTION OF THE ELEMENTS. 417 ;n.2], which r theia tlicir e coclKcicnts each of these .'om these we ], lde.2l . . . rntc the cor- es], and ['»(], Mm- [aaj 1 subtracting 1.2]. A gain, » finally, sub- 3], under which lied, 'vc nuist ■'■jI and 1 ^"1, ^£ placing Iffl under [''J. 4], and by means of five .successive subtrac- tions entirely analogous to what pi'eeedcs, and as indicated by the remaining equations for the auxiliaries, we obtain the values of [//.o], [/y.5], and [//(.5]. The values of [6«.l], [fs.l], [c.s\2], &c. serve to check the calcula- tion of the successive auxiliary coetficionts. Thus we must have [66.1] + [6^1] 4- [6f?.l] + [6e.l] + [6/.1] = [6.y.l] [6c.l] + [c.l] + [crf.l] + Ice.l] + [.f.l] -^ [r..l], Ac, [«-.2] + [crf.2] + [C..2] + H2] =-- [C..2], [«;.2] + [cW.2] + [f?6'.2] + [rf/;2] = lds.2l &c. Hence it appears that when the numerical calculation is arranged as fibove suggested, the auxiliary coiitaining .s must, in each line, be iiiil to the sum of all the terms to the left of it in the same line , 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 [sfi] and [?in] to be determined. These additional auxiliaries will be found by means of the formuhe [SH.l] = [«»] [oa] [as], [8«.3] = [.«H,2]-g^[c..2], [m.o] = [.3,!.4] [CC.2] [e«.4] [..».2]=-[sH.l]-M[6..1], [6«.4] = [sn.3] - [^^^l] [ds.^l (133) and tlio proc [ee.4] i.-t'-ns (81) and (83) [e«.4], isn.6] = [s«.5] [^••5], The arrang>^ment of the numerical ' i Ho similar to that already explained. Tilt v^ : '• uf [.sn.l], [s/..2], &c. check the accuracy of the results for [6n.l], L '■^^, [en. 2], [f//i.3], &c. by means of the equations Ibn.l] -j- [CM.I] + Idn.'" ;- rci.l] + [/«.!] = [s«.1], icn.2] + ldn.2i 'c«.2] + r/".2] r_-_z [.,„.i], [rf«.3] + ['"'.3] + [>.3] - r.-.r,], [c».4J + [>.4] = [..«.4], [/«.5] = Isaf)]. (134) It ui'pi.irs 'iirther, that, in the case of six unknown quantities, since [fa.'}] ■ -J .0], we have [,su.G] = 0. Having thus determined the numerical values of the auxiliaries reqnir .'d, wo arc prepared to form at once the equations (74), by means of wluch the vali;c« of the unknown quantities will be determined 27 418 THEORETICAL ASTRONOMY. by successive siibstitution, first finding t from the last of these equa- tions, then substituting this result '\i\ 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), fii'st clearing them of fractions, we get = iacC] X + ([a61 + [iS.l]) y + ([«c] + [6c.l] + [cc.2]) z + ({aa : • ^hfl 1] -f [cd.2] + idd.^) u + (M -. ■ '] + [cc.2] 4- [rfe.8] + [ce.4])w (135) + ([«/] + I. ■-] + [C/.2] + [r?/.3] + [e/.4] + Um)t + laiq + [6/1.1] + [cu.2] + [rf/i.3] + [e«.4] + [//i.5] ; and this equation must be satisfied by the values of x, y, z, &c. found from (74). 152. Example. — The arrangement of the calculation in the case of any other number of unknown quantities is precisely similar; and to illustrate the entire process let us take the following equation.s, each of which is already multiplied by the square root of its weight:— 0.707.r + 2.052;/ — 2.3723 — 0.221« + 6".58 = 0, 0.471:c + IMly — 1 :\r>z — 0.085it + 1 .63 = 0, 0.2G0.C + 0.770// — 0.356^ + 0.483it — 4 .40 == 0, 0.092.C + 0.343y + 0.2353 + 0.469it — 10 .21 = 0, 0.414.1- + 1.204;/ — 1.5003 — 0.205k + 3 .99 = 0, 0.040.C + 0.150;/ + 0.1043 f 0.20G« — 4 .34 = 0. First, we derive [wi] = 204.313, [an] --=. + 4.815, [aa] = + 0.971, [6w] =z + 12.9()1, [«6] = + 2.821, [66] = 4- 8.208, [cft] = — 2.-).C07, [nc]-^— 3.175, [6c] := — 9.1G3, [cc] = + 11.028, [(/ft] = — 10.2 ;, [arf] = — 0.104, [6rf] = — 0.251, [erf] = + 0.938, [rfd] :-= + 0.594, [sn] =-18.139, [as] = + 0.513, [6,s] = + l.G10, [cs] = — 0.377, [rf,s] =+1.177. The values of [sn], {(is\, [6s], [cs], and [rfe], found by taking the sums of the normal coefficients, agree exactly with the values ooni- puted directly, thus proving the calculation of these coefficiente. The normal equations are, therefore, NUMERICAL EXAMPLE. 419 0.971X + 2.821.7 — 8.1753 — 0.104(( + 4.815 = 0, 2.821.T 4- 8.208// — 9.1()83 — 0.251 a + 12.9(U =. 0, — 3.175j; — 9.1()8// + 11.0283 + 0.9;38(( — 25.G97 = 0, — 0.104.f — 0.251// + 0.9383 -(- 0.594(t — 10.218 = 0. It will be observed tliat the coefficients in these equations are nu- morieally greater than in the cquation.s of condition; and this will generally be the case. Hence, if we use logarithms of five decimals in forming the normal equations, it will l)e expedient to use tables of six or seven decimals in the .solution of these equations. Arranging the process of elim'* ..aon in the most convenient form, the successive results are as follows : — \lb.\] = + 0.0123, [id] = + 0.0562, [cc.l] = + 0.0463, [CC.2J = + 0.3895, [M.l] = + 0.U511, [c(/.l] = + 0.o97'.t, [c ASTRONOMY. found is due to the decimals neglected in the computation of the ninnerical values of the several auxiliaries. The sum of all the ccjuations of condition gives generally [a]x- + [6]y + [c]3 + [d] u + . . . . + [H] - M, (136) which may be used to check the substitution of the numerical values in the determination of v, v', &c. Thus, we have, for the values here given. 1.984.r + 5M6y — 5.6103 + 0.647« — 6.75 = [v] 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 ; may be found directly from the auxiliaries whose values have been computed. Thus, we have p^ = Idd.S-] = 0.0297, P'-m^A^''-^^=^'-'''^' If we now completely revei'se the order of elimination from the normal equations, and determine x first, we obtain the values and also X = — 82."750, [ii.2] = + 0.0425, [ort.3] = + 0.00056, [a«.2] =: + 0.0033, [»m.4] = 14.665, 2/ = + 24."365, z = — 2."699, ti = -{- 1:."272. The small dift'erences between these results and those obtained by the first elimination arise from the decimals neglected. This second elimination furnishes at once the weights of x and y, namely, p^ = [«a.3] = 0.00050, p [««-3] tart.2] [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 and hence ldd.2l = [c?(/.l] — ^^^ Icd.l] = + 0.02977, ^.=E^ -eft*"! =»■»»'*■ 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 tlie auxiliaries obtained in the iirst elimination, we find from the equations (100), (101), and (102), A" = + 16.r)442, B"' = + 0.1202, A' = — 2.9052, B" =-- — 4.5G91, A'" C" — 3.3012, — 0.9356, and then the equations (103) and (108) give x--= — 81".609, y : p^ ^ 0.00057, p^ + 23".977, 2 = — 2".705, « = + 17".316, 0.0074, p^ =. 0.0312, ^„ =:. 0.0297, agreeing with the results obtained by means of the other methods. The weights arc so small that it may be inferred at onee that the values of x, y, z, and u are very uncertain, although they are those whi(;h best satisfy the given equations. It will be observed that if we multiply tlic first normal equation by 2.9, the resulting equation will diifer 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 bo little more than rude approximations. Thus, adopting the value of [vv] obtained by direct substitution, we have and hence ./l»'±i] = jpl| = 2.416, 1".629, which is the probable error of the absolute term of an equation of cond'^^'on whose weight is unity. Then the equations r r.. = r r g — =, r^ = ~^, &c., Vp^ Vp, give r^=±68".25. r =: ± 18".94, r^ = d= 9".22, ± 9".45. It thus appears that the probable error of s exceeds the value obtained for the quantity itself, and that although the sum of the squares of the residuals is reduced from 204.31 to 11.67, the results are still quite uncertain. 153. The certainty of the solution will be greatest when the coef- ficients in the equations of condition and also in the normal equations 422 TIIEOUKTICAI. ASTHOXOMY. / dilfor very considoriibly both in niaj^iiitudo and in sign. In tlie cor- rection of tlio elements of the; orbit of a planet when the observa- tions extend only over a short interval of time, the coeflieienls will generally change value so slowly that the ecjnations for the direct determination of the corrections to bo aj)})lied to the elements will not atford a satisfactory solution. In such cases it will be expedient to form the eciuations for the determination of a less number of (juantities from which the corrected elements may be subsequently derived. Thus we may determine the corrections to be applied to two assumed geocentric distances or to any other quantities which afford the required convenience in the solution of the problem, various formula; for which have been given in the preceding chapter. The (quantities selected for correction should be known functions of the elements, and such that the equations to be solved, in order to combine all the observed places, shall not be subject to any uncer- tainty in the solution. But when the observations extend over a long period, the most complete determination of the corrections to be applied to the provisional elements will be obtained by forming the equations for these variations directly, and combining them as already explained. A complete proof of the accuracy of the entire calcula- tion will be obtained by computing the normal places directly from the elements as finally corrected, and comparing the residuals thus derived with those given by the substitution of the adopted values of the unknown quantities in the original equations of condition. If the elements to be corrected differ so much from the true value? that the squares and jn'oducts of the corrections are of sensible maj^- 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 etpiations first provisionally, and, having applied the resulting cor- rections to the elements, we compute the places of the body directly from the corrected elements, iind the differences between these and the observed places furnish '.lew values of n, n', n", etc., to be used in a repetition of the solution. The corrections which i-esult from the second solution will be small, and, being applied to the eleineiit.s as corrected by the first solution, will furnish satisfactory results. In this new solution it will not in general be necessary to recompute the coefficients of the unknown quantities in the equations of condition, since the variations of the elements will not be large enough to affect sensibly the values of their differential coefficients with respect to the observed spherical co-ordinates. Cases may occur, however, in which it may become necessary to recompute the coefficients of one CORRFXTIOX OF THE J^jEMEXTS. 423 or more of tlie unknown quantities, hut only when tla'so coolVicicnt?! arc vorv consiilcriibly cluingcd by ii small variation in tlio udopti'd values of the oloments employed in the calculation. In such cases the residuals obtained by sui)stitution in the equations of condition will not agree with those obtained by direct calcidation unless the C(»rrections applied to the corres])onding 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 l)e sufli- ciently accurate to relax a little the rigorous requirements of a com- plete solution, and use, instead of the actual coetHcients, ecpiivalent numbers which are more convenient in the numerical operations re- quired. Although the greatest confidence should bo })laced 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 cond)ination of a series of observations, as well as the effect of uncliminated constant errors, may at least warrant a little latitude in the numeric^il application, provided that the weights of the I'esults are not thereby much affected. A constant error may in fact be regarded as an unknown <[Uantity to he determined, and since the effect of the omission of one of the unknown (pmntities 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 l)y the method of least squares from different corresponding series of ol)servations may differ beyond the limits which the probable errors of the different determinations have assigned. Further, it should be ol)served that, on account of the unavoidable uncertainty in the esti- mation of the weights of the observations in the preliminary cond)i- nation, the probable error of an observed place whose weight is unity as determined by the final residuals given by the equations of con\diich renders [vv] a minimum, or M' Afi^ (139) and the corresponding value of the sum of the squares of the residuals is M = Iv^vJ - ^ M- (1-10) The correction given by equation (139) having been applied to u, the result may be regarded as the most probable value of that cle- 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 Si + ^O^/i, i + Bo^n, r. -f CflAAi, &C. CORRECTION OF THE ELEMENTS. 425 the coofficiciits Ag, ^„, C'u, S:r. bcin<; tlioso givoii by the o(iuations (114), and thus the eh'inents may he derived which correspond to any assumed value of // dillerinj^ iVom its most probable vahie. The unknown (|uantity A/i will also be retained in the values of the residuals. Hence, if we assign small increments to /i, it may easily be seen how much this element may ditler from its most probable value without giving results for the residuals which are incompatible with the evidence furnished by the observations. If the dimensions of the orbit are expressed by means of the ele- ments (J and e, it may occur that the latter will not be determined with certainty by the observations, and hence it should be tnvited as suggested in the case of /a; and we proceeti in a similar manner when the correction to be applied to a given value of the semi-transverse axis a is one of the unknown quantities to be determined. 420 tiiix)Ui;tical astkoxomy. CHAPTER Vlir. / INVESTIGATION OK VARIOUS FORMt.'L.i: KOU THK DKTKRMINATION OK THE SPECIAL PERTURBATIONS OF A IIEAVENIA' iiODV. 155. We liiivc thus fur considered the oiroumstnnees of tlie uiidis- tiirhed motion of the lieavenly bodies in their orhits; l)ut si eoini)I('te determination of the elements of the orbit of any body revolving around the sun, requires that wo should determine the alterations in its motion due to the aetion of the other bodies of the system. For this purpose, we shall resume the general equations (18),, namely, dt '^ + /c\l + m)~--=^kKl-\-ni-) ''^U^(i + m)^ = m+^^'^, (It d'z u) (Hi wliich determine the motion of a heavenly body relative to tiiO sun when subjeet to the action of the other bodies of the system. We have, further, m' 11 ■.cx'+yjf + zz' \ m" (I xx" + yf + zz" \ , which is called iha perturbinrj fimdion, of which the partial difforou- tial coefficients, with respect to the co-ordinates, are dP. _ m' Ix' — x /\ m!' lx"—x x" \ d^ ~~ 1 + m \ ' p' r" / + 1 + «i \ '"p" /'' I "^ ' dy~l^ m \ p' r" / "^ 1 + )H \ p" r'" j ^ ' dQ _ m' Iz' — z 3^\ vi" lz" — z £' \ dz~l + m\ p' r'' j'^ l + mV p" 7" } + ' (2) and in wiiich m', m", &c. denote the ratios of the masses of the several disturbing planets to the mass of the sun, and m the ratio of the mass of the disturbed planet to that of the sun. These partial differential coefficients, when multiplied by F(l + ?/i), express the I'KirrrmtA'i'ioNs. 427 TIIK SriXIAL ^ the midis- : a complete Y rcvolviiiif Iterations in ■.stem. For namely, uj ! to ti.o sun stem. ^\ l + ^^'o., al (lift'oron- - &c., - Ac, (2) - &e., sses of tlie :he ratio of ie.se partial express the sutii of the compoiicnt.s of the distiirliiiii;' force resolved in directions ]i;iral!cl to the three re(;taiij>iil:ir axes respectively. When we nei^le(!t the consideration V('r, are functions of the partial differential coefficients of iJ with rospoi't to the elements, and before the integration can be perforn'cd it becomes necessary to find the expressions for these partial difft ential coefficients. For this purpose we expand the function iJ into a con- verging - ries 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 pov/ers of the eccen- tricities and inclinations, and so as to include au indefinite number of revolutions; and the final integration will then give what are called the absolute or general perturbatlo)is. When the eccentricities and irdinations are very great, as in the case of the comets, thir? development and analytical integration, or quadniture, becomes no longer possible, and even when it is possible it may, on account of the magnitude of the eccentricity or inclination, become so dittifnilt that we are obliged to determine, instead of the a1)solute perturbations, what are called the Hpccial perturbatiom, by nittiiods oC approyiiu.i- tion known as mechanical quadratures, acfording to which wo deter- mine the variations of the elements from one epoch ("^ to another epoch f. This method is applicable to any case, and may be advan- tageously employed even when the determination of the absolute perturb;.! ions is possible, and especially when a .series of observatioiirs extending through a period of many years is available and it is desired to determine, for any instant t^^, a system of elements, usuiillj' Called osculating element^', ou 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 tlie disturbing forces, of any system of three co-ordinates by means of which the position of PERTURKATIOXS. 429 i ii;r! body or the elements themselves may be found. We shall, there- fore, derive various Ibrmulie for this purpose before investigating the fbrniula) for the direet variation of the elements. 15'.). Let ;r„, iJq, z^^ be the reetangnlar co-ordinates of the body at tlie time t computed b^ means of tiie osculating elements J/„, "„, ^,|, &c., corresponding to the cj)och ^y. Let x, y, z be the actual co-ordi- nutcs of the disturbed body at the time t; and we shall have o.r, '///, and dz being the perturbations of the rectangular co-ordinates from the epot^h ^^ to the time t. If we substitutf> tliese values of .c, )/, and z in the erpuitions (1), aiid then subtract from each the corre- ipoiiding one of equations (3), we get dx' ~df ^+,.(x+.„)(!4i£-|)=*.a-,.)f. Let us now put r - r^ + or; then to terms of the order o)^, which is ' = (a/ - xr +(ij- yY + (/ - zy, (10) in which, when terms of the second order are neglected, we use tlie values .Tfl, j/q, 2„ for x, y, and z respectively. 157. From the values of 5.r, di/, 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 tlie higher powers of the disturbing forces. The equations (5) may bo written thus: — VARIATIOX OF CO-ORDIXATES. 431 (11) ~di' z + ^'(1 + in which nothing is neglected. In the application of these formula?, as soon as ox, dy, and oz have b?en found for a few successive inter- vals, we may readily derive approximate values of these quantities for the date next followiiig, and with these find 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 1 _^ which will be convenient in the numerical calculation, we have r« = (x, + dxy + (y, + dyy + (z, + .hy = r,^ + 2x,.^x + 2ij,<^y + 2z,oz + d.c^ + ,hf + dz\ and therefore 1 = 1 + 2 Cro+^^^O'^-^- + (?/,+ A'5.y) % + (gp + ■ >^ -J? Let us now put 'o 'o 'o and (12) /9 = 1 = l-(l + 2 and -.,.,-, which include the squares and products of the masses, will he obtained. The; integration of these will give more exact values of S.V, di/, and oz, and then, recomputing q and the other quantities which require correction, a still closer approximation to the exact values of the })erturl)ations will result. Tabh XVII. gives the values of log/ for positive or negative values of q at intervals of 0.000 1 from (/= (i to ry -~- 0.03. Unless the perturl)ations 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 r'" r Sq = ^ ' may be computed directly, using the value of r in terras of r^ and dx, OI/, dz. In the application of the preceding forraulffi, the positions of tiie disturbed and disturbing bodies may be referred to au}' 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 futida- mental plane, the positive axis of x being directed to the vernal equinox. By choosing the plane of the elliptic orbit at the time /,, as the plane of xy, the co-ordinate ;: 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 di' the components of the dis- turbing forces will require more labor than when either the e([uator 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 formula) for he integration of the linear diifercntial equations of the second order which express tlie variation of the co-ordinates, and generally the furmulaj for finding the integrals of expressions of the form j /(.r) dx and j I f{x) dx^ MECHANICAL QUADRATURE. 433 when the values of /(.c) are computed for successive values of ;r in- creasing in arithmetical progression. First, therefore, wo shall find the integral of /(.r) d.v ',vithin given limits. Witiiin the limits for which x is continuous, we have fix) = a + /3.C + yx'' + (Jar" + ex* + (15) and if we consider only three terras of this series, the resulting equa- tion /(a;) — a + /J.r + yx" is that of the common parabola of which the abscissa is x and the ordinate /(.r), and the integral of /(a;) 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 /(.(•) as the equation of a parabo'ic curve whose degree is one less tlian the number of terras taken. Hence, if we take n terras of the scries as the value of/(;c), we shall derive the equation for a parabola whose degree k n~ 1, and which has n points in common with the curve represented by the exact value oi' f{x). If we multiply equation (15) by dx and integrate between the limits and x', wc get J fix) dx ^ ax' -f ^iix'^ + Ir^' 4- -Idx'* + . . (16) If now the values of f{x) for different values of x from to x' are kiuiwn, each of these, by means of equation (15), will furnish an ('([Uiition for the determination of a, /?, y, &c. ; and the number of terms which may bo taken will be equal to the number of different known values oi' f{x). As soon as a, /?, y, &c. have th 'i been found, the equal luii (16) will give the integral required. If tlie values of /(.f) are computed for values of x at equal inter- vals and we integrate between the limits x = 0, and x = riAx, Aa' l)oiiig the constant interval between the successive values of x, and 11 the number of intervals from the beginning of the integration, we obtain f{x) dx ^ anux + ]/J/i'A.c» 4- ?^rn^^3^ + &c. Let us now suppose a quadratic paral)ola to pass through the points of tlie curve represented by f{x), corresponding to x — 0, x 28 /^ £kX, f 434 THEORETICAL ASTROXOMY. and X = 2a;k/ then will the area included by the arc of this parabola, the extreme ordinates, and the axis of abscissas be 2Aj; Cfix) (Ix = Aa; (2a + 2,?Aa; + ir^x"). The equation of the ; urve gives, if we designate the ordinates of the three successive jwints ;;y y^^, y^, and y^, 2ax and hence wc derive 2Aa! ffix)dx 2ax' ^x (2/0 + 4?/, + y,). In a similar manner, the area included by the ordinates 3/2 and y^, — corresponding to x == 2ax and x =-- 4^x, — the axis of abscissas, and the parabola passing through the three points corresponding to t/jj ^3) and y^, is found to be 4Ax fjXic) dx = J Aa; (y, + 4y^ + y^) ; 2Sx and hence we have, finally, nAx (10 Jf(x) dx == J Ax (y„_, + 42/,, _i + yj. (H — 2)Ax The sum of all these gives nSx ffix)dx = J^-^-((2/o + 2/J + 4(2/, + 2/3+2/5 + . "^/n-i) + 2(2/j + ^* + . ..y„-.)), by means of which the approximate value of the integial within the given limits may be found. If we consider the curve which passes through four points corre- sponding to 2/u, 2/1, 2/2> aiid 2/3) we have y =f(,x) = tt + /?a; + yx* + ^ar" for the equation of the curve, and hence, giving to x the values 0, A,r, 2a^, and Sao;, successively, we easily find MECHANICAL QUADRATURE. 435 nates of the 6A.r Therefore we shall have 3Ax "' = ft ^:;5 (2/3 — 3(/2 + 3)/, — 2/0). J/(a;) rfj; = I AX (y„ + 3^^ + 3y, + 2/3). (18) In like manner, by taking successively an additioniil terra of the series, Ave may derive 4A« ffix) dx = ^^^- Cly, + 32^, + 12i/, + 32^3 + 1yd, 6Ak //(.t-) dx = 1^ (1%„ + 75y, + 50y, + 0O//3 + Iby, + 10^5). (19) This process may be continued so as to include the extreme values of X for which /(a;) is known; but in the calculation of perturbations it will be more convenient to use the finite differences of the function instead of the function itself directly. Wc may remark, further, that the intervals of quadrature when the function itself is used, may be so determined that the degree of approximation will be much greater than Avhen these intervals are uniform. 159. Let us put A.r = to, and let the value of x for which 71 = be designated by a; then will the geucrai value be (0 being the constant interval at which the values of f{x) are givcu. Honce we shall have dx = (odn, I /(*) dx = u)l /(a + nut) dn. If we expand the function /(a + nco), we have /(a + n.)=/(a) + n^-^^. + ^--^. -1-^ + ^-^ . -i -^ + fee. (20) THEORETICAL ASTRONOMY. 1 n ■ ■/ 436 and hence J /(a -f- nw) dn = C ■{- nf(a) + h da + ^' l«'/(i^ d^fia) da' + .V^V'^ + &c.. (21) C being the constant of integration. The equations (54)g give w da dy(a) da' - =/'" (a) - 1/^ :«) + liu/'" («)-..., „,dy(a) da^ da^ =r(«)-i/"'(«)+.-., =r(«)-ir "(«) + •••> (22) in whicJi the functional syml)ols in the second members denote the difterent orders of finite differences of the function. Hence we obtain I /(a 4" nut) dn = C + r*^ + I /(« + 'i*") (?H -\-i f(a-^niw) dn, -k -4 1 i-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 + i n — i n = i J/(a + nco) dn = ^/(a -f nm) + ^\ ^/" (« + n^) n = 7! = Let us now consider the functions /(«), /(« + nco), &c. as being thoniselvcs the finite differences of other functions symbolized by '/, tlio first of which is entirely ai'bitrary, so that we may put, in accord- ance with the adopted notation, /(aJ='/(a + >)-'/(a->), /(a + ^) = '/(a + !">) - 7(« + -», /(a + n) - '/(a + (n - i) ) =/"'(a + (i + ^)«^) -/'" (a - >), &c. n = 438 TIIEOUETICAL ASTRONOMY. Furtlior, since tlio quantity 'f{)=:-,'i/'(a-H + 5^L/"'(«->)-«j^V^or («-•'•)+<&€. (20) Substituting these values in (25), it reduces to o + (*' -\- i)«> i + 1 I f{x) rfx = w I /(a -{■ niu) dn = wi'/(a + (i + ^.)«>) + .'J'(« + (/+P'-) ^^'^ -sao/"'(«+('+i)-)+WVi5^o/''(«+('-l-P"')-»^<'-! In the calculation of the perturbations of a heavenly body, the dates for which the values of the functioji arc computed may be so ai'ranged that for 7i = — .|, corresponding to the inferior limit, the integral shall be equal to zero, the epoch of /(« — \io) being that of the osculating elements. It will be observed that the equation (26) expresses this condition, the constant of integration being included in '/(a — \io). If, instead of being equal to zero, the integral has a given value when n = — \y it is evidently only necessary to add this value to '/(a — \io) as given by (26). 160. The interval lo 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 maimer entirely analogous to that already indi- cated. Thus, let it be required to find the expression for the integral taken between the limits — \ and i. The general formula (23) gives 4 J/(a + n«>)dM = i/(a) + ^/(a) + 5), /'v(a 4- ;•«>) =/'"(« + (i + A)'") -/'"(« + (i - i)<"), />'(a + H = p{a + (i + Aj'") - r(a + (i - ^)w), &c. (31) we get a + I'ui i I fix) dx = oji J {a + Hw) dn a-iu — i ("^2) -WoV5ur(«+(^+-D'")-WoVaur(«+(*-i)'")+&c.i, which is the required integral between the limits — | and i. 161. The methods of integration thus far considered apply to the cases in whicii but a single integration is required, and when appli'^d to the integration of the differential equations for the variations jf the co-ordinates on account of the action of disturbing bodic", thev .„,.,, , „ ddx d'hi . ddz , will only give the values oi -5-, -57. and -j-, and another integration becomes necessary in order to obtain the values of o.v, dy, and 3z. We will therefore proceed to derive formulse for the determination of the double integral directly. 440 TUKOUKVIVAL ASTRONOMY. For the double integral i jf{x)(l.i;' we have, since dx- ~ lohln^, fffix) dx^ =-- uj' fffia + nu>) dn\ The value of the function designated by /((/) being so taken that when n '-^ — J, I /(ffl + "'") dn = 0, the equation (23) gives C— i f(,a + noj)dn. Therefore, the general e^^uation is I /(" + "*") dn = I /(a + nut) dn -\- nf(a) the values of a, ,9, ;',... being given by the equations (22). Multi- plying this by dn, and integrating, we get fffia + nio) dn' = C" + nffXa + nu>) dn + in'fia) C" being the new constant of integration. If we take the integral between the limits — ^ and -|- I, we find Jfj/(a + na})dn' = Cf(a + nw) dn + -^^a +TM(jr + asssso^ + &c. From the equation (32) we get, for i = 0, ffia + nio) dn = '/(a) - J./' (a) + ^^Tif" («) - 5^1 hf («) + &c. (33) -i Substituting this value, and also the values of a, y, e, &c., — which are given by the second members of the equations (22), — in the pre- ceding equation, and reducing, we get + i JJ/(a-|-nc«)rfn'=.'/(a)-,'J'(a)+^fJ,r'(a)-4?i|o/^(a)-f-&c.(34) MECHANICAL QUADRATURE. 441 fff(a + "'") '/'i' ^ '/{a + tw) — :IJ' (a + im) ■ -1 and n-i — i n =■ n ^- n-0 (35) + t1?u^/"(« + "<") " T!i^/y30^/''(«+ »'") + &C. n-O Wo may evitlently consider '/(a — Jw), '/(a + Jro), tfec. as the ditfer- cnees of othex* functions, the first of whicli is arbitrary, so that we have 'j(a) = A7(a + » + m«' - I'") = i7(a + ">) - hf" (« - '"). y(a + «;-^ = iy(« + 2«>) + Vfia -f » = Vy(a + 2«>) - A/" (a), -i7(« + 0*-i)'4 Thciofoic n --- Hi n - i Tt - i n==0 Substituting these values in equation (35), and observing that "/(«) + "fia - CO) = 2"/(a - c.) + 7(« - \)) ,„., + 1,8^55 (3/'K«) + 2/"^ (a - 0,)) - Ac, ^'^^^ 442 THEORETICAL ASTRONOMY. tlie integral becoiaes a -i- (l + i)a) <-, + a — ii^ — ■ijfid' + nu>)dn + |/(«) + 4'go which gives + 3i.V^5 + 3 S i li r-\- 4yobo '5 + &c. 4 (38; 5 1 JO /'^'(«)- 3 r. 7 3Ht01' -^r(' 4- ' S7 gu/^'(")-i-'t(: and this again, by means of (28), gives Sf i-ri fia ~\- «<«) dn' ^ i'fia + (i f ^) o,) -^/(a 4- ;'■ ) - Vs/' (« + (^^4 ' ") +3^4.r(«+^«)-f5ijn,r(«-t-(^+y)'") 37 ia-rW^ i*i^^-/'("-l-('-l4)«')- I ■'•< I '3 out) /V^J^af/<.)-|-^c. MECHANICAL QUADKATURE. 443 cn the VuAiii Tlicreforc, since i + i -i aiul '/(« + (i + A)w) :■- - 'y(a + (/ + 1 }o,) - "f(a + M, j" (rt + (i + A) w) z^ f{n + ( / + 1 ) ...) — /(« + /r«), we shall have '',-;. ^-^i (-30) which fiivos the ro(jnire(l intcixral between the limits — \ and /. lG."i. It will be observed that the coetHoients of the several terms of the forninhe of intet^ration eonverire rapidly, and iK.'uee, l)y a jiroper selection of the interval at whieli the values of the fiim^tion ;up eomputed, it will not bo neees.sary to eon.sider the terms whi( h depend on the fonrth and higher orders of differences, and rarely tliusc which depend on the second and third differences. The value ;i>si(' an unn'icessary 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 w smaller than what would appear to be strictly re(piircd, in order that there may be no unceitainty with respect to the accuracy of the integration. On account of the rapidity with which the higher orders of diflerences decrease as we diminish w, ;i limit for the magnitude of the ado{)ted intcj'val will s])eedily be ol)t;iiiied. The magnitude of the interval will therefore be suggested hy the rapidity of the change of value of the function. In the com" I. 444 THEORETICAL ASTUON<»MY. putation of the perturbations of the group of small planets between Mars and Jupiter we may adopt uniformly an interval of forty days; but in the determination of the perturbations of comets it will evi- dently be necessary to adopt different intervals in different parts of the orbit. When the comet is in the neighborhood of its perihelion, and also wlien it is near a disturbing planet, the interval must nocw- s irily be much smaller than when it is in more remote parts of its trbit or farther from the disturbiTii»' bodv. It will be observed, further, that since the double integral contains the factor o/, if we multiply the computed values of the function by cu', 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', and only tlie single integral is sought, the result obtained by the formula of integration, neglecting the factor (or, will be a) times the actual integral rccpiired, and it must be divided by co in order to obtain the final result. 164. In tlie computation of the perturbations of one of the asteroid planet-s for a period of two or three years it will rarely be necessary to take into account the eflFect of the terms of the sei^ond order with respect to the disturbing force. In this case the numerical values of the expressions for the forces will be com])uted by using the valnes of the co-ordiilates computed from the osculating elements fur tlie beginning of the integration, instead of the actual disturbed values of these co-ordinates as recpiired by the formulae (8). The values of the second differential coefficients of ox, o//, and dz with I'cspcct to the time, will be determined by means of the equations (9). H the interval ^y> ^"^^ ^^> ^^■''''''' are the quantities .sought, the subsequent determination of the differ- ential coefficients must be performed by successive trials. Since the integral must in each case be equal to zero for the date t^, it will be admissible to assume first, for the dates t^ — ^to and ^q + ^cj corre- sponding to the arguments a — (o and a, that 8x =- 0, di/ —- 0, and dz = 0, and hence that the three differential coefficients, for oaili VARIATIOX OF CO-ORDINATES. 445 date, are respectively equal to A'^,, }''„, and Z^. We may now by inte- gration derive the aetual or tlu! very approximate values of the variation.s of the co-ordinates for the.^e two dates. Thus, in the ease of each co-ordinate, we compute the value of '/(« — Uo) by means of the equation (26), using only the first term, and the value of "J{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 7(a) =: "f(a - oO + '/(a - W). 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 — o) and a. These operations will 1)0 performed in the case of each of the tlu'ee co-ordinates; and, by moans of the results., the corrected values of the ditl'erential coeffi- cients will be obiained from the equations (9), the value of or being computed by means of (7). With the corrected values thus derived a new table of integration will be commenced; and the values of '/(« — l(o) and "f{a — (o) will also be recomputed. Then we obtain, also, by adding '/{a — |o;) to /(«), the value of '/(a + Uo), and, by adding this to ''/(«), the value of "f(a + w). An api)roximate value of /(a + to) may now be readily estimated, and two terms of the equation (39), putting <== 1, will give an ap- proximate value of the integral. This having Ijeen obtained for each of the co-ordinates, the corresponding complete values of the ditU'rential coefficients may be comj)uted, and these having been introduceil 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 iXv, (5//, and (h for the date represented by the argu- ment a H- 2(0, and then the corresponding values of the differential cootfieients. We may thus by successive partial integrations deter- mine the values of the unknown quantities near enough for the cid- ciilation of the series of diffi'rential coefficients, even when the inte- grals are involved directly in the values of the differential coelfieients. If it be found that the assumed value of the function is, in any case, nnu'h in error, a repetition of the cahndation may become necessary ; Init when a few values have been found, the course of the function will indicate at once an approximation sufficiently close, since what- ovor error remains affects the aj)proximate integral by only one- twelfth part of the amount of this error. Further, it is evident thai, in cases of this kind, when the determination of the values of the dilferential coefficients requires a preliminary approximate into- 446 THEORETICAL ASTROXOMY. gration, it is necessary, in order to avoid the effect of tlie errors in the values of the liigher orders of differences, that the interval (o should he smaller than when the successive values of the function to be integrated arc already known. In the case of the small planets an interval of 40 days will afford the re(|uired facility in the a])j)roxi- mations ; but in the ease 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 (o will be indicated, in the numerical a[)pli(a- tion of the formulte, by the maimer in which the successive assump- tions in regard to the value of the function are found to agree witli tlie 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 ( inplete ecpiations (14) must be employed. In this case the forces X, Y, and Z should not be computed at once for the entire period during which the perturbations are to be determined. The values computed by means of the osculating elements will lie employed only so long as simply the first power of the disturl/ini; force is considered, and by means of the approximate values of J,r, 8i/, and fh which would be employed in computing, for the next plate, the last terms of the equations (9), we must compute also the cor- rected values of A', Y, and Z. These will be given by the second members of (8), using the values of .v, y, and z obtained from X = Xg -{- 6x, y = yo + %, z, + oz. Wc compute also q from (12), and then from Table XVII. find the corresponding value of/. The corrected values o^ —A-, - ,',' and — .— will bo given by the equations (14), and these being introduced, in the continuation of the table of integration, we obtain new values of ox, 01/, and 8z for the date under consideration. If these ditl'er 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 coeificionts of 8x, Sif, and 8z in the formula for q, their values being given with sufficient accuracy by means of the previous assumption ; and geue- VARIATION OF CO-ORDINATES. 447 ^^TI. fiiul the rally a repetition of tlie calculation of X, Y, and Z will not be reqiui'cd. Next, the values of ox, 6i/, and 8z may be determined approxi- mately, as already explained, for the followinj; date, and by means of these the corresponding values of the forces X, V, and / 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 Ibr- mnho of integration which sensibly aifect the result may be taken into account. By thus performing the complete calculation of each successive })lace separately, the determination of the perturbations in the values of the co-ordinates may be cifcctcd in reference to all j)0\vers of the masses, provided that we I'cgard the masses and co-or- dinates of the disturbing bodies as being accurately known; and it is api)arent that this complete solution of the problem re([uires very little more labor than the determination of the pertnrl)ations when only the first power of the disturbing force is considered. But altiiough 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 wili be cx[)('dient to determine the perturbations with all possible rigor. When there arc several distiu'bing 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 IX, ^'F, and -Z instead of A', Y, and Z respectively; and the integration of the expressions „ iP(Xv (I'dy il'ih .,, , . , , . , , tor ,--. ^T^T' ^"^^ "7/F ^^"* u\G\\ give the perturbations due to the iW =^-^-^^7(«) = + 2.21. Then the formula (39), putting first i = — 1, and then i = 0, gives Dec. 12.0 Jan. 21.0 SX: + 2.24 + ^-^^ + 2.21 + 53.7^^ 12 + 6.66, + 6.69. In a similar manner, we find Dec. 12.0 Jan. 21.0 Sy =-. 4- 5.85 dy = + 5.82 dz = — 0.16, fe = — 0.14. By means of these results we compute the complete values of the second members of equations (40), 8r being found from and thus we obtain Date. Dec. 12.0 Jan. 21.0 dr = ^Sx + y-^Sy-\-^^3z, U' ■■ + 53.86 + 54.23 6)5 i -f 47.76 + 47.25 1.45 0.96 6r + 8.85, + 8.63. We now commence anew the table of integration, namely. '/ 7 y 7 '/ "/ +53.86 _ ..^+ 2.26, +47.76 , ..2+ 1-97, -1-45 n 02 -'^•^^' +54.23 , 5^-21 + 2.24, +A7.25'\_^^[^,^ ^ 1.99, -0.96 _Q;9g -0.06, +56.45, +49.26, -1.04, the formation of which is made evident by what precedes. Wc 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 EXAJfPLE. 451 (If -7 df 1864 Oct. 27.0 + 34.84 — 26.32 + -1.44, Dec. 6.0 68.79 47.87 6.86, 1865 Jan. 15.0 + 112.64 — 58.39 -t- 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, &g., and tlie 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, ex])ressed in units of the seventh decimal place, are thus found to be the following: — Berlin Mean Time. 6x - - 4 ((x' - x) Sx+OJ- y) dy + (z' - z) 'h), in Avhioh ^^ is the modulus of the system of logarithms. Thus we obtain, for Sept. 17.0, gration, the resulting values of the integrals are changed so little that a repetition of the ealculation is not required. We now derive approximate values of dv, (\i/, and dz for Oct. 27.0, and in a similar manner we obtain the (jorrected values of the dilfoi- ential coefficients ibr this date; and thus by computing the forces for each pla(^e in succession from approximate values of the perturbations, and repeating the calculation whenever it may appear necessary, wc may determine the perturbations rigorously for all powers of the masses. Tiie results in the case under consideration are the follow- ing:- Date. dt'' ''' di^ 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 0.86, 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 to = + 1772.6, ^2/ = + 1992.3, ^s = — 28.2, agreeing exactly with those obtained when terras of the order of the square of the distiu'bing 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, = cos £ (hj — sin e Sz, (41) Sz, ==: sin £ Sy -\- C09 £ dz, x„ y„ z, being the co-ordinates in reference to the equator as the fun- damental plane. Thus we obtain, for 1865 Jan. 15.0, 0% = + 1772.6, dy, -= + 1838.9, 8z, = -|- 767.2. These values, expressed in seconds of arc of a circle whose radius i;- the unit of space, are dx, = -f 36".562, Sy, = -f 37".930, Sz, = -f- 15".825. VAUIATION OK CO-ORDINATES. 4oo The approximate geocentric place of the planet for the .same date is a = 183° 28', 3 = — f)" ;?9', log A = 0.:522!), luul hence, neglecting terms of the second order, we derive, Ity means of the e(juations (.']).., for the pertnrhations of the geocentric right ascension and declination, Aa .-== — IT'M, A') = + r,".Cy7. I(i7. The values of dx, oi/, and oz, com[)uted hy means of the co- ordinates referred t(» tiie ecli[)tic and mean e(ininox of the date /, must ho added to the co-ordinates given by the undistnrl)e) -f- y" cos {0 + p), (44) z, =z , in which p denotes the precession during the interval t' — t. Elimi- nating x", y", and z" from these ecpiations by means of (43) and (42), observing that, since ;y is very small, we may put cos;y = 1, we get 456 THEORETICAL ASTllOXOMY. 7 X, --■-. X iMsp — ij sriiy> --(- - 2 siu (0 -\- 2>), 'I y, —-. .r siu p -\- y coi*p — ~z cos {0 ■■{- j)), s Ti . JJ z, T=zz — - X SIU 4- •• '/ <'0S 0, tf ,1 ' (4o:t in which .s - 2O(J204.^y -~ h ^7 y + ~ -^ — 7 C^os o—p sin , ^, and f', we hav(! ■p = (50".21120 -f 0".00n24429(!0r") {t' — 0, Tj n=( 0",488<)2 — 0".()<)0()()(q43T) (t'-~t), r.r.- 351° ;Ui' 10" -f 3;)".7i> a - IToO) - 5".21 (1! — t), ill which c ---■■ l{t.' -- t) — 1750, t and t' beint^ expressed in years from the beginnine; of the era. If we add the nutation to the value of p, the co-ordinates will be derived for the true equinox of f. The e((uations (45) and (40) serve also to convert the values of ox, 3>f, and Sz belonging to the co-ordinates referred to the eclij)ric and mean equinox of f into tliose to be a])])lied to the co-(n'diiiates re- ferred to the ecli])tie and mean equinox of t\ For this purpose it is only necessary to write d.r. dy, and oz in place of .c, y, and - re- spectively, and similarly for .i-,, y„ z,. In the eoinputation of the perturbations of a heavenly b(jdy ditriii[r a period of several years, it will be ;,- (Venient to adopt a tixed ecpii- nox and ecliptic throughout the calculation; but when the pcrturlni- tions a''c to hv a})plie(l to the co-ordinates, in the calculation of an epheineris of the body taking into account the [)erturl)atioi s, it will be convenient to comjmte the co-ordinates directly for the cclijitic and mean e(iuinox of the beginning of the year for which tiio ephenieris is re<|uired, and the values of ox, dy, and dz nnist he reduced, by means of tin; equations (45), as already ex[)lained, fiom the ecli})tic and mean equinox to which they belong, to the ecliplii; and mean equinox adopted in the ease of the co-ordinates required. VARIATION OF CO-ORDIXATES. 457 In a similar manner wo may derive Ibrmuliv lor the transformation of the co-ordinates or of their variations referred to the mean e([ninox and equator of one date into those referred to the mean equinox ;uid equator of another date; hut a transformation of this kind will rarely bo refjuired, and, wlienevor required, it may be etfeeted by lirst ciinvertinii; the co-ordinates relerred to the equator into those referred to the ecliptic, reducing these to the equinox of t' by means of (45) or (46), and tinally converting them into the values referred to tho equator of t' . Since, in tho conij)utation of an e]>hemeris for the comparison of observations, tl)e co-ordinates are i'"nerally required ill reference to the equator as the fundamental plan<-. it would ai)i)ear preferable to ado[)t 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 vrill be most conveniently perlbrmed by adopting the (vliptic as tue 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 ecpiator of the l)o'nitude and tlie interval t' — t h also not verv larw, the con'ection of ox, d;/, and o- on account of the change of the j)osition of the ecliptic and of the equinox will be insignificant; and the conversion of the values of these (piantities referred to the ecliptic into the corresponding values for the equator, is cflfected with great ficility. In the determination of the perturbations of comets, c|»hemerides being required only tluring the time of describing a small ])()rti(in of their orbits, it will sometimes be convenient to adopt th(> plane of the uiulisturbed orbit as the fundamental plane. In this ease the posi- tive axis of X should be directed to tho ascending node of this plane on the ecliptic, and the subsequent change to the ecliptic and ecpiinox, whenever it may be required, will be readily etfeeted. I 458 THEORETICAL ASTRONOMY. 1G8. The ])crtiu'l)ations of a licuvcnly l)o(ly may thus bo deter- mined vigorously for a long period of time, ])rovided 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 })recision, a repetition of the cali'ulation of the j)erturbations may become necessary, since changes of the ele- ments which do not sensibly aifect the residuals for the given diifer- 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. ^^^len the calculation of the perturbations is carried forward for a long period, using constantly the same osculating elements, — and those which are supposed to icr|uire no correction, — the secular per- turbations of the co-ordinates arising from the secular variation of the elements, and the perturbativill not again become sinudtaneously equal to zero. Ilenee it appears that even when the ado])ted elements do not diifer much from their mean values, the numerical amount of the perturbations may be very greatly inex'eased by the secular perturbations and by the large perturbations of long period. But when the perturbations are large, the calculation or the complete values oi —jfT> lii > '^"'^' —777- (which is eifected indirectly) cannot be performed with facility, rc|uiring often several repetitions in order to obtain the retfuirod 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 modo- rate period would have no sensible effect, may in the course of a long ])eriod 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 necessaiy 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 e(|iial to zero. The calculation will then be continued until another chan«;o of the ck«»ents is required. CHANGE OF THE OSCULATING ELEMENTS. 459 Tlie transforiaation from a system of osculating elements for one epoch to that for another epoch is very onsilv cfrected by means of the values of tlie i)erturbations of the eo-onlinates in connection with the corresponding values of the variations of the velocities dx dy , rfs rni 1 rfl' 'dt' di' ^^^^^'^" obtained from i\w. values of the second differential coefficients by nieans of a sin, ^in ^, A, /i, and Care determined in reterence to that plane. To transform them still further, we have dr kV'l + m . , -— . -_^ - ^-= enm'u — w), Vp dp Jcyp{i-{m.) _ kVl f '^ dt"" m /- — (1 4 e cos (« — io)V Vp in whk4» lo denotes the angular dLstaiiee of the perihelion from the ascending iKxle. Substit ■ ing i\tt?m .aiues, we obtain, by reduction, 460 THEORETICAL ASTRONOMY, (Ix 'dt dy dt (h dt -jl ((e cos w -f- cos u) cos, A — (e sin w -f- sin u) sin A) sin u, Vp kV^ i'-\-7n_ Vp Vp Let us now put ((e cos u) -{- cos It) cos B — (e sin w -|- sin «) sin JS) sin b, ((e cos w + cos u) cos C — (e sin w -f sin u) sin C) sin c. m and we have kVi + Vp kVl + »t rf.r _ rf?/ _ lit ~ dz 'dt' (e sin w + sin u) = Fsin C7, (e cosw -f- cosft) = Fcos U, = Fsin a cos (J. -|- U), = Fsin b cos (B + U), = Fsinccos(C'4- U). (48) (49) 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, etc. belong. When the ecliptic is the fundamental plane, we have sin c = sin /, C = 0. The sum of the squares of the equations (48) gives y2 ^ ^i(l+!!L) (1 + e^ + 2e cos (« - w)) --. /t»(l -f. m) ( ? - - \ p \t a) and hence it appears that I'is tho linear velocity of the body. Tlie dctonnination of the osculating "Icments corresponding to any date for which the perturbations of the co-ordinates and of the veloci- ties have been found, is thorcfvire efl" «'ted in the folloviug mat>rior: — First, by means of the osculating elements to which the jvrturl)a- tions belong, we compute accur.ii values of )*q, r:„, ?/„, 5,., mid l)y means of the equations (4H) and (49) we compute th*- vaJu*^ of -rn and -jt" Then we apply to these the valw*^ of the perturba- tions, and thus find .r, ]j, z, -.-, J , and - -• Thest- having been (b: dt' dV dt' dt CIIANGK OF TIIK OSCULATING ELEMENTS. 461 ibuiul, the equations (32)j will furnish the values of SI, i, fvnti p; and the remaining elements may be cletermined as explained in Art. 112. Thus, li-om Fr sin 4o = H>{l + m), Frcos+„ = ..^^^+^-- + .-^-, we obtain T';- and i^/dj '-^^^^ from ?• sin ?t = ( — X sin Q, -\- y cos S^) sec i, r cos it := X cos S^ + 2/ si i Q, , we derive r and «t; and hence T^frona the value of Vr. When / is not very small, we may use, instead of the preceding expression for '/•sin u, V sin n = 3 cosec i. Next, we compute a from 2a — r and from 2 ^ni+j'0_i 2ae sin w = — (2a — 7') sin (24.^ -{- '0 — '' sin «, 2aB cos w = — (2a — ?•) cos ( 24^ + «) — r cos «, we find la and c. The mean daily motion and the mean anomaly or tlie mean longitude for the epoch will then be determined by means of the usual fbrmuUe. In the case of a very eccentric orbit, after r and n have been found, -r will be given by equations (48)5, and the values of c and v will be given by the equations (49)^. Then the perihelion distance will be found from V ■" 1 + e and the time of perihelion passage will be found from v and c by means of Table IX. or Table X. In the numerical values of the velocities —^r, — ,-, etc., 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 sijlution. If it be considered necessary, the different parts of the calculation may 1)0 checked by means of various formula; which liave already been Srivon. Thus, the values of SI and i must satisfy the equation We have, also, 462 THEORETICAL AHTRONO^rY. 3 COS i — y sill t COS Q -f- .v sin / sin JJ = 0. r' := x'^ -\- if -\- z\ z = r sin u sin i, ■\vliich must be satisfied by the resulting values of V, r, aud u; and the values of a aud e must satisfy the equation ji = a(l — e'') = a cos" f. 169. When the plane of the undisturbed orl)it is adopted as the fundamental plane, we obtain at once the i)erturbations iJ(?'C0Slt)) S (r sin n), Sz, and from these the perturbations of the polar co-ordinates are easily derived. There are, however, advantages which may be secured by employing fornudai Avliieh give the perturbations of the polar co-or- dinates directly, retaining the plane of the orbit for the date t^ as the fundamental jdanc. Let IV denote the angle which the projection of the disturbed radius-vrctor on the plane of xy makes with the axis of x, arid /9 tlie latitude of the body with respect to the plane of xy; then we shall have x = r cos (3 cos ^v, y = r cos /5 sin w, (60) g = r sin ,?. Let us now denote by X, Y, and /, respectively, the forces which are expressed by the second members of the equations (1), and the first two of these equations give (Ix 4!-!'§=/(^^-^!'>*+^- C being the constant of integration. The equations (50) give dx d(r cos 13) . . dw -77 = COS W — ^- j^ — r cos /? Sm W -rr* at at at and hence dy d (r cos ,?) dw dt = sm w — , + r cos /3 cos to jj, dy dx , ,r, dio X-J7 — y-jT = r cos" /3 -jr- dt ^ dt dt tf VARIATION OF POLAR CO-ORDINATES. Therefore wc have ^ dw 463 r' co3» fi yj^- ^J( Yx — Xy) dt + C. If wo denote by Sq the eoniponetit of the disturbing force in a direc- tion perpendicular to tlie disturbed radius-vector and parallel with the plane of xy, we shall have and Therefore X = — S„ sin IV, Y^=^Sa cos to, Yx — Xy = Sf,r cos /3. dio r' cos' /? -~ = r^S; r cos (3 dt + C. In the undisturbed orbit we have /9 = 0, and , du " dt kVpo{i-{-m); and thus the preceding equation becomes r' cos' ,5 — - = j Sg r cos /? (Zi + ^y^Poi^ + '«■)• The equations (1) also give (51) 1 _ xd\v H - yd'^ y + ^dh P(l-fm ) ^ j^« , y^-^ 4- zi (52) If we denote by R the component of the disturbing force in the direction of the disturbed radius-vector, we have We have, also, R = X-i-Y'l + Zt r r r (53) xd'x + ydhj + zdh = d (xdx + ydy + zdz) — (rfx' + dy^ + dz') = d (rdr) — (rfr' + r'rfy') = rdh' — r'rfi;', D denoting the true anomaly in the disturbed orbit, or, since du- --= con' i3 did" -{- d^\ xd'x -f yd'y + zdh = rd'r — r' cos" ^ dw^ — r'rf/j\ Hence the equation (52) becomes dh- >rfio' -rcos'/5-^.---r^ + di5' , ^'(1-1-7/0 R. (54) 464 TIIEOUKTK'AI. ASTUONOMY. 170. The cq'.iatlons (51) and (54), in coiinoction with the last of equations (1), com])lctely rcproscnt the motion of a hoavculy body about the sun when u(!tc(l upon by disturi)inj; forces, and, when cnni- ph'tely inU'j^ratt'd, they will j^ive the vahies of ?/', /•, and z for any point of the orbit; but, sineo they cannot be integrated directly, we nuist, as in the case of the recitungular co-ordinates, find the equations whicli give by integration tlie vahies of uio, dr, and z, lu the case of the undisturbed orbit, we have r - , (ho„ ° (It kl/p„{l + m), clt^ _ dw„' Ic'd-^m) _ (55) d( ]- + 0. If we denote by (Iw the variation of to arising from the action of the disturbing force, we have ?« = «'y + (Jw; and hence we easily fuid, from (51), d8io 1 r., ou li h' \lVpJlT~m) .rn dt r'cos'iij \ r' cos', if r^' We have, further, which gives Let us now put r' ^ r,' + 2r,dr + Sr', \ 'o and we have f>n' — 1 .S>. J '1 ■•■ !1 1 C' 1 + 25' The equation (56), therefore, becomes d'hv "dV "" r^^'7if^'> '' ^°'^ '^ '^' ~ ^''•^'^'' in which we put ffo dt~ r' (57) (58) (59) (60) If we substitute r,, + dr for r in equation (54), and combine the result with tlie second of equations (55), we get d'; fon^e will be computed. We compute also (/ i'rom the first of eipiations (57), and q" from the first of ((U); then, by means of Table XVII., we derive the corresponding values of log/"' and lug/". The coeiHcients of or in the expressions lor fj and (/' will be given with snfllicient accuracy by means of the approximate values of dr and sin ,% and will not require any further correction. Then we compute > -^^1 '^^<>' Photographic Sciences Corporation 23 WEST MAIN STRKT WIBSTH.N.Y. 14510 (716) •72-4503 V C^ 6^ 168 THEORETICAL ASTKOXOMY. ecliptic, the fui.daniontal piano, ami the plane of the orbit of the dis- turbing body witij the celestial vault, we have sin Usin >, (N + iV") ^ sin A( ft' — R,) sin \ (V + /,), sin }, I cos UN+ N') ---- cos \{Si' ~ Slo) «»" i (*" — »«). cos Usin .J ( iV — iV') =.^ sin ^ ( ft ' — ftj cos .1 (i' + ig), cos .J / cos A ( iV — iV') = cos .1 ( ft ' — ft j) cos ^ (t' — i^), (60) from which to find A', N', and /. Let ,y denote tlu; helio%. Finally, we have Z=.„i'/t'//ir'sin/— ^-X from which to find Z. When we determine the perturbations only with respect to the first power of the disturbing force, the expressions for li, >%, and Z become R = wi'^-' ( h r' cos .r cos (ru' — «'.) — ~-A, Sg ^=^ m'k* h r' cos [^ sin {w' — tt',), Z = jft'P h r' sin (5'. To compute the distance p, we have ,,» = (x' - x)' -4- (y - 2/)' + («' - «)', which gives ^» = /•'' ^ r' — 2r ?•' cos ,J cos /J* oos iw' — w) — 2r >•' sin /S sin ,5', (75) ami, if we neglect terms of the second order, we have /,„» = ,•" + r,' — 2ro r' cos [:f cos (u-' — u\). (76) (74) If we put we have cos Y ^= cos /5 cos (^' cos iw' — w) + sin ;5 sin ,J', p» =; r' -f *** — 2rr' cos y = r'* sinV -f (»• — »*' cos j*)' ; (77) 470 THKOnETI('AI< ASTKOXOMY. and hence wc may readily find ft (Voni p 8in II p cos n r — r' von y, (78) the exact vahic f)f the angle », howcN'er, not heing rccinired. Introducing y into the expression for A', it heeonies by means of which U may be conveniently determined. (79) \T'\. When we neglect the terms dependi.ig on the sqnaros niid higher powers of the masses in the <'ompntation of the pertnrlKitions, the forces /^, »S'y, and /will be computed by means of the e from (78), «Sy from (72), /from (7.'J), and li from (71) or (79). The values of f?H', 5/', and 2, computed to tlie point at which it becomes necessary to consider the terms of tlic scc(»nd onhM", will enable us at once to estimate the values of tlio perturbations for two or three intervals in advance to a degree of approximation sufliicient for the cidculation of the forces; and tlio values of iif, N„, and /thus found will not require any further cur- rcetion. When the places of the disturbing planet are to be derived from an ephemeris giving the heliocentric longitudes and latitudes, the values of ft' and /' will be obtained from two places separated by a considerable interval, and then the values of u' will be deteriniried by means of the first of equations (82), or by means of (85),. Wlicn the inclination V \i very small, it will be sufficient to take «' = r — ft ' 4- « tan' J J' fin 2 (r — ft'), in which « - 206264.8. But when tlie tables give directly the lon- gitude in the orbit, u' \- ft', by subtracting ft' from each of those longitudes we obtain the required values of «'. It should 1)0 observed, also, that the exact determination of the values of the forces requires that the actual disturlMxl values of /', %c\ and /3' should be used. The disturbed radius-vector r' will Iw VAKIATIOX OF l' time to time, l)y means of which the true values may l)e readily interpolated for each date. We may also determine the variations of jN', X', and / arisinj^ from the variation of ft' and /', by means of diiferential formula?. Thus the relations will he siniilar to those f^iven hy the equations (71)^, so that we have ^ i^ >*'" '" ^"O"* 'o ^''"'* fto — ^ '■*'" 'o *^'"*' ft o» "^^' 2, = r cos ,5 sin «» sin /„ + - <'*>8 'o- Introdiiein^ also the auxiliary constants for the ecliptic according to the equations (94), aiul (9<>)i, wo obtain X, = r cos /S sin « sin (A -\- w) + « cos a, y, ^=r cos ,5 sin b sin (/i + "') + 2 cos 6, z, = ?• cos /} sin /, sin »<; -\- z cos /j, (82J by means of which the heliocentric co-ordinates in reference to the ed'ptie may be determined. If the place of the disturbending values of the co-ordinate»s of the sun, will give the required geocentric places of the disturbed body. These equations are jipplictvble to the case of any fundamental [)lane, provided that the auxiliary constants a, A, b, B, ifee. are determined with respect to that plane. In the numerical application of the formuhe, tlie value of w will be found from VARIATION OF POLAR CO-ORDINATES. 473 >ns with the i/„ iH^injr tlio arjjiiinont of tlio latitude for the fuiuliiint'iital (»s<'ul!itiiiK I liiiiciits, and t-jiro iimst* bo taken that the proper alj^i'braie >igii i.s !i«si!.riie(l to voHO, COS 6, and co.sc. If the vahies of z^, Q^, and /„ uxod in the caleuhition of tlie per- tin'l)ationH are referrcnl to the eeliptie antions (llo),. Then J2,i iU'd /„ shouhl be reduced from the (H'liptie and mean e<|uiuo.\ ttf /^' to the ecliptic and mean iipiinox of /„" by means of the second and third of liie oipiatiuns (llo),, and, using the values thus found in the calculation of the auxiliary constants for the eeliptie, the e ill rf?„ dt r coa fi sin h cos (5 -f- w) . -f sec /? sin 6 sin {B + w) j- -f (cos h — tan ^ sin h sin (B -f-'i')) —.-, (^iv • fir r cos /9 sin c cos (C -\- «') -jr -\- sec ,'i sin c sin ( C -f if) jf (84) -|-(co8 (-• — tan (5 sin c sin (C-\- «')) dz dt' by means of which the component* of the velocity of the disturbed body in directions parallel to the co-ordinate axes may l)c determined. diir dz d*„( 1 + »i ) «/''«' dr kV\ + m . S 17 21 44 .2 MmitIi 1.0, 0..'«)(>(i74 22:{ 8 ."1 .9 0.7:{.'Hr) 20 2.'> .') .2 Ai.ril 10.0, 0.:!l(tMf!4 2:<7 .">1 38 .3 0.7:i2:{7 2:{ 28 .W .8 May 20.0, 0.:{242'.>8 2')1 '>'! 47 .« 0.7:Ufi4 2(5 Xi IVl .1 .Iimo 2!t.O, 0.:ni»74.') 2(J4 r)9 :{0 .0 0.7;Ut8(i 29 38 44 .8 Auk. S.O, 0.3.')(il0l 277 10 24 .(5 0.7:UK»3 .•{2 44 41 .2 Si'i.t. 17.0, 0.3724fi9 288 28 4 .1 0.729ir) 3.-) 51 24 .(> Oct. 27.0, 0.388214 298 57 16 .3 0.72823 38 58 57 .5 Doc. (i.O, 0.402894 308 43 48 .7 0.7272(J 42 7 23 .3 18(i.') .Jan. 15.0, 0.41(5240 317 .')3 .39 .1 0.72(525 45 16 43 .9 3' — 0° r38".l 18 9 .1 34 39 .9 51 7 .(1 1 7 -Jll .7 1 23 43 .5 1 .'!!» 4t! .:! 1 55 :\r, :2 2 11 7 .') 2 2(5 20 .:) — 2 41 10 .() The values of p^ may bo found from (76) or (78) a.s already given in Art. 166. The forces R, .S',, and Z may now be detertnined by means of tlic equations (74), h being found from (70), and if we introduce the factor (0^ for convenience in the integration, as already explained, wc obtain the following results: uj S„r„ill -f-' 0".0'2S2 — .2.'i61 Date. 186.3 Dec. 12.0, 1864 Jan. 21.0, «^ '■'''Vo 4-l".4608 +0".M76 -f 1 .4223 — .6757 -f 0".0009 + .0101 N L'.MKH1CA I- K.\ A M I'LK. 478 +- .... Dnle. u'/J ..AS;,r, ..-'/ (.. j ,\r^ll 1W4 Marc h 1.0, -1- I".2«ilO - l".4.-il2 -{- 0".01!»0 — ' 1".:50«5() Apri 10.0, 1 .OOlH 2 .122t; .027:5 :5 .10:5.") May 20.0, .07»tO 2 .tJ47;J .0:547 r> ..^)020 1 <»f tlw jHT- Irt. 1(! ). Wt' liiHl, u' measured nrmuKi' (•)♦]), ivc 5° 9' r,(j".i. w' and ,i' are $' — 0° l'.3S".l 18 !) .1 C' ;m :!!• .',1 ol 7 .tl 1 7 -Jil .7 1 2:$ 4:i ..") 1 ;{!) Jfi .;! 1 55 ;!.") .'J 2 11 7 .") 2 -Jd I'd .:! — 2 41 10 .() Kuly given in moans of tlic introduce the explained, we wf .%'•-/" 4-' 0".02H2 — .2361 The intej;ral ful >\i\,ift is obtained fri>m tlie .suct-cssivo values of'w'*'N^/', l)V ujeans of the formula (•V2). Next we compute the values of the diderential eoetlieiei ts hy means of the formula; ((Jo). For the dates WHi Ikw 12.0 and 1«i»('ct t<» all powers of tlif tlistiirl)iii>; forcr, un*l fur u loii^ scries of years, iisiii\r eoiistantly the same I'lm- (laiiieiital nseiilatiti^ eleinciits. iSiit even when tliese elenient.H are ho neeiimte as not to re<|uirt' eorreetion, on tUT'onnt of the efloet of* the large pertnrhations of lonj; period npon the values of oir anil nr, the iiuniitrieal values of the ]ierturl)ati<)ns will at length Ih; sueh that u • liaiiige of the oseulating elenients heeonies desirahle, so that the integration may again commence with the value xcro for the variation of Oifeh of the co-ordinates. This chang(> from ono system of ele- ments to another system may be readily cU'ected when the values of the perturbations are known. Thus, having found tlii' distuvbei! values of /•, w, and s, we have p being the semi-parameter of the instantaneous orbit of the disturbed hodv. In the undisturbed orbit we have 170 _.,'1 +»h) dt '0 p^r* rfc' and hence we derive Substituting for .. the value above given, there results (87) 1 a by means of which jj may be determined. To find -4ti wc have dt We have, also. rf;J 1 dz tan;J dr dt rcos,? dt r dt (88) lit ~ ~17p k\/l-\- VI iiff • rv v a. ~i~ lit ■ , itOr — e sin V = ^— - €o sin v^ -f- -.-, Vp, and if we put (89) 478 TIIKOUKTKAI. AsriloNOMY. tliix equation hoconicM c sin « — c„ sin I', -f- »/■„ Hin I'o + r- Wo liuvo, flirt lior, (90) P 1 und, putting we obtain J'o I' -- 1 + « *'o ~f" ''' • This c. kVl+m 03 a = ;j sec' l or the jseectiid of ei|iiatioiiH (\i'.\). If we denote l»y ^ the ar^nmcnt of the latitntle of the disturlM-d iMKJy with respect to the adopted funxjin'srtion« for the forces and ditU'rcntial cot'lHcients, the first integrals will l)o Ut' di' and that when these )|nuntitles are expressed in seconds of arc, (hoy ninst he converted i!it() their valnes in parts of the nnit of spaco whenever they are to he eonihined with (piantities which an? not cx- prcsstnl ill seconds. In other words, the homogeneity of the? several terms must be cairefully attended to in the actual applic^ation of the forniuhe. When the elements which correspond to given vahies of the per- turhations have been determined, if we compute the heliocentric longitU(h> and latitude of the body for the; instant to which the ele- ments belong, the results should agree with those obtained by com- puting th(! heliocentric place from the fundamcntul osculating ele- ments and adding the perturbations. 177. The (!omputation of the indirect terms when the perturha- tions of the co-ordinates r, w, and s are determined, is cHected with greater fiicility than in the rase of the rectangular co-ordinates, although {\w liiial results are not so convenient for tin; calculation of an ephemeris for the conip. * on of observations. This indinu't cal- culation, which, when the pertin-bations of any system of t.hre(> co- ordinates are to be computed, cannot in any case be avoidcsd witlioiit impairing the accuracy of the results, may be further simplified l)y determining, in a pecruliar form, the perturbations of {\w niciui anomaly, the radius- vector, and the (M)-ordinate ; perpendicular to the fundamental plane adopted. Let the motion of \\w disturbed body be, at each instant, referrih to the plane of its instantaneous orbit; then we shall have /9 - 0, and the etpiations (51) and (51) become r''^'" = ■Xnrdl-ykVp^a+m), k'a -\- VI) dt d^r div* (fl7) dt dt* --=R, in which R denotes the component of the disturbing force in flic direction of the disturbed radius-vector, and »S' the (!omponent in the plane of the disturlK'd orbit and perpendicular to the disturbed mditis- veetor, being positive iu the direction of tlie motion. The efl'ecl of VAUIATION OF I'OLAK CO-OUDINATKS. 481 ic forces and the ('fimpnnonts II and »S' is to vary the form of tlio orbit and the iiM^nhir distance of the periiiclion ironi the ncxle. If we denote by / the (H>ni|)onent of the distiii'i)in}{ ioree |)(>i'pendi = V,+Xo Co' sin E, (104) If, therefore, we determine the value of dMso as to satisfy the con- dition that ?.,=■- V -{- y, the disturbed value of the true longitude in the orbit, neglecting the effect of the component Z of the disturbing force, will l)c 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 + v), (105) and V will always be a very small quantity. When dM and v have been found, the effect of the disturbing force perpendicular to the plane of the instanianeous orbit may be considered, and thus the complete perturbations will be obtained. VARIATJON OP CO-ORDINATES. 483 In the equations (97), J/*^ ,- expresses the areal velocity in the in- stantaneous orbit, and it is evident that, since the true anomaly is not affected by the force Z perpendicular to the plane of the actual orbit, \i^ ~ must also represent this areal velocity, and hence the equations become r*^ =fSrdt + kl/p,{l + m), d?r -K^)'+^- (1 + m) (106) = R. 179. If we differentiate each of the equations (104), we get ,. „.dE, . dm (l-e,cosi;,)-^-=/x. + -^. dt dr, dv, dt dE, cos V, -—■ — r, sin v, - jf = — a^ sin E, -^ , (107) «/■, , dv, / _, dE, sm v,-T- -\-r, cos V, -^ = a^V 1—eo cos L, —r-, d>., dv, 'dt '"dt' From the second and the third of these equations we easily derive dr, ''dt r, ~,r = (oo^l — V **' 8^" ^1 ^os E, — a^r, cos v, sin E,) -v-. (IE, dt dE, Substituting in this the values of r, sin v„ r, cos v„ and -i-'. and re- ducing, we got dr, ., ' tA , dUfK r,-^ = ao'^„sm£,^/.„ + -^|, or dr, kVT+^i . I, , 1 dS3r\ ..... dr, From the same equations, eliminating — , we get dt ' dt which reduces to dE, r,* --f^ — (Ool/l — ej' r, cos v, cos E, + a^r, sin v, sin E,) -~, .■1=,/,-;a+^)(i + i.«f), (109) 484 THEORETICAL ASTBONOMV. or Combining this with the first of equations (106), we get = I'o ( 7T-T-T, - 1 ) + }l-4'Si • ,--^— =^ fSrdt, (110) mi dt from which dM may bo found as soon as v is known. The equation (105) gives d*r .^ . ..d^r, , _rfr, dv d'v rf^^^^ + ^^W + ^dT'W + '-'.F (111) dv, . 1*^0 Differentiating equation (108) and substituting for -^ its value already found, we obtain dt^~ 6,0031', /ill d<^M\ ' kV^l 4- tn f sin v, d''J3/ 7 ^ ■^";i;;"dr/+ ;i;;^;^ "dt^' and the last of the preceding equations becomes ''df + + dh' rf'v I ^"(1 + m) fjcost', df~''' 'dt' ■ r/ kVl + m <'+'>(>+i-ifr =- — Cg sm rfv , 2 rfv d53I\ The equation (110) gives /L±_!: ^¥ x.9^M- — --\ "'l /^o ' «^f' ^ rf< "^/x/ rf< ■ dt J' 1 rP<5J»f 2 dv 2 rfw Mo rft' ^ (1 + ")' rf< ' (1 + ")' dt k\/p,{l + jh) 1 >Sr / ^rd« which is easily reduced to (i-i-")' k\/p,(i+my and hence we derive rfV_ d'v , Jt'(l + m ' *"' jrt df ■ r, The equation (109) gives O^^c^, / 1 , dmy e^ VARIATION OF CO-ORDINATES. 485 and, siucc this becomes ''[dt)-~-r;^ V'^]:: dt }' r ^ = 1 + ,(l + m)»/ Po ^»(l + «0 P(Mim) / (115) which is the complete expression for the determination of v. It remains now to consider the effect of the component of the disturbing force wliich is perpendicular to the plane of the disturbed orbit. Let x„ y„ z, denote the co-ordinates of the body referred to the fundamental plane to which the elements belong, and x, y the co-ordinates in the plane of the instantaneous orbit. Further, let a denote the cosine of the angle which the axis of x makes with that of x„ and /9 the cosine of the angle which the axis of y makes with that of y„ and we shall have «, = «x + /9t/. (116) If the position of the plane of the orbit remained unchanged, these 486 THEORETICAL ASTRONOMY. cosines a and ^ would be constant ; but on account of the action of the force perpendicular to the plane of the orbit, these quantities arc functions of the time. Now, the co-ordinate z, is subject to two dis- tinct variations: if the elements remain constant, it varies witli 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 di~\dt 1^ \_dtS in which I -.- I expresses the velocity resulting from the constant elements, and y- 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 dz . constant, and hence the velocity -jr in a direction parallel to the axis of z, may be regarded as constant during the same time, and as receiving an increment only at the end of this instant. Hence we shall have dt~\dt) \_dtj~ Differentiating equation (116), regarding a and /9 as constant, we get ldz,\ dz, dx dy , \dtl'-Ht=''di + ^lii' ^^^'^ and differentiating the same equation, regarding x and y as constant, we get Differentiating equation (117), regarding all the quantities involved as variable, the result is d\ Now, we have da dx di3 dy ^** i s ^V 'dt" ~di '^ ~dt "dt '^"dF "^ ' dF' Z, = aX + 13 Y-i-Z COS i, (119) (120) in which Z, denotes the component of the disturbing force parallel to the axis of z„ and i the inclination of the instantaneous orbit to VAniATION OK CO-ORDINATES. 487 the constant the fdiulatnontul phmc. Substituting for A' and y their vahics given by the e(juations (1), and reducing by means of (116), we obtain or Comi>aring this with (119), tliere results da dx , (/,? (hi ., -dt' dt■^-dt'-dt='^''''^ lSl. The equation (120) gives kHl-\- Ml) lit:' z, -{■ Z cos i + ^X + /3 Y. (121) (122) The component of the disturbing force perpendicular to the plane of the disturbed orbit does not alfijct the radius-vector r ; and hence, when we neglect the effect of this com])onent, and consider only the components R and *S' which act in the phu.c of the orbit, we have dt' Jl-'(1 + m) h-\-'*oX-\-,%Y, (123) in which z^ denotes the value of s, obtained when wc put Z~-0. Lot us now denote by oz, that part of the change in the value of z, which arises from the action of the force perpendicular to the plane of the disturbed orbit, so that we shall have Substituting these in equation (122) and then subtracting equation (123) from the I'csult, we get dt* k'il +m) Sz, + Zcos i + XSa. -f- r'+^^--^y^'^\ (127) If we introduce the components R and S of the disturbing force, we have and hence r r r r dr x^l-y^ = ^,VF(l+^^-S^. dt Yx — Xy ■ Sr. Therefore the equation (127) becomes d*Sz, ife'd + m), df -Sz, -\- Zcosi IB 8 dr \ ^j^ We have, further, Sr dSz, (1 + m) ' dt ' dr .. . .dr, , dv (128) which, by means of the equations (108) and (109), gives dr Co sin t), , dv, dv kVpjl + "0 dt ~j9o(l +^)'^ dt '^'^' dt i>o(r+^)~'°^^"'' + ^'-^ Substituting this value in the equation (128), we obtain VARIATION OF CO-ORDINATES. 489 »0 . I « • , /-R Co sin r, „\ ffz, — oz, 4- Z COS I 4- \ ° »S I , -,^— \r, po / 1 + " _5r / (l^Zj^ oz,_ I'undamental plane to which SI and i are referred, it being the argument of the latitude in the case of the disturbed motion. lAit w' denote the lon- gitude of the disturbing body measured from the same origin and in the plane of the orbit of the disturbed body, and let ^i' denote its latitude in reference to this plane. Finally, let N, N', I, 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 (6G). Then we shall have sin Usin • (JV+ N') = sin J (ft' — ft) sin ^(i' + i), sin Ucos A (N+ N') = cos ^ (ft' — ft ) sin J (i' — i), cos5/sini(^— iV')==sini(ft'— Q)t'osi(i'+ 0, cos Ucos ^ (iV— N') = cos A (S^' — ^) cos \ (t' — i), from which to determine N, N', and /. Wo have, also, n^ = u'-N', tan {w' — N) = tan »,' cos /, tan t^ = tan /sin (to' — N), (130) (131) from which to find w' and ,9', w' being the argument of the latl-ade of the disturbing bo Mm ,. ,,^, _. Hill iV , ,., , sill .V ,. dN' = . ,co8iV rf(R — ft)— . ,cort/(/t'4- . , di, mix I sill / bill J J., Hint' „, ,._, _, sill .V ... , siiiiV , ,. (136) fill / » w ^m y jjllj ^ rf/ =cusiV'(/t' — COS iVf/t + sin ('.sill iVr/( Q' — Ji). When / anil /are very small, it will bo hotter to use Hin ' sill .V sill ( !*iii A'' sin/ sin (ft' — Si)' sin/ " 8in( ft' — ft )' (137) in findiiif^ tlio niiinorical valiies nf these cooflicieiits. JJy moans of those f'ornuihe we may derive the values of fh\\ oN'y and o/ corre- spandinp to given values of oft, di, ^ft', and <)/'. The formuho by niean.s of wliioh o(T, fJft, and di may be obtained direetly, will be lirosontly eonsidered. The results for iiX, dN', and fJ/ being applied to the quantities to whieli they belong, we may eomiiute the aetuul values of ir' and ;5'. The val ! of r will be found from the given value of v, and that of w will be given by means of equation (l.'Jo). Then, by moans of the formuhe (132), the forces li, S, and Z will be obtained. The porturbataons will first be computed in reference only to terms do- ponding on the first power of the disturbing force, and, whenever it hooomes necessary to consider the terms of the .second order, the rosults already obtained will enable us to estimate the valu(>s of the jwrturbations for two or more intervals in advance with sutHcient 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 comp. jd at once, and the values of each component for the several disturbing bodies may be united into a single sum, thus using I'R, 1\S, and I'Z in place of li, S, and Z respectively. The approximate values of the porturbations will also facilitate the indirect calculation in the deter- mination of the complete values of the required diirerential coeffi- cients. 183. When only the perturbations due to the first power of the disturbing force are required, the osculating elements ft^ and i^ will be used in finding N, N', and I, and r„, tOg will be used instead of r and ?» in the calculation of the values of li, S, and Z. The equations for the determination of the perturbations dM, v, and dz„ neglecting terms of the second order, are, according to the equations (110), (115), and (129), the following:— m I y 492 THEORETICAL ASTRONOMY. *l'';'o(l+»n) (138) kHl-^m) /fe'Cl+m), Tlio valuo of V is first foinul by intcj^ration from tlm results jjivon by tlio socoiul of tlu'so equations, auil ihcji JJf is found from the first C(iuation. Finally, Sz, is found l)y means of the last e(|nation. The intej^rals are in eaeli ease o(|ual to zero for the dates to which the fundamental oseulating elements belonj;, and the process of inte<;ru- tion is analogous, in all respeet*", to that already illustrated in the case of the variation of the rectangular co-ordinates. It will be ob- 8erve wherein Xq denotes the modulus of the system of logarithms. We may also use v, instead of v^,; 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) v/itli 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 terras becomes troublesorae. VARIATION OK « O-OUIUXATI-X 493 Tlio results may l)o convcincntly oxpiN'ssod in seconds of arc, and afd'rwanis v and th, may Ix; coiivi'rtcd into tlicir values (expressed in units «»(' tlio .seven til decimal place, or, >;ivin^? proper attention to the lutinojjeneity of the sevi-ral terms of the (Mpiations, in the nnmeri<-al operations, rM/" may he expressed in seconds of are, while u and ih, are oi)tained directly in units of the seventh decimal place. It will he advisable, also, to introduce the interval lo into the tormuhe in Hiich a ntanner that this (piantity may be omitted in the ease of the liirmuhe of integration. 184. In the case of orbits of pjreat eccentricity, the mean anomaly and the mean •„ we have r, — 7o sec' h,. For the other cases in wliich the elements M^ and /i^^ cannot be em- ployed, the solution must be offeete')' kVp, or, putting t,='t-\- dt, dt, (143) dt (lH-v)> -1 + (1 + ")' kVp„{l+m) /■ SrdL (144) If we determine 8t by means of this equation, the values of the radius-vector and true anomaly will be found for the time t-\-ot instead of /, according to the methods for the diflereut conic sections, VARIATION OF CO-ORDINATES. 495 usiiip; the fundamental osculatinpj elements. The results thus obtained arc the required values of v, and i', respeetively. 18'). When the values of the perturbations v, 8z„ and (iM, ilT, or ()/ have been determined, it reniains to liud the place of the disturbed body. The heliocentric longitude and latitude will be given by cos b cos (1 — Sl)- r=cos(A— Q), cos b sin (1 — Sl)-- = sin (k — SI) cos i, sin 6 = sin (A — ^) sin i, or, since ^ ■== ^, — — sin(A, — ft„}cos( ^^^ t'*c angle oj)posite to (r — ft„ by jy'. The auxiliary triangle thus formed gives the Ibllowing relations : — 496 THEORETICAL ASTRONOMY. cos(/(o— S^o)— cos(t— SJo)co8(Ji— /0+sin(l,— JJo)sin((T — ^j)sini. Since the actiou of the component of the disturbing force perpen- dicular to the plane of the disturbed orbit does not change the radius- vector, Ave have r sin 6 =^ r sin % sin (A, — Slo) + ^^n (149) and hence the last of these equations gives dz -l = s'm(X, — Slo) (cos (l — ft „) cos ( ^o- ft „) cos ^-cos (X,— ft „) sin {h,— ft „) sinij' dz, smb Sz =sm(^,— ft,)sin /q H ^• r cos T)' r If we multiply the first of these equations by cos(A„ — ft„), and the second by — sin (Ag— ft^,), and add the results; then multiply the first by sin (/to — fto), and the second by cos(A„ — fto), and add, we get cos6 cos(;-fto— (A-/0)-=cos(/l,-fto)+sin(;i„-fto) ,-^^^ • ^, 1 — COS); r ' cos6 sin (^-fto— (/t-/io))=i>in (-t — fto)cos4-cos(Ao— fto) ®'" ''' ''^' dz, smb Let us now put =sin (-*,— fto) sin 4+- p' — s\n(a — fto) sin i, ^ = cos ((T — fto) sin i — sin io, 1 — COS);' r ' (152) and there results, from (149), Sz, -- = q' sin (X, — fto) —p' COS (A, — fto). Comparing this with equation (150), we observe that / = sin)?' sin (/to— fto), q' = sin 1)' COS (/to — fto) COS io — sin % (1 — cos yj'). Therefore, we have (153) (154) sm); 1 — C08)J 7sin(^o— fto) P COS)? sm)? ,, s -ji = -cosft„^-sma dl dt da dt' d,3 (161) 3in$^„^ + cosJi,^^ From the equations (118) and (121), observing that X dy dx 2'-:77 = *^^.P(i + "^)' dt " dt we derive, by elimination, da, r sin ^, cos i „ d£^ r cos ^, cos i „ di~ kVpil +m) ' di kVpUr^'m) 500 THEORETICAL ASTRONOMY. Therefore we shall have rcosisinC^, — ft„) dp^ dt ~ kV'pil -f-m) d hVp (1 + m 1 kVp{\ +m I ddz, df\ )V dt ^'' dty (163) VARIATION OF CO-ORDINATES. Substituting further the vahies ' ^ r cos {^, — Slo), 501 and also y" = r sin {X, — J^,), dt dr lit kVl+m Vp e sin V ■ kVpa + m) esin V 1 -j- e cos V we easily find, since X, — 'W =^ >f> ^' =^ - (cos (-1, — S2o) + e COS (;if - ^„)) -^ + — /.y. --f -"-'^- • 7,. '. 5' = + (sin (;, - Ji„) + e sin {x — fto)) ~ + TT 7-==T ' V' (164) which may be used for the determination of p' and q'. These equa- tions require, for their exact solution, t..«,t the disturbed values c, ;f, and JO shall be known, but it is evident that the error will be slight, especially when e is small, if we use the undisturbed values f^,, ^Ju, and ;fu = ttq. The actual values of ^ and r are obtained directly from the values of the perturbations. When p' and q' have been found, it remains only to find cos /, and 1 — cos r/, in order to be able to obtain F by means of the ecjuation (159). From (153) we get and hence pit _|_ g/j __ gjjjj I — g;jj2 1^ — 2g' sin \, COSl: i/r= ^■^-(3' + sini„n from which cosi may be found. The equation (157) gives 1 — cos 5j' = COS \ (cos \ -\- COS i) — cl sin i„, by means of which the value of 1 — cos r/ will be obtained. (165) (166) If we substitute the values of p', q', -^i and -~ given by the dt " dt equations (153) and (162) in (159), it is easily reduced to Sz, ^-Skz (1 — cosV) kVp{\ -\- m) Zdt, (167J) v/hich may be used for the determination of P. When we neglect terms of the order of the cube of the disturbing force, in finding F we may use po in place of p and put 1 — cos jy' = 2 cos^ Iq, so that the formula becomes 502 THEORETICAL ASTRONOMY. r= 2cos'<;^V/;)o(l + '«) S"'' Zdt. (168) 187. By means of tlie formulae which have tlius been derived, we may find the values of all the quantities recjuired in the solution of the equations (155), in order to obtain the values of / and b for the disturbed motion. From r, /, and 6 the corresponding geocentric place may be found. The heliocentric longitude and latitude may also be determined directly by means of the equations (145), provided that Q,, when the correct values of R, Sf Z, if and p are known. The corrected values of / and p — 504 THEORETICAL ASTRONOMY. which are required only in the case of 8z, — may he easily estimated with sulliflent afcuraoy, since wo reciuirc only cos/, while V p ap- j)cars as the divisor of a term whose numerical value is gcnoraliy insignificant. To obt^iin the actual values of li, >S', and %, the cor- rections to be applied to N, iV', and /must first be determined by means of the formulte (13(J). The values of di' and tlQ' will be found by meaiiK of the data furnished by the tables of the motion of the disturbing l)ody, ano i" p'li^^s of e, ■)(, and p. The results for 81 and 8N' obtained from (173) being applied to the values of /' and iV' as already corrected on account of 8i' and 8^', give the requii'ed values of these quantities. mm NUMERICAL EXAMPLE. When wc considor only di nnd dQ , since sin i' cos N' = cos i sin i + sin i cos / cos N, we easily find ^iV=cos/<5iV' — 5(>".4. Then, I)y means of the data furnislied by the Tahlcfi of Jiijtlfcr, \vc find the values of n', the argument of the latitude ol' Juj titer in reler- encc to the ecliptic of 18G0.0, and from the ecpiations (131) we derive w' and /9'. The values of /•' are given by the Tables of JujiUrr, and the values of ?•„ and t'y are found from the elements given in Art. 166. The results thus obtained are the following: — lll'llill MlHIl Tliiif. loRro '•o lOR I-' IV' /3 18r,;{ Dec. 12.0, 0.291084 354= 20' 18".0 0.73425 14° 18' .>t''.0 — 0° l'38".l 18G4 Jan. 21.0, 0.29-48:{7 10 >> 45 .7 0.73308 17 21 44 .2 18 9 .1 Mnrd 1 1.0, o.;?oor)74 25 24 59 .4 0.73305 20 25 5 .2 34 39 .9 April 10.0, 0.31 0SG4 40 13 31 .8 0.73237 23 28 59 .8 51 7 .6 »ray 20.0, 0.;?24298 54 14 41 .4 0.73104 20 33 32 .1 1 7 29 .7 .June 29.0, 0.3:59745 07 21 23 .5 0.73080 29 38 44 .8 1 23 4:$ .5 Aug. 8.0, 0.350101 79 32 18 .1 0.7.3003 32 44 41 .2 1 39 40 .3 Sept. 17.0, 0..3724()9 90 49 57 .0 0.72915 35 51 24 .6 1 55 35 2 Oct. 27.0, 0.38S214 101 19 9 .8 0.72823 38 58 57 .5 2 11 7 .5 Dec. CO, 0.402S94 111 5 42 2 0.72720 42 7 23 .3 2 20 20 .3 1805 Jan. 15.0, 0.410240 120 15 32 .0 0.72025 45 16 43 .9 ■ -2 41 10 .6 The value of w for each date is now found from w- ^^•o + ^o-«o = n■^197°38'6".5, and the components of the disturbing force are determined by mean.s of the fornuilaj (132), p being foiin.i ^'rom (133) or (134), and h from (70). The adopted value of the mass of Jupiter is m' 1047.879 and the results for the components E, S, and Z are expressed in units of the seventh decimal j)lace. The factor to^ is introduced for conve- nience in the integration, (o being the interval in days between the successive dates for which the forces are to be determined. Thus we obtain the following results: — NUMEniCAL EXAMPr.E, 507 Dutc. (-»« <..'.SV„ '.''/con /„ (.jSr„iU 1863 Dec. 12.0, + 70.82 4- 7.10 + 0.04 +■ 1.37 1864 Jan. 21.0, 68.05 - 32.76 0.49 -11.45 March 1.0, 61.16 70.38 0.92 63.32 April 10.0, 4«..")7 102.01 1.32 150.48 May 20.0, 32.77 128.34 1.68 miio June 2U.0, + ir).41 145.3U 1.96 404.35 Au;r. «.(), 2.1'J 153.44 2.17 554.54 Sept. 17.0, 10.12 152.41 2.29 708.21 Oct. 27.0, 34.81 142.50 2.25 8.')»;.39 Doc. 6.0, 48.95 124.04 2.0!) 990.36 180;') Jan. 15.0, 61.45 — 07.36 + 1.75 - -1101.73 0" V 38".l 18 9 .1 M 39 .9 51 / .6 1 1 29 .1 1 2.3 4:? .5 1 ;«) 4tJ .3 1 r^r, 3.^1 •) 2 11 7 .5 2 2(5 20 .3 2 41 10 .6 Tlic sijif^lo integration to find (oiSffjiU is efl'cotcd by mcan.s of tlie forniula ('>2). Tlio iHniations for the ''"termination of the required differential coefticients are dt dt 4ih:rfi''''^'~^''')' 0>'Ji 2Mk' ' n 'a ^•1 1 C< u ''oS'n^'o 2 , - til I lirM - w'o — ,3 "' (P '"^"^^ 'o *o the equinox and ecliptic of that date; and then, having computed ?,, and r, we obtain by means of the equations (172) the required values of I and b. 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, tiie proper corrections for precession and nutation may be applied. In order to compare the results obtained from the perturbations dMy V, and dz, Avith those derived by the method of the variation of rectangular co-ordinates, we have, for the date 1865 Jan. 15.0, .ron= — 2.5107584, ;/„ = + 0.0897713, z„ = — 0.1400500 ; and for the perturbations of these co-ordinates we havo found (Ja; = + 0.0001773. J^/ :^ + 0.0001992, J? =^— 0.0000028. Hence we derive ar=r — 2.5105811, y^ + 0.6899705, s = — 0.1406618, and from these the corresponding polar co-ordinates, namely, log r = 0.4162182, I = 164° 37' 59".05, 6 = — 3° 5' 32".54, from which it appears that the agreement of the results obtained by the two methods is complete. 190. When the perturbations become so l.i.rge that the terms of the E'cond order must be retained, the approximate values which may be obtained for several intervals in advance by extending the <^'olumn9 of diflferences, will serve to enable us to consider the neglected terms partially or even completely, and thus derive the complete perturba- tions for a ve»y long period. Bui on account of the increasing diffi- culties which present themselves, ari.sing both from the consideration 510 niLuUETICAL ASTRONOMY. / of tlie perturbations due to the action of the componen*^^ Z in com- puting the place of the body, and from the magnitude of tiie 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 terras, be effected with great facility. The mode of effecting the calculation when the higher powers ox' the mass o are taken into account has already been explained, and it will present no difficulty beyond that which is in- separably connected with the problem. The determination of F, p', and q' may be effected from the results for -,. , —-. and —.r by means of the formula; for integration by mechanical quadrature, as alrca'l, illustrated, or we may find F by a direct integration, and the values of p' and q' by means of the equations (164), -^.- being found from -,.j' by a single integration. The other quantities required for the com^jlcte solution of the equations for the perturbations will be obtained according to the directions which have been given; aiul in the numerical application of the formulte, particular attention should be given to the homogeneity of the several terms, especially since, for convenience, we cxpi'ess some of the quantities in units of the seventh decimal place, and others in seconds of arc. The magnitude of the perturbations w'll 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 j)roceed to develop the formuloe for this purpose. 191. The single integration of the vahies of or^-Tp and ft>^-^will dt* d'Sz, dt give the values of w -tt and a> —yf> and hence those of -j. and -t7 '> dm which, in connection with - -, , are required in the determination of dt dv, the ..ew system of osculating elements. Since r" -y~ represents double the areal velocity in the disturbed orbit, we have in*; Z in com- f tiie iiumeri- to determine, nding to the id thus com- the terras of ;t part of the IS, be effected on when tlie already been which is in- ion of r, j9', ^^ by means e, as alread, d the vakios J found from ired for the ons will be ^'en; aiul in ntion should lly since, for tlio seventh CHANGE OF THE OSCULATING ELEMENTS. 511 dv^ __ Wp (1 + m) dt ~ r* The equation (109) gives dv^ ^_ kV^JT+mj I 1 dm \ Hence, since r = r, (1 + v), we obtain ^=^-(' + ^/1fJ^l + ^)^ (176) by means of which we may derive ^3. This formula will furnish at once the value of p, which appears in the complete equation for -^jf. and also in the equations (164); and the value of cosi may be determined by means of (165). In the disturbed orbit we have dr kV\ + m and the equations (108) and (111) give dr _1c\/T+^i . /, , 1 d3M\ , , dv Therefore we obtain _ ,/;: ..;„../ 1 , V dm \ ^ ^ _^ ^^ ^ ^i ^pp^ d. 1 r^ t, .-in v = Vp e^ sin v, ( 1 + - ^I-^i + m* dt' Mo dt ' \vi:H b" .asans of (176), becomes esiu.^e,sin.,(l+l.'^M\\i + ,)a^_iyZ_.4!^.. (l77^ The relation between r and ?•, gives P Po 1 + - COS V 1 + e.j cos V, ' "" ''^' bstituting in this the value of p already found, we get e cos V — i^l + e^ cos v,) \ ^/'o dt } (1 + ")'-!. (178) 512 THEORETICAL ASTRONOMY. Let US now put 13 = r,\/p kV\-{-m c?<' (179) a and /9 being small quantities of the order of the disturbing force, and the equations (177) and (178) become e sin V = e^ sin v, -\- ae,, sin v, -\- /?, e cos V = fo cos V, -\~ oCo cos v, -\- a. - These equation'^^ give, observing that r, (cos v, + e^) =j)o ^^^ ^t e«i - v) = o sini;, — /3 cos v„ e cos (v, — v) = Cj + -^^ cos E,-\- p sm v„ '^ from which e, v, — v, and v may be found; and thus, since X = ^o-\-i^,-'")> (181) we obtain the values of the only remaining unknown quantities in the second members of the equations (164). The determination of p' and q' may now be rigorously effected, and the corresjjonding value of cos i being found from (1 65), ~^t and -^ will be given by (162). Then, having found also 1 — cos;y' by means of (166), F may be determined rigorously b^ the equation (159), and not only the complete values of the perturbations in reference to all powers of the masses, but also the corresponding heliocentric or geocentric places of the body, may be found. If we put / = a sin V, — /9 cos v„ 3' = «^cosJ:, + /9sint;„ ^^^^^ and neglect terms of the third order, the equations (180) give e=e„ + 5' + ^, "» ^^ " ^ 2e; V, — v = —s -s, (183) in which s = 206264".8. These equations are convenient for the CHANGE OF THE OSCULATING ELEMENTS. 513 determination of e and v, — v, and hence X by means of (181), when the neglected terms arc insensible. The values of p, c, and v having been found, we have sm ^ = e, a=p sec'' V, /* = kVl + m ai (184) tan \ E = tan (45° — j ^p) tan A v, M= E—e sin E, from which to find the elements f, a, //, and 31. The mean anomaly thus found belongs to the date t, and it may be reduced to any other ej)och denoted by f^ by addintr to it the quantity fx (<„ — t). When we neglect the terms of the third order, we have V>~ Feos 4' '»- Vdt) ~ — - -^ — e sin V, Vp V.in,',^r{^.)=:^:lLi^-^JI^, in which v denotes the true anojnaly in the instantaneous orhitj and hence there results da__ 2a" (^i ~ k\/p(l-{-m) P c'^ =.(esmvE-\-^S), (198) hy means of which the variation of a may be found. If we introduce the mean daily motion fi, we shall have and hence dfi ~dt if^ da a di ' -ju- = yzj=^=:^== (e sm vE -{• ~ S), dt kVp (1 + w) ^ ^ r ^' (199) (200) for the determination of 5//. The first of the equations (97) gives and hence we obtain dt Hh^" djV'p) _ Sr dt or ^F 1 + m dp ^ Ip r ^ dt kVl + m (201) The equation ^ = a (1 — e^) gives dp p da ^ de dt a dt dt dp da Equating these values of -^-> and introducing the value of — dt already found, we get ^=I7?TO(''""*+f(f-v)^)' (202) / 620 and since THEORETICAL AHTRONOMY. P _ 1 + e COS V, — = 1 — c cos ^, a E being tlie eccentric unonialy in the insttintuneous orbit, this becomes lie dt hV'p(\ + m) {^p sin vR + p (cos v -\- cos E) S), (203) which will give the variation of e. If we introtluce the angle of eccentricity + r) 8iu vS), (205) IVoin which the vnhio of .'; nmv be (lerived. (It If wo introduce tlio ok'incnt (o, or the niiguhir distance of the i)eri- helion from the ascending no(U', it will be neecssury to (lonsider also the contponent Z; and, since (o ----- X — «t, we shall have and hence dx 'i<^ dx . riJi 7lt ~ lit ~ Tit ~ ^^^ ^ dt' dm It — ; • — ( — pco3vE4-(p + r)fimvS) — eos i-,— " ('20()) kVp{l-\-m) e > V7 -r / ^^ v In the case of the longitude of the perihelion, we have di: lit and therefore ~dT k]/j)(i-\-m) e dtu dQ dt "•" iir ( — p cos vR -\- (p -\- r) sin vS) -f2 8inM4^. ' dt (207) The first of the equations (15)2 gives [drl ^ . ldM„ , ,, ,, rfM\ 2r dfi dc . in which J/y 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 -,- and -^ the values already found, we get dt kVpil-{-m) P \ (p cot

(l + m) (/> 4" r) cot ^ pin vS = ^•vXl + »«) ^pcot^ eosvR + COSf dt' by means of which (208) reduces to dM, dt — COS ^ dt 2-0^^ /e-«-0-t' (209) A;l/;>(l + w) (/< which will determine the variation of the mean anomaly at the ejioch. Since the equations for the determination of the place of the Lody in the case of the disturbed motion are of the same form as thode for the undisturbed motion, the mean anomaly at the time t will be given by M=^ M, + oM, + « - g iih + <5/i), in whicli ii^ denotes the mean daily motion at the instant Iq. There- fore we shall have M- = J/. + J '^^1 dt + /.„ {t - t,) + {t -^ QJ fdfji dt, dt "^ ' ''''•'' ''' ' '^'' '"V dt the integrals being taken between the limits t^ and t. The quantity expresses the mean anomaly at the time t in the undisturbed orbit ; and if we designate by 331 the correction to be applied to this in order to obtain the mean anomaly in the dioturbed orbit, so that SM-- /f* we shall have and hence dt "" ^' di "' ' ^'' '"V dt Differentiating this with respect to t, we get dM _ dM, d,. rd,i dt dt dt. VARIATIOX OF COXSTAKTS. 523 (UL Substituting in this the value of *, ° from (209), the result is dM dx -^-=:-COS^- 2r cos

lion distance q and of the time of perihelion passage 7' will be determined instead of those of the elements M h\k\ a or fx. Tiie equtttioa p .:-.-. 5 (1 + e) gives 624 THEORETICAL ASTRONOMY. 1 dq 1 dp <1 de dt ~ 1 -f- e ' dt 1+e dt and substituting in this the value of -jr already found, and neglect- ing the mass of the comet, which is always inconsiderable, we get 7 dq Hi kVp 1+e dt' (218) by means of which the variation of q may be found. In the case of de elliptic motion the value of -j. may be found by means of (202) or (203); but in the case of hyperbolic motion the equation (202) will be employed. It should be observed, also, that when the general formulte for the ellipse are applied to the hyperbola, the senil- cransverse axis a must be considered negative. When the orbit is a parabola, the equation (202) becomes de 1 -77- = — j=- (ps>\jxvR -{- 2p COS^ -}iVS), dt kVp \ i- ^ ^' dq (214) and for the value of —J.- we have dt dq "dt JlL^S-lq^ kV p ~ at (215) It remains now to find the formula for the variation of the time of perihelion passage. The relation between T and M^ is expressed by 360°-J/„ = M(r-fo), the differentiation of which gives dM, dt dIL and, substituting for - ,.- the value given by equation (209), we get dT_2ar aVp dy ^ 1 di k dt _d,^^ fi dt Substituting further the values of -^ and -^ given by the equations (205) and (199), the result is VARIATION OF COXSTAXTS. 525 tlT aR ^^ p ?jk it —T) . . — cos V — — e sin v) e i/p sin V 3/fc^ — T) p ~~7p~ (216) 'v} which may be employed to determine the variation of T wlieiievor the eccentricity is not very nearly ecjual to unity. It is obvious, however, that when a is very large this e(iuation will not be con- venient for numerical calculation, and hence a further transformation of it is desirable. Thus, if we derive the expressions for ^ and . from tlie equations (24)2 ii»''^« (-'3)2> we easily obtain ' — . _ -— (J (2r — — cos V -— — - e sm v) -\ -— — -, cos v, \-\- e de e y j) e {1 -\- ey 2j) dv I p-\-r . -— : — r - ■■ a I — \ e sin V U(t — T) p )-e-(r-b)^(i +];)--• 1 -f e de \ e y p By means of these results the equation (216) is transformed into dT_ qR ^^ dr^ _ q di qR .^ dr q a , 7'*^' o ^'' i '// 1 , '" \ • ^ -oi-rx k' ^^Te-e ''' '^ + k^ '^'' -Te^iV+pr'"'^- ' -^^^ dT dr which may be used for the determination of -., > th dues of .- and -J- being found by means of the various formulse develoj)(d in Art. 50. When a is very large, its reciprocal denoted by/ may often he conveniently introduced as one of the elements, and, for the deter- mination of the variation of/, we derive from equation (108) -^ := ^--=: (e sin vR + ^ S). dt kVp ^ r ^ (218) In the case of parabolic motion we have c = l, and p = 2fj; and dv d tliey may be, will not indicate with certainty the semi-transverse axis <»f the orbit, and hence the periodic time. But when r is known, by eliminating the effect of the disturbing forces, we may determine with accuracy the value of the semi-trans- verse axis a at each epoch, and, from this and the observed places, the other elements of the orbit according to the process already explained. Let /^u be the mean daily motion at the first epoch, and we shall have f^o'^ +/^ dt dt = 2ff, in which - denotes the semi-circumference of a circle whose radius is unity. Hence we obtain. / dM 2.-|-^^-rf« (220) by means of which to determine n^^. Then, to find the mean daily motion ji at the instant of the second return to the perihelion, we have ^-''^+St''' (221) the integral being taken between the limits and r. The provisional value of the mean motion as given by the observed interval r will be sufficiently accurate for the calculation of the variations of M and ji during this interval. The semi-transverse axis will now be derived by means of the formula a=\ \1. VARIATION OF CONSTANTS. 627 from the values of fx for the t>vo epochs. Let r' denote tlie interval which must elapse before the next succeeding perihelion passage of tlie comet, and we have and consequently lz = „z'JrS- dM dt dt, ^dM 1^ = U-p-^^M (222) se raLiius is the integral being taken between the limits 1 = 0, corresponding to the beginning of the interval, and t = r'. We have, therefore, 8t -\P>- (223) for the change of the periodic t5.ne due to the action of the disturb- ing forces. 198. The ct^lculation of the values of the components R, S, and Z of the disturbing force will be effected by means of the formulte given in Art. 182. It will be observed, however, that not only tliese components of the disturbing force, but also their coefficients in the expressions for the differential coefficients, involve the variable ele- ments, and hence the perturbations which are sought. But if we consider only the pertui'bations of the first order, the fundamental osculating elements may be employed in place of the actual variable elements, and whenever the perturbations of the second order have a sensible influence, the elements must be corrected for the terms of the first order already obtained. Then, commencing the integration anew at the instant to which the corrected elements belong, the calculation may be continued until another change of the cL ncnts becomes necessary. The several quantities required in the computation of the forces may also be corrected from time to time as the elements are changed. The frequency with which the elements must be changed in order to include in the results all the terms which have a sensible influence iu the determination of the place of the disturbed body, will depend entirely on the circumstances of each particular case. In the case of the asteroid planets this change will generally be required only after an interval of about a year; but when ^he planet apjiroaches very near to Jupiter, the interval may ne -essarily be much shorter. The n 'ii 528 THEORETICAL ASTRONOMY. magnitude of the resulting values of the perturbations will suggest the necessity of correcting the elements whenever it exists; and if we apply the [)roper corrections and commence anew the integration for one or more intervals preceding the last date for which the per- turbations of the first order have been found, it will appear at once, by a comparison of the results, whether the elements have too long been regarded as constant. The intervals atwi'i! the differential coefficients must be com- puted directly, will also depend on the relation of the motion of the disturbing body to that of the disturbed body; and although the in- terval may be greater than in the case of the variations of the co- ordinates which require an indirect calculation, still it must not be so large that the places of both the disturbing and the disturbed body, as well as the values of the several functions involved, cannot be inter- polated with the requisite accuracy for all intermediate dates. In tlie case of the asteroid planets a uniform interval of about forty days will generally be preferred; but in the case of the comets, which rapidly approach the disturbing body and then again rapidly recede from it, the magnitude of the proper interval for quadrature will be very diiferent at different times, and the necessity of shortening the inter- val, or the admissibility of extending it, will be indicated, as the numerical calculation progresses, by the manner in which the several functions change value. If we compute the forces for several disturbing bodies by using I'R, 1\S, and I'Z in the formulae in place of R, S, and Z, respect- ively, the total perturbations due to the combined action of all of these bodies may be computed at once. But, although the numerical process is thus somewhat abbreviated, yet, if the adopted values of the masses of some of the disturbing bodies are uncertain, and it is desired subsequently to correct the results by means of corrected values of these masses, it will be better to compute the perturbations due to each disturbing body separately, and, since a large part of the numerical process remains unchanged, the additional labor will not be very considerable, especially when, for some of the disturbing bodies, the interval of quadrature may be extended. The successive correction of the elements in order to include in the results the per- turbations due to the higher powers of the masses, must, however, involve the perturbations due to all the disturbing bodies considered. The differential coefficients should be multiplied by the interval w, so that the formulae of integration, omitting this factor, will furnish directly the required integrals; and whenever a change of the inter- NUMERICAL EXAMPLE. 529 val h introduced, thf proper caution must be observed in regard to the process of integration. The quantity s = 200264". 8 should be uitroduced into the fornmlie in such a manner that the variations of the elements which are expressed in angular measure will be obtiiined directly in seconds of ai*c ; and the variations of the otlier elements will be conveniently determined in units of the nth decimal place. It should be observed, also, that if the constants of integration are put equal to zero at the beginning of the integration, the integrals obtained will be the required perturbations of the elements. 199. Example. — We shall now illustrate the calculation of the perturbations of the elements by a numerical example, nnd for this purpose we shall take that which has already been solved by the other methods which have been given. From 1864 Jan. 1.0 to 1865 Jan. 15.0 the perturbations of the second order are insensible, and hence during the entire period it will be sufficient to use tiie \ olues of r, V, and E given by the osculating elements for 1864 Jan. 1.0. The calculation of the forces it, S, and Z is effected precisely as already illustrated in Art. 189, and from the results there given we obtain the following values of the forces, with which we vrite also the values of Eg: — Berlin Mean Time. 40R 40^ AOZ Eo 1863 Dec. 12.0, + 0".0365 + 0".0019 + 0".00002 355° 26' 8' .2 1864 Jan. 21.0, .0356 — .0086 .00025 8 14 57 .8 March 1.0, .0315 .0182 .00047 20 57 55 .1 April 10.0, .0250 .0259 .00068 33 26 47 .6 May 20.0, .0169 .0314 .00087 45 35 25 .3 June 29.0, + .0079 .0343 .00101 •''7 20 3 .8 Aug. 8.0, — .0011 .0349 .00112 68 39 14 .6 Sept. 17.0, .0099 .0333 .00117 79 33 13 .1 Oct. 27.0, .0179 .0301 .00116 90 3 23 .2 Dec. 6.0, .0252 .3253 .00108 100 11 49 .1 1865 Jan. 15.0, — .0317 — .0193 + .00090 110 54 .3 We compute the values of the required differential coefficients by means of the equations dsa r sm u Z, dSi It r cos « Z, dSK dt dt kVp si^*" "' dt jci/p 1 / _ ^cos^ ^ (p+^^ A ^.^, ^.d^^ \/p \ s\n

osculat hlg llowiug : — In order to compare the results thus derived with the perturbations computed by the other methods whicli 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, logr= 0.4162182, Z=:164°37'59".04, E-= 110° 5'14".15, V =-- 120 19 18 .01, 6 = — 3 5 32 .54, agreeing completely Avith 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 rr, Q, 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 d3I, if we denote by JM the value which is obtained when we neglect the last term of the equation for ~ir-> we shall have «53/=- dM +J[j(^< 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, by means of the formula for double integration. //** = + 56-.69, dt Hence we derive dM=-\- 234".74 + 56".59 = + 291".33, agreeing with the result ah'eady 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. dMa = + 165".29, 532 tiip:oretical astronomy. I / ami 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, wc have, for 18G5 Jan. 15.0, and hence (<-goV = 4-126".27, ' dM= 'Uf, -\-(t — g ¥ = + 291".56. The error of this result is — 0".23, and arises chioHy from the in- crease of the accidental and unavoidable errors of the numerical cal- culation by the factor t — /„, 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 831 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 ephcmcris 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 ephcmeris may become sensible. The variation of the geocentric place of the disturbed body arising from the action of the disturbing forces, may be obtained by substituting the corresponding variations of the ele- ments in the differential formula) as dei'ived 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 oil/ directly in the equations for the variations of the geocentric co-ordinates, the coefficient of d/i must be that which depends solely on the variation of the semi-transverse axis. But when the coefficient of d/i has been computed so as to involve the effect of this quantity during the in terval t tuted in the equations t^, the value of oMg must be found from oM and substi- 200. It will be observed that, on account of the divisor e in the expressions for -^. -tt' and -^-> theseelements 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 OP CONSTANTS. 633 account of the divisor sin t in the expression for ,, the variation ' (It of SI will be large whenever i is very small. To avoid the difHciil- tios thus encountered, new elements nuist be introdueetl. Thus, in the ease of SI , let us put a" == sin i 8iu Si, /5" = sin i cos SI ; (224) then we shall have da" . _ .di . dSl - ., - = sm SI cos t-jj- -\- sm i cos SI ,. > dr _ . (// . . . ^dSi -J.- = cos SI COS I ,- — sui I sni SI ,, • at at at Introducing the ^'alucs of -r- and given by the equations (212), and introducing further the auxiliary constants a, b, A, and B com- puted by me.'iiis of the fornuila) (94)i with respect to the fundamental plane to which Si and i are referred, we obtain da." dt dt kVp (1 + m) 1 rZ sin a cos (A -\- u), (225) kV]) (1 + m) — ^TTT^ rZ sin b cos (B + u), by means of which the variations of «" and ft" may be found. If the integrals are put equal to zero at the beginning of the integration, the values of da" and dft" will be obtained, so that we shall have sin i sin Ji = sin ig sin Slo-\' ^*"> sin i cos SI = sin tj cos S^o -|- Sfi", or sin i sin ( Si — SJo) = cos Sio '^«" — sin Slo 'W, sin i cos (Si — J^^) = sin i^ + sin J^o da" -f cos Slo^f^"> by means of which i and Si — Sio '"^y be found. In the case of ;f, let us put and we have e sin X, dr," de f " = e cos x, dx (226) (227) ^dr='''''^-dt-^''''^-'dr d:"^ dt de dx cosx-^-esmx^^ 534 THEOUETICAL ASTUONOMY. Substituting for and f tho values given by the equations (203) and (205), and reducing, wc obtain df^ 1 dt I — p cos (y -\- x) R •\- \(p -\- '■) sin {v -\- x) -l-cramxl^j, ' 1 / " = ,./ ,, , A p sin{v + /)]i-\- Kj) + r) cos (v + jr) 4- e7'co8/(^S'|, (228) by moans of wliich tl>o values of 3r/' and 3!^" may be found. Then ■sve shall have e sin x=^e„ sin r„ + ,^7)", 1 e cos / = fo *^o^ ''o "f ''»"> or e sin f ;^ — rr^) == cos r„ ^jj" — sin rr^ rJC", e cos ix — '^o) = ^0 + s'" ^0 ^v" -\- cos tTq K", (229) from which to find c and ;f. If, in order to find the variation of z, we write t: instead of ;f in these formula', the termrj + 2ccos7rsin'^ i* 7; and — 2e sin TT sin'Hi -,7- must be added to the second members of -^ dt (228), respectively. 201. By means of the four methods which we have develoiicd and illustrated, the special perturbations of a heavenly body may be de- termined with entire accuracy, and the choice of the j)articular method will depend on the circumstances of the case. By computing the perturbations of the elements, correcting these elements as often as may be required, the terms depending on the higher powers of the masses may be included, and no indirect calculation becomes necessary. The frequent correction of the elements will also render insensible the effect of whatever uncertainty remains in regard to their true values. But, since the perturbations of the elements are in general much greater than those of the co-ordinates, the effect of the terms of tlie second order will be much greater upon the values of the ele- ments than upon those of the co-ordinates. Hence, the frequency with which a change of the elements will be required will fully com- pensate the labor of the indirect part of the calculation in the case of the perturbations of the co-ordinates. VARIATIOX OF COXHTANTS. 530 The (Ictermlimtion of tlu; poi'tiirl)ation8 of the pohir eo-onliiiatort r, w, and ;, ami tluit of the |)orturl)iitioii.s d.]f, u, and dz„ arc t'tll-ctt'd with ahiiost etjual facility, espcoially when tin' eH'cct of the dishirl)- in}f forces is to he determined ibr a long interval of lime. If the perturhatioiiH are recjuired oidy for a hrief jteriotl, it will be preicr- ahle to determine r?J/, u, and (h, rather than die, rr, and 2, since tl»e indirect part of the calculation will thus bo effected with less repe- tition. In both of these awes the values of the perturbations are fienerally smaller than in the case of the rectanjjjniar co-ordinates, and hence they are less affected by terms of the second order; but on account of the simplicity of the fornudie, even when we include the terms depending on the higher powers of the masses, so long as the nuigniiude of the values of o.r, rii/, and dz is not so large as to render troublesome the indirect part of the calculation, the method of the variation of rectangular co-ordinates may be advantageously employed when the perturbations are to be determined for a long period. By whatever method the perturbations are determined, if the fun- damentsU osculating elements are correct, the final elements of the instantn 'cous orbit will be the same. But, since the effect of the error of *he elements will differ in degree in the different methods of treating the problem, if these elements are aflected with small errors, the agrcoiuent of the final os(!ulating elements obtained by the different iiiethods, in connection with the corrections derived by the conij)arison of observations, may not be complete. When the disturbed body approaches very near to a disturbing planet, the magnitude of the perturbations will be such as to enable us by means of accurate observations to correct the adopted value of the disturbing mass. In this case the perturbations, computed by means of either of the methods applicable, must be converted into the corresponding perturbations of the geocentric spherical co-ordi- nates. Let the variation of either of the geocenh'ie co-ordinates arising from the action of the disturbing planet be denoted by (W; then, if we suppose the correct value of the disturbing mass to be 1 + n times the assumed value used in con)i)uting (W, the correspond- ing variation of the geocentric spherical co-ordinate will be (1 + n) do. The value dd may be included in the determination of the difference between computation and observation in the formation of the equa- tions of condition for finding the corrections to be applied to the ele- 536 TIIEOKETICAI. ASTIIOXOMY. / rncnts; and, finally, the term nod may be added to each oi the equa- tions of condition, so that we thu.s introduce a new unknown quantity 71. The solution of ali the equations thus formed, by the method of least squai'j, will then furnish the most probable values of the cor- rections to be applied to tlie adopted elements, and also the value of 11, by means of which a corrected value of the mass of tlie disturbing body will be obtained. 202. If the determination of the perturbations of a heavenly body required that all the disturbing bodies in the system should be con- stantly considered, the labor would be very great. But, fortimatoly, it so happens that the nidsses of many of the planets arc so small in comparison with that of the sun, that the si>hcre of their disturbing influence is very much restricted. Thus, in the determination of the perturbations of the asteroid planets, only the action of Max's, Jupi- ter, and Saturn need be considered; and of these disturbing planets Ju]>iter exerts the principal influence. It is true, however, that, on account of the elongated form of the orbits of the periodic comets, tliey may at different times be sensibly disturbed by each of the planets of the system. But since in the remote parts of their orbits they are very distant from many of the disturbing planets, the deter- mination of their perturbations will thi snialJ. If, tlicrefore, we compute the porturbsitions of the motiou relative to tlie sun as far as to the point at which the second nienihers of (231) have not any appreciable influence on the results, it ^\■ill suffice simply to convert the elements which refer to the centre of the sun into those relative to the common centre of gravity of t!io sun c nd disturbing planet, and then to regard the motion as undis- turbrd until the comet again approaches so near that the direct per- turb tions must be considered, at which point the motion will again be referred to the centre of the sun. 203. The reduction of the elements from the centre of gravity of the sun to the common centre of gravity of tlie sun and the disturb- ing planet, may be easily effected by ineans of the variations of the rectiingular co-ordinates and of the corres])onding velocities. To derive the co-ordinates of the comet referred to the centre of gravity of the sun and planet, it is oidy necessary to add to the heliocentric co-ordinates the co-ordinates of the sun referred to this origin, so that, according to (230), we shall liave m' and, also. dx oy=z — in' 1 -f iii :v2/. m d^ 1 + m' ' dt ' dz d~ dij 'lit Sz m' . «~ m' dz' ''dt~'~ 1 + m' ' W 1-f-m' tz', (232) VI dj/_ 1 -f m' ' dt ' (233) If, therefore, from the elements of the orbit of the disturbing j^lnnot we compute the auxiliary constants for the adopted fundamental plane by means of the equations (94)i or (99),, and also V and U' from --=-- — (e sm u>' -f sm u) = V sm U , Vp' kVl + m' . , , »^ T7/ TT' ■ -,-= - (e cos u» -f- cos « ) = V cos U , Vp' the equations (100), and (49), in connection with (232) and (233), give m' Sx = 1 + m' r r' sin a' sin (A' -f «')> (234) PERTURBATIONS OF COMETS. 539 TO' dy = — Jfl-,- / sin h' sin {B' + ;/'), 52=-~ — T r' sin c'sin ( C" + v!) : 1 -f TO ^ I / ' (it m , ,--~Fsina'cosM'4-C7'), 1 4- m ^ ^ (234) m 1 + «i' >- F' sin 6' cos (B' + 6^'). 1 + m' rV sine' cos {C -{-U'), dx ^ (7;/ T ^ dz o~Tz' and r) ,. (W dt to the cor- If we add the values of dx, dy, dz, 3 , ■ 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 formuhe, by moans 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 })lanet, the resulting variations of the co-ordinates and velocities ivill be so small that their squares and products may be neglected, and, there- 540 THEORETICAL ASTRONOMY. fore, instead of using the complete formula) in finding the new ele- ments, it will suffice to employ diflerential formula;. The formulie (100), give dx 'di dy dt dz W ■ sm a sm {A -f- u) -% - -f- r sm a cos {A + «) -,t> : sin b sin (B -\- ii) -j- -{- r sin b cos (B -\- u) -j-, dv dv sin c sin ( (7 + ■**) -TT + »* sin e cos ( C + «) -t,- (235) If we multiply the first of these equations by Sx, the second by dy, dx and the third by dz; then multiply the first by ^ ---> the second by Q'V dz ^-jf' and the third by ^-^> and put dt (236) we P= sin a sin (^4 + u) Sx -f- sin b sin (B -f «) dy -\- sin c sin ( (7 -f u) Sz, Q = sin a cos (^ + «) ^x -f- sin 6 cos (B + «) ''^Z + sin c cos ( C + 'iO ^2 ; dv dv P' = sin a sin (u4 + -«) 3~-\- smb sin CJB + m) d-^ dz -f- sin c sin ( C -|- «) o ,^-, dx dv Q' = sin a cos (^ + «) ^-jj + sin 6 cos {B + «) "' ^ »■' sin v .„ , . . 5v :; ; — (2 + e cos v) 6e, a' cos

" « kV p ,. e sin ut -f- sin « _, , 7' cos u ,., 01 = li -+- — --- ii , /?', 645 (257) by means of which dQ and oi may be found. To find o(o and or we have du} = dx — cos i«5 ft , o;r = J;/ + 2 sin» \ io ft , (258) (ly being found from equation (249). Neglecting the mass of the comet as inappreciable in coini)arison with that of the sun, the attractive force which acts upon the comet in the case of the undisturbed motion relative to the sun is k^, but in the case of the motion relative to the common centre of gravity of the sun and jdanet th's force is Ir {1 -\- m'). Hence it follows that the increment of this force will be la'k^, and we shall have Sk = Im', (259) by means of which the value of this factor, which is required in the formula} for 3{y^p), 5 -> tfcc, may be found. 206. The formula) thus derived enable us to effect the required transformation of the elements. In the first place, we compute the dx , dy , ^ dz values of dx, oij, 8z, d dt' 'P ""^ 'm by means of the formuhe (234) ; then, by means of (236) and (250), we compute P, Q, It, P', Q', and W , the auxiliary constants a, A, &q. being determined in reference to the fundamental ])lane to which the co-ordinates are re- ferred. AVlien the fundamental ]>lane is the plane of the ecliptic, or that to which ft and i arc referred, we have sm c = sm ?, C=0. The algebraic signs of cos a, cos 6, and cose, as indicated by the equa- tions (101),, must be carefully attended to. The formuhe for the variations of the elements will then give the corrections to be ap})lied 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 TIIKOUKTICAL AST1{(»\<)MY. Sinco the equations liuve l)een derived for the viiriations of more than the six eU'inents usually eniph)yed, the acMitionul fonuuhe, as well as those which j^ive ditlereiit relations between the elements em- ployed, may be used to eheek the numerietil ealeulation; and this proof should not be omitted. It is obvious, also, that these differen- tial formulic will serve to convert the perturbations of the rectaufrular eo-ordinates into jx'rturbations of the elcnnents, whenever the terms of the second order may be neglected, observing that in this case t?/; = 0. If some of the elements considered are expressed in angular measure, and some in parts of other units, the quantity 8=^ 200204". 8 should be introduced, in the numerical application, so as to preserve the homogeneity of the formulse. AVhen 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 i' in order to reduce them again to the centre of the sun, will give the total ]u rtur- bations of the elements of the orbit relative to the sun duri the interval t' — t. It should be ob.served, however, that the value of oJ/ for the instant t should be rechiced to that for the instant t', so that the total variation of il/ during the interval t' — t will be In this manner, by considering the action of the several disturbing bodies separately, referring the motion of the comet to the common centre of gravity of the sun and })lauet whenever it may subsequently be regarded as undisturbed about this point, and again referring it to the centre of the sun when such an assumjition 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, i^ Mill at once appear that a very near approach of some of these bodies to a planet may often happen, and that when they ajjproach very near some of the large planets their orbits may be entirely changed. It is, indeed, certainly known that the orbits of comets liavc 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 fonnulaj applicable to this special case in which the ordinary methods of calculating perturbations cannot be applied. rERTITllBATlON.S OF COMKT8. 647 s of more nniilii', as lU'iil.s cin- and this ) (liffm'on- ictiuigular tlie tonus this case n anguhir :062G4".8 ) prt'sorvo )cd about imcnts for ity of the order to al ]»' rtnr- iiri tlie value of taut t', so be listurbiiig ) common •sequontly •ring it to • admissi- •e revohi- orbits of li of some ,'hen tliey s may be orbits of pitcr, and lally thus 3, to eon- 2 ordinary If we denote by .r', //', s', v', the co-ordinates and radius-vector of the ])huiet referred to the centre of the sun, and reiiard its motion rehitive to the sun as disturbed by the comet, we shall have (If + (I'll' d'l' + dh' df + P(l -^ m')z' «.fL;?-,^), (200) m mk Let us now denote by ^, r^, ^ th(! co-ordinates of the comet referred to the centre of gravity of the planet; then will y — y> : =r- 3 — z'. Substituting the resulting values of x' , y' , z' in tlio preceding equa- tions, and subtracting these from the corresponding equations (1) for the disturbed motion of the comet, we derive 7? df "^ ' f>' ~^ \ r'-' ? r A k' (m + m' )r,__,J][^ y'±-'i\ "df + ?' ^ " \ >" r'' /' (2G1) d'" dt 1 + k' (m 4- m") % ^'\?^ ,,3 )• These ecpiations express the motion of the comet relative to the centre of gravity of the disturl)ing planet; and when the comet approacb.es very near to the planet, so that the second mend)er of each of these equations becomes very small in comparison with the second term of the first member, we may take, ibr a first approximation, df'^ (>' --^' iVt) li- (m + '»;.') t; __ rf? "^ f>' ~ "' dK />■' (m + m') : _ (262) 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 TIIEORETirAL ASTIIOXOMY. ( tlioMc c(|iirttions, bcinjx of the? saiiic form as those for tlio undisturlxd iimtioii of lilt! comet relative to tlu! sun, show that when tlie aetimi of the (listiu'hini^ planet on tlio comet exceeds that of tlie sun, the result of the first approxinmtion to the motion of the comet is that it (IcscribcH a conic section around the centre of f:;ravity of the phmet. Further, since — x', — y', — z' are the co-ordinates of the sun re- ferred to the centre of jijravity of the phmet, it apj)ears that tlic second memhers of (201) express the disturbing force of the sun (ui the comet resolved in dire— ^^ )' and when the comet api)roaclies very near the planet this force will he extremely small. It is evident, further, that the action of the sun regarded as the disturbing body will be very small even when its direct action u])on the comet considerably exceeds that of the planet, and, therefore, that we may consider the orbit of tlie 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 })lanet, let us suppose the sun, planet, and comet to be situated on the same straight line, in which case the disturbing force of the sun will be a maximum for a given distance of the comet from the planet. Then will the direct action of the sun be -^, and that of the planet — j-* The disturbing action of the sun will be PERTrKHATIOXS OF COMKTS. k'n 2r -h p 548 {r-\-f>)' r' (/• + /'/ wlik'li, since f> is suppu.scd to bo small in loinpHrisDu with v, may ba put I'quiil to 2k'p and Iicnt'c the ratio of the di.stnrbin}; action of the sun to the direct action of ti»e planet on the ('()mct cannot exceed /? = in f 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^ = «i'»*', and the ratio of the direct action of the siui to its disturbing action cannot in this case exceed 2]^vi'. In the ease of Jupiter this amounts to only 0.06. So long as ft is small, the disturbing action of the planet is very m'k^ nearly - ,^ - in all positions of the comet relative to tlie planet, and hence the ratio of the disturbing action of the planet to the direct action of the sun cannot exceed B' m r At the point for which the value of p corresponds to RR', the coinct, 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 () less than this R will be less than R', and the i)lanct must be regarded as the con- trolling and the sun as the disturbing body. The sui)positi()n that R is equal to R' gives 2,o« mV ?»'?•' and therefore 5 / 1 In p = rv \m , (203) Hence we may compute the perturbations of the comet, regarding the planet as the disturbing body, until it aj)proaches so near the ooO TII i: e small even when// is greater than ri Ini'-, Henee we niav eommenee to consider the sun as tlui disturbing body even before the comet reaches the point ibr which and, sine(> th(> ratio of the distm'bing a<'tion of the planet to the direct action of the sun remains nearly the same tor all values ot' t's wiien !> is within the limits liero assigned tli( sun must in all e;i-ts be so consider(>(l. (Vorresponding to the value of (> given by ecpiatioii (20']), we have and in the case of a near a])proach to Ju{)iter th(! results are ,>=--^)M\r, i?':^. 0.:i3. 209. In tlie actual calculation of the perturbations of any particu- lar comet when very near a large ]>lanet, it will be easy to determine the point at wliicli it will lie advantageous (o commence to regard the sun as the disturbing body; and, having i'oun.l the elementt' of the orbit of the cotnet relative to tlie planet, tin- perturitations of these elements or of the co-ordinates will b" obtaii ed by means ( I' the Ibrmulic already derived, the aoeessary ilistinctions being made in tlu' nottition. When the ])lanet agi.in liecotnes the disturbing body, the elements will be found in reference to the sun; and thus we an enal.>led to t>'aee the motion of the comet before and subscMjuei.' to it.- being considered as subject principally to the ))lanet. In the case of the first trai^sformation, the co-ordinates and \eloeities of the comet and plaiK't in reference to tlu> sun being determined for' the ir taut at uhieh the sun is regarded as ceasing to be the controlling body, we shall have i>Er;TT:ni'..\T[ONs of comets. 551 wlncli, so 1 mu>t lie 11(1 COllU't, ;iuct near r_r- X ■ Z=^z- df dx dt dx' dt' dr, '/,'/ i/' 'inu ,, ' the cIcMncnts oi tlie orl)it of tlio ' • ' dt dt (It coinct about the j)Iiinot are to he (leterniined precisely a.s the elements ill reference to the sun are ibuiul from ,r, (/, ,:, " » ' . and ," . and ■ (It (It dt as explained in ^Vrt. 108. Jlasing eomjinted the perturbations of the motion I'elative to the planet to the [loint at wlii' h the planet is UL!,ain considered as the disturbing l)ody, it only remains to llnd, titr the eorrespoudint!; time, the co-ordinates and velocities of the comet in reterence to the centre of and ' will be obtained directly, and then, having found the corresponding co-ordinates x', y', z' and velocities dx' dy' dz' „ ^ , . ,, , , , > -.; ) -.7- 01 the planet ni reterence to tlie sun, we have ly particu- d( teniiiiie r(\ii'ar(l the •Ml.-- Ill" till' |l-. (it llioe liiis ( I' the liade ill thi' body, the Ins we ai' iuei:' til it- he ease ut |*tlie comet ii' taut ai |; body, \\r x — x + i;, dx rfj;' ._ d? lit "' (It "'' "(It' dti dt dl, dri dt ~^' dt ' 3 =- 2' + ?, dz _ dz' d: (It^ dt'^df l)v means of which the elements of the orliit relative to the sun will he found. If it is not considered necessary to compute rigorously tli^' path of the comet liefore and after it is subject principally (o tin action of the {)laiiet, but simply to llnd the ])rincipal effect of tin I artioii of the planet in changing its eh'inents, it will be suilicicnt, iluriiig the time in which the sun is regarded as the disturbing body, to ^u])pose the comet to move in an undisturl)ed orbit abinit the planet. For the point at which we cease to regard the sun a^ the ilisturbinsr body, the co-ordinates and velocities of the comet relative to the centre of gravity of the planet will be determined from the I'leaients of the orbit in reference to the ])lanet, pri'cisely as the corre- sponding ([(lantities are determined in the case of the motion relative to the sun, the necessary distinctions being made in tiie notation. 552 THEORETICAL ASTRONOMY. 210. The results obtained from the observations of tlic periodic eoniets at their sueeessive returns to the })criholion, render it probable that there exists in s})ace a resisting medium which opposes the motion of all the lieavcnly 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 iluid is so slight that it can have an appreciable etlect only in the case of rare and attennatcd 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 chietly 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 srpiare of the velocity. The density of the fluid may be assuin;xl to diminish as the distance from the Sim increases, and hence it may be expressed as a function oi' the reciprocal of this distance. I^ot (Is he the element of the path of the body, and r the I'adius- vector; then will the resistance be T- Ml)%' (2(34) K being a constant quantity depending on the nature of the body, and ell the density of the ethereal fluid at the distance r. Since the force acts only in the plane of the orbit, the elements which de- fine fliC position of this })lane will not be changed, and hence we have oidv to determine the variations of the elements Jf, c, a, and y. If Ave denote by ^''y the angle which the tangent makes with the prolon- gation of the radius- vector, the components 11 and 8 will be given by and, since Fogs 4'^ ■ we have li = T cos (,'-0 - r. sni V, S: F=inv''o Tsin^'v kVp _.. — , r ds B-=-K,{l) e sm r \ p dt' *=->^(l)^'. rf8 dt' (205) RESISTING MEDIUJI IN SPACE. 553 Substituting tlicso values of 7^ and S in the equation (205), it reduces to edx =^ — 2AV ! I .sin v ds. Now, since we have F=^4 ^ + 2e cos i» + e')~, Vp ds = Vdt =: — (1 + 2c COS u + e'y-dv, and hence e4 = P (i)-ni + 2e cos V + e^ - sin v dv. (2G6) If we supp'jr^e the function m^ Ki' V of which is always positive, to be developed in a scries arranged in reference to the cosines of v and of its multiples, so that we have A> (-)>•'(! + 2e cos v + e'f ^ A -{- B cos v -\- C cos 2v -\- &c., (267) in wliich A, B, Sic. are positive and functions of c, the equation (266) becomes 2 edx ^^ ( A -\- B cos V + • • • .) sin v dv. Hence, by integrating, wo derive 2 eS-/v=— {A cos V -f- \ Bcos,2v -r . . . .)> (268) from which it appears tliat y is subject only to periodic perturbations on account of tlie resisting nu-Alntn. In it <4milar manner it may be nli^wn that the second term of the second member of equation (210j profluces only jKirifxlic terms in the value (»f fiM, so that if we seek only the s«'ular jn-rturbations due to the action of the ethereal fluid, the fifjit and wn.'ond t«>rms of tlie second m»»mber of (210) will not r>e considered, and only the soculiu' perturbatio**' arising from tlw variation (.A // will bi- re(|uinHl. Let us litiext consider the Acmuuia a »ad e. Substituting in the 554 THEORETICAL ASTRONOMY. equations (198) and (202) the values of i? and /S' given by (265), and reducing, we get da de = — ^V^( i ) rHl + 2e cos v + e") ^dv, 2 / 1 \ .' = K(p I — I r' (1 + 2e cos u -|- e°) ? (e + cos v) dv. (269) If we introduce into these the series (2G7), and integrate, it will he found that, in addition to the periodic terms, the expressions for oa and oc contain each a term multiplied by v, and hence increasing with the time. It is to be observed, furtliei', that since A and B are posi- tive, the secular variation of «, and also that of c, will be negative, and hence the resisting medium acts continuously to diminish both the mean distance and the eccentricity. 211. The magnitude of the disturbing force ai'ising from the action of the resisting medium is so small that the periodic terms have no sensible influence on the place of the comet during the period in which it may be observed; and hence, since the effect of the resist- ance will be exhibited only by a comparison of observations made at its successive returns to the perihelion, the effect of the planetary per- turbations being first completely eliminated, it is only necessary to consider the secular variations. Further, since "^ is subject only to periodic changes in virtue of the action of the resistance, and since the mean longitude is subjected to a secular change only through n, it will suffice to employ the formula; for d^ and de or 8ip. The variations of these elements may be computed most conveniently by mechanical quadrature from given values of -; and --,, or ~y , al- though their values for one complete revolution of the comet may he determined directly, the values of the coefficients A and B wliich ap])ear in the series (267) being found by means of elliptic fiuictKnis. The calculation of the effect of the resisting medium will be n\ade in connection with the determination of the planetary perturlwtions. so that there will be no inconvenience in adding to the results ^'^e tjc* m^ depending on this resistance. Since dji It 3 m 2 a da It' d(p W dc the equations (269) give, putting K— h^U, EESISTING MEI)IUJ[ IN SPACE. 555 'dt dtp ~dX r cos

the expression for the action of the ethereal tluid be- comes T^-lVy,, 556 THEORETICAL ASTRONOMY. Since tlio constant L^'dopencls on the nature of the comet, the value obtained in tlie ease of Encke's coniot may be very different from that in tlie case of another comet. Thus, in the ease of Faye's comet the value has been found to be ?7 = 1 . 10.232' and in tlie aj)plication of the formuhc to the motion of any particular body it will be necessary to nudce an independent determination of this constant. 212. The assum])tion that the density of the ethereal fluid varies inversely as the scpiarc of the distance from the sun, is that which appears to be the most probable, and the results obtained in accoi'd- ancc therewith seem to satisfy the data furnished by observation. It is true, however, that the whole subject is involved in great uncer- tainty as regards the nature of the resisting medium, so that the results obtained by means of any assumed law of density arc not to be regarded as absolutely correct. From the formuhc which have been given, it appears that, whatever may be the law of the density of the resisting fluid, the mean motion is constantly accelerated and the eccentricity diminished, and we may determine, by means of observations at the successive appearances of the comet, the amount of these secular changes independently of any assumi)tion in regard to the density of the ether. ]^ct x denote the variation of // during the interval r, which may be approximately the time of one revolution of the comet, and let y denote the correspond- ing variation of (p; then, after the lapse of anv interval t — 7^, we shall have .t-To P = V'o 4- t-i: 'y> (272) ami, since the average variation of // during the interval t — 1[, is t~Z ¥-^^. i»/=j»4 + /x„(<-!r„) + (t-ny (273j If we introduce x and y as unknown quantities in the equations of condition for the correction of the elements by means of the ditll r- cnoes between c- niputation and observation, the secular variations of fi. and ^ may be determined in connection with the corrections to h( RESISTIXG MEDIUM IX SPACE. 557 applied to the olenients. For tliis purpose the partial dift'erential co- efficients of the geocentric sj)herical co-ordinates with respect to x and 7/ must be determined. Thus, if we substitute the values of ft, (f, and Jf given by (272) and (27;3) in the equations (12).j and (14)^,, we obtain -J- = a tan ^ sui v — ^ -' -,3— dx 2t d/j. t-T„ 's, dv a' cos

cty \ cos ^ / T dy in which s = 20G2G4".8, fi being expressed in seconds of arc. Com- binintr the results thus obtained with the differential coefficients of the geocentric spherical co-ordinates with respect to /• and r, as indi- cated by the equations (42).2, we obtain the recpiired coeiHcients 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 >/ and x, or of the secular variations of the eccentricity and mean motion, without any assump- tion being made in reference either to 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 3T, fi, and (S for the subsequent returns of the comet to the perihelion. In all the cases in which the periodic comets have been observed sufficiently, the existence of these secular changes of the elements seems to be well established; and if we grant that they arise from the resistance of an ethereal fluid, the total obliteration of our solar system is to be the final result. The fact that no such inequalities have yet been detc>cted in the case of the motion of any of the planets, shows simply the immensity of the period which must elapse before tlie final catastrophe, and does not render it any the less certain. Huch, indeed, appear to be the present indications of science in re- gard to this important question ; but it is by no means impossible that, as in at least one similar case already, the operation of the simple and unique law of gravitation will alone completely explain these inequalities, and assign a limit which they can never pass, and thus afford a sublime proof of the provident care of the Omxii'otkxt CUE.VTOK. TABLES. 659 18 HI 20 '^1 I •i-i I 23 ( 30 33 34 35 36 TABLE I. Angle of the Vertical and Logarithm of the Earth's Radius. 1 Arijiiriii'iit -t:= f!f'Ofrr:i|iliiciil I,;iiitii(li'. { 'iiiiiiirts>mii l!!t',l.l5 1) (1 1 •i II :< II 1 (1 5 (i 4 II H II » II 10 (1 11 II \i II \.i II 11 II 15 l<> 17 IH 1!) 20 n •il •i'i II 33 21 a.) 'H\ 27 II 28 (1 'id 30 (1 III L'O :;i) ■lU aO 31 10 L'O :;(! 40 50 32 II 111 20 :iii ■(0 .00 33 (1 10 20 :0 40 oO U 10 20 ;)o 40 50 35 0-*' 9 9 10 lO 10 10 10 o.oo 24.02 4S.02 I195 35. So 59-54 23.12 46.54 9.76 32.74; 5547! 17.921 4 40.06 5 ■•S5 5 ^V^^, 5 44-33 6 4.95: 6 25.14, 44.861 4.09 22.80 40.99 58.61 15.66 32.10 47-93 3.12 17.65 31.50 44.66 1 57-»ij 59.12! I.I I 3.07 5.02 6.94 8.85 IO-73 12.59 »4-44 16.26 18.06 19.84 21.60 *3-34 25.05 26.75 Z8.43 30.08 31.71 33.32 34.91 36.48 38.03 39-55 ,}.i.o6 42.54 44.00 45-44 46.86 10 48.25 iiiir. 24.02 24.00 13-93 ^3-^5 23-74 23.5S 23.42 23.22 22.98 22.73 22.45 22.14 21.79 21.43 21.05 20.62 20. 1 9 19.72 9.23 8.7, 8.19 7.62 7-05 6-44 5.83 5.19 4-53 3.85 3.16 2.46 2.00 1-99 1.96 1.95 1.92 1. 9 1 1.88 1.86 1.85 1.82 1.80 ..78 1.76 '•74 1.71 1.70 1.68 1.65 1.63 1.6 1 '-59 '-57 1-55 1.52 1. 51 1.48 1.46 1.44 1.42 '•39 h'lrp Dill. 9.999 9.999 0.000 0000 9.999 9996 99S2i 996 1 9930, 9891! 9X4 3 1 9786' 9721. 9648 1 9^661 9476 9377 9-7': 9'57 9°35; 89051 8768; 8624I 8472! 8314: 8149 7977 7799 7614 7424I 7228: 7027 6820' 6608 i 9-999 6392' 6355 6319; 6287. 6245: 6208 9.999 9.999 9.999 9.999 61 7 1 6134 6096 6059 6021 5984 5946 5908 5X70 5«3i 5794 5755 5717 56-8 5f . - 5(01 55"- 55'-' S4f.^ 54--J 54<'6 5317 5327 5288 9-999 5*48 9-999 9-999 4 '4 21 3' 39 48 57 <'5 73 82 90 99 106 "4 122 130 '37 '44 152 158 165 172 178 185 190 196 201 207 212 216 37 36 37 >7 37 37 37 3** 37 3« 37 38 3S 3X 38 38 39 38 39 ;8 :<') 3S 39 39 39 39 40 39 40 39 Ml I'll .':o III r.ii :iG 11 10 20 ,'SII 1(1 ,00 37 10 20 ;io 40 611 38 10 20 :!ii III 50 39 (I 10 20 ;io 40 50 40 II III 20 30 40 50 41 111 20 ;io 40 50 42 10 20 80 40 50 43 10 20 ;io 40 50 44 10 20 :!0 40 50 45 -/>-0' 10 10 1 1 II II II 48.25' 49.63 50.98 52.31 53.62 54.90 56.16 57-4', 58-63; 59-821 I.OOj 2.15 3.28 i 4-39| 5-47 6.54! 7.58 8-59: 9-59' 10.56 n.51 12.44 '3-34 14.22 15.08 15.92 16.73 17.52 18.29 19.04 19.76 20.46 21.13 21.79 22.42 23.02 23.61 24.17 24.70 25.22 25-7' 26.18 26.62 27.04 27.44 27.S2 28.17 28.50 28.80 29.08 29-34 29.58 29.79 29.98 30.14 30.29 30.41 30.50 30.57: 30.62 3°-65 Hilt. 1. 38 '-35 '•33 '•)' 1.28 1.26 1.25 1.22 1. 19 1. 18 1. 15 '•'3 I.I I 1.08 1.07 1.04 1. 01 1. 00 0.97 0.95 0.93 0.90 0.88 0.86 0.84 o.Si 0.79 0.77 0.75 0.72 0.70 0.67 0.66 0.63 0.60 C.59 0.56 0.53 0.52 0.49 0.47 0.44 0.42 0.40 0.38 0.35 0-33 0.30 0.28 0.26 0.24 0.21 0.19 0.16 0.15 0.12 0.09 0.07 0.05 0.03 lot^p 9-999 9.999 9.999 9.999 9.999 5248 5208 5169 5129 5081) 5049 5009 4909 4929 488S 4848 48071 4767; 4726 4686I 4645 4604' 4563 4522! 4481 4440 1 4399 4358 43'7| 4276! 4234! 4' 93 4152 4110 4069! i 4027! 3985 39441 3902. 3860 3819 9-999 3777, 3735: 3693' 36511 3609' 35671 9-999 3525 3483 3441 1 33991 3357! 33'5j 3273' 3230I 3188I 3'46' 3104: 3062 3019 2977' 2935^ 28921 28501 2808! 9.999 2766! 9.999 9-999 9.999 DIfr. 40 39 40 40 40 40 40 40 4' 40 4> 40 4' 40 4« 4« 4' 41 4« 41 4' 4' 41 4« 42 4' 4' 42 4' 42 42 4' 42 42 41 42 42 42 42 42 42 4*. 4» 42 42 42 42 42 43 42 42 42 42 43 42 42 43 42 42 42 661 IMAGE EVALUATION TEST TARGET (MT-S) fe .^/ ■■'■\^ ^. «v. 'ii.' V % /, ^ = y= 1.25 '• 110 I ITS 22 2.0 U 11 1.6 i^ /a Vl w Hiotographic Sciences Corporation as t'/ESV MAIN STREIT WEBSTER, N.Y. 145S0 (716) S73-4S03 V iV :1>' ;\ \ ^ ^ [V 6^ <6^ TABLE I. Angle of the Vertical and Logarithm of the Earth's Radius. e'--:r, (iiHMH'iitrir Latitixlo. /) - Kurth'H Itiuliud. r J '' t J. I i * 1 ♦ -♦' Uiir. l<'«P iMir. « ♦ -*' j)iir. tf 1 i-l" l"ttp iiifr. o 1 1 11) ' ft II 30.6? If 0.00 9.999 2766 2723 i 43 55 II III 10 49.7A 48.36 9-999 0*75 0235 40 2(1 0.01 0.05 2681 2639 4» 41 20 .'ill 46.97 45-51 , '-39 1.42 0195 0155 4^ ( 4' ID 3o.,-i 30.42 0.07 0.09 ' 0.1 1 2596 »S54 43 4» 41 III 50 44.11 42.65 1.4A 1.46 1.49 0116 0076 ! 39 4' i 3') 40 II 11 30.31 0.4 0.10 9.999 2512 ! 4* 43 , 42 42 43 4» 50 10 41.16 9.999 0037 III 30.17 2470 III 39.65 1. 51 9.998 9998 39 '20 ;i(i III 30.01 li;.tf2 iy.61 0.19 0.21 0.2, 1 0.26 2427 13X5 »143 211 .'111 to 38.11 36.5^ 35.01 1. 52 '•55 '•57 9958 99 '9 9880 40 39 11' 6(1 29. 3 X 2300 40 334' !•" 1.61 9841 39 39 ' 47 II 2C).I2 9.999 2258 57 10 31.80 1.64 1.66 1.67 9.998 9802 111 211 iX.S? 28.54 0.27 0.31 2216 2174 4* 4» 10 20 30.16 28.50 9764 97*5 31* 39 .■(0 i8.ia o*3nt infervniM :;f Sidcrea! Time. I«KP urn. 9.998 ,999 027 V 023s 0195 o»55 0116 0076 .999 o°?7 1.998 9998 995*1 9880I 9841 1 ^.998 9802 9764 91*5! 96861 9648 i 96101 957« 9533 9495 9457 9419 938*1 9.998 9344, 9307; 9169I 9»3»1 9>95i 91581 9.998 91 21 1 8902 8688 8479 8275 8077, 9.998 7884' 7697 75«7 734»| 7174 7013 9.998 6859' 6713 6573 6441 6317 5201 9.998 6093, 5993 5901 i 5818! 5676^ 9,998 5619 557° 553° 549^ 5476 54<^'J, 9.998 5458 40 4^ 4= 39 45 3') V) 40 3') 3'* 3') V) 3^ 3') 3') 3* 3^ 39 3'^ 3" 3* 3!< V 3^ 37 3* 37 37 37 37 211) J14 loq 204 198 '93 18: iSo \bl 161 '54 146 140 '3- 116 100 f, 75 6' 57 49 40 3» 11 '3 5 lluun. M'-nii T. i>lal Tiiii<-. h I 2 3 4 7 8 9 10 1 1 12 "3 '4 '5 16 >7 18 >9 20 21 22 »3 »4 7 8 9 10 II 12 «3 '4 «5 16 •7 18 >9 20 21 22 13 »4 9.856 19.713 29.569 39.426 49.282 59«39 8.995 18.852 28.708 38.5f'5 48.421 58.278 8134 17.991 27.847 37-704 47.560 57.416 7-»73 17.129 26.986 36.842 46.699 56-555 .S r. 'J- St S .5 o •25 = ^ i I S - g <. ti o .S CI J -it 4< V — £ r S 5 I a ^ 1 IS » + MiMiii T. 2 3 4 5 6 7 8 9 10 II 12 «3 '4 >5 16 17 18 >9 20 21 22 23 »4 II 17 28 29 30 3' 3» 33 34 II ]l 39 40 41 4» 43 44 46 47 48 49 50 5« ;a 53 54 II H 11 Miiiutt>ii. »< »-uuibi. UvciuiaU. 8lderml Time Uiaii T. 8iil(>rpiil TInio Mean T. 8('5o 8 1.34 9 '-478 7 8 7,019 8.022 O.IA 0.16 O.IAO 0.160 9 9,025 0.18 o.)8o 10 1.643 10 10.027 0,20 20| II 1.807 II 11.030 0.22 0.221 11 1.971 12 12.033 0.24 0.241 13 2.136 "3 13.036 0.26 261 14 2.300 '4 14-038 0.18 0.281 1 15 1-46-4 I 16 2.628 »5 15.041 16.044 0.30 0.301 16 0.32 0.321 17 2,793 «7 17.047 18.049 O.3A 0.36 0-3.41 0.361 18 2.957 18 19 3.121 >9 19.052 0.38 0.381 ! 20 3.285 20 20.055 0.40 0.401 , »« 3-45° 21 21.057 0.42 0.421 , »» 3-6I4 22 22.060 0-44 0.46 O.4AI 0.461 i 13 3-778 13 23.063 24.066 H 3-943 14 0.48 0.481 25 4.107 25 25.06' 26.071 0.50 0.501 1 26 4-271 26 0.52 0.521 17 4-435 1 28 4 600 17 27.074 0.5 A 0.56 0-541 0.562 28 28.077 I 29 4-76.^ 30 4.928 19 29.079 0.58 0.582 0.602 30 30.082 0.60 31 5x92 3' 31.085 0.62 0.622 i 31 5-157 31 32.088 0.64 o.6a2 0.662 1 33 5.421 33 33.090 0.66 1 34 5-585 34 34-093 0.68 0.682 35 5-750 35 35.096 0.70 C.702 36 5.914 36 36.099 0.72 0.722 37 6.078 37 37,101 0-74 0.76 0-741 0.762 ; 38 6,242 38 38.104 39 6.407 39 39.107 0.78 0.782 40 6.571 40 40. 1 1 0.80 0.802 41 6.735 4' 41.112 0.82 0,822 42 6,899 41 42.115 0.84 0.842 43 7-064 t3 43.118 086 O.S62 , 44 7,228 44 44,120 0.88 0.882 ; 45 7-39* : 46 7-557 Jl 45-'i3 46.126 0.90 0.92 0.902 0.923 47 7-71' 47 47.129 0.94 o.v^3 0.963 i 48 7-885 48 48.131 0.96 i 49 8.049 49 49 '34 0.98 0.983 1 50 8.214 ; 5' 8.378 50 50.137 1. 00 1.003 5> 51.140 52 8.542 51 52.142 1 53 8-707 53 53-145 54.148 1 54 8.871 t 54 ; 55 9-035 55 S5»5' 56.153 S7-'50 1 56 9 "99 56 57 9-364 58 9.52S 57 58 58-159 59 9-691 60 9.856 f,2 59.162 60.164 jua TABLE m. For convertinf{ intcrvalfi of Sulcrcal Tiiiu! into «-(iuivalcnt intervals of Mean Solar Time. 1 — - ■ - - m ■' ■ Hour Iloure. Miiiutva. 1 Socdiiila. 1 DcciuiaU. 1 I >> 4 j 81.1. T. Mean Time, «ia. T. Mmiii TIiiip. Sid. T. Mean Tiino. Sl.l. T. i Meuu Time. h Am 1 m m 1 « f ( I 59 50.170 I 59.836 I 0-997 0.02 0.020 H •> 2 I 59 40.341 2 I 59.672 2 1.995 0.04 0.06 O.OAO 0.000 H 3 2 59 30.511 3 2 59.509 3 2.991 ^1 4 3 59 20.6S1 4 3 59-345 4 3.989 0.08 0.080 H 7 1 4 59 lo-^S* 1 4 59 «X« I 4.9X6 0.10 0.100 H K 5 59 1.023 5 59.017 5.9X4 0.1 2 0.120 ■ i) 7 6 58 51.193 7 6 58.853 7 6.981 0.14 0.16 0.140 I 10 11 8 7 58 4«-3<'3 ? 7 58.6X9 8 7-978 0.160 H 9 « S« 3<-534 9 8 58.516 9 8.975 0.18 0.180 i H 12 lO 9 5*^ 2«-704 10 9 5X362 10 9-973 0.20 0.199 ^1 1:) II 10 58 11.875 II 10 5X.198 II 10.970 0.22 0.219 ^^ 12 II 58 2.045 12 II 58.034 12 11.967 0.24 0.239 H 14 l.'> IK 17 IS «3 12 57 52.216 «3 12 57.X70 •3 12.964 0.26 0.259 ! H 14 13 57 42.3X6 »4 13 57.706 >4 13.962 0.28 0.279 H •S '4 57 3»-557 ;^ «4 57543 '5 14.959 0.30 0.299 ^1 i6 15 57 22.727 «5 57-379 16 15-956 0.32 0.319 ^^ >7 16 5-' 12.X97 17 16 57.215 'Z 16.954 O.3A 0.36 0-339 ^^ li» 20 21 l8 17 57 3.068 18 17 57-051 18 17.951 0.359 H >9 18 c6 53.238 »9 18 56.887 '9 1X.948 0.38 0.379 ^^1 20 19 56 43.409 20 19 56.723 20 19-945 0.40 0.399 ^1 22 21 20 56 33-579 21 20 56.560 21 20.943 0.4.2 0.419 ; ^1 •»:i 22 21 56 ■13 750 2^ 21 56.396 22 21.940 0-44 0.46 0.439 ^1 24 13 22 56 13.910 »3 22 56.232 23 56.068 24 55.904 *3 22.937 0.459 ^1 24 23 56 4.091 *4 *4 11 23.934 24.932 0.48 0.50 0.479 0.499 I 26 25 55.740 25.929 0.52 0.519 ^H ti 2*§ *6 55-577 *7 16.926 0.5A 0.56 0.539 ^1 »7 55-4' 3 28 27.924 0.558 ^H \ -t g 19 28 55.2A9 29 55.0X5 30 54.911 29 iX.921 o.<8 0.60 0.62 0.578 ^1 1 2 i 30 3» 30 3« 29.918 30.915 0.59X 0.618 ■u B "2 3» 31 54.758 32 31.913 0.64 0.66 0.638 ^1 3 'S , 33 32 54.594 33 32.910 0.65X ^1 t 1 =?a 34 33 54-430 34 33-907 0.68 0.67X ^H 2l2 5^5 1^ 34 S4-if'6 35 34.904 0.70 0.69X ^H 35 S4-«02 36 35.902 0.72 0.71X ^1 37 36 53.938 37 36.899 0-74 0.76 0.7 3X H § « :? 3« 37 53-775 3X 37.X96 0.758 ^H S J3 X a — fi ^ rt ?i 39 38 53.611 39 3X.X94 0.78 0.778 ^1 i 40 39 53 447 40 53.283 40 39-S2i 0.80 0.798 ^H < M? 4> 4« 40.X88 0.82 0.818 ^H 4a 41 53.119 42 41.885 0.84 0.83X ^1 i = - 43 42 52.955 43 42.883 086 0.858 ^1 £ g 2 44 43 5*-79» 44 43.8X0 0.88 0.87X ^1 & a s 45 44 52.628 45 44.X77 0.90 0.89X ^H 1 il-3 46 45 5»-464 46 45-874 0.92 0.917 ^H 47 46 52.300 47 46.872 0.94 0.937 ^H s|i 48 47 5»->36 48 47.869 0.96 0.957 H b s S 49 48 51.972 49 S'-«09 49 48.866 0.98 0.977 ^1 t SO 50 49.863 1. 00 0.997 ^H I S « T S> 50 51.645 SI 50.861 ^H s a + Sa 51 51.481 s» 51.858 1 ^H s -^ 53 5» 5«-3>7 53 52.855 ^H s a 54 53 S"-iS3 54 53-853 ^1 2 ■ 11 54 50990 \l 54.850 ^1 S a m S 55 50.826 56 50.662 55-847 ^H li 57 56.844 ^H i 5S 57 50-498 58 57.842 ^H II 58 50.334 it 58-839 ^H 59 50>7o 59.836 I 604 can Solar Time. TABLE IV. For convortinR Ilourst, Miniite.'«, and Sccondft into Dcoimals of a Day. Deciinali. - .Menu Time. , ! O.OIO O.OJ.0 o.ooo 0.080 O.I 00 0.120 0.140 0.160 0.180 ' 0.199 0.119 0.239 0.2S9 : 0.279 0.299 0.319 0.339 0.359 0.379 0.399 0.419 1 0.439 0.459 0.479 0.499 0.519 0.539 0.558 0.578 0.598 0.618 0.638 0.658 0.678 0.698 0.718 0.738 0.758 0.778 0.798 0.818 0.838 0.858 0.878 0.898 0.917 0.937 ' 0.957 0.977 0.997 1 lloiirn. 1 Decimal. .Min. Dttiuinl. Mill. 81 Dtcimal. S.T. Drciinal. 8<-c. Dccinml. 1 0.0416 4- 1 .000694 4- .021527 + 1 .00001 16 31 .00035S8 .0833 + 2 .001388 -| 82 .022222 -)• 2 .0000231 32 .0003704 » .1250 + 8 .002083 f- 88 .022916 i 3 .0000347 33 .0003819 4 .1666 4- 4 .002777 ■■}- 84 .023611 I 4 .0000463 34 .0003935 5 .2083 -i- 5 .003472 -;- 85 .024305 -f 5 .0000579 85 .0004051 « .2500 + « .004166 -] 80 .025000 -|- .0000694 84i .C004167 i 0.2916 -f 7 .004861 r 87 .025694-:- 7 .0000810 37 .0004282 H •3333 ^- H .005555 1- 8H .026388 •- H .0000925 38 .0004398 » .3750-t- » .006250 1 80 .02708^ •• .0001042 30 .0004514 U) .41 6 J t- 10 .006944 4- 40 .02-777-4- 10 .0001 1 57 40 .0004630 11 •45J'3-|- 11 .00-658 f 41 .028472-1- 11 .0001273 41 .0004745 1-2 . 5000 |- 12 •008333 -i- 42 .029166 ■ 12 .0001389 42 .0004861 18 0.5416-)- 18 .009027 -j- 48 .029861 -1 18 .0001505 43 .0004977 14 •5833 + 14 .009722 4 44 •030555 -r 14 .0001620 44 .0005093 1.1 .6150 ;- ir» .010416 -j 45 .031250 -;- 15 .0001736 45 .0005208 in .66664- l(i .OIIIII 4 40 •031944 + 10 .0001852 40 .0005324 17 .7083 + 17 .011805 • 47 .032638 i- 17 .0001968 47 .0005440 is .7500 -f IN .012500 -(- 48 •033333 + 18 .0002083 48 .0005556 1ft 0.7916 4- ID .013194-.- 40 .034027-;- 10 .0002199 40 .0005671 H> ■8333 t- 20 .013X88-:- 50 .034-'22-|- 20 .000231 5 50 .0005787 21 .8750 1- 21 .014583 1- 51 .035416 i- 21 .0002431 51 .0005903 4>i .9166 + 22 •015277 + 52 .036111 : - 22 .0002546 52 00060 1 9 2:1 0.9583 -f- 2» .015972 : 58 .036805 -l- 28 .0002662 53 .00061 34 24 I.OOOO -\- 24 .016666-1- 54 .037500 i- 24 .0002778 54 .0006250 25 .017361 4- 55 .0381944- 25 .0002894 55 .0006366 1 2« .018055 -f 50 .038888 -i 20 .0003009 50 .0006481 27 .018750 -i 57 .039583 -J- 27 .0003125 57 .0006597 2S •019444 + 5") .040277 -f 28 .0003241 58 .000671 3 20 .020138 -1- 50 .040972 -\- 20 .0003356 50 .0006829 ' 80 .020833 ■■• 00 .041666 i- 30 .0003472 (to .0006944 Thu sign -f-, appondiHl to numbiTs in tliis tiible. Kignillpt thiit the Iu8l flgur' ropoatit to intinity. TABLE V. For finding the number of Days from the beginning of the Year. Date. Com. Bia. 1 January 0.0 February 0.0 3» 3« March 0.0 59 60 April 0.0 90 9' May 0.0 120 12! June 0.0 '5> '5» July 0.0 181 182 August 0.0 212 213 September 0.0 »43 244 October 0.0 *73 274 November 0.0 304 305 December 0.0 334 335 5tfa TABLE VI. For finding the Tnip Anomaly or iV.c Timt" from tin- IVriholion in n Pnrnbolic Orbif. V, 0° 1° 2° 3° M. DIIT. 1". u. Dlff. 1". M. Diflr. I". K. DIff. I" 0' o.cooooo iXi.Xi 0.654532 't't^ 1.309263 181.92 1.964393 il'i.05 1 oio<;oS iXi Xi 0.665442 , 1X1.83 1.32017X 1X1.92 1.975316 1X2.06 'i o 021817 iXi.Xi 0.676352 181.83 1.331093 181.92 1.9X62^0 1.997164 1S2.06 :i 0.03271s iXi.Xl 0.6X7262 1X1X4 1.34200X 181.92 1X2.06 * 0.043633 181.81 0.69X172 181.84 i.35»9»3 181.92 1.008087 182.07 5 0.05454a IXI.8I 0.7090X2 181.84 1.363X39 181.93 1.019011 1X2.07 0.0(15450 iXi.Xi 0.719993 1X1. X4 '•374755 1X1.93 1.029916 2.040X60 1X2.07 7 007635S iXi.Xi 0.730903 1X1. X4 1.3X5670 1X1 93 iSi.o: 8 0.0X7167 iXi.Xi 0.741X13 'S'-^ 1.396586 1X1.93 2.<-5"7S5 1X2. cS o.09«i75 IXI.8I 0.752724 1X1.84 1.407502 181.93 2.C62709 182.08 lO o". 1 0908 3 181 Xl 0.763634 181.84 1.418418 181.94 2.073634 1X2.08 11 0.1 199V- iXi.Xi 0.77454 > 1X1. X4 1.429334 181.94 2.084559 1X2.08 \'2 0. 13090U iXi.Xi 0.7X^4^6 1X1. X4 1.440251 18.. 94 1.0954X5 1X2. 09 13 o.r+ixos iXi.Xi 0.7963(10 1X1. X5 1.451167 1X1.94 2.106410 iX2.0C( : >* O.I 52717 181.81 O.X07277 1X1.X5 1.462083 1X1.94 2.117335 1X2.09 i 15 0.163625 181. 81 C.818188 181.85 1.473000 181.95 2.128261 iX:.ic i Vi o.i745!4 iXi.Xi 0.829099 181.85 1.4X3917 181.95 1.139187 iXi.io 17 0.1S5442 iXi.Xi O.X40010 1X1. 85 1.494X34 181.95 1.150114 1X2.10 IH 0.196350 iXi.Xi 0.850921 1X1.X5 1.505751 181.95 1.161040 iXi.ii lU 0.207259 IXI.8I 0.861832 1X1.85 1.516668 1X1.95 1.17 1966 iXi.ii 20 0.21 Si 67 iXi.Xi 0.872743 181.85 «-5i75«5 181.96 2.182894 ix2.II 'Zl 0.229076 iXi.Xi O.XX3654 1X1X6 1.53X503 181.96 1.193810 iS:.i: TZ 0.2399X4 iXi.Xi O.X94566 1X1.X6 1.549420 181.96 1.204747 1S21: 23 0.250X93 IXI.8I o.90^47X 181.X6 1.560338 1X1.96 2.215674 iSi.i: 'Z4 0.261801 181.81 0.9163X9 181.86 1.571256 181.96 2.216602 182.1^ 25 0.272710 iXi.Xi 0.927301 181.86 1.582174 181.97 1.2371:29 1X2., 3 2» 0.2X3619 1X1. Xl 0.93X212 181. 86 1.593092 181.97 1.148457 .S2.,; 27 0.294527 1X1. Xl 0.949124 181.86 1.604010 181.97 2.2593X5 1S1.14 28 305436 1X1.81 0.960036 181.86 1.614928 i.625i<47 181.97 2.270313 1S2.14 2U 0.316345 iXi.Xi 0.970948 181.87 181.97 2.181242 1X2.14 i 30 0.327253 181.81 0.981860 181.87 1.636766 181.98 2.191170 1X1.14 31 0.33X162 1S1.81 0.992772 181.87 1.6476X4 181.98 1.303099 1S2.15 :« 0.349071 iXi.Xi 1.0036X4 181.87 1.65X603 1X1.98 1.314018 1S2.15 33 0.3599X0 181. Xl 1.014596 181.87 1.669522 181.98 2.324957 lX;.i; i a» 0.370XX8 1X1.81 1.025509 181.87 1. 680441 181.99 1.335X87 1S2.16 ! 35 0.381797 181.81 1.036421 181.87 1.691361 181.99 1.346816 1X1.16 30 0.392706 1X1.81 • 047334 1X1.87 1.702280 181.99 1.357746 iXm6 37 0.403615 1X1.81 1.05X246 181. XX 1.713200 181.99 1.36X676 iXi.i- 3H 0.414524 1X1.82 1.069159 181. XX 1.724120 iSi.oo 1.379606 i8:.r i 3U 0-415433 1X1.82 1.0X0072 181.88 1.735039 18a. 00 1.390536 1X2.1- 40 0.436342 181.82 1.0909X5 1.101898 181.88 1.745960 1S2.00 1.401467 iXl.I.*! i 41 0.447251 181.82 1X1.88 1.756XX0 181.00 1.412398 1S2.18 12 0.45X160 181.82 1.1I281I 181.89 1.767X00 182.01 1.423329 lS2.i8 43 0.469069 181.82 1.123724 181.89 1.77X-21 182.01 2.434260 1S2.19 44 0.479979 181.82 1.134637 181.89 i.789('4i 181.01 1.445191 I X 2.19 45 0.490X88 181.82 i.i4i;55o 181.89 1.800562 182.01 1.456113 iX;.i9 4« 0.501797 181.82 1.156464 181.89 1.811483 182.02 1.467055 182.:" ' 47 0.512706 1X1. X2 i-«67377 181.89 1.X22404 182.02 1.4779X7 iX:.2o 48 0,523616 181.82 1.178291 181.89 >->'333i5 181.01 1.48X919 iX;.:o 4U o.S34S»5 181.82 1.189205 181.90 1.844247 181.01 2.499X51 iXi.:i 1 50 0.545435 181.82 1.2001 19 181.90 1.855168 182.03 1.510784 1X2.21 51 0.556344 181.82 1.211033 181.90 1.866090 1X2.03 2.521717 1X2.22 ; 52 0.567254 1X1.82 1.221947 181.90 1. 877012 1 8 1.04 2.532650 1X2.21 53 0.578163 I8I.X3 1.232861 181.90 1.887034 1.898856 181.04 1.543583 1X2.2: 1 »•« 0.589073 181.83 «-»43775 581.91 181.04 i-SS45'7 1X2.23 55 0.599983 0.610892 181.83 1.254689 181.91 1.909779 181.04 2.565450 2. 5 76 3 84 182.23 50 181.83 1.265604 1.276518 181.91 1.920701 181.04 182.23 57 0.621802 181.83 181.91 1.931614 182.05 2.587319 1X2.24 58 0.632712 181.83 1.28^433 1.198348 181.91 1.942547 182.05 2.598253 1X2.24 59 0.643622 181.83 181.91 1.953470 182.05 2.609187 182.24 ! 00 1 0.654532 181.83 1.309263 181.92 1.964393 181.05 1.620111 182.15 50tf iralKilir Orbit. TABLE VI. For ilnding the Tnio Anomaly or tlie Tinii' fn>m the I'prihi'lion in a I'urabolic Orhil. I>11T. 1" ifl o<; iSj c6 iSi 06 iSi 06 1X2 <^7 1S2.0: l!ti 07 iSi.o: iSi cX lii oX 182 cX 1S2 oX 1S2 09 1X2 ov |S2.0I> iS: IC 1S2 .10 1X2 .10 iSl .11 1X2 .11 IX2.II 1X2.11 1X1 12 IX2.I2 1X2.13 1X2. 15 1X2.I3 1X114 1X2.14 1X2.14 1X2.14 1S1.I5 IS2.I5 1X1.13 1X2.16 1X2.16 1X2.16 1X2.17 1X2.1'' 182.17 1X2.1X 1X2.1X 1X2. iS 1X2.19 1X2.19 1X2.19 1X2,:'^ 1X2.20 1X2.20 1X2.21 1X2.21 1X2.22 1X2.22 1X2.22 1X2.23 182.23 182.23 1X2.24 1X1.24 I 182.14 1 182.15 r. 40 5° 6 70 M. 4. 59 19 '7 M. roir. 1". M 3 276651 WIT. I". 1X2.50 .M. 3.934181 Diir. 1". 1X2.80 Ditr. 1". 2.610112 iSi.15 183.17 1 1.631057 182.25 3.2S7602 1X2.50 3-94515' 1X2.81 4.603907 1X3.18 •» 2.64li>9j 181.16 3.29X552 1X75, 3.956119 1X2. 82 4.614S98 183. iX Ti i.652'>l8 182.26 3-309503 1X2.51 3..;07o88 1X2. X2 4.(125X89 1X3.19 4 2.60^86+ 182.16 3.320454 182.51 3.97X058 182.83 4.636880 183.19 5 2.674800 182.17 3.331405 181.51 3.989018 182.83 4.(147872 183.20 » 2.685756 1S2.17 3 3+i35<' 181.53 3.99999X 1X2. X4 4.658864 1X3.11 1 1.696671 182.27 3-3533o>* 1X2.33 4.01 096 S ■.X2.S4 4.669X57 1X3.11 H 2.707609 181.18 3.564260 1X2.54 4021939 1X2. X5 4.6X0X50 1X3.22 V 2.718546 181.28 3-375»«i 1X2.54 4.032911 1X2.86 4.691X43 1X3.23 1«> 2.719483 181.19 3.386165 182.55 4.043882 182.86 4.702X37 183.24 II 1.740410 181.29 3.397118 182.55 4o;4!<54 182.X7 4.713X31 '!<3--4 li i.75i35« 182.19 3.40X071 1S2.56 4.065X26 1X2.X7 4.7i4-^i<' 1S3.25 i:i 1.76219s 182.30 3.419024 182.56 4.07(1799 1X2.88 4.733821 1S3.25 II *-775S33 181.30 3.429978 182.57 4.0X7772 IX2.X8 4.74<''*'6 1X3.26 15 2.784171 1H1.31 3440931 182.57 4-098745 182.89 4-757>''i 1X3.27 III 1.795 no 181.31 3.451887 1 82. 58 4.109718 1X2.90 4.76XX09 1X3.27 17 2..><575 182.61 4.18(1544 182.94 4-^45794 1X3.32 'ii 2.X82617 182.34 3-539531 182.61 4.197520 182.94 4.850793 1X3.33 2.-1 2.893567 181.35 3.550489 182.62 4.20S497 182.95 4.867793 1X3,34 '.:*> 2.904 ',.''8 182.35 3.561447 182.62 4.219474 182.95 4.87X793 1X3.34 27 2.91544^ 182. 36 3.571404 182.63 4.230451 lXj.96 4.XX9794 1X3.3, 28 2.926391 181.36 3.5X3361 182.63 4.241429 1S2.97 4.900795 1X3.30 2U s-93733i i«i.3''' 3-594310 181.64 4.25240;! 181.97 4.911797 1X3.36 3() 2.948274 182.37 3.605179 182.64 4.263386 182.98 4.922799 183.37 31 2.959117 181.37 3.616238 1X2.65 4.274365 182.99 4.933801 't^-'it :i2 2.9701159 181.37 3 627197 1X2.65 4.2X5344 1X2.99 4-944«'H 183.38 :i;t 2.981102 182.38 3.638156 182.66 4.296324 1X3,00 4.955X07 •S3.39 34 2.992045 i8i.3« 3.649116 181.66 4.307304 1X3.00 4.966811 183.40 io 3.001988 181.39 3.660076 182.67 4.318284 183.01 4.977X15 183.4, 36 3.013931 182.39 3.671037 IX2.6X 4.329265 1X3.01 4.9XX820 1X3.41 37 3.024875 182.39 3.6X1997 182.68 4.340246 1X3.02 4.999S25 1X3.42 3N 3.035819 182.40 3.691958 182.69 4.351228 183.03 5.010X30 IXJ.43 M 3.046763 182.40 3.703920 182.69 4.362210 183.03 5.021836 1X3.43 to 3.057707 181.41 3.7i4«fi 181.70 4.373191 183.04 5.032842 1X3..44 U 3.068652 1X2.41 3-7i5«43 182.70 4-3X4'75 1X3.05 5043^49 '■^3-45 42 3-079597 182.41 3.736806 182.71 4-395<5^ 183.05 5.054X56 1X3.4O 13 3.090541 182.41 3-7477''« 182.71 4.40(1141 1X3.06 5.005X64 1X3.46 41 3.101488 I8Z.43 3.751*731 182.72 4.417125 1X3.06 5.076X72 1X3.47 45 3.111433 182.43 3.769694 182.72 4.428109 183.07 5.0878X0 ,83.48 4(1 3-'i3379 182.44 3.780658 182.72 4.439093 1X3.0X 5.09XXX9 1X3.4X 47 3-«343»'; 182.44 3.791612 182.73 4.45007.S 183.08 5.109X9X 1X3.49 4N 3 >4^27a 182.44 3.801586 1S2.74 4.461064 183.09 5.12090X 1X3-50 4U 3.156119 181,45 3.81355, 182.74 4.472049 1X3.10 5.131918 183.51 5(» 3.167166 181.45 3.824515 182.75 4.483035 1X3.10 5.142929 1X3.51 51 3.178113 181.46 3.835481 182.76 4.494022 1X3.11 5.. 153940 1S352 52 3.189061 181.46 3.8464.46 182.76 4.505008 1X3.12 5.164951 1X3.53 53 3.200009 181.47 3.857412 182.77 4-5 '5995 1X3.12 5.1759^') 1X3.54 54 3.210957 182.47 3.868378 181.77 4.5269X3 1X3,13 5.186975 183.54 55 3.221905 182.48 3-879345 181.78 4-53797' '^3-'4 5.197988 ii<3-55 50 3.232854 181.48 3.890312 182.7S 4-541^959 183.14 5.209001 ,83.56 57 3.243803 182.49 3.901179 182.79 4.559948 1X3.15 5.220015 183-57 5« 3-2U75» 182.49 3.911146 181.79 4-570937 183.15 5.231029 183.57 5» 3.265702 182.49 3.913114 182.80 4.581927 183.16 5.242044 ,83.58 60 1 3.276651 182.50 ^.934182 182.80 4.592917 183.17 5.153059 183-S9 467 TABLE VI. Fur fiiuUng llic True Aiioinaly or tlit- Time t'roiii the iVrilit lioii in u Parabolic Orbit. I V, o I •i .-{ 1 it T N lO II 12 l.'i It 15 10 17 18 10 'M 'Zl TZ »3 24 25 20 27 28 2U 30 31 32 33 34 35 30 37 38 30 40 41 42 43 44 45 40 47 48 40 50 51 52 53 54 55 SO 57 58 50 00 8^ M. 253059 264075 275090 2S6107 297124 30X141 3'y»59 n^^>77 341 195 35i2>4 3^-3234 374154 385*75 396296 4073 '7 418339 429361 440384 451407 462431 473455 484480 49550s 506530 5'7556 528583 539610 550637 561665 572693 583722 59475* 605782 616812 627X43 638874 6499 6 660938 671971 683004 694038 705072 71 6 1 06 727141 738177 749213 760250 771287 781325 793363 804401 815440 826480 8375*0 848561 859602 870644 881686 892728 903771 91481S Diir. 1" -59 59 .60 .(>! .62 .62 .63 .64 .65 .66 .66 .67 .68 .69 .69 .70 •7« -72 ■73 -73 •74 ■75 ■75 .76 -77 .78 ■79 ■79 .So .81 .82 .8, .83 .84 .85 .86 .87 .87 .88 .89 -90 .91 .92 .92 •93 -94 ■95 .96 .96 -97 98 -99 84.00 84.01 84.01 84.02 84.03 84.04 84.05 84.06 184.06 9^ .M. 5.914815 V925S59 5.936904 5-947949 5-958995 5.970041 5.981087 5.992134 6.00 5 1 82 6,014230 6.025279 6.036328 6.047378 6.058428 6.069479 6.080530 6.091582 6.102634 6.113687 6.124740 6-135794 6.146849 6.157904 6.168959 6.180015 6.191072 6.202129 6.213187 6.224245 6.235304 6.246363 6.257422 6.268482 6.279543 6.290605 6.301667 6.31 2729 6.323792 6.334855 6.345919 6.356984 ''■36X049 6.379115 6.390181 6.401248 6.412315 6.423383 ''■43445« 6.445520 6.45(1590 6.467660 6.47X731 6.4X9X02 6.500874 6.5 1 1946 6.523019 6.534092 6.545166 6.556241 6.567316 6.578391 10 ItllT. 1". M. 184.06 1X4.Q7 1X4.08 184.09 184.10 6.17839, 6. 5X9467 6.600544 6.61 1622 6.622700 184.11 1X4.11 184.12 '84-13 184-14 6.633778 6.644X57 6.655937 6.667017 6.678098 184.IS 184.16 184.17 184.18 184.18 6.6X9179 6.70026, 6.711343 6.722426 6.733510 184.19 184.20 184.21 184.22 184.23 6.744594 6.755679 6.766764 6.777850 6.788937 184.24 184.25 184.25 184.26 184.27 6.800024 6.Xii,,2 6.822200 6.833289 6.844378 184.28 184.29 184.30 184.31 184.32 6.855468 6.S66559 6.877650 6.S88742 6.X99834 184.32 184.33 184.34 '84-35 184.36 6.910927 6.922021 6.933115 6.9442,0 6.955305 184.37 1X4.38 184.39 1 84.40 184.41 6.966401 6.977498 6.988595 6.999693 7.0,079, 184.4, 184.42 ,84.43 184.44 184.45 7.021890 7.032990 7.044090 7.055,91 7.066292 184.46 184.47 184.48 184.49 184.50 7^077394 7.088497 7.C99600 7.110704 7. ,21808 184.5, 184.52 184.52 •84-53 184.54 7., 32913 7.144019 7^'55i*5 7.166232 7.177340 •84-55 ,84.56 ,84.57 ,84.58 ,84.59 7.188448 7^'99557 7.2,o'i66 7.i2k/76 7.232886 184.60 7.243997 iMir. 1". 84. 60 84. 6 1 X4.62 84.63 84.64 84-65 8466 8467 84.67 84.68 84.69 84.70 84-71 84.72 8473 84.74 84.7s 4.76 84-77 84,78 84.79 84.X0 X4.8, 84.82 84.83 84.84 84-85 X4.86 84.87 84.88 84.X9 X4.90 X4.91 84.92 84.93 84.94 84.95 84.96 84.97 84.98 84.99 85.00 85.0, 85.02 85.03 85.04 85-05 85.06 85.07 85.08 85.09 85. ,0 85.11 85.12 85.13 85.14 85., 5 85. 16 85.17 85.18 185.19 11 M. *43997 255109 16622; *77335 288449 199563 310678 321793 332909 344026 355144 36(1262 37738" 3X8500 399620 4io74« 421862 432983 444106 455*30 354 8 Ditr I" 85-19 8v20 85. 21 Xv2i 85.2, 85-2,- Xvlb 85.27 X5.2.S 85.29 X5.50 Xi.;. 85.3* 85-33 85-3+ 85-35 8^.36 85-37 85-38 85.39 85-40 '.5.41 5-4J X; .466 ■477^t7 .488603 499729 510X55 521982 533110 544*39 555368 5664.97 5776-8 588759 599890 .6,1022 ,622155 , ,633289 7.644423 7-655558 7.666694 7.677830 ,688967 7.700,04 7.7,1242 7.72238, 7.733521 7-74466I- 7.755802 7.766943 7.778085 7.789228 800372 81,516 822661 833807 844953 856100 867247 878396 889545 900694 911845 I IS5.: «5^44 85.46 85^47 85^48 X5.49 85-50 5' 5i 53 54 55 85-57 85.58 85.59 85.60 X5.61 S5.62 85-63 Xv64 X5.65 X5.66 X5.6X 85.69 X5.70 X5.71 85.72 85-73 85-"4 85^75 X5.76 85.78 X5.79 8s. Xo X5.81 85. 82 85.83 568 ubolif Orliii. TABLE VI. For (inilinj; [\w True .Vnoiiiiily or ilu- Tiiiif tniiii tlio Pcrilu'linii in m I'liialnilic ()il)it. 11 \l. nvv7 I 77335 8)i449 99 i'' 3 2 1793 32' 9° 9 44026 55'44 ()(i2(ii 773«' X«500 (^9620 .I07ii .21862 .3i9«3 ^44106 ^55^3o t66354 t7747i< 1.8X603 ^99729 510X55 521982 533110 44139 5536S ,66497 5776-8 5X8759 599890 II I022 ^22155 533289 S44423 'SS55« 366694 &77830 588967 ' 700104 711242 722381 733;*» 744661- 755802 766943 778085 789228 S00372 511516 822661 833807 S449S3 S56100 867247 S78396 58954S 300694 911845 DIff. 1" 185.19 185.20 1X5.21 1X5.21 185.23 1X5.25 1X5 :b 185. ;7 lX5.2,S 1X5.29 1X5.50 1S5.51 1X5.31 "«5.54 185.55 1X5.57 1X5. 58 1X5.59 185.60 185.61 1X5.62 1X5.63 1X5.64 185.65 1X5.66 185.68 185.69 1X5.70 1X5.71 1X5.72 '5*5-"3 i«5-4 185.75 185.76 185.78 185.79 1X5. Xo ; 185.81 185.82 ; i^H \ i»5-84 r. 12" 13 140 16 ( M. Dirr. 1". M. DifT 1'. M, lit!. 1". >i Mti r 0' 7.91 1845 85.84 8.582146 1X6.56 9,255120 X-.35 9.93C9X4 88. Id 1 1 7.922995 185.86 8.593340 lXh.5' 9.26(1360 '87.34 9.9422-4 XX. 18 1 'Z 7-9 34 '47 85.S7 X.604535 1X6.5X 9,277601 87-35 9 9^35's SX,19 1 :i 7.945300 85XX X.O15730 lXh.59 9 288812 9,3000X5 X-,3- 9,964X5- XS.ii , 1 795*'453 85.X9 8.626926 1X6,61 87,38 9.976149 |XS,21 .*> 7.967606 85.90 X. 63X123 1 ^6.62 9.311 32X X-.4C. 9-987+43 XX. ■•1 41 7.97X76. 1X5.91 X.6A9310 X.06051X i)s6.63 9-3ii>7S '87.41 9.99X-3S XX,2- 7 7.9X9916 8 5. 9 J 1X6.64 9.333X17 X-.42 10.01003 3 iXX.iii H 8.00.072 185.93 8.671717 1X6.66 9.345063 8'.44 10.02132c) XX 2X « S. 012228 185.95 X. 682917 186.67 9.356310 87.45 10.032626 1XX.29 Hi X.023385 185.96 8.694,17 1X6,68 9.3675^7 187.46 10.043924 IXX.3I It 8.034543 8597 X. 705318 186.69 9.378805 ,X-.4X loo55.'.23 SS, 32 I'i X. 045702 1X5. 9X 8.716510 186.-1 9.390054 87.49 10,06(1523 '88.34 ' III X. 056X61 1X5.99 8.727723 1X6.72 9.401304 87.50 10.077S13 XX. 3 5 ' II X. 06X021 r 86.00 X.738927 186.73 9.412555 1X7.52 10.0X9125 '88.37 ^ 15 X. 079181 1 86.02 8.750131 186.74 9.423X06 87-53 10.100427 ,xx.38 ; Kl 8.090343 86. 03 X.76I 336 1X6.76 9.43505S 8-. 54 10.1 11730 XX. 39 \ IT S. 101505 186.04 8.775541 1X6.77 9.446311 X7,56 10.123035 X8.41 IN X.1I266X S6.05 S. 7X3748 186,78 9-457^65 87-57 10.134140 10.145646 'l:■/■^* i li) X. 123X31 186.06 8-794955 186.79 9.46XX20 1X7.59 '88.44 ; 'M X. 134995 18607 8.X06163 1X6.X1 9.4S0076 ' iX7..o 10.1 5695- 88.45 ; 'Z\ 8.146160 186.09 8.817372 186,82 9.491332 1X7.61 10. l682<'0 8X,47 'ii 8.157326 86.10 X.X2X5X2 l8(>.X3 9.5025X9 1X-.63 10.17956X iXX,48 , •i:t X. 10X492 rX6.ii 8.839792 1X6.84 9.513847 1X7.64 I0.190X7S lXX,5o 'it X.179659 86.12 8.851003 186.86 9.525106 187.65 10.202188 188.51 'z:. 8.190X26 86.13 X.X62215 186.X7 9.536366 ■ 1X-T.67 10.213499 188.53 w 8.201995 1X6.15 X.X73+27 1 86.88 9 ^47626 1X7.68 10.224X12 '^8.54 XX, 56 27 X.2I3I64 1X6.16 X.XX46'i 186,90 9.5SXXXX 187.70 10.236125 w X. 224334 X6.17 8.895' I. 186.91 9.570150 187,71 10.247439 88.57 •i\i X. 235504 186.18 X. 907070 186.92 9.581413 187.72 10.25X753 88.59 M) 8.246675 186.19 8.918286 186.93 9.592676 1X7.74 10.270069 8X.60 :n 8.257847 1X6.20 S. 929502 186.95 9.603941 '87-75 10,281386 XX. 62 \M X 269020 186.22 8.940719 186.96 9.615207 87.77 10.292703 XX. 63 ' :<:< 8.2X0193 186.23 8.951937 186,97 9.626473 187.78 10.304011 XX. 65 , :m X. 291 367 186.24 8,963156 186.99 9.637740 187.79 10.315341 88.66 ».-> X. 30254* 186.25 8.974376 187.00 9.64900? 187.81 10.326661 88.68 ! 3a S.3137I7 186.26 8.985596 187.01 9.660277 87,82 10.337982 188,69 1 :i7 X.324X93 186.28 8.996817 1X7,02 9.671547 87-84 10.349304 XX. 71 i ;w X. 3 3(1070 86.29 9.00X039 9.019262 187.04 9.682X17 : Sf.X5 10.360627 88.72 I 39 X.347248 86.30 187.05 9.694088 X7.86 10.371951 88.74 : 10 8.358426 ' 186.31 9.030485 187.06 9.705361 : 9,716634 t 187.88 10.383275 88.75 11 8.369605 86.32 9.041709 1X7.08 87.89 10.394601 , 88.77 \'l 8.3X0785 86.34 9.052934 9.064160 187.09 9.727908 : 87.9, 10.405927 88.78 13 8.391966 186.35 1S7.10 9-739'82 1 87.92 10.417255 88, 80 II X.403I47 86.36 9.075387 187.12 9.750458 87 93 10.428583 88, 81 l.-i 8.414329 86.37 9.086614 1X7.13 9.761734 ; 187.95 10.439912 XX. 83 Kl 8.425512 X6.38 9.097842 187.14 9 773°«» i 87-96 10.451242 XX, 84 47 8.436695 86.40 9. 1 0907 1 1X-.16 9.784290 87.98 10.462573 XX. X6 48 X.447879 86.41 9.120301 187.17 9795569 j 87-99 10.473905 XX. 87 41) 8.459064 86.42 9.I3153I 187.18 9.806X49 ; 88.00 10.485238 XX. X9 5(> 8.470250 86.43 9.142763 187.20 9.X18129 ' 88.02 10,496572 XX, 90 51 8.481436 : 86.45 9- "53995 187.21 9.X29410 ' 88.03 10.507907 XX. 92 5'i 8.492623 86.46 9.165228 187,22 9.X40693 88.05 10.519242 88-93 33 8.503811 i 8647 9.176462 1S7.23 9.851977 i 88.06 10.530579 ^!!-95 34 8.515000 86.48 9.187696 187.25 9.863261 88 08 10.541916 88.97 55 8.5261X9 1 86.49 9.198931 187.26 9.874546 88.09 10.553255 ' 88,98 30 8-537379 i " 86.51 9.210167 187.27 9.885832 88.10 10.564594 1 89,00 57 8.548569 ! 1 86.52 9.221404 187.29 9.897118 1 88,12 IO-575934 89.01 3N 8.559761 86.53 9.232642 9.243880 187.30 9.908406 1 88.13 10.587276 1 89.03 59 8.570953 1 86.54 187.31 9.919694 J 88.15 10.598618 1 89.04 eo 8.582146 1 86.56 9.255120 187.33 9.930984 1 88.16 10.609961 1 89.06 1 56» TABLE VI. For fiiuliinf the Tnu> Anomiily or tin- Tinu' fr »iii llie IVrilu'lioii in ii I'lirabolie Orhii. i 16 17" 18 19 M. 11.667850 M. wir. 1". M. 11.191177 WIT. 1". M. 1 1.978 1 62 IHII. 1' iwir. '. ' lo.fioy<^6i 189.06 190.02 191.04 .91.13 1 lo.hll^o^ 189.07 11.303679 190.03 II.989(>15 191.0(1 12.679379 19*. J 5 u i'..f'5it)4'; 1 89.09 11.315082 190.05 12.001089 njl.o.'! 1 2.090908 »9i 1" » io.f)4V;95 189 10 II 326485 190.07 11.012554 191.09 11.701439 IV 2. 19 ' 4 io.<.55j4i 189.12 11. 337889 190.08 11.0140x1 191.11 11.713970 I9i»»i ' 5 lofifififiyo 1X9.14 11.349295 190 10 11.035488 191.13 11.715503 192 21 !f 189.15 1 1.360701 1 90. 1 2 ll.o4(u^5(i 191.15 12.737037 19124 7 189.17 11.372109 190.13 ii.05.>-:4i5 191. 11' i2.74X';73 192 l(> N lo. 7007:5 »l 189.18 II. 3X3517 190.15 12.-j69,-;9(i 191.18 12.760109 192. JS » io.7iioyo 189.20 11.394927 190.17 12.0X13(17 191.20 12.771646 192.3, lO 10.71344.1 189.21 11.406337 1 90. 1 8 1 2.092X40 191.22 12.783185 192.32 1 1 lo.7U79i 1X1J.23 11.417749 190.20 12.104313 191.24 12.794714 192.34 l-^ 10 74614'; 1X9 24 1 1.429161 193.22 12.115788 191.25 12.X06265 l92.-,(. i:i io.7S7?Q5 1 89, id II.44C575 190.23 12.117264 I9'^7 l2..' 10.78011)! 1 89. 19 11.463405 190.27 12.150219 .91.31 11 840X94 192.41 10 10.791 i7h 189.31 il.474'^ 18934 11.497657 19c 32 12.184659 191.3(1 II-X75S34 '9247 lU io.Xi^655 1X9.35 11.509077 190.33 12.I96141 19' I" 12.887081 192.49 w 10 >'^7oi7 1X9.37 11.520497 190.35 12.207624 191.^0 12.89X631 192.51 'il io.K4)i^Xo 189.39 11.531919 190.37 12.219108 191.41 12.910183 192.53 ' ut ic.S5,)'44 189.40 "•';4iuj 190.39 12.230594 >9'43 12.911736 '9255 •i:t 10.S71 loS 189.41 11.554765 190.4c 1 2.2420X0 191.45 12.9332X9 192.56 'Zl io.S)!z474 1S9.43 1 1.566190 190.42 I2.25356X 191.47 12.944X43 192.58 •zr, lo.S9^S40 189.45 11.577'iK' 190.44 12.265057 191.49 12.956399 191.(10 'M 10.90^208 189.47 ii.5'<924» 190.45 12.17654(1 191.50 12.967956 192 62 'i7 10916^76 1X9.4X 1 1.600470 190-47 12.28S037 191.52 12.979514 192.64 'iH 1^.92-946 1X9.50 1 1.611899 190.49 12.299529 191.54 12.991073 191.66 'Z» loy393i(> 1 89. 5 1 ii.623r-!« 190.50 12.311011 191.56 13.002633 191.68 ■M 10.9^06X7 189.53 11.634759 190.51 11.311516 191.58 i>oi4i95 192. "O :ti I0.9620VJ 1X9.55 11.646191 190.54 11.334011 191.60 i3-o»S7';7 192.72 •,ti '0 971413 1S9.56 II.657024 190.5(1 12.345508 191.61 13.037321 192.74 :i:i 10.9S4S07 1 89. 58 11.6(19057 19057 11.357005 191.63 13.048XX6 192.76 :ii 1 0.9961 Si 1X9.59 1 1.680491 190.59 11.36X503 191 65 13.060451 192.78 :i5 1 1.0075^8 1X9.6 1 1 1. 691918 190.61 11.3X0003 191.67 13.071019 192.80 :itt 1 1.01X9;,- 1X9.63 I'. 703365 190.62 12.391504 191.69 13.0835X7 I92.S2 :j7 11.030515 1X9.64 11.714803 190.64 1 2.403006 191.70 13.095157 I9;.S3 :iH 1 1.041691 1X9.66 ii.7i'''i4i 190.66 12.414509 191.72 13.106717 192.S5 :w 11.0^5071 189.67 ii.JVMi 190.68 12.42601 3 191.74 13.118199 I92.il7 10 11.064453 189.69 1 1. 749123 190.69 i»-4r5'7 191.76 13.119871 192. 89 11 11.075X3? 189-71 11.760565 190.71 12.449023 191.78 13.141446 192.91 IJ 11.0X721X 189.71 11.772008 190.73 1 1.460531 191X0 13.1530^2 192.93 , r.i 11.09X602 189.74 11.783452 190.74 12.472039 191.81 13.164598 192.95 II 1 1.1099X7 189.76 11.794897 190.76 11.4X354X 191.83 13.176176 192.97 ; 15 11.121371 189.77 11.X06344 190.78 11.495059 191.85 13.187755 192.99 1(( 11.131759 1X9.79 11.X17791 19 '.Xo 11.506^71 191. 87 i3-'99335 193.01 1 '»7 u.i4+'47 1X9.80 1 1. X 292 39 190.81 11.518083 191.X9 13.210916 193.03 i 1?. 11.155536 189.X1 II.X406X9 190.83 12.519597 191 91 13.212498 193.05 1 4U 11.166925 189.84 II. 852139 190.85 1 2.541 ■ ' ^ 19IV3 13.234082 193.07 ! 50 II. 17X316 189.85 11.863590 190.87 12.552628 191.94 13.145667 193.09 ' 51 1 1.1X970X 189.87 I1.X75043 II.X86496 190.88 11.564145 191.96 I3'2>7i53 193.11 i 52 11.101100 1X9.89 190.90 12.575664 191.98 J 3.168840 193.13 1 5:i 11.211.194 11.213X89 1X9.00 11.897951 190.92 11.587183 192.00 15.2X0428 193.15 1 54 189.91 11.909407 190.94 11.598704 192.02 1 3.. ••.910 1 7 193.17 55 11.235184 189.93 11.910863 190.95 12.610225 191.04 13.303608 193.19 50 II. 246681 189.95 11.931321 190.97 11.611748 192.06 13.315100 193.21 i 57 •1.258078 1X9.97 11.943780 190.99 12.633171 192.07 13.316793 193.23 5H 11.269A77 189.98 II-95S159 1 9 1. 01 11.644797 192.09 13.338387 193.25 50 11.280876 190.00 1 1.966700 191.01 12.656313 1 92. 1 1 13.349982 193.27 00 11.191177 190.01 11.978161 191.04 11.667850 192.13 13.361579 193-^9 570 ilxilic OrMi. TABLE VI. Kor liixliiiK tin- Tnio Anoiimlv or ilif Tim* from the IVrihi-lioii in a I'lirnlxilic OrliiU 19° • 1 . 1 Wif. V 7«5° ,.,!.,, 9179 , \ ')*.>;, 090S 1 \ %'}7o Hjvi.zi 55°1 i.>2 j: 7057 I.;; 24 XS71 . |.J21(. oioi; ii;2 2S l()4h I.J2.V ^S, 192.12 4714 '9i u (il6^ IV2.V- 7X07 'yi57 915° 192.}.; 0894 192.41 2440 192.4, )V>'<6 ";m? 'ilU 192.47 l7ol<» 192.4.J ,X6',2 192.51 oiS, 192.5; 117,6 I9i S? 5,2X9 192.5(1 ^"41 : 19251! 5 ''199 192.60 >79^h 192 1-1 /y-iH 192. ^4 »i"71 19266 u(),3 1 192.68 1419? i 192.70 157S7 192 '2 i71^« . 192-4 vssxfi ; 192.76 .0452 , I92.7S 72019 192.^0 <1'i><7 .92 S2 >siS7 I92.S, 36727 I9I..S5 18299 fV-*7 29S72 192. 89 M44'' I 192.91 S,022 192.95 VH'^** 192.95 76176 191.97 »7755 192.99 )9n? 195.01 1 09 1 6 l95-°5 21498 195.05 54C82 195.07 j<;667 193°') ^-21;? 195.11 (1SS40 195 15 S0428 195.15 [>20I7 195.17 0-^608 195'9 1^200 195.21 26791 ")•,•-) 1«1«7 195.25 49981 , 19527 61579 1 1 93.^9 20' 21 ^ 22 23 1, 5''«f79 IMtT. 1'. 193.19 M 14.059591 IMIT 1". 194 51 M. 14.762133 IHir. 1" M. IMir 1". • 97- 1 7 195.80 •5469459 1 M?71'77 193 31 l4.o7iihi •94 51 • 4771''82 19583 15.481290 197 19 '■i 15 584776 •9111 14.081955 '94 5 5 • 4 7''5632 195.85 15.493122 197 21 :i 15.396576 '93-3S 14.094608 '94 57 •4-7971X4 195.87 15.5C4956 '97 24 •1 " 3.407977 «9V17 14.106283 •94 59 14.809137 195.89 15 516791 197.26 5 15.419580 191 19 14.1 1-960 194.61 14.810891 •95 9^ 15.52X627 • 97 28 n-4i"' 7 1544*788 '91-<1 14.141516 l94-''6 "4-'<-<44 3 195 9'' l5-5>25t4 '9-53 H '3.454194 '9145 14.152996 194-68 14 85( 161 '95 9X 15 564144 '97 35 M i5.4h6ooi >91 47 14.164677 194.70 14.86r92l 19(1.00 I5-5";9X6 19-58 10 15.477^10 •91 49 14.176360 •94 71 I4.X79682 196.03 15.587850 197 40 1 1 1 5 489220 195.51 14.188044 '94 74 14.8,^1444 19(1.05 I5.59V''75 '9743 I'-i i5.5'o85i •93 53 14.199719 19 J -6 14 9c 5 208 196.07 15 611521 •9745 1:1 • i 512443 15.524056 •93-55 14 21 1415 194.78 '4 9 "4971 196 09 15.623569 '97-47 1 1 •91 57 14.213103 i94-«^ 14.926739 196.12 15.635218 197.50 l<'i "3.515''7i "91 59 14.234791 194.83 14.918506 196.14 15.647068 • 97-5» ' Ml '3-54'287 19361 14.2464X1 194K5 14.9^0275 196.16 1 5.658920 '97-54 1 « 15.558904 193.63 ■ 4258174 194.87 14.962045 196.18 1 5-670773 •97 57 IK 15.570522 195.65 14.269867 194 89 14.973X17 196.20 15.6X2628 • 97-59 , IW 15.582141 193.67 14.281561 194.91 14.985590 19(1.23 15.694484 197.61 W l3-5917 156518S3 •9179 14.551752 195.04 15.056256 196.36 15.765651 197.76 •m 15.663511 193.81 •4-163455 195 06 1 5.068039 196.39 ' 5-7775 ' 7 197.78 •i' 15.67^140 193.83 I4-I7';i^9 195.08 15.079823 196.41 15.7X9I-X5 197.80 1 'iX 15.686770 193.85 14.386865 195.10 1 5.091608 196.43 15.801254 '97X3 'M 1 5.698401 193.87 14.398572 195.13 15.103394 196.45 15.813124 197.85 m 13.710034 193.89 14.410280 195.15 15.115182 196.48 15-824996 197.88 :o554 15.161348 196.54 15.860620 '97-95 :577 •91-97 14.457126 195.23 196.57 15872498 •97-97 :t.-> 15768J16 193.99 14.468841 195.26 15.1-4142 196.59 15.884377 198.00 •M\ 15779856 194.01 14.480557 195.28 15.1X591X 196.61 15.S96258 198.02 ;»7 I5.79i49« 194.03 14.492274 195.30 15.19771'' 196.64 15908 140 198.04 :w 13.803140 13.814784 •94-05 14.503991 195.32 15-209535 196.66 15.920025 15.931908 198.07 \Vi 194.07 14.515711 •95-14 15.111335 196.68 198.09 i to 1 3.S26429 194.09 '4-527414 • 4-559»56 •95-16 •5-233«37 196.70 •5 941794 19X.11 II 15.X38075 194.11 •95-19 1 5-244940 196.73 15.955681 19X.14 [ li 15.849713 194.14 14.550880 '954' '5-2^6-44 196.75 15.9675-1 19X.17 III 15.861371 194.16 14.561605 •95-43 15.26X550 196.77 15.979462 19X.19 II 15.873011 194.18 •4-57411' •95-45 15.280357 196.80 '5-99' 154 198.21 Vt 15.88.673 13896315 194.20 14.586059 •95-47 15.191165 196.82 16.603148 19X.24 : la 194.22 i4-5977'*>' 195.50 15-301975 196.X4 16.015143 198.16 IT 1 5-907979 194.24 14.609519 195.52 15.315786 196.87 16.027039 19X.29 IK 15.919634 194.26 14.621250 •95-54 15.32-599 196.89 16.03X937 19S.51 li» 13.931290 194.28 14.631983 195-56 ' 5-1194' 1 196.91 16.050X36 19X.34 :>() 15.942948 194.30 14.644718 195.58 15.351228 196.94 16.062737 19X.36 1 51 13.954606 194.32 14.6^6453 195.60 15.363045 196.96 16 074639 198.38 5i 1 5.966166 •94-14 14.668190 195.63 '5-174X''1 196.98 16.0X6543 19X-4' 5:« > 19779*7 194.36 14.679919 195.65 15.386683 197.00 16.098449 198.43 198.46 i 51 13.989590 194.38 14.691668 195.67 15.398504 197.03 16.110355 5.-> 14.001154 19441 14.703409 195.69 15.410326 197.05 16.122263 198.48 50 14.012919 •9441 14.71515^ 195.71 15.422150 19T.07 16.134173 i9«-5' 57 14.024585 •94-45 14.716895 •95-74 '5-411975 197.1c 16.1460X4 198.53 •9X-56 1 5H 14.036151 •94 47 14.73X640 195.76 15.445X02 197.12 16.157997 51) 14.047911 •94-49 14.750386 195.78 15457630 197-14 16.169911 198.58 GM 14.059591 194.51 14.761133 195.80 '5-4<'9459 197.17 16.181826 19X.60 571 TABLE VI. Vor (\ni\\n(t llu- Triii> Anoiii.ilv nr llir 'rime rniiii the I'lrilitliiiii in ii I'liriilHilic Orliil. t'. 24 M. ') 25 M <.'<5 16,913^16 200. 1 7 17.646954 18.3-6255 20341 ! 3 16.11:5X1 I9X.6X 16.935517 200.19 17.6^9060 201.78 18.3XX461 *f^3 45 1 Ifi. 21950^ I9X.70 16.947539 100.22 17.671168 201.81 18.400669 1034X » ifi.i4i4i() 19X71 l'^";595 5 3 100.24 17.6X317X 201.84 iX. 411879 103.51 1 n 1'' i>n?° t')^-7> 16.9715(18 200.27 176953X9 201.87 18.425090 103 54 7 1(1.1(1^176 I9X.7X 16.9X35X5 200.30 17.707501 201. 89 iS. 437303 iX. 449518 103 57 H l<'U«797 19X.95 17.067748 2C0.48 '7.79»337 202.08 18.521843 203.77 ir> 16.360737 198.97 17-079777 200.50 17.804461 202.11 18.535070 203.80 10 16 371676 I99.OU 17.091 Xc'X 200.53 17.X16590 202.14 18.547199 203 82 17 16.3S4617 199.01 17.103X41 200.56 17.!<1X7'9 202.17 1X5 595 29 203X5 IH 16.3961559 « 99-05 17.115X75 200. 58 17.X40X50 202.19 18.571761 203. 88 lU 16.40X503 199.07 17.127911 200.61 17.X52982 202.22 18.583995 203.91 W 16,41044s 199.10 17.1399.^8 200.64 17.865116 202.25 18.596230 203.94 Ul 16.431395 199.11 I7.I5'9'<7 200.66 17.877151 102.28 1 8,60X467 IC3.9-- Ti 16.444343 199.1.- 1 7.1 6401*' 200.69 17.8X9389 101.30 iX. 620706 IC4.00 T.t 16.456192 199.17 17. 176070 200.71 17.901528 202.33 201.36 iX. 632947 104.03 'H i6.4()Xi43 199 10 17.188114 200.74 17.913669 iX. 645190 104.05 ar. 16.4X0196 199.21 17.1001,59 200.77 17.925811 101.39 iX. 657434 204.08 •M 16.4911 51 199.15 I7.2i2106 200.79 '7-';3-'955 101.41 iX. 669679 204.11 'i7 16. 50 J, 107 16,1; 16064 199.17 17.224154 200.X1 17.950101 201.44 18.681927 104.14 UH 199.30 I7.13630A I7.24«356 200. 85 17.961248 202.47 18.694177 104.17 '^U 16.51X021 199.33 200.87 '7-974397 202.50 IX.70641X 104.20 :m» 16.539983 «9«MS 17.260409 200.90 I7.9«6548 202.52 18.718680 104.13 :ti 16.55194c 16.5(13908 199. 38 17.2-1464 100.93 17.99X700 201.55 18.730935 104. :(, :t'i 199.40 17.2X4520 200.95 200.98 18.010854 101. 5X IX.743I9I 204.19 :i:i 16.575X73 »V'r43 17.396578 18.013010 202.61 '8-7 5 5449 204.31 31 16.587X39 199.45 17.308637 201.00 18.035167 202.64 18.767709 204.35 ' 35 16.599807 199.48 IT. 310698 101.03 18.047316 201.66 18.779971 204,37 ; 30 16.611776 199.50 17.332761 201.06 1X.0591X7 iX. 071649 102.69 18.791234 104.40 37 16.613747 '99-53 17.344X15 201.08 202.72 18.804499 204.43 1 3N 16.635719 >99-55 199.58 17.356X91 201. II 1X.083X13 202.75 18.816767 204.46 ' 30 1 16.647093 17.368959 201.14 18.095979 202.78 18.829036 204.49 40 16.659669 199.60 17.381018 201.16 1X.108146 202.80 18.841305 204.51 ! 41 16.671646 199.63 17.393098 201.19 iX. 120315 202.83 18-853577 204.55 ! 4'i 16.683614 199.65 17.405171 201.12 1X.I32486 202.86 18.X65X51 204.58 43 16.695604 199.68 17.417245 201.24 18.144658 202.89 18.X7X127 204.61 44 16.707586 199.70 17.419310 201.27 18.156831 202.92 1 8.x 90404 204.64 45 16.719569 >99-7? 199.76 > 7-44 '397 201.30 18.169008 202.94 18.902684 204.67 40 16.731553 • 7-4.5 347f' 201.31 18.181186 202.97 18.914965 204.70 47 16.743539 199.78 17.465556 201.35 iS. 193 365 203.00 18.927247 204.73 204.70 48 16.755527 199X1 17.477638 201.38 iS. 105546 103.03 103.06 18.939531 40 16.767516 199.83 17.4X9721 201.41 18.117728 18.951818 204.79 50 16.779507 199.86 17.501807 201.43 201.46 18.119911 203.08 18.964106 204.81 51 16.791499 199.88 17. 5« 3894 18.242098 203.11 18.976396 204.84 52 16.X03493 16.815488 199.91 17.5159S2 201.49 iX. 154286 203 14 18.9886X7 204.87 , S3 >99 94 17.53X072 101.51 18.166475 203.17 19.000981 104.1,1 1 54 16.827485 199.96 17.550163 101.54 18.178666 203.20 19.013276 204.93 , 55 .6.839484 199.99 17.562257 201.57 18.190859 203.23 19.025573 204.96 50 16.8514X4 200.01 '7-574351 201.59 iX. 303053 203.25 203.28 19.037871 204.99 57 16.863485 16.X75488 200.04 17.586448 201.62 18.315249 19.050172 105.02 58 200.06 17.598546 201.65 iX. 327447 203.31 19.062474 205. 0^ 50 16.887493 200.09 17.610646 101.68 IX.339646 203.34 19.074778 105.08 60 16.899499 200.12 17.622747 201.70 18.351847 203.37 19.087084 205.11 , 672 liolicOrliil. TABLE VI. For flMilinff tlu' Triii' Anoinnlv or tlie Tiiiio Irom ilu- IVrihi-lion in n I'nrntK.lii' Orl)il. 27 iHiT. r l8+7 1 »cvr P>0 ' 103.4c »li!^ 10341 1*4(11 103 41; 06 ('9 1034X i«7'J 103.;! 1™;° 103 <+ 7 101 103 r 103 ;., •71? 103 (is V^>1 ioyhf, hl71 103. 6X *vr^ , 103.71 oMX I 203.74 1S43 103.77 ^070 103. So 7Z'J9 103 *i ySi'J 1 103. «^ I7''i 203. XX vm 203.91 (m-\o 103.94 «4"7 103.9-' 070(1 204.00 1947 104.03 SI 90 204.0? 7414 I04.0X 9679 104.11 1917 104.14 4<77 104.17 )64S« 104.20 S680 104.23 o'>1S 204:'< I'V 204.29 5449 104.31 .7709 104.35 997" 104.37 iJiH 104.40 >4499 104.41 67(^7 204.4(1 19036 ■■ 204.49 hMOS ' 204.51 ;3< 44.01 .319 8151 42.87 1 19.1 3'' 3> 5 105.13 19.878417 107.0(1 .314 44^9 44,00 .330 0723 41.8J .'» 19 ivX(i3(, 105 16 19 X90X51 107.09 1.314 - ,5^ 43.98 1.330 3193 41.83 tl 19. 1(10956 205 19 19 9^1279 207.13 .314 9(..,(, 43.96 .330 5XM 41. Xl I 19173*74 105 31 19 9' 5707 207.1(1 ■1^5 Mr. 4194 .330 8431 42.80 H 19.1X5574 205 35 19918137 107.19 .315 49">9 4192 .331 099X 41.-8 it 19.197916 105. 3X 19.940569 107.21 .315 7<'04 43.90 •31^ 15''4 41.76 l«» 19.210240 105 41 19953003 107.15 1.316 0137 43.X8 1-3 r 6119 42-74 II 19.121566 205.44 •9 9''5419 107.18 .316 lS(i(^ 43,X(, .331 8(193 41.71 1'^ 19.234X93 205.47 • 9 9"'7877 107.31 .31(1 5 5'>o 43. X4 •312 1155 41.70 i:i 19.147111 105.50 19.99-J317 107.34 .31'' X130 4i.>;i -112 381- .3 3:' '""8 42.69 II '9*59551 105.53 10 001759 107.3X .317 0759 43.X0 41.67 l.'i 19.271885 105.56 10.015101 207.41 I.317 3386 .317 6013 43.7« 1.332 8»j 41.65 Itl M). 1X1110 105.59 10.017647 107.44 43.76 .333 149'. 41.63 1 1 19.29(1556 105.61 10.040095 107.47 .31-' 863X 41-4 •Ill 10., 3 41.61 IS I9.30SX94 105.65 10 051544 107.50 .31X 1161 4171 "-, (16011 42.59 111 i9.3n»U 105.68 2o.-'i4995 207.53 .318 3X8, 41-70 1 ,3 9>"4 4i.5'< W "9U157fi 205.71 10.077448 107.57 I.31X 6506 41.68 • 314 "71 ' 41.56 'ii 19 1459»o ir ■,- 10.0X9903 107.60 .31X 9127 41''7 •114 4271 42 54 'ii 19.35X165 105. /7 10.102160 20.114818 207.63 .319 '74*' 4; ''5 •314 "Xil 42.52 •i.l 19.370612 105.80 107.66 .319 4364 43.(13 •134 934 41.50 'it 19.18;. . 105.83 10.1 27179 107.69 .319 6981 43-'' I •13 5 '724 41.49 v> 19.395311 19.407665 105. X6 2o.i3974« 107.71 1.319 9597 41-59 "•115 4471 42-47 w 105. X9 10.151106 107.76 .320 2211 41-57 .335 7010 42-45 'i7 19.410019 105.91 10.164671 207.79 .310 4815 41-55 ■il'i 95"7 42.43 'iH •9412175 105.95 10.177140 107.81 .320 7438 41-51 .33(1 2112 41.41 w 19 444714 105. 9S 20.1X9610 107.85 .321 0049 435" .336 4656 42.40 :m •9457094 106.01 10.101081 107.88 1.321 2659 41-49 1.336 7199 41.38 :tl •9-4('945 5 106.04 10.114556 107.91 .311 526X 41-47 .336 9741 41.36 :r^ 19.4X1819 206. oX 10.217031 107.95 .3:1 7X75 41-45 •117 2283 42.34 :i:i 19.494184 206.l 1 10.239510 107.9X .321 0481 41+1 •117 41*23 42.31 :u 19.506551 106.14 20.2519X9 108.01 .322 30X7 41-4^ •337 73''2 41.31 X, • 9-5i*'9H 106.17 20,261471 20.IT6954 108.04 1.322 5691 43.40 1.337 9900 41.19 :m 19.531292 106.10 108.07 .322 8195 43.38 .338 2437 42.27 ;n 19.543664 106.23 106.2(1 20.2X9440 20X.1 1 .313 0897 43.36 .338 4971 42.25 ;w 19.556039 20.301917 208.14 .313 3498 .323 6097 41-14 .338 7507 42-24 :iu 19.568415 206.29 10.314416 108.17 41-12 •3 39 004 « 41..'.2 M 19.580794 106.31 20.326907 208.10 1.323 8696 41-10 "•339 2573 42.20 II • 9. 591 1 74 106.35 20.339400 208.14 .314 1294 43.28 •139 5105 42.18 n 19.605556 106.38 20.351895 108.17 .324 3X90 43.26 •3 39 7'i35 .340 0165 42.17 i:i • 9.617939 206.41 20.364192 20.376891 108.30 .324 ''4K5 4324 42.15 , II 19.630325 106.44 108.33 .324 9079 43.21 .340 1693 42.13 l.-i 19.642713 206.47 20.389192 20.401X95 108.36 1.325 1672 4j.11 1.340 5111 41.11 Id 19.655101 106.50 108.39 .325 4263 43.19 •340 7747 41.10 IT 19.667193 I9.6798.<6 106.53 20.414399 20S.43 .315 f'854 41.^7 .341 0171 42.08 IN 106.57 20.426906 108.46 •125 9441 41'5 .341 1796 42.06 I'J 19.691281 106.60 20.439415 108.49 .316 2032 43-' 3 •34^ 5 3"9 42.04 50 19.704678 106.63 106.66 10.451925 108.51 1.326 4619 43.11 1. 341 784' 42.03 51 19.717076 20.464437 108.56 .326 7205 43.09 .342 0362 .342 2882 42.01 yi «9-729477 106.69 20.476952 208.59 108.61 .326 9790 43-07 41.99 53 19.741879 106.71 20.4X9468 •327 2374 43.05 •342 540" 4«97 51 19.754183 106.75 10.501986 108.65 •327 4957 43.04 -3V2 79 "9 41.96 55 19.766689 106.78 10.514506 108.69 1.327 7538 43.02 1.34, 0436 4»94 5)1 19.779097 106.81 10.517019 108.71 .328 0119 43.00 •343 2951 41.91 57 19.791507 106.84 20.519553 108.75 108.78 .318 1698 41.98 •343 5467 41.90 SH 19.803919 19.816331 106.88 10.552079 .318 5176 42.96 •343 79«o 41.89 5U 106.91 10.564607 108.81 .318 7853 42.94 •344 0491 41.87 QO 19.818747 106.94 20-S77137 108.8s 1.319 0430 41.91 1.344 300$ 41.85 573 TABLE VI. For fimliiiK the True Annmiily or thf 'I'liiu- from tlie Perihelion in .1 Parabolic Orliil. 1 t 1 0' 32 33 34 35 1.344 3oo<; Dlff. 1". loK M. 1.359 1859 I)iff. 1". log SI. Dlff. 1". IngM. Diir. 1". 39.06 41.X5 40.86 '•373 7251 39-93 ..387 941S 1 ■U4 SSI? 4i.l'4 •359 43'° 6760 40.84 •373 9646 39.91 .388 1761 39-'-j5 'i ■in ><'^i5 41.82 •359 40.82 •374 2041 39.90 .388 4104 39-^4 :i ■:?n osu 41.80 •359 9209 40.81 •374 4434 39.88 .388 6446 3902 4 •345 304" 41.78 .360 1657 40.79 •374 6827 39.87 .388 8787 39.01 6 "•345 lU* 4'-77 1.360 4104 40.78 '•374 9218 39.85 1.389 1127 38.99 a •345 X053 4«^75 .360 6550 40.76 •375 1609 39.84 •389 3466 38.98 i I .34(1 oi,-K 4'-73 .360 S995 40.74 •375 3999 6388 39.82 -389 5804 38-97 ! ** .346 3061 41.72 .361 '439 40.73 •375 39.81 -389 8142 38.95 i 1) .346 S564 41.70 .361 3883 40.71 •375 8776 39^79 •390 0479 38-94 10 1.346 806^ 41.68 1. 361 6325 40.70 1. 3^6 1164. 39.78 1.390 2815 3893 II •347 oS<'S 41.66 .361 876(1 40.68 .376 3550 39^77 •390 5150 38.9, Vi •U? io('i 41.65 .362 1207 40.66 •376 5935 39^75 -390 7484 38. 90 > 1:1 •347 55^'3 41.63 .362 3646 40.65 .376 8320 39^74 .390 9817 3S.88 i 11 .347 SoDO 41.61 .362 6084 40.63 •377 0703 39.72 .391 2150 38.87 15 1.34S 05^-7 41.60 1.362 8522 40.62 '•377 3086 39-7' '•39' 4482 38.S6 1({ .348 30^2 41.58 .363 0959 40.60 •377 5468 39.69 •39' 6813 38.84 17 .34X 5546 41.56 .363 3394 40.59 •377 7849 39.68 •39' 9'43 38-83 IH .348 8040 4'^55 .363 5X29 40';7 37« 0230 39.66 •392 1472 38.82 lU •349 o53i 4«-53 .363 X263 40.56 .378 2609 39.65 .392 3801 38. So ao '•349 3013 41.51 1.364 0696 40.54 1.378 4987 39.64 1.392 6128 38.79 '^i •349 S5«3 41.50 .364 3128 40.52 .378 7365 39.62 .392 8455 38-77 'Zt •349 *> •351 7795 41.28 .367 465/ 40.32 .381 8 '94 39-43 -395 8631 3S.60 :i5 '•353 0*7* 41.26 1.367 7076 40.31 1.382 0559 39.42 1.396 0947 38^59 i :i(i •353 ^747 41.25 •3'''7 9494 40.29 .382 2924 39.40 .396 3262 38^57 , i :i7 •353 5"! 41.23 .368 19.1 4028 .382 5288 39-39 •396 5576 38.56 ;w •353 7C'94 41.21 .368 4327 40.26 .382 7651 39^37 .396 7889 38-53 :iu •354 01 "7 41.20 .368 6742 40.25 .383 0013 39-36 •397 0201 38-53 10 1.354 ;.6 58 41.18 1.368 9'57 40.23 1.383 2374 39-35 '•397 25 r 3 38.52 4! ■354 5108 41.16 .369 I 570 40.21 .3S3 4734 39-33 •397 4823 3851 42 •354 757** 41.15 .369 3983 40.20 •3i^3 7093 39^3 2 •397 7'3 , 38-49 4:t •355 004'' 41.13 .369 6394 40.18 •3i*3 9452 39^3° •397 9442 38-48 44 •355 *5'3 41. II •3 ('9 8805 40.17 .384 1809 39^29 -398 1751 38.47 45 1.355 49*'° 41.10 1.370 1214 40,15 1.384 A166 6522 39.27 1.398 >'.o58 38-45 40 •3 55 7445 41.08 .370 3623 40.14 .384 39.26 •398 (',-,65 38-44 47 .355 9909 41.07 .370 6031 40.12 .384 8878 39-25 .398 86', ( 38.43 4H •35'' 2373 41.05 .370 8438 40. n .385 1232 39-23 •399 <97'' 3 .4. lU .356 4836 41.03 •371 0844 40.09 .385 3585 39.22 •399 32' I 3S.40 50 1.356 7297 41.01 1.371 3249 40.08 1.38; 5938 8290 39.20 1.399 55^ 38.39 51 .356 9758 41.00 •37» 5654 40.06 .385 39.19 •399 7887 5^^I 52 .357 i2«7 40.98 ■37' 8057 40.05 .3S6 0641 39.18 .400 0189 38.36 5:1 •357 4''76 40.97 •372 0459 40.03 .386 2991 39.16 .400 2491 38.,- 54 •357 7«34 40.9s .372 2X61 40.02 .3S6 5340 39-15 .400 4791 38^33 55 1.357 9590 40.94 '•37?. 5261 40.00 1.386 7689 39-n 1.400 7091 38-32 5A .358 2046 40.92 •372 7661 39^99 .3S7 0036 39^ 1 2 .400 9390 38-3' 57 .358 A501 •35>^ 6954 40.90 •373 0060 39^97 .387 2383 39.11 .401 1688 ^S'^S 5H 40.89 •373 2458 4855 3996 •3«7 4729 39.09 .401 r>^5 18.18 5» .358 9407 40.87 •373 3994 .387 7074 39.08 .401 6282 38.27 (M) 1.359 '859 40.86 •■373 7251 3993 '•387 9418 39.06 1.401 8578 38.26 m iibolic Orldt. TABLE VI. For fmdinp tlie True Anoiniily or tlic Time from the Perihelion in a Piiraholic Orhh. 35= ?M. Dinr. 1 ". 9418 1 39.06 1761 39'-^5 4104 39-^4 6446 3902 S787 i 39.01 1127 3X99 3466 3X.9X 5«04 ; 3^-97 XI42 38.95 0479 3X.94 281? 3«93 5IS0 3X.9, 74«4 1 3X.90 9817 1 3X.X8 2150 3X.X7 4482 i 3X.X6 6813 3X.X4 9143 3X.X3 , 1472 3X.X2 I 3801 3X.S0 . 6128 38.79 - 8455 3>''77 ! 07X1 3X.76 J 3107 3>*-75 5 543» 3>'-73 5 7755 ■jX.72 ^ 007X 38.71 1 2400 3S.69 ^ 4721 3X.6X h 704' 3S.67 ^ 9361 3X.65 ; 16X0 3X.64 ; 399i< 3X.f,3 ; f'3'5 3S.61 ; «('3' 3X.60 ' 0947 3X.59 ) 3262 3!<-57 S S576 3X.56 S 7XX9 1 ^!!-'' 7 0201 3i<-53 7 25'3 ' 3^'5^ 7 4'<^3 3X5 1 7 713-. : ^'^->'! 7 944^ •;X.4X if 1751 i 3^'47 X .-.OS 8 3!<-45 8 rtift^ ' 3^-44 « X(.', r ! ^U^ 9 ^97' ! 3'^-4' 9 3i'i 3X.40 9 5584 3X.39 9 7««7 38.37 0189 ! 3^-3'^ 2491 i 3X.,- 4791 1 3>'-3i 7091 1 38. V- 9390 1 3«'3> I 1688 1 38.30 I 3985 i ,8.2X I 6282 i 3*''-7 I 8578 1 38. 26 r. 'Z A 4 5 » 7 H 9 10 II 12 i:i II 15 l(i 17 IH lU 20 21 'i'i 'iA 21 2.-1 2« 27 2H 2S» :<() :u \n :ii ;{.■> :i7 :iH :t!) 10 II 12 r.i II 15 i(> ir IS i'.) 50 51 52 5:1 51 55 :>(( 57 5H 5» QO 36^ IngM. 401 8578 ^02 0X73 402 3i()7 401 5460 402 7753 403 004^ 403 2336 403 4626 403 6916 403 9205 404 1493 404 37X0 404 6067 404 X352 405 0637 Diff. 1". 405 405 405 405 40(1 406 406 406 407 407 407 407 408 40X 40X 408 408 409 409 409 409 410 410 410 410 410 4" 411 4" 411 412 412 412 412 4«3 413 2392 413 4649 413 6905 413 9161 414 1416 414 3670 414 5924 414 8176 415 0420 415 2680 2921 5205 748X 9769 2051 433' 6()ii 8X89 116X 3445 5721 7997 0272 1547 4820 7093 93''5 1636 3907 6177 8446 0714 2981 5248 7S«4 9780 2044 430X 6571 8833 1095 3356 5616 7«75 0134 1.415 4930 37.50 38.26 3X.24 3X.23 38.22 38.20 38.19 38. 18 38., 7 38.15 3X.14 3>*-i3 38. 12 3X.10 38.09 38.08 38.06 38.05 3X.03 38.02 38.01 38.00 37-99 37-97 37.96 37.95 37-94 37.92 37-91 37.. o 37-89 37-87 37.86 37-85 37.84 37.82 37.81 37-80 37-78 37-77 37.76 37-75 37-74 37.72 37-7« 37.70 37.69 37.68 37.66 37-65 37.64 37.63 37.61 37.60 37-59 37-58 37-56 37-55 37-54 37-53 37-5< 37= loK M. 4«5 4'5 415 416 416 416 416 4'7 4'7 4«7 4«7 4'7 41X 4.x 41X 418 419 419 419 419 4930 7180 9429 167X 39^5 6172 84 > 9 0664 2909 5'53 7:96 9639 18X1 4122 6362 X602 0841 3079 53'7 7554 419 9790 420 2026 420 4260 420 6494 420 8728 421 0960 421 3192 421 5423 421 7654 421 9884 422 422 422 422 423 423 423 4-3 423 424 414 424 4^4 425 4^5 425 4^5 4^5 426 426 426 426 427 427 4*7 427 4i7 42X 428 428 428 2113 434« 6569 X796 1022 3248 5473 7697 9920 2143 4365 65X6 8X07 1027 3246 5465 7683 9900 21 17 4333 6548 8762 0976 3189 5402 7613 9824 2035 4244 6453 8662 1)1 ft. 1". 37-50 37-49 3747 37.46 37-45 37-44 37-43 37-4' 37.40 37-39 37-3^ 37.37 37-36 37-35 37-33 37-3* 3;-3' 37-30 37-29 37-17 37.26 37-25 37.24 37-23 37.22 37.20 37-19 37-18 37-17 37-16 i 37-15 37-13 37.12 37-11 37-10 37.09 37.08 37.06 37-05 37.04 37.03 37.02 37.01 36.99 36. 9X 36.97 36.96 I 36-95 ! 36.94 i 36-92 I 36-91 36.90 1 36.89 I 36.8;! I 36.87 36.86 36.85 36.83 36-82 36.X1 36.80 av3 38= loK M. Diff. 1" 1.428 .429 .429 .429 .429 1.429 .430 .430 .430 .430 I.431 •431 -431 -431 •43' 1.432 •432 .432 .432 •433 1-433 •43 3 -43 3 •433 •434 1-434 434 4.^ 435 435 1-435 -435 •435 -436 -436 1.436 -436 .436 ■437 •437 1-437 -437 .438 •438 .438 1-438 438 439 439 439 1-439 440 440 440 44° 1.440 -44' •441 •441 •44' 1.44! 9943 8662 0X69 3076 5281 7488 9693 1X97 4101 6304 X506 0708 2909 5 1 09 730X 9507 1705 3903 6100 X296 0491 26X6 4X8 1 7074 9267 '459 3651 5X42 So 3 2 0221 2410 4598 67X6 ^97 3 1159 3345 5530 7714 9X9X 20X1 4263 6445 X626 0X06 29S6 5165 7 344 9522 i6q9 3^75 6051 8226 0401 2575 4748 6921 9093 1264 3436 5605 7774 36.80 36.79 36-78 36-77 36-75 36-74 36-73 36.72 36.71 36-70 36.69 36. 6X 36.66 3"-6,- 36.64 36.63 36.62 36.6 1 36.60 36-59 56-57 36-56 3 ''-5 5 36-54 36-53 36-52 36.51 36-50 36-49 36.48 36-47 36.46 36.44 36.43 36.42 36.41 36.40 36-39 36-38 36-37 36.36 36-35 36-34 36.32 36.31 36.30 36.29 36.28 36-27 36.26 36.25 3(1.24 36.23 36.22 36.20 36-19 36.18 36.17 36.16 36.15 36.14 39= '-447 447 447 448 448 1.448 -448 -448 -449 •449 1.449 449 ■1-V9 450 .450 1.450 •450 451 -451 .451 1.451 •451 •452 .452 •452 I 452 .4,-2 •453 ■453 •453 1^453 •454 •454 ■454 •454 loK M. '•44' 9943 .442 21 1 1 .442 .1279 .442 6446 .442 8612 1.443 0778 •443 2943 •443 5 '07 ■443 727' •443 9434 1.444 1597 I -444 3758 -444 5920 .444 8cXo •445 0240 i 1-445 2400 , •445 4558 I -445 671" I .445 8874 .446 1031 1.446 3187 ; •446 5343 ; .446 749 X ' .446 9652 •447 1806 I Diff. 1". 3959 (1112 8263 0415 2565 4715 6S65 9014 1162 3309 5456 7603 9749 1X94 4038 61X2 8325 0468 2610 4752 6X93 9033 1173 3312 5450 7588 9725 1X62 3998 6134 X269 0403 2537 4670 6802 36.14 36.13 36.12 36.1 1 36.10 36.09 36.08 36.07 36.06 36.05 36.04 36.03 36.02 36.00 3 5^99 35.98 3 5-97 3 5-';6 35-95 35-94 35 93 55 92 5-91 3 5 90 35.89 35-88 35-87 3 5 ^'6 35-85 35-84 3583 35-82 35.81 35.80 35-79 35-78 35-77 3v76 35-75 35-74 35^73 35-72 35-7' 35^70 3569 3 5-68 35-67 35-66 3565 35.64 35.63 35.62 35.61 35-60 35-59 35-58 35-57 35-56 35-55 35-54 1-454 8934 I 35.53 TABLE VI. For finding the True Anonisily or tin- Tinip from the Perihelion in a Paralxjlic <)rl)it. .1 1 40 ° 41 l<'i. 42 M. IMff. 1". 34-41 43 loK M. DIff. 1". 35-53 1..K M. 1.467 5781 DIff. I".. I..R M. '•492 3597 Diff. 1", 0' 1.454 8934 34^95 1.480 0627 33-9' 1 •45 5 «°''5 35^52 .467 7879 34^94 .480 2691 3440 .492 5631 33.90 2 ■455 3*')(' 35-51 .467 9976 34^93 .480 4755 34.40 .492 766,- 3389 3 •45? 5326 35-5° .46 S 2071 34-92 .4>'o 6819 34-39 .492 969N 33-88 4 •455 7456 35-49 .468 4ib6 34-9' .^8o 8882 34.38 •493 '73' 33-87 3 '•455 9585 35.4« 1.468 6261 34.90 1.481 0944 34^ ^7 '•493 3764 33-87 .456 1713 35-47 .468 83 5 5 34.90 .48, 3006 34^36 •493 5796 33.Sf, 7 ■45'^ 3SA1 •456 59"** 35-4<' .469 0448 3489 .481 5068 34^35 •493 7827 3 3-><5 8 35-45 .469 2541 34.88 .481 7129 34^U •493 9858 3)-"i4 .456 >lo.;4 35-44 .469 4634 34^87 .481 9189 34-33 •494 1888 33-83 1 10 1.457 0220 3543 1.469 6725 34.86 1.482 1249 34-33 '494 39 '8 33-83 11 •457 *34<' 35-42 .469 8S 17 34.S5 .482 3308 34-32 •494 5948 33.82 12 •457 U70 •457 6595 35^4« .470 0907 34.84 .482 5367 34-3' •494 7977 33-81 ]» 3 5-4^ .4-70 2998 34-83 .482 7425 34-30 ■495 0005 33-80 M •457 87' « 35-39 .470 5087 34-82 .482 9483 34-29 •495 2033 33-79 13 1.45X 0841 35-38 1.470 7176 34.81 1..V83 1540 34.28 '•495 4061 '>3-79 : 10 .458 2964 35-37 .470 9265 34.80 •483 3 597 34.28 •495 608S 33--8 17 .458 50S6 35-3*' ■47' 1353 34--^9 ■4'' 3 5653 34-27 •495 8114 33-77 18 .45S 7207 35-35 -47' 3440 34-79 ■48' 7709 34.26 .496 0140 33-76 M> .458 9328 3534 -47' 5527 34-78 .483 9764 34-25 .496 2166 33-75 20 1.459 '44« 3533 1.471 7613 34-77 1.484 1819 34.24 1.496 i"'l 33 75 21 •459 3567 35-3* .471 9699 34.76 -484 3873 3423 .496 6216 33-74 22 •459 5'''**' 353« ■472 1784 34-75 -484 592- 34-22 .496 8240 33-73 2;» •459 71^05 35^3o -472 3869 34-74 -484 7980 34-22 •497 0264 33-72 24 •459 992i 35^29 -472 595 3 34-7 3 -485 0033 34.21 •497 2287 33-7' 23 1,460 1040 35.28 1-472 8037 34-73 1.485 2085 34-20 1.497 4310 33-71 20 .460 4156 35^27 .473 0120 3472 •485 4'37 34- '9 •497 6332 33-70 27 .460 6272 35^*6 -473 2203 34-7' -485 6188 34- '8 •497 8354 33.09 28 .460 8388 35-15 -473 4285 34-70 •485 8239 34-'7 •498 0370 3r(-S 1 2» 1 .461 0503 35-24 .473 6366 ^4.69 .486 02S9 34.16 .498 2396 3 3.(,,< 1 ao 1.461 2617 35-23 1.473 8447 34.68 1.486 2338 34.16 1.498 44' 7 33-67 ! :ii .461 A731 .461 6844 35-23 .4-4 0527 34^''7 .486 4388 34 '5 •498 643- 33-66 1 :{2 35-22 .474 2607 34.66 .486 6436 34' 4 •498 8456 33-65 j Xi .461 8957 35-2« .474 4686 34-65 .486 8484 34-13 ■499 0475 33-''3 34 .462 1069 35-20 •474 6765 34.64 •487 0532 34.12 ■499 2494 33-6+ 35 1.462 3180 35-19 1.474 8843 34-63 '-487 2579 34.12 1.499 4512 33^63 1 30 .462 5291 35^i8 •475 0921 34.62 .487 4626 34.11 -499 6530 33-62 1 37 .462 7401 35-17 •475 2998 34.61 -487 6672 34.10 -499 8547 33.62 j 38 .462 9511 35.16 -475 5075 34.61 .487 8718 34.09 .500 0563 33-61 3» .463 1620 35-'5 •475 7«5' 34.60 .488 0763 34.C8 .500 2580 33.60 i lO 1.463 3729 35-«4 1.475 9227 34-59 1.488 2807 34-07 1.500 4595 33-59 1 *» .463 5837 35-13 .47I'' 1302 34-58 .488 4852 34-°7 .500 6611 3 5-5|; 1 ■»*■« •46: 7944 35.12 .476 3376 3457 .488 6895 34.06 .500 8625 33-58 : »3 ,464 0051 35.11 .476 5450 34-56 .488 8939 34.05 .501 0640 33-57 44 .464 2158 35.10 -476 7524 34-55 .489 0981 3404 .501 2654 33-56 43 1.464 ^263 .464 6369 35.09 1.476 9596 34-54 1.489 3023 34-03 1.501 ll'o^ 33-55 40 35.08 .477 1669 34-54 .489 5065 34.02 .501 6680 33-55 47 .464 8473 35-07 •477 374' 34-53 -489 7106 34.02 .501 8693 33-54 48 .465 0577 35.06 .477 58'2 34-52 -489 9'47 34.01 .502 0705 33-53 4U .465 1681 3S-°5 •477 7883 34-5' -490 1187 34.00 .502 2716 3352 50 1.465 4784 .465 6886 35-04 '•4-'7 9953 34.50 1.490 3127 33-99 33.98 1.502 i''^l 3351 31 35^04 .478 2023 34-49 -490 5266 .502 ^7 3,^ 33-5" 52 .465 8988 35-03 .478 4092 3448 .490 7305 33-97 -502 8748 33-5° 33 .466 1090 35-02 .478 6161 34-47 .490 9343 3396 .503 0758 33-49 34 .466 3190 35-01 .478 8229 34.46 •49' 1381 33-95 .503 1767 33-48 35 1.466 5290 35.00 1.479 0197 34.46 1.491 3418 33-95 1.503 ml 33.48 50 .466 7390 34-99 34-98 •479 2364 34-45 -49' 5455 33 94 .503 33-47 57 .466 9489 -479 4-^30 -479 649'' 34-44 -49' 749' 3 3-93 .503 8792 33.46 ' 58 .467 1587 34-97 34-43 -49' 9527 3392 .504 0800 33-45 I 50 .467 3685 34.96 •479 8562 34.42 .492 1562 339« .504 a8o7 33-44 00 1.467 578Z 3495 1.480 0617 34-41 1.492 3597 35-9« 1.504 48.3 33-44 576 Ixdif Orbit. TABLE VI. For finding the True Anomaly or tiie Time from tiie Perihelion in a Paralwlic Orhit. 43^ M. j Diir. 1". 3597 33-91 S''3> r,.')o 766 q r.^') 96.>S yyXi 1731 33-X7 37<''4 33''7 5796 33^'' 7X17 33.!*^ ,,s,x 3>'M iXXS 33-^3 3ViX 33-^3 594X 33.X2 797-^ 33.S1 000; 33.80 2033 3379 4061 ^3'79 608S 3 3-7X i!li4 3 3- '7 2166 33-7'' 33-75 4191 33 75 6216 33-74 8240 3 3-" 3 0264 33-7 = 2287 33-7« 4310 33-71 f'33z 33"3 i*3>4 33.(19 03 7(1 33. hS 2396 33.(,S 44>7 33.6- ''437 33-''(' 8450 3)-<'5 047 i 33-"5 2494 33-''4 4511 33-'^3 6530 33.1.2 8547 3V(u 05 'n 33.6, 2580 33.1-0 4i9S 33->9 6611 3^-^** 862i 33^'* 0640 33-57 2654 33-51' ¥)V 33-55 67 33-55 80 33-';5 9 3 3'v^+ 05 33-53 lb 3 5-5- i7 33=;' ^0 33-5' 4X 33-50 5X 33-49 67 33-4-^ 76 33.48 84 3 3-47 91 33.46 00 33-45 07 33-44 '3 ' 33-44 r. O' I •i » 1 5 A 7 H U 10 II Vi 1:) II 15 10 17 IN H) W •il 'i'i •Z\l •il !i5 'Hi •Z7 'iH 'i9 30 31 M 33 31 35 3U 37 3N 3U 40 II i'i 43 41 45 411 47 4N 41) 5(> 51 5,4 53 51 55 5(1 57 58 5U no 44 1..K M. niff. 1". 504 4813 33-44 i;o4 6X19 33-43 504 8825 33-4» 505 0830 33-42 505 2835 33-41 505 4839 33-40 505 6843 33-39 505 884(1 33-39 506 0849 33-3« 506 2852 33-37 506 A854 506 68^1; 33-36 33-36 506 88i;() 33-35 507 0857 33-34 507 2857 33-33 507 4^57 33-33 i;o7 6856 33-32 507 8855 33-3' 508 o8i;3 33-30 508 -^851 33.29 508 4849 33-29 508 6846 33.28 508 8843 33-27 509 0839 33-27 509 2835 33.26 509 4830 33-25 509 6825 33-24 509 8819 33-24 510 0813 33-23 510 2807 33.22 510 4800 33.21 510 6792 33-21 510 8785 33.20 511 0776 33-»? 33.18 511 2768 5'« 4759 33.18 511 6749 33-17 511 8739 33-16 512 0729 33-»5 512 2718 33->S 512 4707 33-'4 512 6695 33-«3 i;i2 8683 33-13 513 0670 33 12 5«3 2657 33-11 5' 3 4644 33.11 513 6630 33.10 5'3 J"H5 33.09 514 0601 33.08 514 2586 33.07 514 4S70 33.07 5'4 6554 33.06 5'4 'S-37 33-05 i;!'; 0520 33-05 5«5 1503 33.04 5»5 44«5 33-04 515 6467 33-03 5«5 i<449 33-02 516 0430 33.01 516 2410 33.01 516 4390 33.00 45 ° l..g M. DIPT. 1". 5.6 4390 33.00 516 6370 32.99 516 8349 32.98 517 0328 3298 517 2306 32.97 517 4284 32.96 517 6262 32.96 5'7- .8239 32-95 518 0216 3294 518 2192 32-93 518 .J 1 68 6143 32.93 ,18 32,92 S.8 8118 32.91 5'9 0093 32.91 5«9 2067 32.90 5«9 4041 32.89 519 6014 32.89 519 7987 32.88 519 9960 32.87 520 1932 32.86 520 3904 32.86 S20 5875 32.85 520 7X46 32.84 S20 9816 32.84 521 1786 32.83 521 3756 31.82 521 5725 32.82 521 7694 32.8. 521 9662 32.S0 522 1630 32.80 522 3598 32-79 522 55<'5 32-78 522 7531 32.78 522 9498 32.78 523 1464 32-77 5*3 3429 32-76 523 5 394 32-75 523 7359 32-74 523 9323 32-73 524 1287 32-73 524 3251 32.72 524 5214 32.71 524 7176 32.71 524 9138 32.70 525 1 100 32.70 525 3062 32.69 525 5023 32.68 525 6983 32.67 525 8944 32-67 52b 0903 32.66 ,26 2863 32.65 ^26 4822 6780 32.64 526 32.64 ,26 8739 32.63 527 0696 32.62 527 2654 32.62 527 527 461 1 6567 32.61 32.61 527 8524 32.60 528 0479 32.60 528 2435 32.59 46= log M. Dlff. 1". 28 2435 28 4390 28 6344 28 8299 29 0252 29 2206 29 4159 29 61 12 29 806J. 0010 30 30 1967 30 3918 30 5869 30 7819 30 9769 31 1719 3« 3668 If 5616 31 7565 3' 9513 32 1460 32 34°7 32 5354 32 7300 32 9246 33 33 33 33 33 34 34 34 34 34 35 35 5 5 35 35 36 36 36 36 37 3' 37 37 37 37 38 38 ^i 38 38 39 39 39 39 539 1192 3'37 50X2 7027 8971 0914 285X 4801 67.n 8685 0627 2568 4509 6450 8 390 0330 2270 4209 614X 8o86 0024 1962 3X99 5836 7772 9708 1644 3579 55'4 7449 9383 1317 3250 5183 71 16 9048 32-59 32.58 32.57 32.57 32.56 32.55 3255 32.54 32.53 32.53 32.5a 32.5" 32.51 32.50 32.49 32.49 32.48 32.48 32-47 32.46 32.46 32.45 32.44 3244 32-43 32-43 32.42 32.42 32.41 32.40 32.39 32-39 32-38 32-37 52.37 32-36 32-35 32-35 32-34 32-33 32-33 32-32 32.32 32.3' 32.30 32.30 32.29 32.28 32.28 32.27 32.26 32.26 32.25 32-25 32.24 32.23 32.23 32.22 32.21 32,21 32.20 47c )«K M. 5 39 9048 540 0980 540 2912 540 4843 540 6774 540 8705 541 0635 541 2564 54 « 4494 54' 6423 541 8352 542 0280 542 2208 542 4135 542 6063 542 7989 542 99 "6 543 18^2 543 3768 543 5693 543 7618 543 9543 544 '467 544 3391 544 53 '5 544 7238 544 9161 545 1083 545 3005 545 4927 545 6849 545 8770 546 0690 546 7.6 1 1 546 4531 546 6450 546 8370 547 0289 547 2207 547 4125 547 6043 547 7961 547 9878 548 1795 548 37«i 548 5627 548 7543 548 9458 549 '373 549 328X 549 5202 549 7"6 549 9030 550 °943 550 2856 550 A769 550 6681 550 8593 55' 0504 55' 2416 551 4326 I 31.85 Dlff. 1". 2. 20 2.20 2.19 2.18 2.18 2.17 2.17 2.16 2.15 2.15 2.14 2.14 2.13 2.12 2. 1 I 2.1 I 2.10 2.10 2.09 2.09 2.08 2.08 2.07 2.06 2.06 2.05 2.04 2.04 2.03 2.03 2.02 2.02 2.01 2.00 2.00 .99 .98 .98 .97 .97 .96 .96 .95 ■94 -94 -93 93 .92 .91 .91 .90 .90 .89 .^)^ .88 .87 .87 .86 .86 .85 ■il 577 TABLE VI. For fiiuling the True Anomaly o> tlic Time from the Perihelion in a Parabolic Orbit. *♦. O' I a » 1 3 G 7 8 U lU 11 12 i:i 14 13 IG 17 18 10 SO '^1 22 2.-1 21 23 2G 27 28 2a :¥) :ii :i2 33 34 33 3ii 37 38 30 40 41 42 43 44 43 4G 47 48 40 50 31 32 33 34 33 30 37 38 30 GO 48 l..g M. DllT. 1". '•55" 43*6 ■55' 6*37 31.85 31.84 •55" 8147 31.83 .552 0057 3'-«3 .552 19(16 31.82 1.552 3876 31.82 •552 5784 31.81 .552 7693 31.80 .552 9601 31.80 •553 '5<^8 3'^79 1.553 34'6 3'79 •553 5323 31.78 •553 7230 31.78 •553 V'3'' 3'^77 •554 '042 31.76 •554 4948 31.76 •554 4853 •554 ''758 3'-75 3'^75 •554 8663 3i^74 •555 o5''7 3'^74 1.555 2472 3i^73 •555 4375 31.73 •555 6279 31.72 .555 81X2 31.71 .556 0084 1.556 1987 31.70 .556 388X 31.70 .556 5790 31.69 .556 7691 31.68 .556 9592 31.68 '•557 1493 31.67 •557 3393 3'^67 •557 5293 31.66 •557 7«93 31.66 .^57 9092 3'^65 1.558 0991 31-65 .558 2S90 3'.64 .558 4788 31.64 .558 6686 31.63 •558 8584 31.62 '•559 0482 31.62 •5 59 2379 31.61 •559 4275 31.61 •559 6'72 31.60 .559 8068 31.60 1.559 9963 3«-59 .560 1859 3>-59 .560 3754 3>^58 .560 5648 3'^S7 .560 7543 3I-57 1.560 9437 31.56 .561 1331 31.56 .561 3224 3>^5S .561 S117 3«-5S .561 7010 3>54 1.561 890; 3>-54 .562 070A .562 2686 3'-S3 3i^53 .562 4578 .562 6469 3«-S2 31.52 1.562 8360 3'5' 49' loK M. »I(T. 1 ' 562 8360 563 0250 563 2140 563 4050 563 5920 563 7S09 563 9698 564 15S6 564 347 5 564 5363 564 564 565 565 565 565 565 566 566 566 566 566 566 567 567 56- 567 567 568 568 568 '568 568 569 569 569 569 569 569 570 7250 9' 38 10^5 291 1 4798 66X4 8569 °455 2340 4225 6109 7993 9877 1761 3644 5527 7409 9291 "73 3°55 4936 6817 8698 0579 2459 4338 6218 8097 9976 1854 570 3733 57° 56" 570 7488 570 9366 571 1243 1.571 3119 57' 4996 ,571 6872 571 8748 572 0623 572 2499 572 4373 572 6248 572 8123 572 9997 573 '870 573 3743 573 5616 573 7489 573 9362 1.574 1234 5» 5' 50 50 49 48 48 47 47 46 46 45 45 44 44 43 43 42 4' 4> 40 40 39 39 38 38 37 37 36 36 35 35 34 34 33 33 32 32 3' 3° 30 29 29 28 28 28 27 27 26 26 25 25 24 24 23 23 22 22 21 .21 31.20 578 50' lot? M. 1.574 I23A 574 3'°6 574 4977 574 6849 574 8720 Dtir. 1" 575 575 575 57 5 575 575 576 576 576 576 576 577 577 577 577 577 578 57i! 578 578 578 578 579 579 579 579 579 580 580 580 580 580 581 58, 581 58, 58, 581 582 582 582 582 582 583 583 584 584 584 584 584 584 585 58s 0590 2461 433' 6201 8070 9939 180X 3677 5546 74'4 9281 "49 3016 4883 6749 I 8615 i 0481 I 2347 I 4213 I 6078 ! 7942 ' 9807 I 1671 3535 i 5399 I 7262 9125 0988 2851 47'3 6575 8436 0298 2' 59 4020 5880 7740 9600 1460 ' 33'9 ; 5179 7037 8896 ! 0754 I 2612 j 447° 1 6327 1 8184 I 0041 j 1898 3754 i 5610 7466 9321 1176 3031 .20 .20 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 O o .09 .09 .08 .08 .07 .07 .06 .06 .06 .05 .04 .04 .03 .03 .03 .02 .02 .01 .01 .00 .00 30.99 30.99 30.98 30.98 30.97 30.97 30.96 30.96 3<"-95 30.9s 30.94 30.94 30.94 30.93 3"^93 30.92 30.92 30.91 51 Ion M. 1.585 3031 585 48X6 585 6740 585 859^ 5X6 0J4S 586 586 5X6 5X6 586 587 587 587 587 587 588 1:88 5XX 5X8 588 589 589 589 589 589 589 59° 590 590 59° 590 591 59« 59' 591 591 59' C92 59' 592 592 592 593 593 593 593 593 593 594 594 594 5429 594 7270 594 91" 595 °952 595 a792 595 4633 595 6473 595 83'2 596 0151 596 1990 596 3829 2302 I 4' 5 5 ' 6008 7859 i 97' 3 i '565 I 3417 I 526X 7120 8971 0821 2672 4522 6372 8222 0071 1920 3769 5618 7466 I 93'4 j 1162 I 3009 I 4857 i 6704 I 8550 I 0397 I 2243 I 4089 ; 5935 I 7780 j 9625 I 1470 I 33'5 ] 5'59 I 7003 ! 8847 ! 0690 j 2534 4377 6219 8062 9904 1746 3588 DIff. 1". 30.91 30.91 30.90 30.00 30.89 30.89 30.89 30.88 30.87 30.87 30.87 30.86 30.86 30.85 30.85 30.8+ 30.84 30.83 30.83 30.83 30.82 30.82 30.81 30.81 30.80 30.80 30.79 30.79 30.78 30.78 30.78 30^77 30.77 30.76 30.76 3°^75 30^75 3°^75 30^74 3°^74 30.7] 30^73 30.72 30.72 30.72 30.71 30.71 30.70 30.70 30.69 30.69 30.68 30.68 30.68 30.67 30.67 30.66 30.66 30.65 30.65 30.65 bolic Orbit. TABLE VI. For finding the True Anomaly or (he Time from the Perihelion in a Parabolic Orbit. 51^ M. 5031 ■ 48X6 6740 i OJ48 ' 2302 4«55 600S 7«59 i 97' 3 I 1565 34»7 5268 7120 8971 0821 2(171 4S-* 6372 8222 I 0071 I 1920 I 3769 ( 561X I 7466 93J4 1162 3009 4857 6704 8550 0397 2243 4089 5935 \ 7780 I 9625 1 1470 I 33' 5 1 S'59 I 7003 ; 8847 I 0690 ! 1534 ; 4377 I 6219 i 8062 9904 1746 3588 54»9 7270 9111 0952 2792 >5 4633 )5 H7 3 ,5 8312 j6 0151 96 199° 96 3829 Diff. 1". 30.91 3c..) I 30.90 30.90 30.89 30.89 30.89 3C.X8 30.87 30.87 I 30.X6 i 30.86 1 30'i*> I 3°-i'5 i 30-i'4 \ 30.84 ! ^°-^ I 30.83 1 30.83 1 3°-8^ I 30.82 I 30.X1 . 30.S1 1 30.80 30.80 30.79 30.79 30.78 30.78 30.7X 30'77 30.77 30.76 30.76 3°-7') 30.75 I 3°'7S i 3°-7+ I 3°-74 ; 3°'73 I 30'73 ! 3°-7^ i 3°-7i I 30'7i i 3°-7' - 30.71 30.70 30.70 30.69 30.69 30.68 30.68 30.68 30.67 30.67 30.66 30.66 30.65 30.65 30.65 r. 52 53 54 55 loK M. Diflr. 1". l"t M. DllT. 1". logM. Via. 1". logM. Diff.l". ' 0' 1.596 3829 30.65 1.607 3703 30.40 1.618 2724 30.17 1.629 0959 29.96 1 .596 5668 30.64 .607 5527 30.39 .618 4534 30.17 .629 2757 29.96 , •i .596 7506 30.64 .607 7350 30.39 618 6344 30.16 .629 4554 29.96 3 ■59<' 9344 .597 1182 30.63 .607 9 '74 30.39 .618 8153 30.16 .629 6351 29^95 4 30.63 .608 0-97 30.38 .618 9963 30.16 .629 8148 2995 j 5 1.597 3020 30.62 1.608 2820 30.38 1.619 1772 30.15 1.629 9945 29.95 •597 4**57 30.61 .6c8 4642 30.38 .619 3581 30.15 .630 1742 29.94 7 •597 6694 30.62 .6c8 6465 30-37 .619 5390 30,15 .630 353S 29.94 . H •597 ii53' 30.61 .608 8287 30.37 .619 7199 30.14 ,630 5335 29-94 1 9 .598 0368 30.61 .609 0109 30.36 .619 9007 30.14 ,630 7131 29-93 1 10 1.598 2204 30.60 1.609 '93' 30.36 1.620 0816 30.14 1.630 8927 29.93 ' II .598 4040 30.60 .609 3752 30.36 .620 2623 30.13 .631 0722 29-93 1*4 .598 5876 30.59 .609 5573 30.35 .620 4431 30.13 .631 2518 29.92 : i:i .598 77 1 1 30.59 .609 7394 30.35 .620 6239 30.12 .631 4313 29.92 : II •598 9547 30^59 .609 9215 30^34 .620 8046 30.12 .631 6108 29.92 15 1.599 >3»a 30.58 1.610 'S^^ 30.34 1.620 9853 30.12 1.631 7903 29.91 la •599 31 '7 30.58 .610 2856 30.34 .621 1660 30.11 .631 9698 29.91 17 •599 505' 30-57 .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 l» •599 87 '9 30-57 .610 83,5 30.32 .621 7080 30.10 .632 5081 29.90 20 1.600 0553 30.56 1.611 0135 30.32 1.621 8886 30.10 1.632 6875 29.90 •il .600 2387 30.56 .611 '954 30.32 .622 0692 30.10 ,632 8668 29.89 •i'Z .600 4220 30^55 .611 3773 30.31 .622 2497 30.09 .633 0462 29.89 •a .600 6053 30^55 .611 559' 30.31 .622 4303 30.09 .633 2255 29.89 •z\ .600 7886 30^55 .611 7410 30.31 .622 6108 30.09 .633 4048 29.88 ' 25 1.600 9718 30.54 1.611 9228 30-30 1.622 7913 30.08 1.633 5841 29.88 2i( .601 1551 30.54 .612 1046 30.30 .622 9718 30.08 .633 7634 29.88 27 .6oi 3383 30-53 .612 2864 30.29 .623 1523 30.08 ,633 9427 29.87 2H .601 5214 30-53 .612 4681 30.29 .623 3327 30.07 .634 1219 29.87 29 .601 7046 30.52 .612 6499 30.29 .623 5131 30.07 .634 3011 29.87 30 1.601 8877 30.52 1.612 8316 30.28 1.623 6935 30.06 1.634 4803 29.86 31 .602 0708 30.52 .613 0132 30.28 .623 8739 30.06 ,634 6595 29.86 32 .602 2539 30.51 .6,3 1949 30.28 .624 0543 30.06 ,634 8387 29.S6 33 .602 4370 30-5' .6.3 3765 30.27 .624 2346 30.05 ,635 0178 29.86 3t .602 6200 .30.50 .613 5582 30.27 .624 4149 30.05 .635 1969 29.85 : 35 1.602 8030 .602 9860 30.50 1.613 7398 30.26 1.624 5952 30.05 1.635 3760 29.85 30 30.50 .6.3 9213 30.26 .624 7755 30.04 -635 555' 29.85 37 .603 1690 30.49 .614 1029 30.26 .624 9557 30.04 .635 7342 29.84 38 .603 3519 30.49 .6,4 2844 30.25 .625 1360 30.04 .635 9132 29.84 39 .603 5348 30.48 .614 4659 30-25 .625 3161 30.03 .636 0922 29.84 , 40 1.603 7«77 30.48 1. 614 828^ 30.25 1.625 4964 .625 6765 30.03 1.636 2713 29.83 11 .603 9005 30-47 .614 30.24 30.03 .636 4502 29.83 42 .604 0834 .604 2662 30.47 .6,5 0103 10.24 .625 8567 30.02 .636 6292 29.83 43 30.47 .615 1917 30.23 .626 0368 30.02 .636 8082 29.82 41 .604 44 . :o.46 .615 373' 30.23 .626 2169 30.02 .636 9871 29.82 15 1.604 ''3'7 30.46 1.615 5545 30.23 1.626 3970 30.01 1.637 1660 29.82 4» .604 8145 30.45 .615 7358 30.22 .626 S77I 30.01 ■637 2449 29.82 47 .604 9972 30.45 .6,5 9171 30.22 .626 7571 30.01 .637 5238 29.81 48 .605 1799 30.45 .616 0984 30.22 .626 9372 30.00 .637 7027 29.81 49 .605 3626 30.44 .616 2797 30,21 .627 1172 30.00 .637 8S15 29.81 50 1.605 5452 3044 1.616 4610 6422 30,21 1.627 2972 30.00 1.63? 0603 29.80 .'il .605 7278 30.43 .616 30.20 .627 4771 29.99 .638 2391 29.80 :)2 .605 9104 30.43 .616 8234 30,20 .627 6571 29.99 .638 4179 29.80 ■>:i .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 .■|5 1.606 4581 30.41 1. 617 3669 30.19 1.628 1968 29.98 1.638 9542 19.79 29.78 1 50 .606 6406 30.41 .617 5481 30.19 30,18 .618 3766 2998 .639 1329 57 .606 8230 30.41 .617 7292 .628 5565 29.97 .639 31 16 29.78 58 .bo7 0055 30.41 .617 9101 30,18 .628 7363 29.97 .639 4902 29.78 59 .607 1879 30.40 .618 0913 30.17 .628 9161 29.97 .639 6689 29^77 6(1 1,607 3703 30.40 1.618 2724 30.17 1.629 0959 29.96 1.639 8475 29.77 _ 1 670 TABLE VI. For finiiing tlic True Anomaly or tlie Time from the Perihelion in a Parabolic Orbit. li V. 56 57 3 58 59 1 1 loK M. Wfr. 1". 29.77 l(.(l M. mir. 1". )..« .M. Diff. 1". loK M. Diff. 1'. ! o 1.639 8475 1.650 5336 29.60 1.661 1601 29.44 1.671 7331 29.30 1 .640 0262 29.77 .650 7112 29.60 .661 3368 29.44 .671 9089 29.30 a .640 204X 29.77 .650 8887 29.59 .661 5134 29-44 .672 0846 29.30 :i .640 3X33 29.76 .65. 0663 29.59 .661 6900 29.43 .672 2604 29.19 4 .640 5619 29.76 .651 2438 29.59 .661 8666 29.43 .672 4362 29.19 5 1.640 7405 29.76 1. 651 4213 29.58 1.662 0432 29-43 1.672 6119 29.29 » .640 9190 19-75 .651 5988 29.58 .662 2197 29.43 .672 7876 29.29 , T .641 097 s 29.75 .651 7763 .651 9538 29.58 .662 3963 .662 5728 29.42 .672 9634 29.2X 8 .641 2760 29.75 29.58 29.42 .673 1391 29.28 1 ^ .641 4545 29.74 .652 1312 »9-57 .662 7493 29.42 .673 3147 29.18 ! 10 I.641 6329 29.74 1.652 3086 29-57 1.662 9258 29.42 1.673 4904 29.28 I 11 .641 8114 29.74 .652 4861 i9-57 .663 1023 .663 2788 29.41 .673 6661 29.28 1 1« .641 9898 *9-74 .652 6635 *9.57 29.41 .673 8417 29.27 ; 13 .642 1682 29.73 .652 8408 29.56 .663 4553 29.41 .674 0174 29.27 1 »• .642 3466 29.73 .653 0182 29.56 .663 6317 19.41 .674 1930 29.27 : 15 1.642 5250 29.73 1.653 1956 29.56 1.663 8082 29.40 1.674 3686 29.27 ' V: .642 7033 29.72 .653 3729 *9-55 .663 9846 29.40 .674 5442 29.27 17 .642 8Xj6 29.72 .653 5502 i9-S5 .664 1610 29.40 .674 7198 29.26 18 .643 0599 29.72 .653 7275 29-55 .664 3374 29.40 .674 8954 29.26 1» .643 2382 29.71 .653 9048 *9-55 .664 5137 29.39 .675 0709 29.26 20 1.643 4165 29.71 1.654 °*'*' 29.54 1.664 6901 29.39 1.675 2465 29.26 { ^1 .643 5948 29.71 •654 2593 .654 4366 29.54 .664 8664 .665 0428 29-39 .675 4220 29.25 22 .643 7730 29.71 29.54 29.39 •675 5975 29.25 2:i .643 9513 29.70 .654 613S 49-54 .665 2191 29.39 29-38 .675 7730 29.25 21 .644 129s 29.70 .654 7910 29-53 .665 3954 .675 9485 29.25 25 1.644 3077 29.70 1.654 9682 29-53 1.665 5717 29.38 1.676 1240 29.25 2(( .644 4858 29.69 .655 1454 29-53 .665 7480 29.38 .676 2995 29.24 27 .644 6640 29.69 .655 3225 29-53 .665 9242 29.38 .676 4749 29.24 28 .644 8421 29.69 .655 4997 29.52 .666 1005 29-37 .676 6504 29.24 2« .645 0203 29.69 .655 6768 29-52 .666 2767 29.37 .676 8258 29.24 30 1.645 '984 29.68 ••655 8539 29-52 1.666 4529 29.37 1.677 0012 29.24 31 ■64s 3765 29.68 .65b 0310 29.51 .666 6291 29-37 .677 1766 29.23 32 •645 5545 29.68 .656 2081 2951 .666 8053 29.36 .677 3520 29.23 1 33 .645 7326 29.67 .656 3852 29.51 .666 9815 29-36 .677 527.1 .677 7028 29.23 j 3.1- .645 9106 29,67 .656 5622 29.51 .667 1577 2936 29.23 35 1.646 0886 29.67 1.656 7392 29.50 1.667 3338 29.36 1.677 8781 29.23 30 .646 2666 29.67 .656 9163 29.50 .667 5100 29-35 .678 0535 .678 2288 29.22 37 .646 A446 .646 6226 29.66 .657 C933 29.50 .667 6861 29.35 29.22 38 29.66 .657 2703 29.50 .667 8622 29.35 .678 4041 29.22 30 .646 8005 29.66 .657 4472 29-49 .668 0383 29-35 .678 5794 29.22 10 1.646 9785 29.65 1.657 6242 29.49 1.668 2144 29-35 1.678 7547 29.22 41 .647 1564 29.65 .657 8011 2949 .668 3904 29.34 .678 9300 29.21 42 •647 3343 29.65 .657 9781 29.49 .668 5665 29.34 .679 1053 .679 2806 29.21 43 .647 5122 29.65 .658 1550 29.48 .668 7425 29.34 29.21 44 .647 6900 29.64 .658 3318 29.48 .668 9185 29.34 .679 4558 29.21 45 1.647 8679 29.64 1.658 5087 29.48 1.669 0945 29.33 1.679 6310 29.20 4G .648 0457 29.64 .658 6855 29.48 .669 2705 29.33 .679 8063 29.20 47 .648 2235 29.63 .658 8624 29.47 .669 A465 .669 6225 29-33 .679 9815 29.20 48 .648 4013 29.63 .659 0393 29.47 29-33 .680 1567 29.20 40 .648 5791 29.63 .659 2161 29-47 .669 7984 29.32 .680 3319 29.20 50 1.648 7569 29.63 1.659 3929 29-47 1.669 9744 29.32 1.680 5070 29.19 51 .648 9346 39.62 .659 5697 29.46 .670 1503 29.32 .680 6822 29.19 52 .649 1123 29.62 .659 7465 29.46 .670 3262 29.32 .680 8574 29.19 53 .649 2901 29.62 .659 9232 29.46 .670 5021 29.32 .681 0325 29.19 54 .649 4677 29.61 .660 1000 29.46 .670 6780 29.31 .681 2076 29.19 55 1.649 6454 29.61 1.660 2767 29-45 1.670 8539 29.31 1.681 3827 29.18 5G .649 8231 29.61 1^° i"' .660 6301 29.45 .671 0298 29.31 .681 5578 29.18 57 .650 0007 29.61 29.45 .671 2056 29.31 .681 7329 29.18 58 .650 1784 29.60 .660 8068 29.45 .671 3814 29.30 .681 9080 29.1S 50 .650 3560 29.60 .660 9835 29.44 .671 5573 29.30 .682 0831 29.18 GO 1.650 5336 29.60 1. 661 1601 29.44 1.671 7331 29.30 1.682 2581 29-17 680 ibolic Orbit. TABLE VI. For finding the True Anomaly or the Tiiue trom the IVriliulion in a Parabolic Orbit. 59° K M. 1 Diff. v. ; 7331 i 9089 ' 0846 . 2604 ■ 436* ' 29.30 29.30 29.30 29.19 29.19 '. 6119 I 7876 ! I 9634 ; 5 »39« i 3 3>47 I 3 4904 ! 3 6661 3 84«7 4 0174 4 1930 4 3686 4 5441 4 7'9» 4 8954 5 0709 5 2465 5 4220 '5 5975 '5 7730 '5 9485 r6 1240 id 2995 I ?6 4749 76 6504 i 76 8258 1 77 0012 77 1766 ! 77 3520 1 77 5V4 ' 77 7028 ! 77 8781 I 78 0535 ' 78 2288 ! 78 4041 ; 78 5794 ; 178 7547 178 9300 179 1053 179 2806 >79 4558 )79 6310 179 8063 J70 98' 5 580 1 567 38o 3319 680 5070 58o 6822 680 8574 681 0325 681 2076 681 3827 681 5578 681 7329 681 9080 682 0831 682 2581 29.19 29.19 29.28 29.18 29.28 29.18 29.18 29.17 29.17 29.27 29.27 29.27 29.16 t 29.16 I 29.26 i 29.16 19.25 I 29.25 I 29--5 I 49'^5 ! 29.15 I 29.14 29.14 ; 29.14 1 19--4 i 29.14 I 29.23 j 29-23 i 29-23 ! 29-23 I 29.13 i 29.11 ! 29.12 ; 29.11 29.11 29.11 29.11 19.11 29.11 29.21 29.10 29.10 29.10 29.10 29.10 29.19 29.19 29.19 29.19 29.19 29.18 29.18 29.18 29. 18 29.18 29' '7 V, O' I 'Z 3 4 5 G *r 4 8 U lU 11 \'Z 13 11 15 Hi 17 18 10 20 21 22 23 24 25 2G 27 28 20 30 31 32 33 34 35 36 37 .38 30 40 41 42 43 44 45 40 47 48 40 50 51 52 53 54 55 50 57 58 50 60 60^ li.g M. I Diff. 1" 7568 93'6 1064 2812 4560 6308 8c55 9803 1.682 2581 .682 4332 .682 6082 .682 7832 .682 9582 1.683 133a .683 3082 .683 4832 .683 6581 .683 8331 1.684 0080 .684 1830 .684 3579 .684 5328 .684 7077 1.684 8826 .685 0574 .685 2323 .685 4071 .685 5820 1.685 .685 .686 .686 .686 1.686 .686 .686 .687 .687 1.687 .6S7 .687 .688 .688 1.6S8 .688 .688 .688 .689 1.689 .689 .689 .689 .689 1.690 .690 .690 .690 .690 1.690 .691 .691 .691 .691 1. 691 .692 .692 .692 .692 1.692 29.17 19.17 29.17 29.17 29.17 29.16 29.16 29.16 29.16 29. 1 6 29.16 29.1 c 29.15 29.15 29.15 29.14 29.14 29.14 29.14 29.14 29.14 29.13 29.13 29>3 29- 1 3 *9-i3 29.13 29.12 1550 29.12 3297 29.12 5044 29.12 6791 29.12 8538 29.11 0285 29.11 20J2 29.11 3778 29.11 5525 29.11 7271 29.10 9017 29.10 0764 29.10 2510 29.10 4256 29.10 6001 29.09 7747 29.09 9493 29.09 1238 29.09 2984 29.09 4729 29.09 6474 29.09 8219 29.08 9964 29.08 1709 29.08 3454 29.08 S«99 29.08 6943 29.08 8688 29.07 0432 29.07 2176 29.07 3920 29.07 5664 29.07 7408 29.07 61^ l(>K M. 1.692 .692 .693 .693 .693 1.693 .693 .693 .694 .694 1.694 .694 .694 .695 .695 1.695 .695 .695 .695 .696 i.6g6 .696 .696 .696 .696 1.697 .697 .697 .697 .697 1.697 .698 .698 .698 .698 1.698 .699 .699 .699 .699 1.699 .699 .700 .700 .700 1.700 .700 .700 .701 .701 1. 701 .701 .701 .701 .702 1.702 3174 .702 4913 .702 6651 .702 8389 .703 0128 1.703 1866 7408 9152 0896 2640 4383 6127 7870 9613 1356 3099 4842 6585 8328 0070 1813 3555 5298 7040 8782 0524 2266 4008 5750 749" 9233 0974 2716 4457 6198 7939 9680 1421 3162 4902 6643 8383 0124 1864 3604 5345 7085 8824 0564 2304 4044 5783 7523 9262 ICOI 2741 4480 6219 7958 9697 1435 Diff. 1". 29.07 29.06 29.06 29.06 29.06 29.06 29.05 29.05 29.05 29.05 29.05 29.04 29.04 29.04 29.04 29.04 29.04 29.04 29.03 29.03 29.03 29.03 29.03 29-03 29.02 29.02 29.02 29.02 29.02 29.02 29.02 29.01 29.01 29.01 29.01 29.01 29.01 29.00 29.00 29.00 29.00 29.00 29.00 29.00 28.99 28.99 18.99 28.99 28.99 28.99 28.98 28.98 28.98 28.98 28.98 28.98 28.98 28.97 28.97 28.97 23.97 581 62= loK .M. 5293 7029 7 *' 5 0501 2237 3972 5708 7444 9«79 0914 2650 4385 61 20 7855 959° Diff. I" 1.703 1866 703 3604 ! 703 53-f2 703 7080 703 .S818 I 704 0556 i 704 2293 I 704 4031 704 576S i 704 7506 I 704 9243 I 705 0981 j 705 2718 , 705 4455 I 705 6192 I 705 7929 ; 705 9666 , 706 1402 706 3139 I 706 4875 706 6612 I 706 8348 j 707 0085 I 707 1 821 i 707 3557 I 707 707 707 708 708 708 708 708 708 709 709 709 709 709 709 710 1325 710 3060 710 4794 710 6529 710 8263 710 9998 7U 1732 711 3467 711 5201 711 6935 711 8669 712 0403 712 21 37 712 3871 712 5605 7«2 7339 712 9072 713 0806 713 2539 7>3 4273 713 6006 28.97 28.97 18.97 28.97 28.96 28.96 28.96 28.96 28.96 28.96 28.96 28.95 28.95 28.95 28.95 28.95 28.95 28.95 28.94 28.94 28.94 28.94 28.94 28.94 28.94 28.93 28.9- 28.93 28.93 28.93 28.93 2893 28. 92 28.92 28.92 28.92 28.92 28.92 28.92 28.92 28.91 28.91 28.91 28.91 28.91 28.91 28.91 28.90 28.90 28.90 28.90 28.90 28.90 28.90 28.90 28.90 28.89 28.89 28.89 28.89 28.89 63= log .M. 6006 7739 9473 1206 I 2939 i 4672 ' 6405 8138 9870 1603 3336 i 5068 6801 8533 0266 1998 3730 5462 7'94 8926 ; 0658 i 2390 ; 4122 i 5853 : 7585 : 9317 \ 1048 2780 1 45" ! 6242 I 7974 ' 9705 1436 . 3«67 i 4898 I 6629 8360 0090 1821 3552 i 5282 . 7o>3 i 8743 0474 . 2204 : 721 3934 721 5665 721 7395 721 9125 722 0855 7 8 8 8 8 8 8 9 9 9 9 9 720 710 720 720 720 720 721 721 Mff 1". 28.89 28.89 28. 89 28. 88 28.88 28.88 28.88 28. 88 28.88 28.88 28.S8 28. 88 28.87 28.87 28.87 28.87 28.87 28.87 28.87 28. 87 28.86 28.86 28.86 28.86 28.S6 28.86 28.86 28. 86 28. 86 28.85 28.85 28.85 28.85 28.85 28.85 28. 85 28.85 28.85 28.84 28.84 28.84 28.84 28.84 28.84 28.84 28.84 28.84 28.84 28.83 28.83 722 2585 28.83 722 4315 28 83 1 722 6044 28.83 1 722 7774 28.83 ,22 9504 28.83 723 1233 28.X3 723 2963 28.83 723 4693 28.82 723 6422 28.82 i 723 8151 28.82 1 723 9881 28.82 TABLE VI. For findinp; tlic True Anoniiily or tlic Tiiiie from llie IVrilii-lioii in a Parabolic Orbit. '/ 1 V. i 1 0' 64 65 66 67 logM. Via. 1". 28.82 log '•'34 M, 3539 I)lff. 1". 28.77 log M. KitT. I". 28.73 \ov [M. IMIT. 1". 1.723 9881 1.744 7031 '•755 0405 28.70 ! 1 •71+ 1610 28.82 •734 5265 28.77 •744 8755 28.73 •755 2127 28.70 a .724 3339 28.82 •^34 6y.;l 28.77 •745 0479 28.73 •755 3849 28.70 1 3 .72+ <;o68 28.82 •734 8718 28.77 •745 2202 28.73 •755 5 57' 28.70 4 .724 6798 28.82 •735 0444 28.77 •745 3926 28.73 •755 7293 2X.70 5 1.724 8527 28.82 '•735 2169 28.76 '•745 5650 28.73 '•755 9015 28.70 ■7-S 0256 28.82 •735 3895 28.76 •74; 7373 28.73 .756 0737 28.70 ; 7 •715 1984 28.81 •735 5621 28.76 •745 9097 28.73 .756 2459 28.70 « •725 37'3 28.81 •735 7347 28.76 .746 0820 28.72 .756 4181 28,70 I •725 544* 28.81 •735 9073 28.76 .746 2544 28.72 .756 5903 28.70 ! 10 1.725 7171 28.81 1.736 0798 28.76 1.746 4267 28. 72 1.756 7625 28. 70 II • 72'; 8900 28.81 • 736 2524 28.76 ,746 599' 2S.72 .756 9347 28.70 vz .726 0628 28.81 .736 425° 28.76 .746 77'4 28. 72 •757 1C69 28.70 1 13 .726 2357 28.81 .736 5975 28.76 .746 9437 28.72 •757 2791 28.70 ' 14 .726 40X5 28.81 .736 7701 28.76 •747 Il6i 28.72 •757 45'3 28.70 Kt 1.726 5814 28.81 1.736 9426 28.76 '•747 2884 28.72 '•757 6235 28.70 1(( .726 7542 28.81 •737 1152 28.76 •747 4607 28.72 •757 7957 28.70 17 .726 9270 28.81 •737 2877 28.76 •747 6330 28.72 •757 9679 28.70 IH •727 0999 28.80 •737 4602 2S.76 •747 8054 28.72 •758 1401 28.70 l« •727 2727 28.80 •737 6328 28.75 •747 9777 28.72 .758 3'23 28.70 «0 1.727 6183 28.80 '•737 8053 28.75 1.748 1500 28.72 ..758 4844 6566 28.70 21 •727 28. 80 •737 9778 28.75 .748 3223 4946 28.72 .758 28.70 aa •727 79" 28.80 .738 '5°3 28.75 •748 28.72 .758 8288 28.70 23 •727 9639 28.80 .738 3228 28.75 .748 6669 28.72 •759 0010 28.70 24 .728 1367 28.80 .738 4953 28^75 .748 8392 28.72 •759 1731 28.70 25 1.728 309- 28.80 1.738 6679 28.75 '•749 0115 28.72 '•759 3453 2 8. 70 26 .728 4823 6551 28.80 .738 X404 28.75 •749 1838 28.72 •759 5'75 28.70 27 .728 28.80 •739 0129 28.75 •749 35'" 28.72 •759 6897 28.70 28 .728 8279 28.80 •739 1853 3578 28.75 •749 5284 28.72 •759 8618 28.69 21) .729 0006 28.79 •739 28.75 •749 7007 28.71 .760 0340 28.69 1 30 1.729 '734 28.79 '•739 5303 7028 28.75 1.749 8730 28.71 1.760 2062 28.69 31 .729 3461 28.79 •739 28.75 •75° 0453 28.71 .760 3783 28.69 32 .729 5189 28.79 •739 8753 28.75 .750 2176 28.71 .760 5505 2S.69 1 33 .729 6916 28.79 .740 0477 28.75 •75° 3898 28.71 .760 7227 28.69 34 .729 8644 28.79 •74° 2202 28.74 •75° 5621 28.71 .760 8948 28.69 35 1.730 037' 28.79 1.740 3927 28.74 1.750 7344 9067 28.71 1.761 06 '/O 28.69 30 .730 2099 28.79 •74° 5651 28.74 •750 28.71 .761 =392 28.69 37 .730 3826 28.79 •74° 7376 28.74 •75' 0789 28.71 .76, 4113 28. 69 38 .730 55S3 28.79 .740 9101 28.74 •75' 2512 28.71 .761 5835 28.69 30 .730 7280 28.79 •74' 0825 28.74 •75' 4234 28.71 .761 7556 28.69 ; 40 1.730 9007 28.78 '•74' 2550 28.74 '•75' 5957 28.71 1.761 9278 2S.69 41 •73« 0735 28.78 •74' 4274 28.74 •75' 7680 28.71 .762 0999 2S.69 42 •73' 2462 28.78 •74' 5998 28.74 •75' 9402 28.71 .762 2721 28.69 43 •73> 4189 28.78 •74' 7723 2X.74 .752 1125 28.71 .762 4442 28.69 44 •731 59'S 28.78 •74' 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 40 •73' 9369 28.78 •742 2896 28.74 •752 6292 28.71 .762 9607 28. 69 47 •732 1096 28.78 •742 4620 28.74 .752 8015 28.71 .763 1328 28.69 48 •732 2823 28.78 •742 6344 28.74 .752 9737 28.71 •763 3050 28.69 40 .732 4549 28.78 .742 8068 28.74 •753 1460 28.71 .763 477' 28.69 50 1.732 6276 28.78 1.742 9792 28.74 '•753 3182 28.71 '•763 6493 28.69 51 •732 8002 28.78 •743 1516 28.73 •753 4904 28.71 .763 8214 28.69 52 •732 9729 28.77 •743 3240 4964 28.73 •753 6627 28.71 •763 9936 28.69 53 •733 1455 28.77 •743 28.73 •753 8349 28.71 .764 1657 28.69 54 •733 3182 28.77 •743 6688 28.73 •754 0071 28.70 •764 3379 28.69 55 '■733 4908 28.77 '•743 8412 28.73 '•754 1794 28.70 1.764 5100 28.69 1 50 •733 663s 28.77 •744 0136 i860 28.73 ■754 3516 28.70 .764 6821 28.69 i 57 •733 8361 28.77 •744 28.73 •754 6960 28.70 •764 8543 28. 69 58 •734 0087 28.77 •7^14 3584 28.73 •754 28.70 .765 0264 28.69 SO •734 1813 28.77 •744 5308 28.73 •754 8682 28.70 •765 1985 28.69 60 '•734 3539 28.77 1.744 7031 28.73 '•755 0405 28.70 1.765 3707 28.69 582 abolic Orbit. TABLE VI. For rmdinigr the True Aiioinsily or tlii' Tiiiii' from tin- Periliflion in a Par.ibolic Orhil. 67^ kM. 0405 I 2127 ' 5571 7293 9015 0737 2459 41S1 5903 7625 93+7 1069 2791 45«3 6235 7957 9679 1401 3>*3 4844 6566 8288 0010 1731 WIT. 1 ". 28.70 28.70 28.70 28.70 28.70 28.70 28.70 28.70 28.70 28.70 28.70 I 28.70 ' 28.70 i »>*-70 ; 28.70 ' 28.70 : 28.70 i 28.70 28.70 28.70 28.70 28.70 28.70 28.70 28.70 o o 60 60 61 61 61 61 61 61 )9 3453 59 5 '75 59 ^**97 9 8618 0340 2062 3783 5505 7227 60 8948 0670 -392 ; 4113 5835 7556 9278 62 0999 62 2721 62 4442 62 6164 ■62 7885 ■62 9607 1328 3050 '63 477« 763 6493 763 8214 763 9936 764 1657 764 3379 764 5100 764 6821 764 8543 765 0264 765 1985 765 3707 28. 70 28.70 28.70 28.69 28.69 28.69 28.69 28.69 28.69 28.69 28.69 28.69 28.69 28.69 28.69 28.69 28.69 28.69 28.69 i 28.69 i 28.69 28.69 28.69 28.69 28.69 28.69 2S.69 28.69 28.69 28.69 i 28.69 28.69 28.69 28.69 28.69 ! 28.69 V. 68 69 70 7P 1 1<.« M. DIff. 1". IokM. i DIff. 1". li>ii .M. DIff. 1". IokM. IHff. 1". ' i-7''5 3707 28.69 '•775 ''9^5 28.69 1.786 028J. 2006 28.70 - - f.796 36^0 28.73 1 .765 5428 28.69 •775 8706 28.69 .786 28.70 •796 5374 28.73 a •765 7150 28.69 .776 0427 28.69 .786 3728 28.70 .796 7097 28.73 , 1 3 .705 ****7> 28.69 .776 2149 28.69 .786 5450 28.70 .796 8821 28.73 , ' 4 .766 0592 28.69 .776 3870 28.69 .786 717* 28.70 ■-97 0545 28.73 1 5 1.766 2314 28.69 1.776 ^591 28.69 I. -86 8S94 28.70 1.797 2268 28.73 1 .766 4035 28.69 •77'' 73'3 28.69 .787 0017 28.70 ■797 3992 28.73 : T .766 5756 28.69 .776 9"34 28. 69 ■787 2339 28.70 ■797 57 '6 28.73 H .766 7478 28.69 •777 0755 28.69 .787 4061 28.70 •797 7440 28.73 .766 9199 28.69 •777 i477 28.69 .787 5783 28.70 •797 9 '64 28.73 10 1.767 0920 28.69 1.777 4198 28,69 1.787 7506 28.70 1.798 0888 28.73 II .767 2642 28.69 •777 5920 28.69 ■7!<7 9218 28.71 .798 261 1 28.73 I'Z .767 4363 28.69 •777 7641 28.69 .788 0950 28.71 •798 4335 .798 6060 28.73 V.l .767 6084 28.69 •777 9363 28.69 .788 2673 28.71 2X.73 11 .767 7805 28.69 .778 1084 28.69 .788 4395 28.71 .798 7784 28.73 15 1.767 95*7 28.69 1.778 2806 28.69 1.788 6117 28.71 1.798 9508 28.73 i 10 .768 1248 .768 2969 28.69 •778 45*7 28.69 .788 7840 28.71 •799 >J32 28.74 ! 17 28.69 .778 6248 28.69 .788 9562 28.71 •799 2956 2Sf^74 , IH .768 4691 28.69 ■778 7970 28.69 ■789 1284 28.71 .799 4680 28.74 10 .768 6412 28.69 .778 9691 28.69 .789 3007 28.71 •799 6404 28.74 «() 1.768 8133 28.69 1.779 i4«3 28.69 1.789 4730 28.71 1.799 8128 28.74 21 .768 9854 28.69 •779 3'40 28.69 .789 6452 28.71 •799 9? 5 3 28.74 Tt .769 1576 28.69 •779 4862 28.69 .789 8175 28.71 .800 1577 28.74 T.I .769 3297 28.69 •779 6578 28.69 .789 9897 28.71 .800 3301 28.74 •u .769 5018 28.69 ■779 8299 28.69 .790 1620 28.71 .800 5026 28.74 , 25 1.769 6740 28.69 1.780 0021 28.69 1.790 3341 28.71 1.800 6750 28.74 , 20 .769 8461 28.69 .780 1742 18.69 .790 5065 28.71 .800 847; 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 I 2U .770 3625 28.69 .780 6907 28.69 .791 0233 28.71 .801 3648 28.74 :io 1.770 S3a6 .770 7067 28.69 1.780 8629 28.69 1.791 1956 28.71 1.801 5373 28.74 j 31 28.69 .781 0350 28.69 .791 3678 28.71 .801 7107 28.74 1 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 1.771 395* 28.69 1.781 7237 28.69 1.792 0570 28.71 1.802 3996 28.75 30 .771 5673 28.69 .781 8959 28.69 .792 2293 28.71 .802 5721 28.75 1 37 •77« 7395 28.69 .782 0680 28.70 .792 4016 28.72 .802 7446 28.75 , 38 .771 9116 28.69 .782 2402 28.70 .792 5738 7461 28.72 .802 9 17 1 28.75 1 39 .772 0837 28.69 .782 4124 28.70 .792 28.72 .803 0896 28.75 i 40 1.772 2559 28.69 1.782 5845 28.70 1.792 9184 28.72 1.803 2^21 28.75 41 .772 4280 28.69 .782 7567 28.70 •793 0907 28.72 .803 4346 .803 6071 28.75 42 .772 6001 28.69 .782 9289 28.70 ■793 2630 28.72 28.75 43 .772 7722 28.69 .783 lOll 28.70 ■793 4354 28.72 .803 7796 28.7s 44 •77a 9444 28.69 .783 2732 28.70 ■793 6077 28.72 .803 9521 2^,75 45 1.773 "65 28.69 i^783 4454 28.70 1-793 7800 2^.72 1.804 1246 28.75 40 .773 i886 28.69 .783 6176 28.70 •793 9523 1246 28.72 .804 2971 28.75 47 •773 4607 28.69 .783 7898 28.70 •794 28.72 .804 4697 28.75 48 •773 6329 28.69 .783 9620 28.70 •794 2969 28.72 .804 6422 28.76 49 •773 8050 28.69 •784 134a 28.70 •794 4693 28.72 .804 8147 28.76 50 1.773 977" 28.69 1.784 3064 28.70 1.794 6416 28.72 1.804 9873 28.76 51 •774 »493 28.69 .784 4786 28.70 •794 8139 9862 28.72 .805 1598 28.76 52 •774 32>4 28.69 .784 6508 28.70 •794 28.72 .805 3324 28.76 , 53 •774 4935 28.69 .784 8230 28.70 •795 1586 28.72 .805 5049 28.76 : 1 54 ■774 6657 28.69 •784 9952 28.70 •795 3309 28.72 .805 6775 28.76 ; 1 55 1.774 8378 28.69 1.785 1674 28.70 "•795 mi 28.72 1.805 8500 28.76 ; 50 •775 0099 28.69 •785 3396 28.70 •795 28.72 .806 0226 28.76 ! 57 •775 '8*« 28.69 .785 5118 28.70 •795 8480 28.72 .806 1952 28.76 i 58 •775 354* •775 5463 28.69 .785 6840 28.70 .796 0203 28.73 .806 3677 28.76 ' , 59 28.69 .785 8562 28.70 .796 1927 28.73 .806 5403 28.76 1 1 00 1.775 6985 28.69 1.786 0284 28.70 1.796 3650 28.73 1.806 7129 28.76 am :-'^ TABLE VI. For finding Iho True Anomaly or the Tinu" fnini ilu' IVriliolion in a I'.iriiljolic Orbit. V, 0' 72 73 74 IHff. 1". 75 DilT V 28.95 InK M. 1.806 7129 Diir. 1". 28.76 l<>K M. 1.817 0765 Dur. 1 ". 1..K M. 2X.81 1.827 a6o2 .827 6315 .827 8068 28.XX 1.X37 X6X6 1 .Hob XXs? 28.76 .817 2494 2X.X1 2X.X8 .X3X 0423 2X.9, 'i .Xo; ojXi 28.77 .X17 4222 2X.X2 2X,S8 ,838 2160 28.9^ 3 .X07 2307 28.77 .817 5951 28.X2 .827 9800 28.88 ,838 3X98 28.9^ 4 .807 4033 28.77 .817 7680 2X.82 .828 1533 28.88 .838 5635 28,96 5 1,807 5759 28.77 I.817 9410 28.82 1.828 3266 28.88 1.838 737f 28,96 .807 7485 28.77 .81X 1139 2X.X2 .X2X 4999 ,828 6732 28,88 .83X 9i:.y 28,9(1 7 .807 9211 28.77 .818 286X 2X.X2 28, XX .X39 0847 2X,9li 8 .XoX 0937 28.77 .XI 8 4597 ;X,X2 .828 8465 2X,X8 .839 25X5 28.9(1 .X08 2663 28.77 .XiX 6326 28.x?. .829 01 98 28.89 .839 4323 28.96 10 .XoX 4389 .XoX 6116 28.77 1. 818 8056 2X.S2 1.X29 1931 .X29 3(165 28.89 1.X39 6060 28.96 11 28.77 .818 97X5 2X.X2 28.89 .X39 7798 28.97 i >» .XoX 7X42 28.77 .XI9 1515 2X.X3 ,X29 5-,9X 28.89 .X39 9536 28.97 1 13 .808 9568 28.77 .XI9 3244 2X.X3 ,X29 7131 2X.89 .Xp 1274 28.97 1 14 .X09 1295 28.77 .8 1 9 4974 2X.X3 .X29 X865 28.89 .X40 3012 28.97 15 .809 3021 28.78 1.8 1 9 6704 Z8.83 1.830 0599 28.89 1.840 4751 .840 64X9 28.97 10 .X09 4748 28.78 .819 8433 28.83 .830 2332 28.89 28.97 17 .X09 6474 28.78 .820 0163 2X.S3 .830 4006 28,90 .840 8227 28,97 18 .X09 Xioi 28.78 .820 1X93 28. S3 .830 5800 28.90 .X40 9966 2X.97 10 .809 9928 28.78 .820 3623 28.83 .830 7533 28.90 .841 1704 28.98 «0 .810 1655 28.78 1.820 53S3 28. 83 1.830 9267 28.90 1.841 3443 28,98 Ul .Xio 33X1 28.78 .820 70X3 28. 83 831 ICOI 28. 90 .841 5182 28,98 U'Z .810 s'loX 28.7X .X20 XX 1 3 28. 84 .831 273; 2X,90 .841 6921 28,98 23 .810 6X35 28. 78 .821 0543 28.X4 .831 4470 28,90 .X41 8(159 28,98 •u .810 8562 28.78 .X21 2273 28. S4 .831 6204 18,90 .842 039X 28,98 25 .811 02X9 28.78 1.X21 4003 28.X4 1.831 7938 28.91 1.842 2138 28,98 , 20 .Xii 2016 28.78 .X21 5734 28,84 .831 9672 2X.91 .842 3X77 28,99 ' 27 .811 3743 28.78 .821 7464 ^li* .83a 1407 28.91 .842 5616 28,99 28 .8-1 S470 28.79 .821 9194 28. X4 .832 3141 28.91 .842 7355 2X.99 20 .1. I 7197 28.79 .822 0925 28.84 .83- •X76 28.91 .842 9095 28,99 30 .Xii 8924 28.79 1.822 2656 28.X4 1.832 66ii 28.91 1.843 0834 28.99 31 .812 0652 28.79 .X22 43X6 .822 6117 Hi* .832 8345 28.91 •843 i574 28,99 »i .812 2379 2X.79 28.85 .833 0080 28.92 •843 43>3 .843 6053 29,00 33 .812 4106 28.79 .822 784X 28.85 .833 ,X.5 28,92 29,00 34 .812 5834 28.79 .822 9578 28.85 •833 3550 28,92 •843 7793 29,00 35 .812 7561 28.79 1.823 '3°9 28.85 1.833 52X5 28,92 ■•843 9533 29.00 30 .812 9289 28.79 .823 3040 28.85 .833 7020 28.92 .844 1273 29.00 37 .813 1016 28.79 .823 4771 28. 85 •833 «755 28,92 .844 3013 29.00 1 38 .813 2744 28.79 .823 6502 28.85 .834 0491 28,92 •844 4753 29.00 1 30 .813 4472 28.79 .823 8233 28.85 .834 2226 28,92 .844 6494 29.01 40 .813 6199 28.80 1.823 9965 28.85 1.834 3961 2X.92 1.X44 8234 29.01 41 .813 7927 28.80 .824 1696 28.85 .834 5697 28.93 .844 9974 29.01 42 .813 9655 28.80 .824 3427 28.86 ■834 743» 28,93 •845 >7I5 29.01 43 .8,4 1383 28.80 .824 S159 28.86 .834 9168 28.93 •845 3456 29.01 44 .814 3111 28.80 .824 6890 28.86 .835 0904 28.93 .845 5196 29.01 45 .814 4.839 .814 6567 28.80 1.824 8622 28.86 1.835 2640 28.93 1.845 6937 29.01 40 28.80 .825 0353 2X.86 .835 4376 28.93 .845 X678 29.02 47 .814 8295 28.80 .825 20X5 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 40 .815 1751 28.80 .825 5548 28.86 .835 9584 28,94 .846 3901 29,02 50 .815 3479 .815 5208 28.80 1.825 7280 28.86 1.836 1320 28.94 1,846 5643 29.02 51 28.81 .825 9012 28.87 .836 3056 28.94 .846 7384 29.02 ' 52 .81C 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 2I2I 28.81 1.826 5940 28.87 1.837 0002 28.94 1.847 4350 29.03 50 .816 3850 28.81 .826 7673 28.87 .837 1739 28.95 .847 6092 29.03 57 .816 5578 28.81 .826 9405 28.87 •837 3475 28.95 .847 7834 29.03 58 .816 7307 28.81 .827 1137 28.87 .837 5212 28,95 .847 9576 29.03 50 .816 9036 28.81 .827 2870 28.87 .837 6949 28.95 .848 1318 29.04 00 |i .817 0765 28.81 1.827 4602 28.88 1.837 8686 28.9s 1.848 3060 29.04 684 iiholic Orbit. TABLE VI. For ffniliiiK tlic Tnic Anoiimly or lla- Tiinc from tin- Pcrilielion in a Purulxilio Orbit, V, 0' 76 D 77° 78 79 log M. wir. 1". 19.04 l»K M. 7769 mrr. i". 29.14 i.>g M. Wff. 1". 29.15 l->K .M. iMir. y. 2937 1.X4S 3060 1.858 1.869 ?,8 57 '•879 8369 1 .X4S 4803 29.04 .85,X 95'7 29.14 .869 a6|2 6367 29.25 .880 0131 2937 , 3 .X4X 6545 29.04 .859 1266 29.14 .869 29.25 .880 1894 19.38 » .X4S 8287 29.04 ■ 8^9 3014 29.14 .869 8122 29.25 .880 36 5f' 19.38 4 .849 0030 29.04 .859 4763 29.15 .869 9878 29.16 .880 54' 9 29.38 1 5 1.84., '773 29.04 1.859 6512 29.15 1.870 1613 19.26 i.88o 7182 19.38 ' .84.; 35'5 29.05 .859 8260 29. 1 5 .870 i^h 29.26 ,880 8945 19. 38 T .849 5258 29.05 .X60 0009 29.15 .870 5'44 19.16 .881 0708 19.39 8 .849 7001 19.05 .86d 1758 29.15 .870 6900 19.16 .881 2471 19.39 .849 8744 29.05 •.860 3507 29.15 .870 8656 19.26 .881 4235 29. 39 10 1.8^0 0487 29.05 . 1.860 5256 29.15 1.871 0412 19,17 1.881 5998 29.39 11 .8^0 2231 29.05 .860 700S 29.16 .X71 2168 19.27 .881 7762 19.39 Vi .S50 3974 29.06 .860 8755 29.16 .871 3914 19.17 ,88 1 9516 19.40 13 .850 S7'7 29.06 .861 0505 29.16 .871 5O81 19.17 .882 1290 19.40 ; 14 .X50 7461 19.06 .8bi 2254 29.16 .87. 7437 19.28 .882 3°54 29.40 : 15 1.8^0 9204 29.06 1.861 4r.04 29. 16 1.871 9 '94 29.18 1.882 4818 29.40 ; lU .85, 0948 29.06 .8f I 5754 29.16 .871 0950 19.1 s .882 6581 19.41 n .85, 2692 29.06 .861 7504 29.17 .872 1707 19.18 .882 8347 19.41 18 .85, 4436 6i8o 29.07 .861 9»54 29.17 •2'" 4464 19.18 .883 01 11 19.41 10 .85, 29.07 .862 1004 29.17 .872 6221 29.29 .883 1876 29.41 20 1.8^1 79*4 9668 29.07 1.862 »754 29.17 1.872 7979 29,29 1.883 3641 29.42 •il .851 29.07 .862 45°5 29.17 .872 9736 29.29 .S83 5406 29.41 'Z'Z .85a 1412 29.07 .862 6^55 29.18 .873 '493 29.29 .883 7171 19.41 33 .852 3157 29.07 .862 8006 29.18 .873 325' 29.29 .883 8937 29-42 34 .85Z 4901 29.07 .862 9756 29.18 .873 5008 19.30 .884 0702 19.42 1 35 1.852 6646 29.08 1.863 1507 29,18 1.873 6766 29.30 1.884 2468 29-43 1 3» .8,2 8391 29.08 .863 3258 29.18 .873 8524 29.30 .884 4233 2943 37 .8,3 0135 29.08 ■l^J 5009 29,18 .874 0282 29.30 .884 5999 2943 38 .853 1880 29.08 ■It^ 6760 29.19 .874 2041 29.30 .884 7765 19.43 30 .853 3625 29.08 .863 8512 29.19 •874 3799 29.31 .884 953' 19.44 30 ••853 5370 29.09 1.864 0263 29.19 ..874 5557 29.31 1.885 1297 29.44 31 .853 711S 29.09 .864 2015 29.19 .874 7316 29.31 .X85 3064 29.44 ' :i3 .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 o8t3 29.31 .885 ''597 2945 31 .854 2351 29.09 .864 7270 29.20 .875 2592 29.32 .885 8364 29.45 35 ..854 4097 29.09 1.864 9022 29,20 1-875 4351 6111 29.32 1.886 0131 29.45 30 .854 5843 29.10 .865 0774 2526 29.20 -875 29.32 .886 1898 29.45 3T .854 7588 29.10 •^^5 29.20 ■875 7870 29.32 .886 3605 29.45 38 .854 9334 29.10 .865 4278 29,20 .875 9629 29.32 .886 5432 7 30 .855 1080 29.10 .865 6030 29.21 .876 1389 29-33 .886 7200 29t6 40 1.855 2826 29.10 1.865 778; 953^ 29.21 1.876 3148 29.33 1.886 8967 29.46 41 .855 4572 29.10 .865 29.21 .876 4908 19.33 .887 0735 19.46 i 43 .855 6319 29.11 .866 1288 29.21 .876 6668 29-33 .887 2503 29-47 1 43 .855 8065 29.11 .866 3041 29.21 .876 8428 29.33 .887 4271 29-47 1 44 .855 9811 29.11 .866 4794 29.22 .877 0188 29-34 .887 6039 29-47 i 45 1.856 1558 29.11 1.866 6547 29.22 1.877 '949 29-34 1.887 7807 29.47 ' 40 •^56 3305 29.11 .866 8301 29.22 .877 3709 29,34 .887 9576 19.48 47 .856 505* 29.11 .867 0054 29.22 .877 5470 29.34 .888 '344 19.48 48 .856 6799 29.12 .867 1807 29.22 •877 7230 29.34 .888 3113 19.48 40 .856 85+6 29.12 .867 3561 29.23 .877 8991 29.35 .888 4882 29.48 50 1.857 0293 29,12 1.867 53J4 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 53 .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 .87S 6036 29-35 ■It^ '959 29.49 54 .857 7282 29.13 .868 2330 29.24 .878 7797 29.36 .889 3728 29.49 ; 55 1.857 9030 29.13 1.868 4084 29.24 1.878 9559 29.36 1.889 5498 29.49 50 .858 0777 29.13 .868 5839 29.24 .879 1321 29.36 .889 7168 29.50 57 .858 2525 29.13 .868 7593 9348 29.24 .879 3082 2936 .889 8038 29.50 58 .858 4173 602l 29.13 .868 29.24 .879 4844 29.36 .890 0808 19.50 50 .858 29.13 .869 1102 29.25 .879 6606 29-37 .890 2578 29.51 GO 1.858 7769 29.14 1,869 2857 29.25 1.879 8369 29,37 1.890 4349 29.51 585 TABLE VI. For liiiilinK llii^ True Aiioimilv ur tin- Tiiiu' I'ntni tlio IVrilii'lioi) in a I'liraliolic Orliit. € I'. i.Xi^o 80 ° I..V 81 [M. Dinr. 1". 29.66 82 fj 83 !..»; .M. 1.922 5548 M. 4U'> IHIT. 1". ^9-51 Ion 1.9 I I M. 7893 iMir. 1". 29. Xi wrr. 1". o 1.901 0X41 19.99 1 .X.>o M19 29 SI ,901 2(121 29.66 .911 96X2 19.x 1 ■92* 7347 19.99 u .S.;o 7X90 29.51 .901 4400 29 66 .911 1471 19. Xi .912 9147 30. OJ :i .Syo 9661 19-51 .901 61X0 29 66 .911 3261 29-83 -923 0947 30.00 4 .S91 I43» 29. SI .901 7960 19.67 .911 5050 29.83 .923 1747 30.00 a l.Xi;l -5103 »9-5» 1.90 1 9740 19.67 1.9 1 2 6X40 ^.9.83 1-923 4548 30.01 n .Xyi 4V74 19.52 .9^2 1521 29.67 .912 X630 2984 .923 6348 30.01 7 .X9I ^745 29.51 .901 3301 29.67 .913 0420 29. X4 -92? 8149 30.01 1 N .XVI 8^17 »953 .901 50X2 19.68 .913 2211 29-84 -913 9950 30.01 1 .X.;z 0189 »9-53 .901 ()862 19.68 .913 4001 2984 .924 1751 30,02 10 I.S92 1061 »9-53 1.901 8643 19.68 1.913 579* 29-85 1.924 3552 30.01 ! li .Xyi 3«^3 *9-53 .903 0424 29.69 ■913 7583 29-85 -924 5354 30.03 1'^ • Syl t,(■! 44''7 *9-55 .904 "4 1896 29.70 .914 833> 29.X7 .925 6166 3004 IH .X.;3 6240 1955 .904 29.70 .915 012A 1916 29-87 • 925 7969 30.05 10 .X93 8013 »955 .904 4678 19.71 •9'S 19.87 .925 9771 30.05 •zo i.X.n 9787 19.56 1.904 6461 19.71 1-915 3708 29-87 1.916 1575 30,05 1 *-*' .X,H 1560 29.56 .904 X243 29.71 .915 5 5o< 19.88 .926 337X 30.06 a« •X'H ^3U 29,56 .905 0026 29.71 -915 7294 29.S8 .926 51X1 30.06 1 «3 .X94 S loX 29.56 .905 1X09 29.72 .915 90X7 29.X8 .926 69X6 30.06 'H .Xy4 6X82 19-57 .905 359* 29.72 .916 0880 19.89 .926 87X9 30.07 j '^5 I.X.H 86^6 29-57 1.905 5376 29.72 1.9 1 6 2673 19.89 1.927 0591 .927 2398 30.07 1 2U • X'^i 04^0 29-57 .905 7 '59 29-73 .9 1 6 4466 19.89 30.07 i «7 .X.,s 2204 »9-57 .905 8943 29-7 3 .916 6i6o 29.90 .927 J.202 .927 6007 30.08 1 «H .X9<; 3979 29.58 .90(1 0726 29-73 .916 ^t^t 29.90 30.08 1 at) .X95 5753 29.58 .906 2510 29-73 .916 9848 29.90 .927 7X11 30.08 :i(> 1.X95 7,-28 19.58 1.906 4294 29.74 1.917 1642 29.90 1.927 9616 3r.rS :ii .X9S 9303 29.58 .906 6o'9 2974 .917 3436 29.91 .928 1422 3-9 1 :w .X96 1078 19.59 .906 7X63 29.74 .917 523' 29-91 -928 3227 30.09 »a .X96 2854 19.59 .906 964X 29-74 .917 7025 29.91 .92S 5032 30.09 34 .X96 4628 19.59 .907 .432 *9-7S .917 8820 19.91 .918 6838 30,10 35 1.X96 6404 29.59 1.907 3217 29-75 1. 918 0615 29.92 1.928 8644 30. JO 3U .896 8180 29.60 .907 5002 29-75 .918 2410 29.92 -929 0450 30.10 37 .X9O 995 5 29.60 .907 6787 29-75 .918 A206 6001 29.92 .929 2256 3 c. 1 1 38 .X97 1732 29.60 .907 «573 29.76 .918 29-93 .929 4063 -,c.ll 30 .X97 3508 29.60 .908 0358 29.76 .918 7797 2993 •929 5869 30.11 4U 1.X97 5284 29.61 1.908 2144 29.76 1.918 9593 2993 1.929 7676 3c.ll 41 .X97 7060 29.61 .90X 393" 29-77 .919 1389 29.94 .929 9483 50.12 4'Z .X97 8837 19.61 .908 5716 29-77 .919 31X5 29-94 .930 1 291 50.12 43 .X9X 0614 29.61 .908 7502 29-77 .919 4982 29.94 .930 309X -,c.i3 44 .898 2390 29.62 .908 9288 29-77 .919 6778 29.94 .930 4906 30.13 45 1.X98 4168 29.62 1.909 .075 29.78 1. 919 8575 29.95 1.930 6713 3c, 13 40 .89X 5945 19.62 .909 2X62 29.78 .920 0372 29-95 .930 8521 3C.13 47 .89X 7722 29.62 .909 4648 29.78 .920 2169 29.95 -93' 0330 30 14 48 .X98 9500 29.63 .909 6436 29.78 .920 3966 29.96 .931 213X 30.14 40 .899 1277 29.63 .909 8123 25.79 .910 5764 29.96 -93 « 3946 30,14 50 1.899 3055 19.63 1. 910 0010 29.79 1.920 7561 19.96 '-93' 5755 30.15 51 .899 66 1 1 29.63 .910 1798 29.79 .920 9359 29.97 -93' 7564 3c. 15 52 .X99 29.64 .910 35!<5 29.80 .921 "57 29.97 •93' 9373 30.15 53 .899 8389 29.64 .910 5373 29.80 -921 2956 29.97 .932 1183 30.16 ■ 54 .900 0168 29.64 .910 7161 29.80 .921 4754 29.98 -932 2992 30.16 . 55 1.900 1946 29.64 1. 910 8949 29.80 1. 911 6552 29.98 1.932 4802 30.16 50 .900 3715 29.65 .911 0738 29.81 .911 835" 29.98 .932 6612 30,17 57 .900 5504 29.65 .911 2526 29.81 .922 0150 29.98 .932 8422 3C.17 58 .900 7283 29.65 .911 43 > 5 29.81 .921 1949 29-99 -933 0232 30.17 50 .900 906a 29.66 .911 6104 19.82 .911 3748 19.99 •933 2043 30.18 60 1. 90 1 0841 29.66 1.911 7893 19.82 1.911 5548 19.99 1-933 3853 30.18 580 nilxtlic Orltit. TABLE VI. Kiir liiiiliii),' till' TriK' Atiom^ily nr llir 'I'iiiii' iVkiii tlu' Pcrilirlion in a Par.ilMilii' ()rlii(. 83 ) •kM. j iMir. r I sux 29.9., » 7U7 1 29.99 1 <>i47 ' 30.0 J 1 0';47 30.03 i *747 ' 30.00 ? 454i< ' 30.01 \ f'U» 30. '11 X K149 1 30.', I 1 9750 ' 30.02 4 '751 1 30,01 4 l^-?* 30.02 4 ^1H 30.03 4 7i>S 30-C'l 4 !<957 30.03 5 °759 30.03 5 »5''l 30.04 i; 4lf'4 5 6166 30.04 3004 5 7969 30.05 5 977* 30.05 6 M75 6 3V» 30,05 30. c6 6 5181 30.06 6 6.)86 30.06 6 8789 30.07 7 OS93 7 1V>» 30.07 3C,07 7 4101 7 6007 3o.o.>f 30.0X 7 7«n 30.08 7 9616 30.08 8 1422 30.C9 8 3227 30,09 8 ^032 3o.o<) 8 6838 30.10 8 8644 30. iO 9 0450 30.10 9 2256 3c. 1 1 9 4061 3c. 11 9 S«69 30.H 9 7676 30.12 9 94*' 3 30.12 D I 29 1 30.1a 3098 30.13 4906 30.13 6713 30.13 8521 30.13 I 0330 30 14 I 2138 30.14 I 3946 30.14 I 1755 30.15 « 7564 3C.I5 « 9373 30.15 2 1183 30.16 2 2992 30.16 2 4802 30.16 2 6612 30.17 2 8422 3C.I7 ' 3 0232 3"'i 3 2043 j 30.18 3853 i 30.18 84 85 86 87 r, 1'..: M. •933 3><53 IMfl. 1". 30.18 l"K M. nitr. f 30.38 I..K M. f.955 1602 iiiir 1". 30.59 l"K M. Kill 1". 30.U2 ' '>44 2856 1.966 3140 1 •933 S<'"4 30.18 •944 4678 30.38 •955 ■f43'' 30.60 .966 4990 30.82 u •933 747; 30.19 •94 4 <'50» 30 39 •955 •'2T4 30.60 .966 6839 30.83 :i 933 91X7 30.19 •'U4 8}2< 0148 30.39 .955 8110 30.60 .966 8689 30.83 1 9 34 '09'* 30.19 •945 30.39 95 5 9946 30.61 ,967 0539 30.84 ^ 5 1.934 19 >o 30.20 "•945 1972 30.40 1.956 1783 30.61 f.967 2,89 3084 n ...34 4711 30.20 •945 3796 30.40 .956 3619 30.61 .967 424-3 30.84 7 934 <'^33 30.20 •945 5620 30.40 .956 5456 3C.62 .967 6090 30.85 N 9 34 X34<' 3021 •945 -444 ♦130.41 .956 7294 30.62 .967 794 « 30.85 U .935 0158 30.21 •945 9269 30.41 .956 9131 30.63 .967 9792 30.85 10 1.935 1971 30.21 ■ ••)46 1094 30.41 "•957 '^9'^9 30.63 i.93 .9h8 30.86 n •935 5597 30.22 .746 4744 30.42 •957 4<>45 30.64 .968 5 347 30.87 1:1 •93 5 74 "o 30.22 .946 <'5''9 30.42 •957 64X3 30.64 .968 7200 30,87 II •93 5 9113 30.22 .946 I* 395 30.43 957 83'H 30.64 .968 9052 30.87 i.'i 1.936 1037 30.23 '947 0221 30-43 1.958 0160 30.65 1.969 0905 30.88 lit .93(1 2851 30.23 •'n7 2047 3044 .958 1999 30.65 .9(19 2757 30.88 '■ 17 .936 4665 30.23 •947 3X73 3044 .958 3839 30.66 ■'>'P 4610 30.89 IH .936 6479 3o^i4 •947 5699 30.44 .958 5678 30.66 •9''9 30.89 l» .936 8293 30.24 947 7ji6 30-45 .958 7518 30.66 .969 8317 30.89 '^0 1.937 0108 30.24 ' 947 9353 30.45 1.958 9358 30.67 1.9-0 0171 30.90 •il •9 37 192=1 30.25 .948 iiSd 30.45 .95<) 1198 30.67 .970 2025 30.90 •ii •937 3737 30.25 ■'n^ 3007 30.46 •959 3038 50.67 .970 3879 30.91 'i'.i •937 5551 •9 37 73"i< 30.25 .948 4«34 30.46 •959 4'*79 30.68 -V 5734 75S9 30.91 •a 30.26 .948 6602 30.46 •959 <'720 30.68 .970 30.91 •i't 1.937 91S4 JO. 26 1.948 8490 30.47 1.959 8561 30.69 1.970 9443 30.92 ; 'U\ .938 0999 30.26 •94'> 0318 30.47 .960 0402 30.69 •97' 1 299 30^92 I 'i7 .938 2815 3- i7 •949 2146 30-47 .960 2243 30.69 •97' 3'54 30.93 i !iH .938 4632 v^.27 •949 397 5 30.48 .960 40S5 30.70 .971 5010 3093 •i\i .938 6448 30.27 •949 5804 30.48 .960 5927 30.70 •97' 6866 30.93 .10 1.938 8264 30.28 1.949 7633 9462 30.48 1.960 7769 30.70 1.971 8t22 3094 31 .939 0081 30.28 •949 30.49 .960 9612 30.71 •972 0578 30.94 M .939 1898 30.28 .950 1291 30.49 .961 1454 30.71 •972 2435 30<)5 ;i:i •939 37 'S 3C.29 •950 3»n 30.50 .961 3297 30.71 .972 4292 30.95 34 •939 5533 30.29 •950 495" 30.50 .961 5140 30.72 •972 6149 30^95 j 35 1.939 7350 30.29 1.950 6781 30.50 1.961 6983 30.72 1.972 8006 30.96 j 3» .939 9168 30.30 .950 8611 30.51 .961 8827 30^73 .972 9864 30,96 37 .940 0986 30.30 •951 0441 30.51 .962 0671 3073 •97 3 1 -ti ■* 30.97 3H .940 2804 30.30 ■ 951 2272 30.51 .962 2515 30.73 •97 3 3580 30.97 3U .940 4623 30.31 •95 • 4103 30.52 .962 4359 30^74 •97 3 5438 30.97 10 1.940 6441 30.31 1. 951 lUt 30.52 1.962 6203 30^74 '•97 3 7297 30.98 11 .940 8260 3031 •951 30.52 .96. 8048 30^75 •97 3 9156 30.98 Vi .941 0079 30.32 •95' 9597 30^ 5 3 .962 9S93 30^75 •974 1015 30.99 13 .941 1898 30.32 .952 1429 30-53 .963 1738 30^75 •974 2X74 30.99 1 II •94« 37" 7 30.32 .952 3261 30-53 .963 3583 30.76 •974 4734 30.99 ; 15 '•94' 5537 ■3o^33 1.952 5093 30.54 1.963 5429 30.76 '•974 6593 31.00 : 1(1 •94" 7357 3033 •952 69 ■ 30-54 .963 7275 30^77 •974 8454 31.00 17 .941 9177 30-34 .952 8758 30^55 .963 9121 30^77 •975 0314 3;. 01 IH .942 0997 3034 -95 3 0591 30.55 .964 0967 3077 -975 2174 31.01 lU .942 2817 3o^34 •953 2424 30-55 .964 2814 30.78 -975 4035 31.01 50 1.942 4638 30.35 '•953 4257 30.56 1.964 4660 30.78 '•975 5896 31.02 51 .942 6459 30.35 •953 6091 30.56 .964 6507 30.78 •975 7757 31.02 5'i .942 8280 30-35 •95 3 7924 30.56 .964 8354 30.79 •975 9619 31.03 53 .943 01 01 30.36 •95 3 9758 30.57 .965 0202 30.79 .976 1481 31.03 54 •943 '913 30.36 ■954 1592 30^57 .965 2050 30.80 .976 3343 31.04 55 «-943 3744 •943 5566 30.36 "•954 3427 3057 1.965 3897 30.80 1.976 5205 31.04 . 50 30^37 •954 5262 30.58 .965 5746 30.80 .976 7067 31.04 57 •943 7388 3037 •954 7096 30.58 •965 7594 30.8, ,976 8930 31.05 58 •943 9m 30-37 •954 0766 30-59 .965 9442 30.81 -977 0793 31.05 59 •944 >033 30.38 •955 30.59 .966 1291 30.81 -977 2656 31.06 00 1.944 1856 30.38 '•955 2602 30.59 1.966 3140 30.82 «-977 4520 31.06 587 f5 /■«?>;■ V TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Oibit. 588 V. 0' 88 S9 90 91 logM. Diff. 1". log M. 1.988 6789 Diff. 1". log M. Diff. 1". logM. Diff. 1". 1.977 4520 31.06 31.31 2.000 0000 31.50 2.01 1 4203 1 ■977 6383 31.06 .988 8668 31.32 .000 1895 3'-59 .011 61 15 3'-87 2 •977 8247 31.07 .989 0548 31.32 .000 3790 3'-59 .Oil 8027 3i.S,S 3 .978 0112 31.07 .989 2427 3'-33 .000 5686 31.60 .01 1 9940 31.88 ' 4 .978 1976 31.08 .989 4307 3«-33 .000 7582 31.60 .012 1853 31.89 1 3 1.978 3841 31.08 1.989 6187 3>-34 2.000 9478 31.60 2.012 3766 31.89 6 .978 5706 31.08 .9X9 8067 31-34 .001 >375 31.61 .012 5680 31.89 7 .978 7571 31.09 .989 9948 3«34 .001 3272 31.61 .012 7594 31.90 8 .978 9436 31.09 .990 1829 3>^35 .001 5169 31.62 .012 9508 31.90 U .979 1302 31.10 .990 3710 3«-3S .001 7066 31.62 .013 1422 31.91 ! 10 1.979 3168 31.10 1.990 5591 31.36 2. Co I 8963 31-63 2.013 3337 31.91 11 •979 S°U 31. II ■99° 7473 3'^36 .002 0861 31.63 .013 52^2 31.92 ! Vi .979 6901 31.11 .990 9355 3>-37 .002 2759 31.64 .013 7167 31.92 i;t .979 8768 31.11 •99' «i37 3«-37 .002 4658 6557 31.64 .013 9083 31-93 11 .980 0635 31.12 .991 3119 31.38 .002 31.65 .014 0999 3193 15 1.980 2502 31.12 1.991 5002 31.38 2.002 8456 31.65 2.014 2915 31-94 10 .980 4369 31.13 .991 6885 31.38 .003 0355 31.66 .014 4831 3 '-94 17 .980 6237 3'-i3 .991 8768 31.39 .003 2254 31.66 .014 tJf^ 3'-95 18 .980 8105 31.13 .992 0651 3'-39 .003 4»54 3i.r^7 .014 8665 3'-9; 19 .980 9973 31.14 .992 2535 31.40 .003 6054 31.67 .015 0582 31.96 20 1.981 1842 31.14 1.992 A419 .992 6304 31.40 2.003 7955 31.68 2.015 2500 31.96 21 .981 3710 3i^»5 31.41 .003 9855 31.68 .015 4418 3'-97 22 .981 5579 3i-»S .992 8188 31.41 .004 'I^o 31.(58 .015 6336 3' -97 23 .981 7449 31.16 .()()^ 0073 31.42 .004 3658 31.69 .015 8255 31.98 24 .981 9318 31.16 •993 '95« 31.42 .004 5559 31.69 .016 0174 31.98 25 1.982 1188 31.16 1.993 38*3 31.42 2.004 7461 3'-70 2.016 2093 31.99 26 .982 3058 31.17 •993 5729 3i'43 .004 93''3 31.70 .016 4012 31.99 27 .982 4928 3'-'7 •993 7615 3'-43 .005 1165 31.71 .016 5932 32.00 28 .982 6798 31.18 •993 9501 3I-44 .005 3168 31.71 .016 7852 32.00 29 .982 8669 31.18 •994 '387 3 1 44 .005 5071 31.72 .Oiu 9772 32.01 1 30 1.983 0540 31.18 1-99+ 3274 31-45 2.005 6974 31.72 2.017 1693 32.01 31 .983 24 1 1 31.39 •994 5'6i 31-45 .005 8878 3>-73 .017 3614 32.02 32 .983 4283 31.19 •994 7048 31.46 .006 0781 3«-73 .017 5535 32.02 1 33 .983 6155 31.20 ■994 8936 31.46 .006 2685 3»-74 .017 7456 32.03 1 34 .983 8027 31.20 •995 08*3 31.46 .006 4590 3'-74 .017 9378 32^03 ! 35 1.983 9899 31.21 1.995 2711 31-47 2.006 6494 31-75 2.018 1300 32.04 1 36 .984 1772 31.21 .995 4600 3'-47 .006 8399 31-75 .018 3223 32.04 1 37 .984 3644 31.22 .995 6488 31.48 .007 0304 1 31.76 .018 5'i5 32.05 1 38 .984 5517 31.22 •995 8377 31.48 .007 2210 1 31.76 .018 7068 32.05 ! 39 .984 7391 31.22 .996 0266 31.49 .007 41 '6 1 3'-77 .018 8992 32.06 40 1.984 9264 31-13 1.996 2155 3«-49 2.007 6022 3'-77 2.019 0915 32.06 41 .985 1138 3i-i3 ■')')" 4°45 31.50 .007 792S 1 31-77 .019 2839 32.07 42 .985 3012 31.24 ■996 5935 31.50 .007 9835 31.78 .019 4763 32.07 43 .985 |886 31.24 .996 7825 3'-5i .008 1742 ' 31.78 .019 6688 32.08 44 .985 6761 ii-H .996 9716 31-51 .008 3649 1 31-79 .019 8613 32.08 45 1.985 8636 31.25 1.997 1606 31.51 2,oo8 5556 i 31-79 2.020 0538 32.09 4fi •"^o^ °5'5 3i^2S •997 3497 31.52 .008 7464 31.80 .020 2463 32.09 47 .986 2386 31.26 •997 5389 3'^5i .008 9372 31.80 .020 4389 3-"^ 48 .986 4262 31.26 .997 7280 31-53 .009 1280 ! 31.81 .020 6315 32.10 49 .986 6138 3i'i7 •997 917a 3»-53 .C09 3189 j 31.81 .020 8241 32.11 50 1.986 8014 31.27 1.998 1064 31-54 2.009 5098 i 31.82 2.021 0168 31. 1 1 51 .986 9890 31.28 .998 2956 31-54 .009 7007 ! 31.82 .021 2095 32.11 52 .987 1767 31.28 .998 4849 31-55 .009 8917 31.83 .02 1 4022 32.12 53 .987 3644 31.28 .998 6742 31-55 .010 0826 31.83 .021 5949 32.13 54 .987 5521 31.29 .998 8635 31-56 .010 2736 31.84 .021 7877 : 32^i3 55 1.987 7398 31.29 1.999 0529 31.56 2.010 4647 i 31.8+ 2.021 9805 ' 32.14 56 .987 9276 31.30 •999 2421 31.56 .010 6557 1 31.85 .022 '734 32.14 57 .988 1154 31.30 •999 43 '6 31-57 .010 8468 31.85 .022 3662 32.1,- 58 .988 3032 31.31 .999 621 1 3«-57 .oil 0380 31.86 .022 5591 32.15 59 .988 49H 31-31 .999 8105 31.58 .011 2291 31.86 .022 7521 32.16 eo 1.988 6789 3I-3' 2.000 0000 31.58 2,011 4203 31.87 2.022 9450 3».i6 in a Parabolic Oibit. TABLE VI. For tliiding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. 91^ loK M. Diff. 1". 2.01 I 4203 I .oil 6115 ' .on 8027 ! .011 9940 \ .012 1853 ; 2.012 3766 1 .012 5680 I .012 7594 .012 9508 .013 1422 2.013 3337 .013 52^2 .013 7167 .013 9083 .014 0999 291S 483. 6748 8665 0582 2500 4418 6336 8255 0174 2093 4012 5932 7852 9772 1693 3614 55 35 7456 9378 1300 3223 5145 7068 8992 0915 1839 47^' 3 6688 8613 0538 2463 43«9 6315 8241 0168 2095 4022 5949 7877 2.014 014 014 1014 ,015 2.015 .015 .015 .015 .016 2.016 .016 .016 .016 • Oku 2.017 .017 .017 .017 .017 2.01S .018 .018 .018 .018 2.019 .019 .019 .019 .019 1.020 .020 .020 .020 .020 2. 021 .021 .02 1 .021 .021 3'-S7 3i.8,>< 31.88 31.89 31.89 31.89 31.90 31.90 31.91 31.91 31.92 3'-9i 3 •■93 31-93 3«94 3«-94 3>-95 3 '9 5 31.96 31.96 31-97 3''97 31.98 31.98 3'-99 i 3 '99 I 32.00 i 31-0° I 32.01 i 32.01 I 32.02 ' 32-Oi 32.03 32.03 32.04 32.04 32.0^ 31.0^ 32.06 32.06 32.07 32.0? 31.08 32.08 32.09 32.09 ^ 5 5 6 6 2.021 9805 ' .022 1734 ,022 3662 .022 5591 .022 7521 7 2.022 9450 l\ 920 93° 94° 95 i let: -M. iHfr. 1". 1„K M. Dim 1". 32.48 2.046 3296 Diff. 1". 1 kM. Diff. 1". 33. '5 1.022 9450 32.16 2.034 5797 32.80 2.058 2005 1 .023 1380 ! 32.17 .034 7745 32.48 .046 <;264 32.81 .058 3994 33-' 5 2 .023 3311 j 32.17 .034 9694 32.49 .046 7233 32.82 .058 59S3 33.16 3 .023 5241 i 32.18 .035 1644 3249 .046 9202 32.82 .058 7973 33.6 4 .023 7172 1 32.18 .035 3593 32.50 .047 1172 i 32.83 .058 9963 33.17 i 5 2.023 9103 1 32.19 2.035 5543 32.50 2.047 3141 32.83 2.059 1953 33- « 8 6 .024 1035 32.19 •035 7494 32.51 .047 5111 32.84 .059 3944 33-'8 7 .024 2967 32.20 .035 9444 32.51 .047 7082 32.84 •059 593 5 33-'9 8 .024 4899 1 32.20 .036 «395 32.52 .047 9053 32.85 .059 7927 33. '9 9 .024 6831 32.21 .036 3 347 , 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 3321 li .025 0697 32.22 .036 7250 32-53 .048 4967 32.87 .060 3904 33.21 Vi .025 2630 32.22 .036 9202 32.54 .048 6939 32.87 .060 5897 33.22 III .025 4564 32.23 .037 1155 3254 .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 3324 16 .026 0367 32.24 .037 7015 32.56 .049 4,(31 32.89 .061 3872 33-24 17 .026 2301 32.25 .037 8969 32.57 .049 6805 32.90 .061 5867 3325 18 .026 4236 32.26 .038 0923 32-57 .049 8879 32.90 .061 7862 3325 10 .026 6172 32.26 .038 2877 32.58 .050 0753 32.91 .061 9857 33.26 20 2.026 8108 32.27 2.038 4832 32.58 2.050 2728 32.92 2.062 1853 33.27 21 .027 0044 32.27 .038 6787 32.59 .050 4703 32,92 .062 3849 33.27 22 .027 19S0 32.28 .038 8743 32.59 .050 6679 32.93 .062 5.X46 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 2.030 461 1 32.61 2.051 260S 32-95 2.063 1S3-/ 33-3° 20 .027 9729 3-3° .039 6568 32.62 .051 45S5 32.95 ''3 3835 33-30 27 .028 1667 3i'3° .039 8525 32.62 .c'l 6562 32.96 .063 5S3; 33-3' 28 .028 ,3605 32-3« .040 0482 32.63 .051 8539 32.96 .063 7832 33-3' 2» .028 5544 32.31 .040 2440 32.63 .052 0517 32.97 .063 9831 33.32 :io 2.028 7483 32.32 2.040 4399 32.64 2.052 2496 32.97 2.064 1 83 1 33-33 :u .028 9422 32.32 .040 6357 32.64 .052 4474 32.98 .064 3830 33-33 ;j2 .029 1361 3*-33 .040 8316 32.65 .052 6453 32.98 .064 5830 33-34 33 .029 3301 3^-33 .041 0275 32.65 .052 8432 32.99 .064 7S31 33-34 34 35 .029 5241 2.029 7182 3»-34 32.3.: .041 2.04; 2234 4194 32.66 32.67 .053 0412 2053 239-. 33.00 3 3-00 .064 9832 33-35 2.065 '833 33.36 30 .029 9123 32.35 .041 6154 32.67 .053 4372 33-01 .065 3834 33.36 37 .030 1064 31-35 .041 8,14 32.68 .053 6353 33.01 .065 5836 33 37 38 .030 3005 32.36 .042 0075 32.68 •053 8334 33.02 .065 7839 33 37 ■ 39 .030 4947 32.36 .042 2036 32.69 .054 0315 33.03 .065 9841 33-38 10 2.030 6889 32.37 2.042 3998 32.69 2.054 2297 33-03 2.ob6 1844 3^39 4« .030 S831 32-37 .042 5960 32.70 .054 4279 33.04 .066 3847 53-39 12 .031 0774 32.38 .042 7922 52.70 .054 6262 33.04 .066 5851 33-40 13 .031 2717 32-39 .042 98x4 3 ■■.7 1 .054 8244 33.05 .066 78155 33-40 14 .031 4660 32-39 .043 1847 32.71 .055 0227 33-05 .066 9860 33-4' 45 2.03 1 6604 32.40 2.043 3810 32.72 2.055 2211 33.06 2.067 '865 33-42 10 .031 8548 32.40 .04-; 5773 32-73 .055 4195 33.07 .067 3S70 33-42 17 .032 0492 32.41 .043 7737 32-73 .055 6179 33.07 .067 5875 33.43 48 .032 2437 32-4> -043 9701 32.74 .055 8163 33.08 .067 7881 33-43 ; 49 .032 438'2 32.42 .044 1665 32.74 .056 0148 33-08 .067 9887 33-44 .50 2.032 6327 32-42 2.044 3630 32.75 2.056 2133 33-09 2.068 1894 33.45 51 .032 8272 32-43 .044 5595 32-75 .056 4119 33.10 .068 3901 33.45 52 .033 0218 32-43 .044 7561 32.76 .056 6105 33.10 .068 5908 33.46 53 .033 2164 32.44 .044 9=;26 32.76 .056 8091 33.11 .c68 7916 33-47 51 ^33 4HI 32-44 .045 '492 32.77 .057 0078 33.11 .068 9924 33-47 55 2.033 6058 32-45 2.045 3459 32.78 2.057 2065 33.12 2.069 1933 33.48 50 .033 8005 32.45 .045 5426 32-78 .057 4052 33.12 .069 3942 33-48 i 57 .033 9952 32.46 .045 7393 32.79 .057 6040 33->3 .069 5951 33.49 58 .034 1900 32.47 .045 9360 32.79 .057 8028 33.'4 .069 7960 33-50 i 59 .034 3848 32-47 .046 1328 32.80 .058 0016 33.'4 .069 9970 33-50 I 60 2.034 5797 32.48 2.046 3296 32.80 2.058 2005 33.'S 2.070 1980 33-51 -- 589 TABLE VI. For finding the True Anomaly or llie Tinu- from tlie Perihelion in a Parabolic Orbit. V. 96 ,97 98 99 lo^ M. IHff. I". 33-5« logM. Diff. 1". los M. Diir. 1". logM. 2. 1 07 0109 Diir. 1". 34.69 2.070 1980 2.0S2 3181 33.88 2.094 5971 34.28 1 .070 399 « 33o« .082 53'6 33.89 .094 8028 34.29 .107 2190 34.70 •z .070 6002 33'52 .082 7349 33 90 .095 0085 34-29 .107 4171 34-70 3 .070 8014 33-53 .082 9 3 '"'3 33.90 -095 2143 34-30 .107 6355 34-71 4 .071 0025 33-53 .083 1418 33.91 .095 4201 34-31 .107 8437 34-72 5 1.071 2037 33-54 1.083 5+53 33-91 2 095 6260 34- 3' 2.108 0521 34-72 6 .071 40,-0 33-54 .0S3 5488 3392 .091; 8318 3432 .108 2604 34-73 7 .071 6063 33-55 .0S3 7523 33-93 .096 0378 34-33 .108 46S9 3-V74 8 .071 8076 33-5*' .083 9559 33-94 .096 2438 3433 .108 6773 34-'' 5 .072 0090 33-56 .084 1596 33-94 .096 4498 j4-34 .108 8858 34-75 10 2.072 2104 33-57 2.0S4 3633 33-95 1 096 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 3358 .084 7707 33.96 .097 06S1 34.36 .109 5 1 16 34-77 i i;i .072 8148 33-59 .084 9745 33-97 -=97 2742 34-37 .109 7201 34-78 14 .073 0163 33-59 .085 '783 33-98 .097 4804 34-37 .109 9289 34-79 15 2.073 2179 33.60 2.0S5 ^Sr- 33.98 2.097 6867 34-38 2.110 1377 34-80 , 16 .073 4195 33-61 .08 s S86i 33-99 .097 8930 34-39 .110 3465 34-80 17 .073 6212 33-61 .08 i; 7901 33-99 .09X 0993 ^ 34-39 -«io 5553 34-81 18 .073 8229 33.62 .085 994' 34.00 .098 3057 34-40 .110 7642 34-82 lU .074 0246 33-63 .086 1981 34.01 .098 51 21 34-4» .110 9731 34.82 20 2.074 2264 33-63 2.086 4021 34.01 2.098 7186 34-4« 2. Ml 1821 34-83 21 .074 4282 33-64 .086 6062 3-V.02 .098 9251 34-42 .111 39II 34-84 22 .074 6301 3364 .086 8104 34-03 .099 I 316 34-43 .111 6001 34-85 23 .074 8320 33-65 .087 0146 34-03 .099 3382 34-43 .111 8091 34-85 24 .075 0339 33.66 .087 2188 34.04 .099 5449 34-44 .112 0184 34.86 25 2.075 2358 33.66 2.C'^7 4231 34-05 2.099 7515 34-45 2.112 2275 34-87 26 .075 4378 33-67 .oXy 6274 34-05 .099 9581 34-45 .112 4368 34-87 ■ 27 .075 6399 33.67 .087 83,7 34.06 .100 1650 34.46 .112 6460 34-88 : 28 .075 84.9 33.68 .08 S 0361 34-07 .100 3718 34-47 .112 8553 34-89 ' 29 .076 0440 33-69 .08 .S 2405 34-07 .100 5786 34-48 .113 0647 34-90 30 2.076 2462 3369 1.088 4449 3408 2.100 7855 3448 2.113 2741 34-?'> 31 .076 4484 33-70 .0S8 6494 3409 .100 9914 34-49 .113 4835 34-91 32 .076 6507 33-71 .08 8 8540 34.09 .101 1993 34-50 .113 6930 34-'.2 33 .076 8529 33-71 .089 0586 34.10 .101 4063 34-50 .113 9025 34-92 34 .077 0552 33-71 .089 2632 34.11 .101 6134 34-51 .114 1121 34-y3 35 2.077 *57S 33-73 2.0S9 4678 34.11 2.IOI 8204 34-52 2.114 3*17 34-94 \ 30 .077 4599 33-73 .0S9 6725 34.12 .102 0276 3452 .114 5313 34-95 i 37 .077 6623 33-74 .089 8772 3412 .102 2347 34-53 .114 7410 34-95 , 38 .077 8647 33-74 .090 0820 34- « 3 .101 4419 34-54 .114 9508 34-96 : 39 .078 0672 33-75 .090 2868 34.14 .102 6492 34-54 .115 1605 34-97 40 i.078 2697 33.76 2.090 4917 34->5 2.102 8564 34-55 2.115 3704 34-97 41 .078 47 •4 3 33-76 .090 6966 34-15 .103 0638 34-56 .115 5802 34-98 42 .078 6749 33-77 .090 9015 34.16 .103 27II 34-56 .115 7901 34-99 43 .078 8775 33.78 .091 1065 34-17 .103 47X5 34-57 .1 16 OOCl 35.00 44 .079 0802 33.78 .091 3115 34- « 7 .103 6860 34.58 .116 2101 35.00 45 2.079 2829 33-79 1.091 5165 34.18 2.103 893s 34-59 2.116 4201 35-01 46 .079 4857 33.80 .091 7216 34-19 .104 lOIO 34-59 .1 16 6301 35.02 ! 47 .079 6885 33.80 .091 9268 34.19 .104 3086 34.60 .ii'S 8403 35.01 48 .079 8913 33-81 .092 1319 34.20 .104 5162 34.61 .117 0505 35-03 49 .080 0942 33. M .092 3371 34.20 -'04 7239 14.61 .117 2607 35-04 50 7. 080 2971 3382 2.092 5444 34.21 2.104 93 '6 34.62 2.117 4710 35-05 51 .080 5000 3383 .092 7477 3422 .105 1393 3463 .117 6813 3505 1 52 .080 7030 33,83 .092 953° 34.22 .105 3471 34-63 .117 8916 35.00 53 .080 9060 33-84 .093 1584 34-23 .lOf 5549 34.64 .118 1020 35-07 54 .081 1091 33-85 .093 3638 34-24 .105 7628 34.65 .118 3124 35.08 ^ 55 2.081 3122 33-85 1.093 5692 34-»5 2.105 9707 34.66 2.118 5229 35.08 . 50 .081 5153 33-86 .093 7747 34-25 .106 1786 34-66 .118 7334 35-09 ; 57 .081 7185 33-87 .093 9^°3 34.26 .106 3866 3467 .118 9440 35-'o ! 58 .081 9217 "•^Z .094 1858 3427 .106 5947 34.68 .119 1546 35-'o 59 .082 1249 33.88 .094 3914 3427 .106 8027 34.68 .119 36^2 35-" 60 2.082 3282 33-88 2.094 5971 34.28 2.107 °io9 34.69 1.119 5759 35.U 590 mA ibolie Orbit. TABLE VI. For finding the True Anomaly or tiie Time from the Periiielion in a Parabolic Orbit. 99^ 3217 53'3 7410 9508 1605 3704 5802 7901 con 2101 4201 6301 Diff. 1". 34.69 34.70 3470 34-71 34-72 34-72 34-7) 34-74 34- ■'5 34-7i 34.76 34-77 34-77 34. 7 S 34-79 34.80 34, So 34.S1 34.S2 34.S2 34-5=3 34->*4 34-i*5 34.i<5 34.86 54-i<7 34-!<7 34.88 34.89 34.90 34-9'' 34-91 34-'72 34-'^- 34-'^) 34-94 3495 349i 34-'''' 34-97 34-9'' 34-9*f 34-99 3vOO 35.00 35.01 35.01 3^- *^403 0505 3v03 2607 35.04 4710 6813 8916 1020 3124 5229 7334 9440 1546 36-2 5759 35-°5 35-°i 3voO 35.08 35.08 35-°9 35.10 35''° 35" 35.12 l\ O 1 a :t 4 5 8 » 10 II vz 1:} 11 15 1(> 17 18 19 -zo •il Ti 23 24 25 2(S 27 28 21) 30 31 32 33 31 35 3U 37 38 3« 10 II 12 43 41 45 40 47 48 4U 50 51 52 53 54 55 5(i 57 58 5» 00 100° 101° loR M. Dlff. 1". logM. Diir. 1". .119 5759 35.12 2.132 2989 35-57 .119 7867 35-'3 .132 5123 35-57 .119 9974 35'3 .132 7258 35-58 .120 2083 35-'4 .132 9393 35-59 .120 4191 35-15 -'33 '529 35.60 .120 6301 35.16 S.I33 3665 35.61 .120 8410 35.16 .133 5802 35.61 .121 0520 35-'7 •133 7939 35.62 .121 2630 35.18 .134 0076 35-63 .121 4741 35'9 .134 2214 35-64 .121 6853 35-'9 J. 134 4352 35-64 .121 8965 35.20 -'34 649' 35-65 .122 1077 35.21 .134 8631 35.66 .122 3190 35.21 -'35 0770 35-67 .122 5303 35.22 .135 2910 35-67 ..122 7416 35-23 2.135 5051 35-68 .122 9530 35-24 .135 7192 35-69 .123 1644 35-24 -'(5 93 34 35-70 •123 3759 35-25 .i}6 1476 35-7' .123 587s 35.26 .136 3619 3i7i .123 7990 35-27 2.^36 5762 35-72 .124 0107 35-27 .136 7905 35-73 .124 2223 35.28 -137 0049 35-74 .124 4340 35-9 .137 2193 35-74 .124 6458 35.30 -'37 4338 35-75 .124 8576 35-30 2.137 6484 35-76 .125 0694 35-3' .137 8630 35-77 .125 2813 35-32 .138 0776 35-77 .125 4933 35-33 .138 2922 35-78 .125 7052 35-33 .138 5070 35-79 .125 9173 35-34 2.138 7217 35.80 .126 1293 35-35 .138 9365 35-81 .126 341^ .126 5536 35-35 .139 1514 35.81 35-36 •139 3663 35-82 .126 7658 35-37 -139 5813 35-83 ..126 9780 35-38 2.139 7963 35-84 .127 1903 35-39 .140 0113 35.84 .127 A027 .127 6151 35-39 .140 2264 35-85 35.40 .140 4415 35-86 .127 8275 35-4' .140 6567 35-87 .128 0400 35.42 2.140 8720 35-88 .128 2525 35.42 .141 0873 35-88 .128 4650 35-43 .141 3026 35.89 .128 6776 35-44 .141 5180 35-90 .128 8903 35-45 .141 7334 35-9' .129 1030 35-45 2.141 9A89 .142 1644 35.92 .129 3157 35.46 35.92 .129 5285 35-47 .142 3799 35-93 .129 7414 35.48 .142 5955 35-94 .129 9542 35-48 .142 81 12 35-95 L.130 1672 35-49 2.143 0269 35-96 .130 3801 35-50 • '43 2427 35-96 .130 5931 .130 8062 35-5' .143 4585 35-97 35-51 .143 6743 35-98 .131 0193 35-52 .143 8902 35-99 .131 2325 35-53 2.144 '062 36.00 .131 A457 35-54 .144 3222 36.00 .131 6589 35-54 .144 5382 36.01 .131 8722 35-55 35-56 •144 7543 36.02 .132 0855 •'44 9704 36.03 ..132 2989 35-57 2.145 '866 36.03 102 IobM. Diff. 1". 36.03 36.04 36.05 ^6.06 35.07 .145 1866 .145 4028 .145 6191 -'45 8354 .146 0518 .146 2682 .146 4847 .146 70.' 2 .146 9 1/8 -'47 '344 36.07 36.08 36.09 36.10 36.11 -'47 35'o -'47 5677 •'47 7845 .148 0013 .148 2182 36.11 36.12 36.13 36.14 36-15 .148 4351 .148 6520 .148 8690 .149 oS6i .1.^9 3032 36.17 36.18 36.19 -'49 5203 -'49 7375 -'49 9547 .150 1720 .150 3893 36.19 36.20 36.21 36.22 36.23 .150 6067 .150 8242 .151 0417 .151 2592 .151 4768 36-23 36.24 36.26 36-27 .151 6944 .151 9121 .152 1298 .152 3476 .152 5654 36.28 36,28 36.29 36.30 36.31 -152 7833 .153 0012 .153 2192 -'53 4372 -153 6552 36.32 36.32 36.33 36-34 36.35 -'53 8734 .154 0915 .154 3097 .154 5280 .154 7463 5^3| 36.36 36.37 36.38 36.39 • 154 9647 • 155 '831 .155 4015 .155 6200 .155 8386 36.40 36.41 36.41 36.42 36.43 .156 0572 .156 2759 .156 4946 .156 7133 .156 9321 36.44 36.45 ^^^^ 36.46 36.47 ■'57 i5'o •'57 3699 •'57 5889 .157 8079 .158 0269 36.48 36.49 36.50 36.50 36.51 .158 2460 36.52 103^ log M. 1 ! .158 2460 .158 4652 .158 6844 .158 9030 , .159 1229 : ■'59 3423 i -'59 5617 1 .159 7811 i .160 0006 ; .160 2 202 .160 4398 160 6594 ; .160 8791 1 .161 09X9 1 .ibi 3187 .161 5385 .161 75*>4 i .161 9784 ! .162 ,984 ' .162 41X5 1 .162 6386 .162 8587 .16-! 0789 ! .163 2992 i .163 5 '95 1 .163 7398 ..63 9602 1 .164 1807 j .164 4012 1 .164 6218 I .164 8424 1 .165 0630 i .165 2837 1 .,65 5045 .165 7253 .165 9462 .166 1671 .166 3881 .166 6091 .166 8301 .167 0513 .167 2724 .167 4936 1 .167 7149 .167 9362 .168 1576 .168 3790 .168 6005 .168 8220 .169 0436 1 .169 2652 .169 4869 .169 7087 .-69 9304 .170 1523 .170 37-J2 5961 .170 .170 8181 .171 0401 2622 .171 .171 4844 nifl. I". 36.52 36.53 36- 54 36.55 36.55 36.56 36.57 36.58 36.59 36.60 36.60 36.61 36.62 36.63 36.64 36.65 36.65 36.66 36.67 36.68 36.69 36.70 36.70 36.71 36.72 36.73 36.74 36.74 36.75 36.76 36.77 36.78 36.79 36. Xo 36.81 36.81 36.82 36.83 36.84 36.85 36.86 36.87 36.87 36.88 36.89 36.90 36.91 36.92 3693 36.93 36.94 36.95 36.90 36.97 36.98 36.99 36.99 37.00 37.01 37.02 37.03 591 TABLE VI. For finding the True Anomaly or the Time from the Perihelion in a Parabolic Orbit. |7 V. 1 104° , 105° 106° 107° log M. Diff. 1". log M. Diir. 1". 37-56 logM. 2.198 5282 Diir. 1". log M. Diff. 1". ! 38.68 2.I7I 4844 1 37-03 2.184 9092 38.11 2.212 3493 1 .171 7066 37.04 .185 1346 37^57 .198 7568 38.12 .212 5814 38.69 ; 2 .171 9288 37^05 .185 3600 37^57 .198 9856 38-'3 .212 8136 38.70 i 3 .I7Z 5" 37-05 -'85 5855 37^58 .199 2144 38.14 .213 0458 38.7. 1 4 .172 i73S 37.06 .185 8110 37^59 -'99 4432 38.14 .213 2781 38.72 1 5 2.171 P^"^ 37.07 2.186 0366 37.60 2.199 6721 38-'5 2.213 5104 38^73 1 6 .172 8184 37.08 .186 2622 37^6i .199 9010 38.16 .213 7428 3^-74 . i 7 •»73 0409 37-09 .i?6 4879 37.62 .200 1 300 3!''Z •2'3 9753 38-75 1 ^ .173 2634 37.10 .186 7137 37^63 .200 3591 38.18 .214 2078 38.76 9 .173 4860 37.11 •'86 9395 37.64 .200 5882 38.19 •214 4404 3'<^77 : 10 2.173 7087 37.12 2.187 1653 37-65 2. zoo 8174 38.20 2.214 6730 38.78 11 •'73 93'4 37^ia .187 3912 37-66 .201 0467 38.21 .214 9057 3S.79 12 .174 1542 37-13 .187 6172 37-67 .201 2760 38.22 .215 1385 3S.80 13 .174 3770 37^'4 .187 8432 37-67 .201 5053 38.23 -2'5 37'3 38.81 14 .174 5999 37^i5 .188 0693 37-68 .201 7347 38.24 .215 6042 38.82 15 2.174 8228 37.16 2.188 2954 37.69 2.201 9642 38.25 2.215 8371 38.83 ! 16 •«75 0458 37^i7 .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.S5 18 •«75 49'9 37^i8 .188 9741 37-72 .202 6529 38.28 .216 5363 3S.X6 - 19 ••75 7150 37^i9 .189 2005 37-73 .202 8826 38.29 .216 7694 38.87 . 20 ^•>7S 9382 1615 37.20 2.189 4269 .189 6533 37-74 2.203 1123 38.30 2.217 0027 38.88 21 .176 37.21 37^75 .203 3421 38.31 .217 2360 38.?i 22 .176 384X 37.22 .189 8798 37^76 .203 5720 38.31 .217 4693 3S.90 23 .176 6081 37.23 .190 1064 37-77 .203 8019 38.32 .217 7027 3S.91 24 .176 83'5 37-24 .190 3330 37-77 .204 0319 38.33 .217 9362 3S.92 25 2.177 0550 37-25 2.190 5597 37^78 2.204 2619 38.34 2.21 8 2697 38.93 2« •'77 2785 37-25 .190 7864 37^79 .204 4920 38-35 .218 4033 3S.94 2T •'77 5020 37.26 .191 0132 37-80 .204 7222 38.36 .218 6369 3S.95 28 •'77 7256 37.27 .191 2401 37-81 .204 9524 38-37 .218 8706 38.96 •^9 •'77 9493 37.28 .191 4670 37-82 .205 1826 38-38 .219 1044 38-97 , 30 2.178 1730 37.29 2.191 6939 37-83 2.205 4129 -205 6433 38-39 2.219 3382 38.98 31 .178 3968 3730 .191 9209 37.84 38.40 .219 5721 3S.99 32 .178 6206 37-3' ,192 1480 37-85 .205 8737 38.41 .219 8o6i 39.00 33 .178 844s 37-32 .192 3751 37-86 .206 1042 38.42 .220 0401 39.01 34 •'79 0684 37-33 .192 6023 37-87 .206 3348 38.43 .220 2741 39.02 35 2 173 2924 37-33 2.192 8295 37.88 2.206 5654 38.44 2.220 5082 3y-°3 , 36 .179 5164 37-34 .193 0568 37.88 .206 7961 38^45 .220 7424 39-°4 37 •'79 7405 37-35 .193 2841 37.89 .207 0268 38.46 .220 g7'57 39-°5 ; 38 .179 9646 37-36 -'93 5"5 37.90 .207 2575 38.47 .221 2110 39.06 39 .180 1S88 37-37 •'93 7389 37.91 .207 4884 38.48 .221 4453 39.07 40 2.i8o 4'3i 37-38 2.193 9664 37.92 2.207 7193 38.49 2.221 6797 39.08 : 41 .180 6374 37-39 .194 1940 37^93 .207 9502 38.50 .221 9142 39-09 , 42 .180 ^^J7 37.40 .194 4216 37-94 .208 1812 38-51 .222 14X8 39.10 43 .181 0861 37-41 -194 6493 37-95 .208 4123 .208 6434 38.52 .222 3834 39.11 ; 44 .181 3106 37^4' .194 8770 37-96 38.53 .222 6180 39.12 45 2.181 5351 37.42 2.195 1048 37-97 2.208 8746 38.54 2.222 8528 39-'3 46 .181 7597 37^43 .195 3326 37.98 .209 1058 38-54 .223 0876 39.14 47 .181 9S43 37-44 .195 5605 37-99 .209 3371 38-55 .223 3224 3'>'5 48 .182 1089 37-45 -'95 7885 38.00 .209 5685 38.56 •223 5573 39.16 49 .182 4337 37.46 .196 C165 38.00 -209 7999 38-57 .223 7923 3'>'7 50 2.182 6584 37^47 2.196 2445 38.01 2.21c 0314 38-58 2.224 0273 39.18 51 .182 8833 1082 37.48 .196 4726 38.02 .210 2629 38-59 .224 2624 3J-'9 52 .183 3 7 '49 .196 7008 38.03 .210 4945 38.60 .224 4975 39.10 53 .183 3331 37-49 .196 9290 38.04 .210 726t 38.61 -224 7327 39.21 54 •'S3 5581 37-50 -'97 '573 38.05 .210 9578 38.62 .224 9680 39.22 55 2.183 7831 37-51 2.197 3856 .197 6140 38.06 2.21 I 1896 38.63 2.225 2033 39.23 56 .184 0082 37.52 ^l-°l .211 4214 .211 6533 38.64 •225 4387 39.24 57 .184 ^2U 45X6 37-53 .197 8425 38.08 38-65 .225 6741 3925 58 .184 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 ».226 3808 39.28 592 ' bolic Orbit. TABLE VL For finding the True Anomaly or the Time from the Periiielion in a Parabolic Orbit. 107= M. Diff. 1" 3493 1 38.M 5814 1 38. 69 8136 I 3S.70 0458 38.71 1 2781 3X.72 1 5104 38-73 7428 3i<-74 9753 38-75 2078 38.76 4404 38-77 6730 38.78 9057 38-79 13X5 38.80 3713 38.x, 6042 38.82 8371 38-83 0701 38. X4 3032 ! 38. S5 > 53<'3 1 i'^-^(> ' 7694 1 T,^'^! J 0027 1 38.88 J 2360 ! jS.^'l 7 4693 38-90 7 70^7 38-91 7 936^ 3S.92 8 3697 38.93 X 4033 1 3S.94 8 6369 1 38.95 8 8706 3S.96 9 1044 38.97 9 338* 38.98 9 5721 38.99 9 8061 39.00 0401 39.01 2741 39.02 5082 39-°3 7424 39.04 97'i7 39.05 I 21IO 39.06 I 4453 39.07 I 6797 39.08 I 9142 39.09 2 1488 1 39-10 2 3>'34 i 39-" 2 6180 i 39.12 2 8528 : 39.13 3 0876 ; 39.14 3 3124 1 39-'5 3 5573 1 39. 11) 3 79^3 1 39->7 4 0273 ! 39.18 4 2624 1 3;.i9 4 4975 ' 39-^-° 4 73^7 i 39'2i 4 9680 - 39.12 5 2033 1 39-23 5 4387 ; 39-^4 5 6741 ' 39 ^-5 5 9096 ' 39-26 6 1452 . 39-27 6 3808 3928 V. o 1 'Z :i 4 5 7 8 9 10 II la 14 ir> 10 17 18 19 20 21 22 23 24 25 20 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 40 47 48 49 50 51 52 53 54 55 5» 57 5H 59 60 108^ logM. Difr. 1". 2.234 .234 -235 .235 •^35 2.235 .236 .236 .236 .236 2.237 •^37 •237 ■*37 .23S 2.238 .238 .238 .238 .239 2.239 •^39 .239 .240 .240 3808 I 6165 8523 ; 0881 : 3240 j 5599 ! 7959 ; 0320 i 2681 j 5043 7405 9768 : 2131 I 4496 I 6861 9226 I 159a ! 3959 1 6326 i 8694 ■ 1063 343a 580Z 8172 0543 2.232 2915 232 5287 232 7660 233 0033 233 2407 2.233 4782 .233 7157 -a33 9533 .234 1910 .234 4287 2.226 .226 .226 .227 .227 2.227 .227 .228 .228 .228 2.228 .228 .229 .229 .229 2.229 .230 .230 .230 .230 2.231 231 231 231 232 6665 9043 1422 3802 6183 8563 0945 3327 5710 8093 0478 2862 5247 7633 0020 2407 4795 7284 9573 1962 4353 6744 9235 1528 3921 2.240 6314 39.28 39.29 39.30 39-3« 39.32 39-33 39-34 39-35 39.36 39-37 39-38 39-39 39.40 39.41 39.4a 39-43 39-44 39-45 39.46 39-47 39.48 39-49 39.50 39-5« 39-5a 39-53 39-54 39-55 39-56 39-57 39.58 39-59 39.60 39.61 39.63 39.64 39-65 39.66 39.67 39.68 39.69 39.70 39-71 39.72 39-73 39-74 39-75 39.76 39-77 39.78 39-79 39.80 39.81 39.82 39.83 39.84 39.8^ 39.87 39.8X 39.89 39.90 2.240 240 241 241 241 109= logH. 6314 8708 1 103 3498 5894 Dim 1". a. 241 8291 ,242 0688 .242 3086 .242 5485 .242 7884 2.243 0284 .243 2685 .243 5086 •243 7488 .243 9890 2.244 •'•'-44 .244 •244 •a45 2.245 .245 .245 .246 246 2.246 246 247 247 a47 2.247 .248 .248 .248 .248 2.249 .249 .249 .249 .250 2.250 .250 .250 .250 .251 2.251 ■25" .251 .252 .252 2.252 .252 -253 .253 .253 a-253 .254 .254 .254 .254 2a93 4697 7101 9506 191 2 4318 6725 9132 i54> 3949 6359 8769 1180 359« 6003 8416 0829 3243 5658 8073 0489 2906 53^3 774« 0159 2578 4998 74'9 9840 2262 4684 7107 953« 1955 4380 6806 9232 1659 4087 6515 8944 1374 3804 6235 8666 2.255 '°99 38 39.90 39.91 39.92 39-93 39-94 39-95 39.96 39-97 39-98 39-99 40.00 40.01 40.02 40.03 40.05 40.06 40.07 40.08 40.09 40.10 40.11 40.12 40.13 40.14 40.15 40.16 40.17 40.18 40.19 40.21 40.22 40.23 40.24 40.25 40.26 40.a7 40.28 40.29 40.30 40.31 40.32 40.34 40.35 40.36 4 37 40.38 40.39 40.40 40.41 40.42 4°-43 40.44 40.46 40.47 40.48 40.49 40.50 40.51 40.52 40.53 40.54 5U3 110= log M. Wff. 1", 2.255 .255 ■a55 -a55 .256 2.256 256 256 257 257 2.257 -257 .258 .258 .258 2.258 .259 -259 ■259 .259 2.259 .260 .260 .260 .260 2.261 .261 .261 .261 .262 2.262 .262 .262 .263 .263 2.263 .263 .264 .264 .264 2.264 .265 .265 .265 .265 2.266 .266 .266 .266 .267 i.267 .267 .267 .268 .268 2.268 .268 .269 .269 .269 2.269 1099 353a 59'' 5 8399 0834 3270 5706 8143 0580 3019 5458 7897 0337 2778 5220 7662 0105 2548 4992 7437 9883 2329 4776 7223 9671 aiao 4570 7020 947' 1922 4374 6X27 9281 «735 4190 6645 9102 »559 4016 6474 8933 «393 3853 6314 8776 1238 370« 6165 8629 1094 3560 6026 8493 0961 3430 5899 8369 0839 33'° 5782 8255 40.54 40.55 40.56 40.58 40.59 40.60 40.61 40.62 40.63 40.64 40.65 40.66 40.68 40.69 40.70 40.71 40.72 40.73 40.74 40.75 40.76 40.78 40.79 40.80 40.81 40.82 40.83 40.84 40.85 40.86 40.88 40.89 40.90 40.91 40.92 40.93 40.94 40.95 40.96 40.98 40.99 41.00 41.01 41.02 41.03 41.0A 41.06 41.07 41.08 41.09 41.10 41.11 41.12 41.13 41.15 41.16 41.17 41.18 41.19 41.20 41.21 111= logM. I Diff. 1". 8255 0728 3202 5676 8152 0628 3104 5582 8060 0538 3018 5498 7979 0460 2942 54a5 7909 0393 2878 5364 7850 0337 2825 53«3 7802 0292 2783 5a74 7766 0258 2.277 a75a 277 5246 277 7740 278 0236 278 2732 2.278 5229 .278 7726 .279 0224 .279 2723 .279 5223 2-279 .280 .280 .280 .280 2.281 .281 .281 .281 .282 2.282 .2X2 .282 .283 .283 .269 .270 .270 .270 .270 2.271 .271 .271 .271 .272 2.272 .272 .272 .273 .273 2.273 .273 -274 .274 .274 2.274 -a75 -275 .275 .275 2.276 276 276 276 277 2.283 .283 .2X4 .284 .284 2.284 7878 7723 0224 2726 5228 773' 0235 2740 5245 775' 0258 2765 5273 7782 0291 2801 5312 7X24 0336 2849 5363 41.21 41.23 41.24 41.25 41.26 41.27 41.28 41.29 41.30 41.32 4'-33 41.34 4'-35 41.36 41.38 41.39 45.40 41.41 41.42 4'-4: 4 '-44 41.46 41.47 41.48 41.49 41.50 41.51 4'-53 4'-54 4'-55 41.56 4'-57 41. 58 41.60 41.61 41.62 41.63 41.64 41.65 41.67 41.68 41.69 41.70 41.71 41.72 4'-74 4'-75 41.76 4'-77 41.78 41. So 4 1 . X I 41.82 41.83 41.84 41.85 41.87 41.88 41.89 41.90 41.91 TABLE VI. For finding the True Anom ily or tlie Time from the Perihelion in a Parabolic Orbit. If V. 112 # 113 114° 115° log M. Dinr. 1". ItiK M. Diff. 1". log M. Dim 1". log M. Dim 1". 44,18 0' 1.284 7878 41.91 2.300 0067 42.64 2-3«5 4927 43.40 2-33' 2564 1 .285 0393 41.93 .300 2626 42.65 •3«5 753' 43-41 -33' 5216 44.20 2 .285 2909 41.94 .300 5186 42.67 .3.6 0136 43.42 .331 7868 44.21 3 .285 5415 41.95 .300 7746 42.68 .316 2742 43-44 -332 0521 44-2 2 4 .285 7943 41.96 .301 0307 42.69 .316 5348 43-45 •332 3'75 44- -4 5 1.286 0461 41.97 2.301 2869 42.70 2.316 7956 43.46 2-:,32 ^Jr 44-2^- G .286 2979 41.99 .301 543« 42.72 -3'7 0564 43-47 •332 8485 44- ^f- 7 .286 5499 42.00 .301 7995 •i-'-73 -3'7 3173 43-49 •333 1141 44. 2 s H .286 8019 42.01 .302 0559 42.74 -3'7 5782 43-50 -333 3799 44.29 .287 0540 42.02 .302 3123 42-75 -3>7 8393 43-5' -333 6456 44-3' 10 1.287 3062 42.03 2.302 5689 42.76 2.318 1004 43-53 2-333 9115 44.32 11 .2S7 5584 42.04 .302 S255 42.78 .318 3616 6229 43-54 -334 '775 44-3 3 12 .287 8107 42.06 .303 0821 42.79 .318 43-55 •334 4435 44-3+ 13 .288 0631 42.07 •303 3390 42.80 .318 8842 43-56 -334 7096 44-3<' 14 .288 3'55 42.08 •303 595S 42.81 -3'9 1456 43.58 -334 9 '58 44-37 15 1.288 5680 42.09 2.303 8528 42.83 2.319 4072 6687 43-59 2-335 2421 44 39 ! 16 .288 8206 42.10 .3C4 1098 42.84 .319 43.60 -335 5084 44.40 1 IT .289 0733 42.12 .304 3668 42.85 .319 9304 43-62 -335 7749 44-4' 1 18 .289 3260 42.13 •3°4 6240 42.86 .320 1921 43.63 .336 0414 44-43 I 11) .289 5788 42.14 •3=4 8i<;2 42.88 .320 454° 43.64 -336 3080 44-44 i 20 t.289 8117 0847 42.15 2.305 1385 42.89 2.320 7'59 43.66 2.336 5747 44-45 1 21 .290 42.16 .305 3959 42.90 .320 9778 43.67 .336 8414 44-47 i 22 .290 3377 42.18 .305 6533 42.91 .321 2399 43.68 •337 1.083 4448 23 .290 5908 42.19 .305 9109 42.93 .321 5020 43.69 •337 3752 6422 44-49 24 .290 8440 42.20 .306 1685 42.94 .321 7642 43.70 •337 44-5' 25 :.29i 0972 42.21 2.306 4261 42.95 2.322 0265 43.72 2-337 9093 44.52 26 .291 3505 6039 42.22 .306 6839 42.96 .322 2889 43-73 -338 1765 44-53 27 .291 42.24 .306 9417 42.98 .322 55»3 43-75 .338 4437 44-55 i 28 .291 8574 42.25 .307 1996 42.99 .322 8139 43-76 -338 7111 44.56 29 .292 1 109 42.26 .307 4576 43.00 •323 0765 43-77 -338 9785 44.58 1 30 1.292 6182 42.27 2.307 7"7 43.02 2-323 339' 43-79 2-339 2460 44-59 ; 31 .292 42.29 .307 9738 43°3 -323 6019 43.80 -339 5'35 44.60 1 32 .292 8719 4230 .308 2320 43-04 •323 8647 43-81 •339 7812 44,62 i 33 .293 1258 42.31 .308 4903 7486 43.05 .324 1277 43-83 .340 0490 44.63 ' 34 .293 3797 42.32 .308 43-07 -324 3907 43.84 .340 3168 44.64 35 1.293 6336 42.33 2.309 0071 43.08 2.324 6537 43.85 2.340 5847 44.66 36 .293 8877 42.35 .309 2656 43.09 -324 9169 ^Hl .340 8527 44.67 ! 37 .294 1418 42.36 .309 5242 43.10 -325 1801 43.88 -34' '^27 44.69 38 .294 3960 42.37 .309 7828 43.12 .325 4434 43-89 •3'r' 3^89 44.70 39 .294 6503 42.38 .310 0416 43-'3 •325 7068 43.91 .341 6571 44-71 40 1.294 9046 42.40 2.310 3°34 43.14 2-325 9703 43.92 2-34' 9255 44-73 41 .295 1590 42.41 .310 5593 43-»5 .326 2339 43-93 •342 1939 44-74 . 42 .295 HP 42.42 .310 8182 43-'7 .326 4975 43-94 .342 4623 44-75 43 .295 6680 42.43 •3«i 0773 43.18 .326 7612 43.96 -342 7309 44-77 I 44 .295 9227 42.44 .311 3364 43.19 -327 0250 43-97 -342 9995 44-78 i 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 55J! 44.00 -343 PV 44.81 1 *t7 .296 6870 42.48 .312 1142 43-23 .327 8168 44.01 •343 8060 44.82 48 .296 9419 42.49 .312 3736 6331 43-24 43.26 .328 0809 44.0. •344 0750 44-84 49 .297 1969 42.51 .312 .328 345' 44.04 •344 3440 44.85 50 2.297 4510 42.52 2.312 8927 43-27 2.328 6094 44.05 2.344 6132 ++■^1' 51 .297 7071 4*-S3 •313 1524 43-28 -328 8737 44.06 -344 8824 44. 8 S 52 .297 9623 42.54 •3>3 4121 43.29 •329 1382 44.08 -345 1517 44.S9 53 .298 2176 42.55 •3»3 6719 43-3» -329 4027 44.09 -345 4211 44.91 54 .298 4730 4*'S7 •3«3 9318 43-32 ■329 6672 44.10 •345 6906 44.9a 55 2.298 7284 42.58 2.314 1917 43-33 2-329 9319 44.12 2-345 9601 44-93 56 .298 9839 42.59 .314 4518 43-35 -330 1967 44.13 •346 2298 44-93 57 .299 = 395 42.60 .314 7119 43-36 .330 4615 44.14 44.16 .346 4995 44.96 58 .299 4952 42.61 .314 9721 43-37 .330 7264 •346 7693 44-97 59 .299 7509 42.63 •3«S 2323 43-38 .330 9914 44.17 •347 0392 44-99 60 2.300 0067 42.64 2.315 4927 43.40 2-33' 2564 44.18 2-347 3092 45.00 694 TABLE VI. Vor finding tJie True Anoniiily or the Tinu' from the Perihelion in ii Parabolic Orbit. l\ 1 'i a 4 5 6 7 H 10 11 12 13 14 15 lA 17 18 lU 20 21 22 23 24 25 20 27 28 29 30 31 32 33 34 35 30 37 38 3» 40 41 42 43 44 45 40 47 48 4» 50 51 52 53 54 55 50 57 58 59 GO 116^ l"K M. 2-347 347 347 34« 348 34X 349 34'; 349 3^0 3'io 350 350 35' 351 351 35« 352 35* 352 352 353 353 353 354 354 354 354 355 355 355 355 356 356 356 357 357 357 357 3091 579* »494 1 196 3*199 6603 9308 Z014 4720 7428 0136 2845 5554 8265 C977 3689 6402 91 16 1831 4547 7263 9981 2699 5418 8138 0859 3581 6303 9027 1 75 1 4476 7202 9928 2656 53«S 8114 0844 3575 6307 9040 Diff. 1" 358 1773 358 4508 358 7243 358 9979 359 2716 359 359 360 360 360 360 36, 361 361 362 362 362 362 363 363 363 5454 8193 0933 3673 6415 9157 1900 4644 7389 0134 2881 5628 8376 1 1 26 3876 6626 45.00 45.02 45.03 45.04 45.C6 45-07 45.09 45.10 45.11 45-«3 45-»4 45.16 45-«7 4518 45.20 45-21 45-23 45-24 45-25 45-27 45.28 45.30 45-3> 45-33 45-34 45-35 45-37 45.3« 45-40 45-41 45-42 45-44 45-45 45-47 45.48 45.50 45-5' 45-52 45-54 45-55 45-57 45-58 45.60 45.61 45.62 45.64 45-65 45-67 45.68 45.70 45-7» 45-72 45-74 V5-75 45-77 45.78 45.80 45.81 45.82 45.84 45.86 117^ I<>K M. 363 6626 363 9378 364 2131 364 48X5 364 7639 365 365 365 365 366 366 366 366 367 367 367 368 368 368 368 369 369 369 370 370 370 370 371 371 371 371 9559 372 2337 372 51 16 372 7896 373 0677 373 373 373 374 374 °394 3'5o 5907 8665 1423 4183 6944 9705 2467 5230 7994 0759 3525 6291 9059 1827 4596 7367 0138 2909 5682 8456 1230 4006 6782 3459 6241 9024 1809 4594 7380 0167 2955 5744 8533 1324 4115 6908 9701 2495 374 375 375 373 375 376 376 376 376 377 377 5290 377 8086 378 0883 378 368? 378 6479 378 9279 379 2079 379 4881 379 7683 380 0486 380 3290 Dlff. 1". 45-86 45.87 45.88 45.90 45-9« 45-93 45-94 45.96 45-97 45-99 46.00 46.01 46.03 46.04 46.06 46.07 46.09 46.10 46.12 46.13 46-15 46.16 46.18 46.19 46.21 46.22 46.24 46.25 46.26 46.28 46.29 46.31 46.32 46.34 46.35 46.37 46.38 46.40 46.41 46-43 46.44 46.46 46.47 46.49 46.50 46.51 46.53 46-55 46.56 46.58 46.59 46.60 46.62 46.64 46.65 46.67 46.68 46.70 46.71 46.73 46.74 118= loK M. Dlff. I". 2.380 3290 .380 6095 .380 8901 .381 1708 .381 4515 2.381 .382 .3X2 .3S2 .382 2-383 •383 .38, .383 -384 2.384 -384 •385 •385 .385 2.385 .386 -386 .386 .387 2.387 -387 .387 .388 .388 2.388 -389 -389 .389 .389 2.390 .390 -390 -39- .391 2.391 .392 .392 .392 .392 2-393 -393 -393 -393 •394 2-394 -394 -395 •395 •395 2-395 .396 .396 .396 -397 7324 o'33 2944 575 5 8567 1380 4 '94 7009 9825 2642 5460 8278 1098 3918 6739 9562 2385 5209 8034 0860 3687 651.1 9343 2173 5003 7835 0607 3500 6335 9170 2006 4843 7681 0519 3359 6200 9042 1884 4728 7572 0417 3264 61 1 1 8959 1808 4658 7509 0361 0067 8922 1778 4634 7492 0350 2-397 3210 46-74 46.76 46.77 46.79 46.80 46.82 46.83 46.85 46. 86 46.88 46.89 46.91 46,92 46.94 46.95 46.97 46.98 46.99 47.01 /I •'.03 ..-05 47.06 47-08 47.09 47.11 47.12 47.14 47- '5 47-'7 47.18 47-20 47.21 47-23 47.24 47.26 47.28 47.29 47-3' 47-32 47-34 47-35 47-37 47.38 47.40 47.41 47-43 47-45 47.46 47.48 47-49 47-5' 47.52 47-54 47-55 47-57 47-59 47.60 47.62 47-63 47.65 47.66 119^ logM. i97 397 397 398 398 398 399 399 399 399 400 400 400 401 401 3210 6070 8931 1794 4657 7521 0386 325^ 6119 8987 1856 4725 759'! 046S 3340 401 6214 401 90S 8 402 1964 402 4840 402 7718 403 0596 403 3475 403 6356 403 9237 404 2 1 1 9 404 5002 404 7886 405 0771 405 3657 405 6544 405 406 406 406 407 407 407 407 408 408 9432 2321 521 1 8102 0993 3886 6780 9674 2570 5467 408 8364 409 1263 409 4162 409 7063 409 9964 410 2866 410 5770 410 8674 411 1579 411 4486 4" 7393 412 0301 412 3210 412 6120 412 9031 413 1944 413 4857 413 7771 414 0686 414 3602 414 6519 Dlff. 1". 47.66 47.68 47.70 47-7' 47-73 47-74 4776 47-77 47-79 47.81 47-82 47.84 47.85 47-87 47-89 47.90 47.92 47-93 47-95 47-97 47.98 48.00 48.01 48.03 48.04 48.06 48.08 48.09 48.11 48.12 48.14 48.16 48.17 48.19 48.20 48.22 48.24 48.25 48.27 48.28 48.30 48-32 48.33 48.35 48-37 48.38 48.40 48.41 48-43 48-45 48.46 48.48 48.49 48.51 48-53 48.54 48.56 48.58 48.59 48.61 48.62 6V5 TABLE VI. For finding tlie True Anomaly or the Time from tlie Perihelion in a Parabolic Orbit. V. o 1 •z 3 4 5 7 8 O 10 11 1'^ 13 14 15 16 17 18 lU 20 'Zl TZ 23 24 23 2A 27 28 2U 30 31 32 33 34 30 37 38 39 40 41 42 43 44 45 40 47 48 49 30 51 32 53 54 55 36 37 58 59 00 120^ log M. Dlir. 1". 4'4 4'4 4'5 4' 5 4>S 416 416 6519 941 7 5276 8197 1119 4042 416 6965 416 9890 417 2816 417 5743 417 8671 418 1600 418 4529 418 7460 419 0392 419 3325 419 6258 419 9193 420 2129 420 5066 ^zo 8003 421 0942 48.62 48.64 48.66 4X.67 48.69 48.71 48.72 48.74 48.76 48.77 48.79 48.81 48.82 48.84 48.85 48.87 48.89 48.90 48.92 48.94 48.95 48.97 48.99 421 3882 . 49.00 421 6822 I 49.02 421 9764 i 49.03 422 2707 I 49.05 422 5650 \ 49.07 422 8595 I 49.09 423 1541 j 49.10 423 4488 I 49.12 423 7435 424 0384 4^4 3 334 424 6284 424 9236 425 21X9 425 S'42 425 8097 426 1053 426 4010 426 6967 426 9926 427 2886 427 5847 49.14 49.15 49.17 49.19 49.20 49.22 49.24 49.25 49.27 49.29 49.30 49-3* 49-34 49-35 427 8808 I 49.37 428 1771 i 49.39 .28 .28 4735 I 49-40 7700 i 49.42 429 0665 I 49.44 429 3632 : 49.46 429 6600 i 49.47 429 9569 I 49.49 430 2539 [ 49.51 430 5510 ! 49.52 430 8482 I 49.54 43« «455 49-56 431 442!- 49-57 43> 7403 ! 49-59 43* 0379 I 49-6i 43* 3356 j 49-62 121' lot? M. Via. 1", 2.432 431 432 433 433 433 434 434 434 435 \l 3356 6334 93'3 2293 5274 8257 1240 4224 7209 0195 182 I7« 9160 2150 5141 8134 1127 4122 7117 01 14 3111 6110 9109 435 435 43 5 436 436 436 437 437 437 438 438 43« 43« 439 2IIO 439 5««2 439 81J4 440 1118 440 4123 440 7129 441 01 36 441 3143 441 6152 441 9162 442 2173 442 5185 442 8199 443 1 21 3 443 4228 443 7244 444 0261 444 3280 444 6299 444 9320 445 2341 445 5364 445 8387 446 1412 446 4437 446 7464 447 0492 447 3521 447 6551 447 9582 448 2614 448 5647 448 8681 449 '7«6 449 47 53 4-V9 7790 45' 0828 450 3 868 49.62 49.64 49.66 49.68 49.69 49.71 4973 49-74 49.76 49.78 49.80 49.81 49.83 49.S5 49.86 49.88 49.90 49.92 49-93 49-95 49-97 49.98 50.00 50.02 50.04 50.05 50.07 50.09 50.1 1 50.12 50.14 50.16 50.18 50.19 50.21 50.23 50.24 50.26 50.28 50.30 50.31 50-33 50-35 50.37 50.38 50.40 50.42 50.44 50-45 50.47 50.49 50-5 « 50-53 50-54 50.56 , -58 50.60 50.61 50.63 50.65 50.67 122' lo« M. I 450 3868 450 6908 450 9950 451 2992 451 6036 ' 451 9081 452 2127 452 5>74 452 8222 . 453 J27« I 453 453 454 454 454 454 455 455 455 456 456 456 457 4321 I 7372 ■ 0424 3477 . 6532 : 9587 i 2644 5701 8760 1820 { 4881 I 7943 1006 i 45X 3 l68 Dim 1". 50.67 50.68 50.70 50.72 50.74 50-75 50-77 50.79 50.81 50.83 50-84 50.86 50.88 50.90 50.92 50-93 50.95 50-97 50.99 51.00 457 4070 , 5 457 7»35 I 5 458 0201 ' 5 5 458 6337 I 5 458 9406 j 5 459 2477 I 5 459 5548 i 5 459 8621 , 5 460 1695 5 460 4770 j 5 460 7846 I 5 461 0923 5 461 4001 I 5 461 7080 i 5 462 0161 ; 5 462 3242 I 5 462 6325 j 5 462 9408 ' 5 463 2493 ; 5 463 5579 ' 5 463 8666 5 464 1754 464 4843 464 7933 465 1024 465 4116 465 7210 466 0305 466 3400 466 6497 466 9595 467 2694 467 5794 467 8895 468 1997 468 5101 468 8205 .04 .06 .08 .09 .11 -'3 -«5 -17 .18 24 26 28 33 35 37 .40 .42 48 49 51 53 ■55 .60 .62 .64 .66 .68 ■70 -7» •73 51-75 123° 10(5 M. I Dlir. 1". I 470 3744 470 6856 470 9968 471 3081 471 6196 468 469 469 469 470 8205 1311 4418 7526 0634 471 472 4.72 •472 473 473 473 474 474 474 93" 2428 5546 8665 1785 4906 8028 1152 4276 7402 475 0529 475 3657 475 6786 475 99«6 476 3047 476 6180 476 9313 477 2448 477 5584 477 8721 478 1859 478 4998 478 8138 479 1280 479 4422 479 7566 480 07 1 1 400 3857 480 7004 481 0152 481 3301 481 6452 481 9604 482 2756 482 5910 482 9065 483 2222 483 5379 483 8537 •1-84 1697 484 4858 48.1. 8020 485 1183 485 4347 485 75«3 486 0679 486 3847 486 7016 487 0186 487 3357 487 6529 «-75 '-77 1.79 I. Si 1.82 1.84 1.86 1.88 1.90 1.92 1.94 '-95 '•97 •-99 2.01 2.03 2.05 2.07 2.09 2.10 2.12 2.14 2.16 2.18 2.20 2.22 2.23 2.25 2.27 2.29 2-33 2-35 2.37 2.39 2.40 2.42 2-44 2.46 2.48 2.50 2.52 2.54 2.56 2.58 2-59 2.61 2.63 2.65 2.67 2.69 2.71 2.73 2-75 2.77 2.78 2.80 2.82 2.84 2.86 52.; 590 rabolic Orbit. TABLE VI. For finding tlif True Anomaly or tiie Time from the Perihelion in a Parabolic Orbit. 123 > OB M. DIfT. 1". 5«-75 8 8205 9 »3>i 5'-77 9 44»8 5 ••79 9 75i6 51. Xi 0634 , 51.82 ° 3744 1 51.84 6856 51.86 9968 51.X8 I 3081 51.90 I 6196 1 51.92 1 9311 S>-94 2 2428 51-95 2 5546 51-97 2 8665 51.99 3 1785 52.01 3 4906 52.03 3 8028 52.05 4 «i52 52.07 4 4276 52.09 4 740* 52.10 5 0529 52.12 5 3657 52.14 5 6786 52.16 5 99'6 52.18 6 3047 52.20 6 6180 52.22 6 9313 52.23 7 2448 52.25 7 5584 52.27 7 8721 52.29 r8 1859 52-3' 8 4998 52-33 8 8138 52-35 9 1280 52-37 r9 4422 52-39 r<) 7566 52.40 0711 52.42 ,0 3857 52-44 !o 7004 52.46 !i 0152 52.48 !i 3301 !i 6452 52.50 52.52 ii 9604 52-54 iz 2756 52.56 fz 5910 52.58 ?2 9065 52-59 !3 2222 52.61 |3 5379 52-63 «3 8537 52-65 <4 1697 52.67 , ^ 4858 52.69 ?.1. 8020 52-71 ' ^5 "83 52-73 . '5 4347 52-75 j is 7513 52-77 1 S6 0679 52.78 1 i6 3847 52.X0 ! i<6 7016 1 52-82 1 S7 oi86 i 52-84 ! ^7 3357 52.86 I 87 6529 52.88 V. O' 1 :i 4 5 7 8 9 10 11 Vi 13 14 15 10 17 18 10 20 21 22 2:1 24 25 20 27 28 20 30 31 32 33 34 35 30 37 38 30 40 41 42 43 44 45 40 47 48 40 50 51 52 53 54 55 50 57 58 50 60 124^ IngM. 487 487 488 488 488 489 489 489 490 490 490 491 491 491 492 492 492 493 493 493 494 494 494 494 495 495 495 496 496 496 497 497 497 498 498 498 499 499 499 500 500 500 501 501 501 502 502 502 503 503 503 503 504 504 504 5°S 5°5 506 506 6529 9702 2877 6053 9230 i 2408 I 5587 8767 ' «949 5132 ^ 8315 ; 1 500 i 4686 '. 7874 1063 I 4252 I 7443 ; 0635 j 3828 I 7023 ' 0218 34«5 6613 9812 3012 6213 9416 2619 5824 9030 2238 5446 8656 1867 5079 8292 1506 4721 7938 1156 4375 7595 , 0817 4°39 ' 7263 ! 0488 I 37«4 ' 6942 0170 i 3400 j 6631 9863 3096 633' 9567 2804 6042 i 9282 I 2522 5763 DIff. 1". 2.88 2.90 2.92 2.94 2. 90 2.98 3.00 3.02 3.03 3.05 3.07 3.09 3.11 3-'3 3-J5 3->7 3.19 3.21 323 325 3-27 3.29 3-3' 3-33 3-35 3 3 506 9006 37 39 3-41 3.42 3-44 3-46 3.48 3.50 3-52 3-54 3.56 3.58 3.60 3.62 3.64 3.66 3.68 3.70 3.72 3-74 3.76 3.78 3.80 3.82 3.84 3.86 3.88 3-9° 3.92 3-94 3.96 3.98 4.00 4.02 4.04 54.06 125^ loK M. 2.506 .507 .507 .507 .508 2.508 508 "509 509 509 2-5 5 5 5 5 2-5 5 5 5 5 2-5 5 5 5 5 2-5 5 5 5 5 2-5 5 5 5 5 2-5 5 5 5 5 9006 2251 5496 8742 1990 5239 84S9 •74> 4993 8247 1502 4758 8016 1274 4534 7795 1057 4321 2 7586 3 0852 3 4««9 3 7387 4 0657 4 3927 4 7«99 5 0473 5 3747 5 7023 6 0300 6 3578 6 6857 7 0138 7 3420 7 6703 7 9987 8 3273 8 6559 8 9847 9 3'37 9 6427 2.519 9719 520 3012 520 6306 520 9601 521 2898 2.521 6196 .521 9495 .522 2795 .522 6097 .522 9400 2.523 2704 j .523 6009 I .523 9316 i .524 2624 ■: -524 5933 ' 2.524 9243 •525 2555 .525 5867 .525 9181 •5*6 *497 2.516 5813 I 55.29 Dlflr. 1". 4.06 4.08 4.10 4.12 4.14 4.16 4.18 4.20 4.22 4.24 4.26 4.28 4.30 4.32 4-34 4.36 4.38 4.40 4.42 4-44 4.46 4.48 4.50 f52 4-54 4.56 4.58 4.60 4.63 4-65 4.67 4.69 4-71 4-73 4-75 4-77 4-79 4.81 4.83 4.85 4-87 4.89 4.91 493 4-95 4-97 4-99 5.02 5.04 5.06 5.08 5.10 5-12 5.14 5.16 5.18 5.20 5.22 5.24 5.26 126^ li>K M. DilT. 1". 526 5813 526 9131 527 2450 527 577« 527 9092 528 2415 528 5739 528 9065 529 2391 529 5719 529 9048 530 2379 530 5710 5 3° 9043 53> 2378 531 5713 53' 9050 532 2388 532 5727 532 9068 533 533 533 534 534 534 535 535 536 536 536 537 537 537 53! "! 538 539 539 2410 5753 9097 2443 579° 9138 2487 5838 9190 "-5+3 5898 9254 261 1 5970 9329 2690 6052 9416 2781 6147 54' 54' 542 542 542 543 5-^3 544 544 544 539 95'4 540 2883 540 6253 i 540 9625 ! 54' 2997 6371 9746 3123 6500 9880 3260 6641 0024 3409 6794 545 0181 545 3569 545 6959 546 0350 546 3742 546 7135 56.57 55-29 5 5-3' 55-33 55-35 55-37 55-39 55-41 55-43 55-45 55-48 55-5° 55-52 5 5-54 55-56 55-58 55.60 55-62 55-64 55-67 55.69 55-7' 55-73 55-75 55-77 55-79 55.8, 55-84 55.86 55-88 55.90 55.92 55-94 55.96 55.98 56.01 56.03 56.05 56.07 56.09 56. II 56-»3 56.15 56.18 56.20 56.22 56.24 56.26 56.29 56-31 56-33 56-35 5637 56-39 56.42 56-44 56.46 56.48 56.50 56.52 56.55 127 luK M. Diir. 1". 2 546 7'35 56-57 ■547 0530 56.59 -547 3926 56.61 547 7323 56.63 5.!8 0722 56.65 2.548 4122 56.68 .548 7523 56.70 •549 0926 56.72 -549 4330 5^'+ -549 7735 56.76 2.550 1141 56.79 -55° 4549 56..S1 -55° 7958 56-83 -55' '369 56.S5 •55' 4781 56-87 2.551 8194 56.90 .552 1608 56.92 -552 5024 56.94 56.96 .552 8441 .553 '859 56.98 2.553 5279 57-01 -553 8700 57^o3 •554 2122 57^os .554 5546 57^07 •554 897' 57^io 2.555 2398 •555 5825 57.12 57.T4 57.16 .555 9254 .556 2685 57.18 .556 61 16 57.21 2.556 9549 57-23 .557 2984 57-25 •557 6420 57-27 •557 9857 57-29 .558 3295 57-32 2.558 6735 57-34 •559 °«76 57-36 ■559 36'8 57-38 •559 7062 57.4' .560 0507 57.43 2.560 3953 57-4S .560 7401 57.47 .561 0850 57.50 .561 4301 57-52 .561 7753 57-54 2.562 1206 57.56 .562 4660 57-59 .562 8116 57-61 .563 1574 57-63 .563 5032 57-65 2.563 8492 57-68 .564 1953 .564 5416 57-70 57-72 .564 8S80 57-74 .565 2345 57-77 2.565 5812 57-79 .565 9280 57-81 .566 2750 57.8a 57.86 .566 6221 .566 9693 57.88 2.567 3166 57.90 597 TABLE VI. For fmiHnjt the Triio Aiiomiily (ir llie I'iiiio rioiii tlic Purihelidii in a Paraljolio Orbit. 1 1 128 •129 130 \° 131 V, 2.^(17 5ifi6 DifT. 1". 57-90 1<>K M. ! _ 1 2.5S8 4112 Dlff. 1". 1.>K M. Dlff. 1". 60.75 loR M. 2.632 1622 Diff, 1". 59.30 2.610 0188 62.28 : 1 > .567 6641 57-9 3 .588 7670 59-32 .610 3834 60.78 .632 5360 62.30 I 1 1 2 .?(>S 0117 57-95 .589 1230 ' 59-35 .610 '7481 60.80 .632 9099 ^*-33 , 1 2 'i •56« 3595 , 57-97 .589 4792 59-37 .611 1110 .611 4781 60.83 .633 2839 62.35 1 3 ! * .568 7074 57-99 .589 831:5 59-39 60.85 .633 6581 62.38 1 i 4 ' 5 2.56,) 0554 58.02 2.590 1919 59-4- 2.6 1 1 8433 60.88 2.634 0325 62.41 1 i 5 » ■5''9 403'' 58.04 .590 5485 59-V4 .612 2086 60.90 .634 4070 62.43 I 7 , 7 .S(>9 75 "J 58.06 .590 9052 59-47 .612 5741 60.93 .634 7817 62.46 1 1 H .S70 1004 58.09 .591 2620 59-49 .612 9397 60.95 .635 1565 62.48 H H 1 » .S70 449° 58,11 .591 6190 59-5' .613 3055 60.98 -635 53«5 62.51 ■ 10 1-570 7977 58.13 2.591 9762 1 59^54 59.56 2.613 6715 61.00 2635 9066 62.54 ■ ' 10 11 •57" H65 58.15 -592 33 35 .614 0376 61.03 .636 2819 62.56 H ; 11 12 ■57" 4955 58.1S .592 6909 59.58 .614 4038 61.05 .636 6573 62.59 I 12 13 .571 8447 58.20 •593 0485 59.61 .614 7702 61.08 .637 0329 62.61 H 13 It .572 1939 58.22 •593 4062 1 59.63 .615 1368 61.10 .637 4087 62.64 ■ ' 14 15 2-572 5434 58.25 2.593 7641 59.66 2.615 5035 61.13 2.637 7846 62.67 I 1 15 1» -572 «929 58.27 -594 •"! 59.68 .615 S703 61.15 .638 1607 62.69 ■ 10 17 -573 1426 58.29 -594 4803 ■ •594 83S6 59.70 .61(1 2373 61.78 .638 5369 62.72 H 17 18 •573 59^4 58-3* 59-73 .616 6045 61.20 .638 9133 62.75 ■ 18 10 -573 94*4 58.34 •595 '970 59-75 .616 9718 61.23 .639 2899 62.77 ■ 10 20 2-574 2925 58.36 2-595 5556 59.78 2-617 3392 61.25 2.639 6666 62.80 ■ 20 21 -574 6427 58.38 •595 9>43 59.80 .617 70(18 61.28 .640 0435 62.82 H 21 22 -574 993« 5S.41 .596 2732 59.82 .618 0746 61.30 .640 4205 62.85 H 22 23 •575 3436 -575 6943 58-43 .596 6322 59^85 .618 4425 6«-33 .640 7977 62.88 H 23 24 58.45 .596 9914 59-87 .618 8105 61.36 .641 1750 62.90 ■ 24 25 2.576 04s I 58.48 2-597 3507 59.90 2.619 '787 61.38 2.641 5525 62.93 1 25 20 .576 3960 58.50 -597 7'02 59.92 .619 5471 61.41 .641 9302 62.96 ^1 2(» 27 .576 7471 58.52 .598 0698 59-95 .619 9156 61.44 .642 3080 62.98 H 27 28 -5 77 09** 3 58-55 •598 4295 59-97 .620 2843 61.46 .642 6860 63.01 S 28 2» j -577 4496 58-57 .598 7894 59-99 .620 6531 61.48 .643 0641 63.04 H 20 30 1.577 8oi I 58.59 2.599 1494 60.02 2.621 0220 61.51 2.643 4424 63.06 1 30 ! 31 .578 1528 58.62 •599 5^96 60.04 .621 3911 61.53 .643 8209 63.09 ^^ 31 32 .578 5045 58.64 •599 8699 60.07 .621 7604 61.56 .644 1995 63.1a ^1 32 ; 33 .578 8564 58.66 .600 2304 60.09 .622 1298 61.58 .644 5783 63.14 ^1 33 34 •579 »o8S 58.69 .600 5910 60.12 .622 4994 61.61 .644 957a 63.17 H 34 35 2-579 5607 58-71 2.600 9518 60.14 2.622 8691 61.63 61.66 2.645 3363 63.19 H 35 30 -579 9130 58-73 .601 3127 60. 1 6 .623 2390 -645 7>55 63. -".i ^1 30 37 .580 2655 5^7^ .601 6738 60.19 .623 6091 61.68 .646 0949 63.25 H 37 38 .580 6181 58.78 .602 0350 60.21 .623 9793 61.71 .646 4745 63.27 SB ! 38 39 .580 9708 58.80 .602 3963 60.24 .624 3496 61.74 .646 8542 63.30 H 30 10 2.5S1 3237 58.83 2.602 7578 60.26 2.624 7201 61.76 2.647 2341 63.33 H 40 41 .581 6768 58-85 .603 1 195 60.29 .625 0907 61.79 .647 6142 63-35 ^1 41 42 .5 82 0299 58.87 .603 4813 60.31 .625 4615 61.81 .647 9944 63.38 ^1 42 43 .582 3832 58.90 .603 8432 60.34 .625 8325 61.84 61.86 .648 3748 63.4. ^1 43 44 .582 8267 58.92 .604 2053 60.36 .626 2036 .648 7553 63.44 H 44 45 2.583 0903 58.94 2.604 5675 60.38 2.626 5748 61.89 2.649 1360 63.46 H 45 1 4G .583 4440 58.97 .604 9299 60.41 .626 9462 61.91 .649 5168 63.49 ^1 40 47 -583 7979 58-99 .605 2924 60.43 .627 3178 61.94 .649 8978 63.52 ^H 47 48 .584 1519 59.01 .605 6551 60.46 .627 6895 6i.>.- .650 2790 63-54 ^H 48 40 .584 5061 59.04 .606 0179 60.48 .628 0614 61.99 .650 6603 63-57 H 40 50 2.584 8604 59.06 2.606 3S09 60.51 2.628 433J. .628 8056 62.02 2.651 0418 63.60 H 50 51 .585 2148 ^9.09 .606 7440 60.53 60.56 62.04 .651 4235 .651 8053 63.62 ^H 51 i 52 .585 5694 '59.11 .607 1073 .629 1780 62.07 63.65 : ^H 52 53 .585 9241 S9-'3 .607 4707 60.58 .629 5505 62.09 .652 1873 63. 6X ^H 53 54 .586 2790 59.16 .607 8343 60.61 .629 9231 62.12 .652 5695 63.70 ^H 54 55 2.586 6340 59.18 2.608 1980 60.63 2.630 2959 62.15 2.652 9518 63.73 ^M 55 50 .586 9891 59.20 .608 5618 60.66 .630 668g 62.17 •653 334* 63.76 HI 50 57 -587 3444 59.23 .608 9258 60.68 .631 0420 62.20 .653 7168 63.79 ■■ 1 57 58 .587 6999 59-^5 .609 2901 60.70 .631 4152 62.22 .654 0096 63.>!i ^H 58 50 .588 05S5 59.27 .609 6544 60.73 .631 7887. 62.25 .654 4826 63-84 ^M 50 00 60 2.588 4112 59.30 2.610 0188 60.75 2.632 1622 62.28 1 2.654 8657 63.87 ■ 598 iliolic Orbit. TABLE VI. For fiiidinp the Tnie Anoniiily or tlif Tiiiu' from the Perilielioii in a I'uriiholic ()ri)il. 131 M. Diff. 1". 1621 62.28 5360 62.30 i)0<)l) 62.33 i 2S39 62.35 6581 1 62.38 1 0325 1 62.41 4070 62.43 7X17 62.4(1 »5''5 62.48 5315 62.51 9066 62.54 2X19 i 62.56 6573 62.59 0329 62.61 4087 : 62.64 7846 ! 62.67 1607 61.69 5369 62.72 9133 62.75 2899 62.77 6666 62.80 "435 62.82 4205 62.85 7977 62.88 1750 62.90 55*5 62.93 9302 62.96 3080 62.98 6860 63.01 0641 63.04 4424 63.06 8209 63.09 «';95 63.12 57i<3 63.14 957* 63.17 3363 63.19 715s < 63. '.Z 0949 . 63.25 4745 I 63.27 8542 ' 63-30 2341 ' 63.33 6142 OOAJ. 1 63-35 374« 7553 ( I 360 I 5>68 1 8978 I 2790 I 6603 0418 4235 8°53 1873 5695 9518 ' "^» ; 7'68 0096 4826 ^ 8657 63-4> 63.44 63.46 63.49 63.52 63-54 63-57 63.60 63.62 63.65 63. 6X 63.70 63-73 63.76 63.79 63. Si 63.84 63.87 i V, 132 133 Dlff. 1". 134 I.-kM. 1)1 IT. 1". 135 Ion .M. Dlff. 1". ' log M. Dlff. 1". ' 2.654 8657 1 63.87 2.678 1547 ■ 65-53 1.702 0562 67.27 1.726 5990 69.09 1 .655 2490 63.89 .678 5480 65.5b .702 4600 67.30 .727 0137 69.12 'Z .655 6324 1 63.92 .678 94<4 1 65.59 .702 8638 '1:^:11 .727 A285 •7*7 8435 .728 25^7 69.15 u .656 0160 63.95 .679 3350 65.61 .703 2679 69.19 1 * .656 3998 63.97 .679 7288 1 65.64 .703 6721 67-39 69.22 i 5 2.656 7837 64,00 2.680 1227 65.67 2.704 0766 j 67.42 1.728 6741 69.25 1 <^ .657 1678 64.03 .680 5168 1 65.70 .704 4812 , 67-45 .729 0897 69.28 7 .657 5521 64.06 .680 9111 65-73 .704 8860 67.48 ■7-') 5055 69.31 8 •657 9365 64.08 .681 3056 65.76 .705 2909 67.5' .719 9215 6934 i I) .658 3211 64.11 .681 7002 65-79 .705 6961 67-54 .730 3376 69-37 1 ' 10 2.658 7058 64.14 2.682 0950 65.81 2.706 1014 67-57 2.730 7539 69.40 1 i 11 .659 0907 64.17 .682 4900 65.84 .706 5069 67.60 .731 1705 69.44 vz .659 A758 .659 861 1 64.19 .681 8851 65.87 .706 9126 67.63 •73' 5^*7^ 69.47 13 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 «3«5 69.56 16 .661 0178 64.30 .684 .674 .684 8634 65.99 .708 5371 67-75 •733 2559 69.59 17 .661 4037 64-33 66.01 .708 9436 67.78 •733 6736 69.62 1 18 .661 7897 64.36 .685 2596 66.04 .709 3504 67.81 •7 34 0914 69.66 19 .662 1760 64.38 .685 6559 66.07 .709 7573 67.84 •734 5°94 69.69 J 20 2.662 5623 64.41 2.686 0524 66.10 2.710 1645 67.87 i-734 9*77 69.72 ' 21 .662 9489 64.44 .686 4491 66.13 .710 5718 67.90 •735 3461 69-75 , 22 .663 3356 64.47 .686 8460 66.16 .710 9792 67.93 •735 7647 69.78 23 .663 7225 64.49 .687 2430 66.19 .711 3869 67.96 .736 1835 69.81 1 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 1 20 .664 8842 64,57 .688 4352 66.27 .712 61 10 68.05 •737 44«° 69.91 27 .665 2717 64.60 .688 8329 66.30 .713 0194 68.08 •737 X605 69,94 28 .665 6594 64-63 .689 2308 66.33 •713 4»79 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 1203 •7 39 5406 70.04 31 .666 8236 64.72 .690 4256 66.42 •7'4 6547 68.20 70.07 32 .667 2120 64-74 .690 8242 66.45 .715 0640 6S.23 •739 9612 70.10 33 .667 6005 64.77 .691 2230 66.48 •7>5 4735 68.26 .740 3819 70.13 70.16 34 .667 9892 64.80 .691 6219 66.51 .715 8832 68.29 .740 8027 35 2.668 3781 64.83 2.692 0210 66.54 2.716 2930 68.32 2. ■'41 2238 70.20 3G .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 .717 1133 68.38 .742 0666 70,26 38 •669 5457 64.91 .693 2194 66.62 .717 5237 68.41 .742 4882 70.29 39 ,669 9353 64.94 -693 6193 66.65 .717 9342 68.44 .742 9101 70.32 40 2.670 3250 64.97 2.694 0193 66.68 2.718 3450 68.48 2.743 33*' 70.36 41 .670 7149 65.00 .694 4194 .694 8198 66.71 .718 7560 68.51 •743 7543 •744 1768 70.39 42 .671 1050 65.02 66.74 .719 1671 68.54 70.42 43 .671 491:2 65.05 .695 2203 66.77 .719 5784 68.57 •744 5994 70-45 S 44 .671 8856 65.08 .695 6210 66.80 .719 9899 68.60 .745 0222 70.48 45 2.672 2761 65.11 2.696 0219 66.83 2.720 4016 68.63 68.66 2.745 445* 70.52 46 .672 6668 65.13 65.16 .696 4229 66.86 .720 8135 -745 8684 70-55 47 .673 0577 .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 1391 70.65 1 50 2.674 2314 65.25 2.698 0289 66.97 2.722 4628 68.78 2.747 5631 70.68 51 .674 6230 65.2§ .69S 4308 67.00 .722 87<;6 68.81 •747 9873 70.71 1 52 •675 o«47 65.30 .698 8330 67.03 .723 2885 68.84 .74? 4n6 70.74 1 53 .675 4066 65.36 •699 2353 67.06 .723 7017 68.88 .748 8362 70.78 54 .675 7987 •699 6377 67.09 .724 1150 68.91 .749 2609 70.81 53 2.676 1909 6539 2.700 0404 67.12 2.724 5286 68.94 2.749 6859 70.84 56 .676 5833 65.42 .700 4432 .700 8462 67.15 .724 9423 68.97 .750 mo 70.87 1 57 ■676 9759 65.44 67.18 .725 3562 69.00 .750 5364 .750 9619 70.90 58 •677 3687 65.47 .701 2494 67.21 .725 7703 69-03 69.06 70.94 59 .677 7616 65.50 .701 6527 67.24 .726 1846 .751 3876 70.97 60 2.678 1547 65-53 2.702 0562 67.27 2.726 5990 69.09 2.751 8135 71.00 599 TABLE VI. For tiiitlinK llic Triio Aiiomiily or lliv Tiiiif tVnin tin- I'i'riliclioii in a I'liniliolic Orhit. V. 0' lot 136" « 137 138° 13£ l-K .M. i.831 8224 Din. 1". 77^31 M. Dili. I". 71.00 Inn 2.777 M. 73" inn. 1". 73.01 loK M. 2.804 3895 .804 8403 Dirr. 1". 75.11 X.75I Hi J 5 1 •7<;i 2396 71.03 •778 1703 73.04 75.'.^ 75.18 .832 »86i .832 7506 77^35 •z •751 6659 71.07 .778 6087 73^o7 .805 2912 77^39 a •753 0925 71.10 ■779 0472 7311 •8oi 7424 75.21 .833 2151 77^43 4 •753 5192 71.13 •779 4859 73'4 .806 1938 75-15 .833 6798 77^47 5 ^•753 9461 71.17 2.779 9249 73.'8 2.806 6454 7 5^i9 2.834 "447 77.50 <) •754 373* 71.20 .780 1641 8034 73^21 .807 0973 75^3i .834 6098 77.54 77.58 7 •754 H004 71.23 71.26 .780 73.*4 73-18 .807 5493 75.36 .835 0752 H •755 2279 .781 2430 .808 0016 75.40 •835 5408 .836 0066 77.62 U •755 6556 71.30 .781 6828 73.31 .808 4541 75^43 77.66 10 ^•75'> 0X3; 7i^-13 71.36 2.782 1228 73^35 2.808 9068 75^47 2.836 4727 77.69 11 .756 ;i 16 .782 5630 7338 .809 3597 .809 8128 75.50 .836 9390 11-11 lU •756 9399 71.40 .783 0034 73-42 75.54 75^58 •837 4055 11-11 i:t •757 3683 7i^4; 71.46 •783 4440 73-45 .810 2662 .837 8722 TI.U 11 •757 7970 •783 8848 73-49 .810 7197 75^6 1 •838 3391 77.85 15 1.758 2259 71.49 2.784 3158 73^51 73.56 2.81, ,735 75^65 2.838 8064 77.89 10 .75S 6549 7i^51 71.56 .784 7('7i .811 6275 75.69 •839 1738 77^9i 17 •759 0S42 •785 2085 73-59 .812 0817 75.72 •839 74'4 77^96 ; IH •759 5»37 7»^59 .785 6502 73.63 73^66 .812 5362 75.76 .840 2093 78.00 lU •759 9433 71.63 .786 0920 .812 9908 li-T) .840 6774 78.04 uo 2.760 3732 8032 71.66 2.786 534' 9764 73^7o i.813 4457 75^83 2.841 '458 78.08 'Zl .760 71.69 .786 73^7; 73.76 .813 9008 75-87 .841 6144 78.11 'i'Z .7fii 1335 7^73 71.76 .787 4189 .814 3561 75.90 .842 0832 78.15 'Z-A .761 6639 .787 8615 73.80 .814 8117 75.94 75.98 .842 5522 78.19 'Z\ ./bz 0946 7 ••79 .788 3044 73-83 .815 2674 .843 0215 78.23 25 1.7(12 5*55 71.83 2.788 7476 73^87 2.815 7134 .816 1796 76.01 1.843 4909 78.27 20 .762 9565 71.86 •789 1909 73.90 76.05 .843 9607 78.31 27 .763 3878 8192 71.89 .789 6344 73-94 .816 6360 76.09 .844 4306 78.35 28 .763 71-93 71.96 .790 0781 7 3-97 .817 0927 76.12 .844 9008 78.38 20 .764 2509 .790 5221 74.01 .817 5495 76.16 .845 3712 78.42 30 2.764 6827 71.99 2.790 9662 74.04 2.818 0066 76.20 2.845 8419 78.46 31 .765 1148 72.03 .791 4106 74.08 .818 4639 76.23 .846 3128 78.50 :i2 .765 5470 72.06 .791 8552 74." .818 9214 76.27 .846 7839 78.54 33 .765 9795 72.09 .792 3000 74-' 5 .819 3792 76.31 •847 2553 78.58 34 .766 4121 72.13 .792 7450 74.18 .819 8371 76.34 .847 7268 78.62 35 2.766 8450 72.16 2.793 1902 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 76.46 •849 '43° 78.73 38 .768 I44» 5784 72.26 •794 5271 74-32 .821 6712 76.49 .819 6155 78.77 30 .768 72.29 •794 9731 74.36 .87,2 1302 76.53 .850 0882 78.81 40 2.769 0123 72.33 ^•795 4194 74.40 2.8:dJ, ,895 76^57 2.850 5612 78.85 41 .769 4464 72.36 •795 8659 74-43 .ti\ 0(91 76.60 •85 ' 0344 78..S9 42 .769 8806 72.39 .796 •jl26 74^47 .9i>, ;j88 76.64 .851 5079 78.93 43 .770 3151 72^43 72.46 .796 7595 74-50 .■:? 9688 76.68 .851 9816 78.97 44 .770 7498 •797 2066 74-54 .i,z\ .^289 76.72 .852 4555 79.01 45 2.771 1846 72.50 2.797 6539 74.58 2.824 8894 76.7s 2.852 9297 79.05 40 •77' 6197 72.53 .798 1015 74^6 1 .825 3500 76.79 •853 404' 79.08 47 .772 0550 72.56 .798 5491 74.64 .825 8108 76.83 .853 8787 79.12 48 .772 4905 72.60 •798 9972 74.68 .826 2719 76.87 •854 3535 79.16 40 .772 9262 72.63 •799 4454 74.7' .826 7332 76.90 .854 8286 79.20 50 2.773 3621 72.67 2.799 8938 74^75 2.827 1947 76.94 2.855 3040 79-14 : 51 •773 7982 72.70 .800 3424 74^79 •827 6565 76.98 •855 7795 79.28 52 •774 2344 72.73 .800 7912 74.82 .828 1185 77.01 .856 2553 79-3J 53 •774 6709 72.77 .801 2402 74.86 .828 5807 77.05 .856 7314 79-36 54 •775 1077 72.80 .801 6895 74.89 .829 0431 77.09 .857 2077 79.40 55 ^•775 5446 9817 72.84 2.802 1390 74-93 74.96 2.829 5058 77.13 2.857 6842 79-4-t 50 •775 72.87 .802 5886 .829 9686 77.16 .858 160.7 79.48 ; 57 .776 4190 8565 72.90 .803 0385 4886 75.00 .830 4317 77.20 .858 6379 79-5; 58 .776 72.94 .803 75^04 .830 8951 77-14 77-18 .859 1151 79.56 50 •777 2942 72.97 .803 9390 75.08 .831 3586 .859 5926 79.O0 GO 2.777 7322 73.01 2.804 3895 75.11 2.831 8224 77.32 2.860 0703 79.64 \ 600 ■ • liolic Orhit. 5612 0344 5079 9816 4555 3040 7795 ^555 73>4 2077 6842 leoT 6379 5926 0703 lllff. I". 77.3» 77-15 7;-i9 77-45 7 7-47 77-5° 77-54 77-55 77.62 77.66 77.69 77-73 77-77 77.X1 77-«5 77.89 77-9» 77.96 78.00 78.04 78.08 78.11 78.15 78.19 78.23 78.27 7«-1" 7«-35 78.38 78.42 78.46 78.50 7«-54 78.58 78.62 78.66 78.69 7«-73 78-77 78.81 78.85 78.89 78.93 78.97 79.01 79.08 79.12 79.16 79.20 79-'4 79.28 79-3J 79.36 79.40 79-44 79.48 79-55 79.56 79. Co 79.64 TABLE VI. l''(ir limliin,' the Tnio .Vriornnlv or lln- Tiim- li-cuii tlio IVrilu-linn in n, Parabolic Orhit. 1 ♦'• 14C )" 141 loK M. DIPT. 1". 14S l"Kll. 5^ Dinr. 1". !•>« 142 M. r : DIff. 1". Diir. I". 2.860 0703 79.64 79.6)5 ..8«9 1754 82.08 2.919 iBjI 84.65 1.950 1420 87-37 1 .860 5482 .889 6680 82.12 .919 6911 84.70 ■ 950 6664 87.4, [ 'Z .861 0264 79.7» .890 1609 82.16 .920 1994 84.74 ■95" 1910 87.46 i :> .861 5048 79.76 .890 6540 81.20 .920 7080 84.78 •951 7' 59 87,50 '1 .X61 9835 79.80 .891 1473 82.25 .921 2169 84.83 .952 1411 87-55 n 2.8(-i 4624 79.8)5 2.X91 6409 82.29 2.921 7260 84,87 2.952 7665 87.60 » .861 9415 .892 1348 81.33 -9" »353 84.91 •95 3 2923 87,65 7 .863 4209 79.92 .892 6289 82.37 .922 74;;o 84.96 -953 8183 87.69 N .863 9005 79.96 .893 1233 82.41 .923 2549 85.01 -954 344" 87-74 11 .864 3803 80.00 .893 6179 82,46 .923 7650 85.05 -954 8711 87.79 10 1,864 8604 80.04 80.08 2.894 1127 82.50 i.924 2755 85.10 1-955 3980 87.81 87.88 11 .86^ 3408 .894 6078 82.54 82.58 .924 7861 85.14 -955 92^1 Vi .861; 8213 80. 1 2 .895 1032 .925 2972 85.18 .956 4515 87-93 ! 1:1 .866 3021 80.16 .895 5989 82.63 .925 8084 85.23 ,956 9802 ti-''"^ II .866 7832 80.20 .896 0948 82.67 ,926 3199 85.27 -957 5082 88,o2 15 2.867 2645 80.24 2.896 5909 82.71 Z.926 8317 85.32 2.958 0365 88.07 10 .867 7460 80.28 .897 0873 82.75 •917 3437 85.36 .958 5651 88.11 17 .868 2278 80.32 .897 5839 82.79 .927 8560 85.41 -959 0939 X8.16 IH .868 7098 80.36 .898 0808 82.84 82.88 .928 36S6 85-45 •959 6230 88.21 Ml .869 1921 80.40 .898 5780 .928 8814 85.50 .960 1524 88.26 'ZO 2.X69 6746 80.44 80.48 2.899 071 : 82.92 1.919 3945 85-54 2.960 6821 ^!!-3° 'Zl .870 1573 .899 5730 82. 96 .919 9079 85.59 .961 2120 ^^35 •Z'Z .870 6403 80.52 .900 0709 83.CI .930 4116 85.63 85.68 .961 7423 88.40 •ZA .871 1235 80.56 .900 5691 83.05 -93° 9355 .962 2728 88.45 : •Zi .871 6070 80.60 901 0675 83.09 ■93' 4497 85.71 .962 7036 88.49 'zrt 2.872 0907 80.64 1.901 5662 83.13 1.931 9641 85-77 2,963 3347 !!-5+ •za .872 5747 80.68 .902 0651 83.18 .932 47S8 85.81 ,963 8661 88,, 9 'Zl .873 0589 80.72 .902 5643 83.21 .932 9938 85.86 .964 3978 88,64 'ZH •i*73 5433 80.76 .903 0638 83.26 •93 3 509' 85-91 .964 9»97 88,68 ' a« .874 0280 80.80 .903 5635 83.3, •9 34 0*47 85.95 .965 4620 88.73 ;io 2.874 5129 80.84 2.904 0635 83-35 2-934 S4°5 85-99 2.965 9945 88.78 :ii .874 9981 80.88 .904 5637 83.39 -935 0565 86,04 86,08 .966 5173 88.83 A'Z .875 4835 80.92 .905 0642 «3-43 -935 5729 .967 0604 88.87 AA .875 9692 80.96 .905 5649 83.48 .936 0895 86., 3 .967 5938 ll-'f"- 31 .876 4551 81.01 .906 0659 83.51 .936 6064 86.17 .968 1275 88.97 :i5 2.876 9413 81.05 2.906 5672 83.56 2.937 1236 86.22 2,968 66:5 89.02 »6 .877 4277 8 1 .09 .907 0687 83.61 .937 6410 86,26 .969 1957 89.07 37 .877 9143 8. .13 .907 5704 83.65 .938 1587 86.31 .969 7303 89.12 38 .878 4012 8,., 7 .908 0725 83.69 .938 6767 86.35 .970 265. 89.17 39 .878 8883 81.21 .908 5748 «3-74 •939 >95o 86.40 •97° 8002 89.11 10 2.879 3757 81.25 2.909 0773 83.78 2.939 7135 86,45 2.971 3356 89.26 41 .879 8633 81.29 .909 5801 83.82 .940 2323 86.49 •97' 8713 89,31 VZ .880 3512 81.33 .910 0832 83.87 -94° 75 > 4 86,54 .972 4073 89.36 43 .880 8393 81.37 .910 5865 83.91 .941 2708 86.58 .971 9436 89.-40 44 .881 3277 81.42 .911 0901 83-95 .941 7904 86.63 ■973 4801 89.4s 45 2.881 8163 81.46 2.911 5940 83.99 2.942 3103 86,67 1.974 0170 89,50 4» .882 3052 Si. 50 .912 0981 84.04 .942 8305 86.71 •974 554' 89-55 : 47 .882 7943 81.54 .912 6024 84.08 •943 35 «° 86.77 •975 0916 89,60 48 .883 2837 81.58 .913 1070 84.13 •943 8717 86.81 ■975 6193 89.05 4« -«83 773 3 81.62 .913 6119 84.17 -944 3927 86.86 .976 1673 89.69 50 2.884 ='*'3i 81.66 2.914 1171 84.22 2.944 9140 86.90 1,976 7056 89.74 51 .884 7532 81.70 .914 6225 84.26 •945 43 5 5 86.95 •977 1442 89,79 5« .885 2436 81.75 .915 1282 84.30 •945 9574 87.00 •977 7831 89.84 53 .885 7342 81.79 .915 6341 84.34 -946 4795 87.04 .978 3223 89.89 ; 54 .886 2251 81.83 .916 1403 84.39 .947 0019 87.09 ■978 8618 89,94 ' 55 2.886 7162 81.87 2.916 6468 84.43 *-947 5^45 87.11 2.979 4015 89.99 5U .887 2075 81.91 •9«7 1535 84.48 .948 0475 87,18 •979 9416 90.03 57 .887 6991 81.95 .917 6605 84.52 .948 5707 87.23 .980 4820 90.08 58 .888 1910 81.99 .918 1678 84.56 -949 °94a 87.27 .981 1226 90.15 90.18 59 .888 6831 82.04 .918 675J 84.61 .949 6180 87.3* .981 6636 (K) 2.889 1754 82.08 2.919 1831 84.6s 2.950 1420 87^37 1.982 1048 90.a3 1 601 TABLE VI. Tor finding flic Tnte Anomaly or the Time Iroiu the Perihelion in a Paraboli'' Orbit. t i ^- i 144° 145° 146° 147° log M. 1048 Diff. 1". lofi M Uiff. 1". 93.26 log M. I)ilT. 1". loK M, DilT, I". 2.982 90.23 3.015 1281 3,049 2733 96.47 3.084 6070 99,87 ' 1 .9X2 '^('■i 90.28 .015 6X7X 93^3> ,049 8522 96.52 .0X5 2064 99,92 2 .9X3 1XX2 90.33 .016 247X 93-36 .050 43'5 96.58 .085 8061 99.9S ■ 3, .9X3 7303 go. 3 8 .016 X082 93-4^ .051 01 1 2 96.63 .0X6 4062 ir^ ni 4' .9X4 2727 90.43 .017 36X8 93-47 .051 5911 96.69 .087 0066 100.10 1 5 2.9X4 8.54 90.48 3.017 9298 93.52 3.052 1714 96,74 3.087 60-, 3 100,16 G .9X.; 3584 90-53 90.158 .oiS 4911 93^57 .052 7520 96,80 .oXX 20X5 100,22 7 .9X,- 9017 .019 0526 97.(12 -^53 3329 96.85 .088 8099 100.2S 8 .9X0 4453 90.63 .019 6145 93.68 .053 9142 96.91 .089 411X 100.33 .9X6 9X9.. 90.67 .020 1768 93^7 3 .054 4959 96.96 .090 0140 100,39 J 10 2.987 5334 90.72 3.020 7393 93^78 3^055 0778 97,01 3.090 6165 100,45 11 .9XX 0779 90.77 .021 3021 93.X3 .055 6601 97.07 .091 2194 100.51 Vi .9XX 6227 9Q.X2 .021 8653 93.89 .056 2427 9/^>3 .091 8226 100.57 i:i .989 1678 90.X7 .022 42XX 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 10 .990 8c49 9 1 .02 .024 1211 94.10 .05X 5765 97^35 •094 2392 100, Xl 17 •99 > 3512 9- 07 .024 r-!?59 94^«5 ,059 1608 97^4i .094 8. -2 100.87 18 .991 8977 91.12 .025 2509 94.20 ,059 7454 97^47 .095 4496 100.93 10 .992 4446 91,17 .025 8163 94.26 ,060 3304 97.52 .096 0553 100,98 20 2.992 9918 91.22 3.026 3820 94.31 3,060 9157 97.58 3.096 66,+ 101.04 21 •993 5 393 91.27 .026 94X0 94.36 ,061 5o«3 97-63 •097 2678 101,10 22 •994 0X71 91.32 .027 5 "43 94.41 .062 0873 97.69 .097 8746 101.16 23 •99+ 6351 9«^37 .028 oXio 9447 .062 6736 97^75 .098 4818 IOI.12 24 •995 1835 91.42 .02X 6479 94.52 .063 2602 97.80 -"99 0893 101.28 25 ^•99 5 7322 91.47 3.029 2152 94^5" 3.063 8472 97.86 3.099 6972 101.34 26 ■99'' 2X12 91.52 .029 7S2S 94.6 1 9.; 6,? .064 4345 97,91 .100 3054 IOI..).0 27 •99" 8305 9^57 .030 3507 .065 0222 97-97 .100 9140 101.46 28 •997 3X01 91.62 .030 9190 /4^:3 .06 c 6101 9X.03 .101 5230 101.52 ' 29 •997 9300 91.67 .031 4875 "■+•79 .066 1985 98.0X .102 1323 101. 58 30 7.99X 4X02 91.72 3.032 0564 94.84 3.066 7872 98,14 3,102 7420 101.64 31 •999 0307 91.77 .032 6256 94.89 .067 3762 98,20 .103 3520 101.70 32 •999 5X15 91. X2 .033 1951 9494 .067 9''5 5 98,21; .103 9624 101.76 33 3.000 rf-'' 91.87 .033 7650 95.00 .06S 5552 98-31 .104 5732 101.82 34 .000 6X40 91.93 .034 3351 95.05 .069 '45 3 98.37 .105 1843 101.88 ■ 1 35 3.001 2357 91.98 3^034 9056 95.11 3.069 7357 9X.42 3.105 7958 101.94 30 .001 7877 92.03 .035 4704 95.-6 .070 3264 9X.4X .106 4076 102.00 37 .002 3400 92. oX .036 0475 95.22 .070 917-1 98,54 .107 0198 102.07 38 .00 7. X926 92.13 .036 6190 95.27 .07. 50X8 98.60 .107 6324 102.13 3« .003 4456 92.18 •037 1908 95,32 .072 1006 98.65 .108 24:4 102.19 , 40 3.003 9988 92.23 3^°37 7629 95.38 3.072 6927 98,71 3.108 8587 102.25 41 .004 55^3 92.28 .038 3353 95^43 •073 2851 98.77 .109 4723 102.31 42 .005 1062 91^33 .038 90X0 95.4X .073 8779 98, X2 .1 10 0864 102.37 43 .005 6603 92.38 •039 4811 95^54 .074 4710 9X,88 .110 7008 102.43 44 .006 214X 92.44 .040 0545 95.60 .075 0645 98.94 ,111 3«55 102.49 45 3.006 7696 92.49 3.040 6282 95^65 3-°75 65S3 99,00 3,111 9306 102.55 40 .007 3246 92.54 .041 2023 95.70 .076 2524 99,05 .112 5461 102.61 47 .007 XSoo 9^59 .041 7767 95.76 .076 8469 99.11 •'»3 1620 102.67 48 .ooX 4357 92.64 .042 35«4 95.8. .077 441 X 99,17 .113 7782 102.73 49 .008 9917 92.69 .042 9264 95,86 .078 0370 99,23 .114 3948 102. Xo 50 3.009 5480 92.74 3.043 5017 95.92 3.078 6325 99.28 3. 115 0118 102.86 51 .010 1046 92.79 .04.., 0774 95^97 .079 2284 99-34 .115 6291 102,92 52 .010 foi5 92.85 .044 6534 96.03 .079 8246 99,40 .116 2468 102,98 53 .011 2188 92.90 .045 2297 96.08 .080 4212 99.46 .116 8649 103,04 54 .oil 7763 92.95 .045 8064 96,14 .081 ri8l 99.52 .117 4833 103.10 55 3.012 3342 93.00 3.146 3834 96.19 3.081 6154 99-57 3. 118 1022 103, '6 50 .012 8923 93.05 .o;6 9607 96,25 .0X2 2130 99.63 .uS 7213 103,25 57 .013 4508 93.10 .047 5383 96.30 .oX-.', 8110 99.69 .119 3409 103,29 58 .014 ;6 93.16 .04X 1163 96.36 .083 4093 99-75 .119 9608 103-3; 59 .014 5687 93.21 .048 6946 96,41 .084 00X0 99-81 .120 5811 103.41 GO 3-o«5 1281 93.26 3.049 2733 96.47 3.084 6070 99-87 3.121 2018 ,03.48 602 )olii' Orbit. TABLE VI. For finding tlie True Anoinnly or the Time fnini the Poriholion in a Parabolic (^rbit. DilT. 1". 99.87 99.9- 99. 9S IC'-' n\ 100.10 6o', 3 1 100.16 2085 ', 100.22 8099 100.2S 4118 : 100.33 0140 i 100.39 6165 ; 100.45 2194 1 100.51 8226 , 100.57 4262 100.63 0302 1 100.69 6US 100.75 2392 I 100. Si 8j.;2 i 100.87 4496 100.93 °553 100.98 6614 101.04 2678 1 101. 10 8746 i 101.16 4818 101.22 0893 1 loi.2!S 6972 1 101.34 3054 1 101.40 9140 101.46 5230 1 101.52 1323 ' 101. 58 7420 101.64 3520 1 101.70 9624 101.76 573^ 101.82 1843 101.88 7958 ! 101.94 4076 j 102.00 0198 ' 102.07 6324 102.13 24:4 102.19 8587 102.25 4723 102.31 0864 i 102.37 ) 7008 i 102.43 3«55 102.49 9306 102.55 - 546' 102.61 I 1620 102.67 , 7782 102.73 ^ 394» 101.80 5 0118 ■ 102.86 5 6291 . 102.92 5 2468 1 102.9S ft 8649 1 103.04 7 4«33 103.10 8 1022 103. '6 S 7213 103.13 9 3409 103.29 9 9608 103.35 5811 103.41 I 2018 { 103.48 V. 0' 148° 149° 150° 151 lot? M. 3.239 3820 ' 1 1«K M. I)i(T. 1". 103.48 log M. Diff. 1". loK M. I)ilT. 1". 111.41 Diff. 1". j 115-77 < 3.I2I 2018 3.159 1367 I 107.31 3.198 4984 I 1 .121 8228 103.54 .159 y^o^ 107.38 •'99 1671 111.48 .240 0768 115.85 'i .122 4AA2 0660 103.60 .160 4253 107.45 ■'99 8361 III. 55 .240 7722 115.92 3 •'i3 103.66 .161 0702 107.51 .200 5056 111.62 .241 4680 116.00 ' 4 •«J3 6882 103.72 .161 7154 107.58 .201 '755 111.69 .242 1642 116.08 1 1 5 3«24 3107 103.79 3.162 3611 107.65 3.2)1 8459 111.76 3.242 8608 116.15 ■ 1 « .124 9336 103.85 .163 0072 107.71 .2 12 5166 111.83 .243 5580 116.23 ! 1 7 .125 5569 103.91 .163 6J36 107.78 .2,3 1878 111.90 •244 255''^ 116.30 ] 8 .126 1805 103.97 .164 3005 107.85 .203 8594 111.97 •244 9536 116.38 : 1 1> .126 8045 104.04 .164 9478 107.91 .204 53'5 1 12.04 .245 6521 116.45 lU 3.127 4289 1 04. 1 3'"5 5955 107.98 3.205 2040 112.11 3.246 3511 116.53 1 11 .128 0537 104.16 .166 2435 108.04 .205 8769 112.18 .247 0505 116.61 ; 1'^ .128 6789 104.22 .166 8920 108.11 .206 5502 112.26 ■247 7503 116.68 ; 13 .129 3044 IOJ.29 .167 5409 108.18 .207 2^39 112.33 •248 4507 116.76 1 14 .129 9303 104.35 .168 1901 108.25 .207 8981 1 12.40 •2+9 •5'S 116.84 15 3-'3° 5566 104.41 3.168 8398 108.31 3.208 5727 112.47 3.249 8i27 116.91 16 .131 1833 104.48 .169 4899 108.38 .209 2478 112.54 .250 5544 1 16.99 17 .131 8103 104.54 .170 1404 108.45 .209 9232 1 12.61 .251 2566 117.07 18 .132 4377 104.60 .170 7913 108.51 .2IO 599' 112.69 .251 9592 117.14 lU .133 065s 104.67 .171 4426 108.58 .211 2755 112.76 .252 6623 117.22 20 3'33 6937 104.73 3.172 0942 108.65 3.211 9522 112.83 3^253 3658 117.30 21 •«34 3223 104.79 .172 7463 108.72 .il2 6294 1 12.90 .254 0698 "7-37 i 1 22 •34 9512 104.86 •'73 39XX 108.78 .213 3070 112.97 •254 7743 "7^45 i i 23 •135 5805 104.92 •'74 °5'7 108.85 .213 985. 113.05 •255 4792 "7^53 i 24 .^36 2102 104.98 •'74 705' 108.92 .214 6636 113.12 .256 1846 117.60 i 25 3.136 8403 105.05 3,175 3588 108.99 3.215 3425 113.19 3.256 8905 117.68 ! 2G ••37 4708 105.11 .176 0129 109.06 .216 0219 113.26 .257 596*; 117.76 i 27 .138 1016 105.17 .176 (1674 1 09. 1 2 .216 7017 "3-34 .258 3036 117.84 ; 28 .138 7329 105.24 •'77 3224 109.19 .217 38,9 113.41 .2s9 oic; 117.91 29 .139 3045 105.30 •'77 9777 109.26 .218 0626 113.48 .259 7186 117.99 30 3-' 39 9965 105.36 3.178 6335 10^.33 3.2. ^^ 7437 "3-55 3.200 4268 118.07 31 .140 6289 i°5-43 • '79 ^^97 109.40 .219 4252 113.63 .2C1 1354 118.15 32 .141 2616 105.49 ■'79 94'^i 109.46 .220 1072 113.70 .261 8446 118.23 33 .141 8948 105.55 .iSo 6032 109.53 .220 7896 "3-77 .262 5542 118.30 34 .142 5283 105.62 i8i 2606 109.60 .221 4724 113.84 .263 2642 118.38 35 3143 1622 105.68 3. 181 9184 109.67 3.222 '557 113.92 3.263 9747 11 8. 46 3C •«43 7965 105.75 .182 5766 109.74 .222 8395 1 1 3.99 .26^ 6857 118.54 37 .144 43'2 105.81 •'X3 2353 109.81 .223 5136 ll.i c6 .265 3972 118.62 I 38 •'45 0663 7018 105.87 .183 8943 109.87 .224 2081 114.14 .266 1091 118.70 30 .145 105.94 .184 5538 109.94 .224 8933 1 14.21 .266 8216 118.77 40 3.146 3376 106.00 •> 185 2136 110.01 3.225 5788 114.2S 3^267 5345 118.85 41 ..46 9739 106.07 ^185 S7?9 : 10. cs .226 2647 114.36 .268 2478 118.93 42 .147 6105 106.14 .186 5346 1 10. 1 5 .226 95 II "4-43 .268 9616 1 1 9.01 43 .14>* 2475 106.20 .187 1957 I 10.22 .227 6379 114.51 .269 6759 119.09 44 .148 8849 106.27 .187 SsvL 110.29 .22S 3252 114.58 .270 3907 119.17 45 3'i49 5227 106.33 3.188 5192 110.36 3.229 0129 114.65 3.271 1060 119.25 40 .150 1609 106.40 .189 1815 '110.43 -•19 7010 114.73 .271 8217 119.33 47 .150 7995 106.46 .189 8443 110. c .230 3896 114.80 ■272 5 379 119.41 48 .151 43X5 106.53 .190 5075 ''■'■57 .231 0786 114.88 ■273 2546 119.49 4!> .152 0778 106.59 .191 1711 110.64 .231 7681 114.95 .273 9717 119.57 50 3.52 7176 106.66 3.191 8351 110.71 3.232 458' 115.03 3.274 6894 119.65 51 •'53 3577 106.72 .192 4906 110.77 •^33 1484 115.10 ■275 4^75 11973 ' 52 •'53 9983 106.79 •'93 '^44 110.84 .233 8392 115.17 .276 1261 119.81 53 •'54 6392 106.85 •'93 *'-97 110.91 .234 S3°5 "5-25 .276 8452 119.89 54 •«55 2805 106.92 •'94 4954 110.98 •235 2222 115.32 .277 56.^7 119.97 55 3^'55 9222 106.99 3.195 1615 111.05 3^235 9'44 115.40 3.278 2848 120.05 50 .156 5643 107.05 .195 8281 I : i.I2 .236 6070 "5-!7 .279 0053 120.13 57 •157 20O8 107 12 .196 4950 111.19 .237 3001 "5 55 .279 7263 120.21 58 •'57 8497 107.18 .197 1624 111.26 .237 9936 115.62 .280 4477 120.29 59 .158 4930 107.25 .197 8302 "'•34 .238 6876 115.70 .:'.8i 1697 120.37 00 3'59 1367 107.31 3.198 4984 111.41 3^239 3820 "5^77 3.281 8921 120.45 f03 TABLE VI. For finding the True Anoinnly ny t\-2 Ti.nc Tium the i'orilielion in a P:.rabolic Orbit. V. 152° « "' 163° 154° 155 1 lo). •M. 1)1 ir. 1". 120.45 I"). M. Di/r. I". 125.46 loB M. iiifr. 1". lof. M. oiir. 1". O' 3.181 8921 3.326 I'^'^i 3.372 2684 130.85 3.420 4064 115.66 1 .2X2 6151 120.53 .326 8978 i25^55 •373 o53'< 130.94 .421 2266 136.76 a .2^ 33>*5 120.61 .327 6513 125.63 •373 *'397 131.04 .422 0475 136.86 3 .284 0624 120.69 .328 4054 125.72 .374 6262 131. 13 .422 8690 136.96 4 .284 7868 120.77 .329 1600 125. Si •375 4'33 131.22 .423 6910 137.06 5 3.285 51 16 120.85 3329 9151 125.89 3.376 2009 i3'-32 3.424 5'37 137.16 .286 2370 120.93 .330 6707 125.98 .376 9890 131.41 .425 3370 137.26 7 .286 9028 121.01 .331 4268 126.07 •377 7778 131.50 .426 1609 '37-37 8 .287 6891 121.10 .332 1835 126.16 .378 5671 131.60 .426 9854 8105 '37-47 » .288 4160 121.18 -33^ 9407 126.24 •379 3570 131.69 •427 '37-57 i 10 3.2S9 H33 121.26 3-333 6984 126,33 3.380 1474 '3'-I2 3.428 6362 137.67 1 11 .289 8711 121.34 •334 .567 126.42 .380 9384 131.88 •429 4626 '37-77 i la .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 •43' 1171 137.98 1 ** .2 )2 0574 121.59 .336 7346 126.68 •3«3 3'48 132.16 •43' 9452 138.08 i 15 3.292 7872 121.67 3-337 4949 126.77 3.384 1081 132.26 3-432 7740 138.18 i 1« .293 5'74 121.75 •33** 2558 126.86 .384 9019 132.35 •433 6034 138.29 17 .294 2481 121.83 -339 0172 126.95 .385 6963 '32-45 -434 4334 '38-39 18 .294 9794 121.91 -3 39 7792 127.03 .386 4913 '32-54 -43 5 2641 138.49 19 .295 7HI 122.00 .340 54>7 127.17. .387 2869 132.64 .436 0953 138.59 20 3.296 4433 122.08 3-34' 5?F 127.21 3.388 0830 132.73 3-436 9272 138.70 i 21 .297 1761 122.16 •342 0682 127.30 .388 8797 132.83 -43- 7597 138.80 1 aa .297 9093 122.24 .342 8323 127.39 .389 6770 132-93 .438 5928 138.90 1 83 .298 6430 122.33 -343 51(09 127.48 -39 4749 133.02 •439 4266 139.01 1 24 .299 3772 122.41 -344 3020 I27^57 -391 2733 133.12 .440 2609 •I 39.11 25 3.300 1119 122.49 3-345 1277 127.66 3.392 0723 133.22 3-44' 0959 139.22 2« .300 847 F 122.58 -345 8939 '27-75 .392 8719 '33-3' •44' 93'5 139.31 27 .301 5828 122.66 .346 6606 127.84 -393 6720 '33-4' .442 7677 1 39 42 28 .302 3190 122.74 ■^H 4279 127.93 •394 4728 '33-50 •443 6046 '39-53 1 29 •303 0557 122.83 .348 1958 128.02 •395 274' 133.60 •444 4421 '39-63 30 3-303 7929 122.91 3.3.18 9641 128.11 3.396 0760 133.70 3-445 2802 ' 39-74 31 .304 53q6 122.99 •349 7330 128.19 .396 8785 133-79 -446 1189 139.84 32 .305 268S 123.08 .350 5024 128.28 .397 6815 133-89 •446 9583 '39 95 ! 33 .306 0075 123.16 •351 2724 12S.37 .398 4852 '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 '23-33 3-35i 8140 128.55 3.400 0942 134.19 3-449 4802 140.26 30 .308 2267 123.41 •353 5856 128.65 .400 8996 134.28 .450 3221 140.37 37 .308 9674 123.50 •354 3577 128.74 .401 7056 134.38 •45' 1646 140.47 38 .309 7086 123.5& •355 1304 128.S3 .402 5122 134.48 •452 0077 '4° 57 39 .310 4504 123.66 •355 9037 128.92 •403 3193 '34-57 •452 851s 140.68 40 3. 311 1926 113-75 3-356 6774 129.01 3.404 1270 134.67 3453 6959 140.79 41 .311 93 i4 123.83 •557 45'7 129.10 -404 9354 '34-77 •454 5410 140.90 42 .312 (1786 123.92 .358 2266 1 29, 1 9 -405 7443 134.87 •455 3867 141.00 43 -313 4:24 124.00 -359 0020 129.28 .406 5538 134-97 .456 2330 141. II 44 -3'4 1667 124.09 -359 7780 129.37 .407 3639 135.07 •457 0800 141.21 45 3-314 9115 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 47 .316 4025 124.34 .362 l°J- i:;.6 5 •409 7977 '35-36 •459 6248 141.54 48 -3>7 1489 124.43 .362 8873 129.74 .410 6102 135.46 .460 4743 141.04 49 .317 8957 124.51 -363 6660 129.83 .411 4233 135.56 .4.61 3245 141.75 50 3.318 6430 124.60 3364 4453 129.92 3.412 2369 135.66 3.462 '753 141.86 51 -3'9 3909 124.68 .365 2151 130.01 .413 0512 135.76 .463 0268 141.97 52 .320 1392 124.77 .366 0055 130.11 .413 8660 135.86 .463 87S9 142-07 . 53 .320 8881 124.86 .366 7864 130.20 .414 6815 '35^96 .464 73'7 585' 142.18 54 .321 O375 124.94 .367 5679 130.29 ■4>5 4975 136.06 .465 142.29 55 3.322 3!-', 4 125.03 3.368 3499 130.38 3.416 3142 136.16 3.466 4392 142.40 50 .323 1379 8888 125.11 .369 1325 130.48 .417 1314 136.16 .407 2939 141.51 57 •323 125.20 .369 9156 '3057 .417 9^92 .418 7677 136.36 .468 1492 142. bl 58 •314 6403 125.29 .370 6993 130.66 136.46 •469 0051 U2.'2 59 ■325 3923 125.37 •37« 483b 130.76 419 5S67 136.56 .469 86it> .;...8l 00 3.326 1448 125.46 3-37* 2684 130.85 3.420 4064 136.66 3^47- 7192 1 1.' - ■ 6U1 ibolic Orbit. 155 M. oiir. 1". 4064 115.66 2266 136.76 0475 I 136.S6 8690 i 136.96 6910 137.06 5«37 137.16 3370 i 137.16 1609 «37-37 9«S4 8105 137-47 137-57 6362 4626 137.67 »37-77 i«95 137.88 1171 137.9*! 9452 138.0H 774° 138.18 6034 138.29 4334 138.39 1^8. IQ i3«-59 138.70 138.80 138.90 139.01 •139.11 139.22 139.32 1 39 41 139-53 139.63 > 39-74 139.84 139-95 140.05 140.16 140.26 140.37 140.47 14057 140.68 140.79 140.90 141.00 141.11 141.21 141.31 141.43 141.5+ 141.64 141.75 141.86 141.97 142.07 142.18 142.29 142.40 142.51 142.61 TABLE VI. for finding the True Anomaly or tlie Time from tlie Perihelion in a Parabolic Orbit. I •J ! 7192 I|.T V. o 1 2 3 4 5 G 7 8 10 11 vz 13 14 15 14( It •Zi TZ 23 24 25 20 27 28 29 30 31 32 33 31 35 30 37 38 30 ^o 41 43 43 44 45 40 47 48 40 50 51 52 53 54 55 50 57 58 59 60 156^ log M. I I 470 7192 i 471 5772 472 4358 473 ^95" 474 JSSO 475 o'S6 475 8769 476 7388 477 6014 478 4646 479 3185 480 1931 481 0583 481 9242 482 7907 483 6579 484 5258 4S5 3944 i86 2636 ■'■''7 133s 488 0040 4S8 8752 489 747a 490 6198 491 4930 492 3670 493 i4>6 494 1168 494 9928 495 8695 496 7468 497 6248 498 S°3S 499 3828 500 2629 501 1436 502 02,0 502 90 !■ 503 7''-.>.; SCI ' '34 }''\ 5o''' 507 320- 508 2143 509 1012 509 9889 510 8772 511 7662 512 6560 513 5464 514 4375 515 3294 516 2219 1 ! I <, I ■■;'■: cc9r 3-5''^ N- ,519 r,o .51c 6951 .52? 5918 .52i 4893 3-5-3 3875 Diff. 1". »4349 143.60 >43-7i 143.82 »43-93 144.04 't4-«5 144.26 '44-37 144.48 144-59 144.70 144-81 144-93 145.04 '45-15 145.26 '45-37 '45-49 145.60 145-71 145.82 '45.94 146.05 146.16 146.28 146.39 146.50 146.62 146.73 146.85 146.96 1+7.08 JA7.19 ' -/-31 •.•.-.42 -t' .-^4 J. 47.1-5 ' - 7 .^■,.^» 148.00 148.11 148.23 148.34 148.46 148.58 148.70 148.81 148.93 149.05 '4;- '7 149.28 149.40 149-51 149.64 '49-/5 157= loK M. 3875 2864 i860 0863 98:3 88 JO 79'5 6947 59\5 5031 4085 3 '45 2213 1288 0370 9459 8556 7660 6771 5890 5015 4148 3289 2430 1591 0754 9924 9101 8285 7477 550 6677 55' 5883 552 5097 553 43'9 554 3548 2785 2029 i.'.8o 523 5^4 525 526 526 527 528 529 530 53' 532 533 534 535 536 536 537 538 539 540 541. 542 5^3 544 545 546 546 547 548 549 555 556 557 558 558 559 560 561 562 5''3 564 565 566 567 568 569 570 571 572 572 573 574 57? .76 577 3.578 0539 9806 9080 8361 7650 6947 C-251 5562 4882 4209 3543 2885 2235 1592 °9:7 0330 9710 9098 8494 7897 7308 6727 6154 Diff. 1". 49-75 49.87 j.9.99 50.1 1 50.23 50-35 50.47 50.59 50.71 50.83 5°-95 51.07 51.19 51.31 5'-43 5'-S5 51.67 5'-79 51.91 52.04 52.16 52.28 52.40 5252 52.65 52.77 52.89 53.01 53-'4 53.26 53-38 53-5' 53-63 53-75 53-88 54.00 54- '3 54-25 54.38 54.50 54.63 54-75 54-88 55.01 55-'3 ,-6 s-3^- 5:1 55- •+ 55-7^) 55.89 56.02 56.15 56.27 56.40 56-53 56.66 56.79 56.92 57.04 '57-'7 60.^ 158= loB M. 3-578 -579 .580 .581 .582 3-583 -584 .585 .586 .587 3.588 -589 .589 .590 •59' 3.592 •593 -594 •595 •596 3-597 •598 •599 .600 .601 3.602 .603 .6-- 1 .605 .606 3.607 .608 .609 .610 .6ti 3.612 .613 .614 .615 3.616 .617 .618 .619 .620 3.621 .622 .623 .624 .625 3.626 .627 .628 .629 .630 3^63' .632 •633 •634 -635 Diff. 1". 57-'7 57^30 57^43 57-56 57.69 57.82 57-95 58.08 58.21 58-34 58.47 58.61 58.74 58.87 59.00 59-'3 59.26 59.40 59-5 3 59.66 59-79 59-93 60.06 60.19 60.33 60.46 60.60 60.73 60.87 61.00 61.14 61.27 61.41 61.54 61.68 61.81 61.95 62.09 61.22 62.36 62.50 62.63 62.77 62.91 63.05 6^.18 63-32 63-46 63.60 «-'37-i- 63.88 64.02 64.16 64.30 64-44 64.58 64.72 64.86 65.00 65.14 3.636 6351 165.28 6154 5588 5030 4480 3937 34°3 2876 2357 1846 1342 0847 0359 9880 9408 8944 8488 ; 8040 } 7600 7167 6743 6327 I 59'9 5518 5126 4742 4365 3997 I 3637 i 3285 2941 j 2605 2277 '957 1646 1342 1047 0760 04S1 0210 9948 9693 9447 9209 8980 8758 8545 8340 8.44 795" 7776 7604 744' 7287 7140 7002 6873 6751 6638 6534 6438 159= log M. 3-636 •637 .638 •639 .640 3.641 .642 -643 .644 -645 3-646 .647 .648 .649 .650 3.651 .652 •653 .654 •655 3.656 .657 .658 •6j9 .650 3.661 .662 .663 .664 .665 3.666 .667 .668 .669 .670 3.671 .672 .673 .674 .675 3.676 .678 -679 .680 .68: 3.68 , .68 ; .68, .68; '^26 1.687 .688 .689 .690 .691 1.692 .693 •694 .695 .696 635' 6272 6202 6140 6087 6042 6006 5978 5959 5948 5946 595 3 5968 5992 6025 6066 6116 6175 6242 6318 6403 6497 I 6599 ! 6710 j 6830 6959 1 7096 I 7243 i 7398 I 7562 I 7735 i 79'7 i 8108 I 8308 1 8516 i 8734 I 8961 I 9196 I 9441 ': 9694 j 9957 0228 I 0509 0799 1098 1406 1723 20.) 9 , 2384 , 2728 3082 I 3445 I 38,7 ; 4' 98 4588 j 4988 i 5397 ; 5815 624? 6680 3.697 7126 Diff. 1". 65.28 65.42 65.56 65.71 65.85 65-99 66.13 66.28 66.42 66.56 66.71 66.85 66.99 67.14 67.28 67.42 67-57 67-72 67.86 68.01 68.15 68.30 68.45 68.59 68.74 68.89 69.03 69.18 60,33 6^.48 69.62 69.77 69.92 70.07 70.22 70.37 70.52 70.67 70.82 70.97 71.12 71.27 71.42 7'^57 71.72 71.87 72.03 72.18 72.33 72.48 72.64 72.79 72-94 73.10 73-25 73.40 73-56 73-7' 73-87 74.02 74.18 TABLE VI. For finding the True Anomaly or the Time from tlie Perihelion in a Parabolic Orbit. i V. 0' 160° • 161 162 163° log M. Diff. 1". log M. Di(T. 1". log M. Diir. 1". logM. Diflf. 1". \ 207.00 ; 5.697 7126 174.18 3.762 1539 183.99 5.830 3147 194.87 3.902 6107 1 .698 7581 »74-34 .763 2584 184.16 .831 4845 195.06 •9°3 8534 207.21 ' 1 '2 .699 8046 174.49 .764 3639 184.34 .832 6554 195.25 .905 0973 207.43 3 .700 8520 174.65 .765 4704 184.51 .833 8275 '95^44 .906 3425 207.64 4 .701 9003 174.S0 .766 5780 184.68 .835 0008 195.64 .907 5890 207.86 5 5.702 9496 174.96 3.767 6867 184.86 3.836 1752 195.83 3.908 8368 208.08 G .703 9999 175.12 .768 7963 185.03 .837 3508 19G.02 .910 0859 208.29 7 .70s 05 1 1 175.28 .769 9070 185.20 .838 5275 196.22 .911 3363 208.51 8 .706 1032 '75^43 .771 0187 185.38 •839 7054 196.41 .9.2 5880 208.72 9 .707 1562 I75^59 .772 1315 185-55 .840 8844 196.60 .913 8410 208.94 10 5.708 2102 i75^75 3^773 2454 185-73 [.842 0646 196.80 3.915 0953 209.16 11 .709 2652 175.91 •774 3603 185.90 .843 2460 196.99 .916 3509 209.38 12 .710 321I 176.07 •775 4762 186.08 .844 4286 197.19 .917 6078 209.60 13 .711 3780 176.22 V6 5932 186.25 .845 6123 197.38 .918 8661 2og.8i 11 .712 4358 176.38 • ■■, "712 186.43 .846 7972 197.58 .920 1256 210.03 15 5.713 4946 176.54 3-V/' 186.60 5.847 9833 197.78 3.921 3865 ■iio •».; 16 •7»4 5543 176.70 .779 186.78 •849 '705 '97-97 .922 6487 210.48 17 .715 6150 176.86 .781 c 186.96 .850 3589 .98.17 .923 9122 210.70 18 .716 6766 177.02 .782 1940 .87.14 .851 5486 198.37 .925 1770 210.92 10 .717 7392 177.18 •783 3»74 .87.3, .852 7394 198.57 .926 4432 211.14 20 3.718 8028 I77^34 3.784 4418 187.^9 187.67 5.853 9314 198.76 3.927 7107 211.36 21 .719 8673 177.50 .785 5672 •855 «245 198.96 .928 9795 211.58 22 .720 9328 177.66 .786 6938 '^Z-^5 .856 3189 199.16 .930 2497 211.81 23 •7i« 9993 . 177.83 .787 8214 188.03 -857 5»45 199.36 .931 5212 212.03 24 .723 0668 178.00 .788 9501 J88.21 .858 7»12 199.56 .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 20 .725 2045 178.3. .791 2108 188.57 .861 1084 199.96 •935 3438 .936 6207 212.70 27 .726 2749 178.47 .792 3427 188.75 .862 3087 200.16 21293 28 .727 3462 178.63 •793 4757 188.93 .863 5103 200.36 •937 8989 213.15 29 .728 4185 178.80 .794 6098 189.11 .864 7131 200.56 •939 '785 213.38 30 3.729 4918 178.96 3-795 7450 .796 8812 189.29 3.865 9171 200.77 3.940 459c 213.61 31 .730 5661 179.13 189.47 .867 1223 200.97 •94' 74'8 213.83 32 .731 6413 179.29 .798 0186 189.65 .868 3287 201.17 •943 0254 214.06 33 .732 7176 >79^45 •799 '571 189.83 .869 5363 201.37 944 3'05 214.29 34 •733 7948 179.62 .800 2966 100.01 .870 7452 201.58 •945 5969 214.52 35 5.734 8730 179.78 3.801 4372 l'^O.20 3.871 9552 201.78 3.946 8847 214.74 ■ 36 •735 95ii '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 1136 180.28 .804 8657 190.65 .875 5928 202.39 .950 7563 .952 0496 215.43 39 •739 >957 180.45 .806 0108 190.93 .876 8078 202.60 215.66 40 3.740 2789 180.61 3.807 1569 191.11 3.878 0240 202.80 3^953 3443 216.90 41 .741 3631 180.78 .808 3041 191.30 .879 2iI4 .i<80 4601 203.01 •954 6403 •955 93/8 216.13 42 .742 4482 180.94 .809 4525 191.48 203.22 216.36 • 43 •743 5344 181.11 .810 6020 191.67 .881 6800 203.42 ■957 2366 216.59 44 .744 6216 181.28 .811 7525 191.86 .882 9012 203.63 .958 5369 216.82 45 3.745 7097 181.45 3.S12 9042 192.04 5.884 1236 203.84 3.959 8385 217.06 40 .746 7989 181.61 .814 0570 192.23 •885 3473 204.05 .961 1416 217.29 47 ■747 8891 181.78 .815 2110 I92.AI 192.60 .886 5722 204.26 .962 4460 217.5^ 217.76 48 .748 9803 181.95 .816 3660 •887 7983 204.46 204.67 .963 7519 49 .750 0725 182.12 .817 5222 192.79 .889 0257 .965 0592 218.00 50 3.751 1657 182.29 3.818 6795 192. 9S 3.890 2544 204.88 3.966 3678 21V-3 1 51 •752 ^599 182.46 182.63 .819 8379 193.16 .891 4843 205.09 .967 6779 218.47 52 •753 3552 .820 9974 «9335 .892 7155 205.31 .968 9895 218.70 ; 53 •754 45H 182.80 .822 1581 «93-54 .893 9480 205.52 .970 t024 .971 6168 218.9+ 54 •755 5487 182.97 •823 3«99 193.73 .895 1817 205.73 219.18 55 5.756 6470 183.14 3.824 4829 '93-9* 3.896 4167 205.94 3.972 9326 219.66 56 •757 7464 183.31 .825 6470 194.11 .897 6529 .898 8905 206.15 •974 2498 -975 5684 57 .758 8467 183.48 183.^5 .826 8122 >94-3° 206.36 219.90 58 •759 948' •827 9785 194.49 194.68 .900 1293 206.57 .976 8885 220.13 50 .761 0505 183.82 .829 1460 .901 3694 206.79 .978 2100 220.37 GO 3.762 1539 183.99 3.830 3147 194.87 3.902 6107 207.00 3-979 5330 220.61 606 )olic Orbit. TABLE VI. For fincUng the True Anomaly or tlie Time from the Pcriiielion in a Parabolic Orbit. 163^ M. 5io7 8534 0973 i 5890 ! 8368 0859 33''3 5880 8410 0953 3509 6078 8661 1256 3865 6487 9122 1770 4432 Diff. 1". 207.00 207.21 207.45 207.(14 207.86 208.08 208.29 208.^1 1 208.72 208.94 209.16 209.38 209.60 209.81 210.03 210.48 210.70 210.92 211. 14 7107 9795 2497 5212 794° 0682 3438 6207 7 8989 ) 1785 211.36 211.58 211. 81 212.03 212.25 212.48 4595 7418 0254 3105 5969 8847 .8 1738 9 4644 ;o 7563 ;2 0496 13 3443 14 6403 57 2366 1 58 5369 59 8385 61 1416 62 4460 63 7519 65 0592 66 3678 67 6779 68 9895 70 J024 171 6168 ,71 9326 174 2198 »75 5684 ,76 8885 (78 2100 >79 533° 212.70 21293 213.15 2I3.3S 213.61 213.83 214.06 214.19 j 214-5^ : 114.74 114-97 215.20 215.43 215.66 216.90 216.13 ; 216.36 * 216.59 216.82 217.06 217.29 217-5? 217.76 218.00 218^23 218.47 218.70 218.94 219.18 0- 164'' 165° 166° 167° 1 1 I"(- M. Diff. 1". log M. Diff. 1". log M. Diff. 1". lOB M. Diff. 1". 3-979 533° 220.62 4.061 6673 236.01 4.149 7198 253-57 4.244 5537 273.78 1 .9X0 8574 220.86 .063 0842 236.28 .151 2422 253.88 .246 1975 274.14 •i .982 1833 221.10 .064 5027 236.56 .152 7664 254.19 .247 8434 274.51 I 3 •983 5106 221.34 .065 9229 236.83 .154 2925 254.51 .249 4916 274.87 1 4 .984 8394 221.58 .067 3447 237.11 -'55 8205 254.83 .251 1419 275-24 1 5 3.986 1696 221.83 4.068 76S2 237-39 4- '57 35°4 25S-'4 255.46 4.252 7944 275.60 ! » .987 5013 222.07 .070 '933 237.66 .158 8822 .25A .256 4491 275-97 ' 7 .9S8 8345 222.31 .07? 6201 237.94 .160 4159 255.78 1061 276.34 8 .990 1691 222.56 .073 0486 238.22 .161 9515 256.10 •257 7652 276.71 .991 5051 222.80 .074 4787 238.50 .163 4891 256.42 .259 4266 277.08 10 3.992 8427 223.05 4.075 9106 238.78 4.165 0285 256.74 4.261 0902 277-45 i 11 •994 1817 223.29 .077 3441 239.06 .166 5699 257.06 .262 7560 277.82 ' vz •995 5222 223.54 .078 7792 239-34 .168 1132 257-38 .264 4240 278.20 1 13 .996 86j-, 223.79 .080 2161 239 fl2 .169 6585 257.70 .266 0943 278.57 1 11 •998 2077 224.03 .081 6546 239.^0 .171 2056 258.02 .267 7669 278.95 15 3-999 55^7 224.28 4.083 0948 240.18 4.172 7547 258.35 4.269 44' 7 279-32 10 4.000 8991 224.53 .084 5368 240.46 -'74 3058 258.67 .271 1187 279.70 17 .002 2471 224.78 .085 9804 240.75 -'75 85S8 259.00 .272 7981 280.08 1 18 .003 5965 225.03 .087 4257 241.03 -'77 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 225.53 4.090 3215 241.60 4.180 5296 259-98 4^277 8497 281.22 1 21 .007 6538 225.78 .091 7720 241.89 .182 0905 260.31 260.64 .279 538' 281.60 ! 22 .C09 C093 226.04 •093 2242 242.08 ..83 6534 .281 ?''^i) 281.98 '■ 1 23 .010 3663 226.29 .094 6781 242. i;6 .185 2182 260.97 .282 9219 282.36 : 1 2-1 .011 7248 226.54 .096 1337 242.75 .186 7850 261.30 .284 6173 282.75 : ! 25 4.013 0848 44^3 8093 226.79 4.C97 59" 243.04 4.188 3538 261.63 4.286 3 '49 283.14 ' 20 .014 227.05 .099 0502 24333 .189 9246 261.96 .288 0149 283.52 ■ 27 .015 227.30 .100 5110 243.62 .191 4974 262.30 .289 7172 283.91 28 .017 '739 227.55 .101 9736 2.-.3.91 -'93 0722 262.63 .291 4218 2S4.30 20 .018 S400 227.81 .103 4379 X^4 20 •'94 6490 262.97 •293 1288 284.69 30 4.019 9077 228.06 4.104 9040 244.4. 4.196 2278 263.30 4.294 .296 8381 285.08 31 .021 2769 228.32 .106 37'8 244.78 •'97 8086 263.64 5498 285.47 32 .022 6476 228.58 .107 8414 245.08 .199 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 31 .025 3937 229.09 .110 7858 245.67 .202 5633 264.66 •3^- 6990 286.66 ; 35 4.026 7691 229^35 4.112 2607 245.96 4.204 1523 265.00 4^3='3 4201 287.05 ! 36 .028 1460 229.62 .113 7374 246.26 .201; 743 5 3363 lint .305 1436 287.45 37 .029 5245 229.88 .115 2158 246.55 .207 .306 8695 287.85 38 .030 9045 230.14 .116 6960 246.85 .208 93'4 266.02 .308 5978 288.25 39 .032 2861 230.40 .u8 1780 247.15 .210 5286 266.37 .310 3285 288.65 40 4.033 6693 230.66 4.119 6618 247^45 4.212 1278 266.71 4.312 0616 289.05 41 .035 °540 230.92 .121 1+^t 247^75 .213 7291 267.06 .313 7971 289.45 : 42 .036 f^'^'*- 231.18 .122 6348 248.05 .215 3325 267.40 -3'5 535° 289. X6 43 •037 8283 231.45 .124 1239 248.35 .216 9379 267.75 -3'"' 2753 290.26 1 44 .039 2177 231.71 .125 6149 248.65 .218 5455 268.10 .319 0181 290.67 ; 45 4.040 6088 231.97 4.127 1077 248.95 4.220 '55' 268.44 4.32c 7633 291.07 ' 40 .042 0015 232.24 .128 6023 249.25 .221 7668 268.79 .322 51 10 291.48 47 .043 3957 237,51 .,30 0988 249.56 .223 3806 269.14 .324 261 1 291.89 : 48 •044 .046 7915 1890 232-77 .131 5970 249.86 .226 9965 269.50 .326 °IE 292.30 49 233-04 •'33 0971 250.17 6:46 269.85 -327 7688 292.71 50 4.047 5880 233.31 4-134 .136 5990 250.47 4.228 2347 270.20 4-329 ^l^J 293.13 51 .048 9887 233-57 1028 250.78 .229 8570 270.55 -33' 2863 293-54 52 .050 3909 7948 233.84 •'37 6084 251.08 -23' 4814 270.91 -333 0487 293-95 53 .051 234.11 •'39 1158 251.39 -233 1079 271.27 -334 .336 8137 294-37 54 -053 2003 234-38 .140 6251 251.70 -234 7366 271.62 5812 294.79 55 4.054 6074 234.65 4.142 1362 252.01 4.236 3674 271.98 4^338 3511 295.20 50 .056 01 1 234-92 .143 6492 1641 252.32 .238 0003 272.34 •34° 1236 295.62 57 .057 ^384 ^35.19 •'45 252.63 239 6354 272.70 •34' ^986 296.04 : 58 .058 235.46 ..46 6808 252.94 .241 2727 273.06 •343 6762 296.47 296.89 59 .060 2520 235-73 .148 '994 253-25 .242 9121 273.42 •345 4562 \ 00 • 4.061 6673 236.01 4.149 7198 253-57 4.244 5537 273.78 4-347 2388 297.31 ' » 607 TABLE VI. For finding tlie True Anomiily or the Time from the Perihelion in a Parabolic Orldt. V. i 0' 16J log M. 4.347 2388 Diff. 1". , 169° 170° 171° log M. 4.459 1242 Diff. 1". 325.07 log M. Diff. I". log M. DUT. 1". , 398.87 297.31 4.581 9445 358^3' 4-7'7 9835 1 .349 0240 ^97^74 298.16 .461 0761 32^57 326.08 .584 0962 358.92 .720 3790 399.62 3 •35° i*'"? .463 031 1 .586 2516 359-53 .722 7790 400.38 3 .352 6019 298.59 .46^ 9891 .466 9501 326.59 .588 4106 360.15 .725 1835 401.14 4 •354 3948 299.02 327.10 •590 5734 360.76 •7*7 59*6 401.90 5 4.356 1902 *99-45 4.468 9142 327.61 4.592 7398 361.38 4-730 0063 402.66 •357 9««2 299.88 .470 8814 328.12 •594 9'oo 362.00 .732 4*45 403.43 7 •359 7888 300.31 .472 8517 328.64 •597 0838 362.62 -734 8474 404.19 8 .361 5919 300.75 •474 8250 329.15 •599 *6i5 .601 4428 5^^o5 -737 2749 404.96 •363 3977 301.18 .476 8015 329.67 363.88 -739 7070 405.74 • 10 4.365 2061 301.62 4.478 7811 330.19 4.603 6280 364.50 4.742 1438 406.52 11 .367 0171 302.05 .480 763; 330.71 .605 8169 365-'4 -744 5852 407.30 13 .368 8308 302.49 .482 7495 331.23 .608 0096 365^77 •747 0314 408.08 13 .370 6470 302.93 .484 7385 33'-75 .610 2061 366.40 •749 4822 408.87 14 .372 4659 303.37 .486 7306 332.28 .612 4064 367.04 •75' 9378 409.66 15 4-37'4 2875 .376 1117 303.81 4.488 7258 332.81 4.614 6106 367.68 4^754 3981 410.45 16 304.26 .490 7242 333^33 .616 8186 368.32 .756 X632 411.24 17 •377 93*^6 3O4^70 .492 7258 333.86 .619 0304 368.96 •759 3330 412.04 18 •379 7681 305.15 -494 7306 334.40 .621 2461 •623 4657 369.61 .761 8077 412.84 10 .381 6003 305-59 .496 7386 334-93 370.26 •764 2872 413.65 20 4-3»3 435^ 306.04 4.498 7498 335-46 4.625 6892 370.91 4.766 7715 414.46 21 .385 2728 306.49 .500 7642 336.00 .627 9166 37'-56 .769 2606 415.27 ' 22 .387 1131 306.94 .502 7818 336-54 .630 1480 372.21 •771 7547 416.08 23 .388 9561 307-39 .504 8026 33708 .632 3832 372.87 •774 *536 416.90 34 .390 8019 307^85 .506 8267 337.62 .634 6224 373^53 .776 7574 417.72 35 4.392 '650? 308.30 4.508 8541 338.16 4.636 8656 374-19 4-779 2662 418.54 ! 26 •394 5015 308.76 .510 8847 338.71 .639 1127 374.86 •7^ 7799 4'9-37 37 •39<> 3554 309.21 .512 9186 339.26 .641 3639 37552 .784 2986 4.70.20 28 .398 2121 309.67 .514 9558 339.80 .643 6190 376.19 .786 8222 421.03 20 .400 0715 310.13 .516 9962 340.35 .645 8781 376.86 .789 3509 421.86 30 4.401 9337 310.59 4.519 0400 340.91 4.648 1413 377^53 4-791 8846 422.70 31 .403 7986 311.06 .521 0871 341.46 .650 4085 378.21 -794 4*33 4* 3- 54 32 .405 6663 311.52 .523 376 342.02 .652 6798 378.89 .796 9671 424,39 33 .407 5368 311.99 .525 1913 342^57 •654 9552 379^57 -799 5160 42.^24 34 .409 4102 312.45 .527 2484 343- '3 .657 2346 380.25 .802 0700 4^.6.09 1 35 4. 411 2863 312.92 4.529 3089 34369 4.659 5182 380.93 4.804 6291 426.95 1 36 .413 1652 3«339 •531 37*8 344.26 .661 8059 381.62 .807 '934 427-81 , 37 .415 0469 313.86 •5 33 4400 344.82 .66a 0977 .666 3936 382.31 .809 762S 42S.67 , 38 .416 9315 3>4^33 •535 5«o6 345-39 383.00 .812 3374 4*9-53 39 .418 8189 314.80 •5 37 5846 345-95 .668 6937 383.70 .814 9172 430.40 , 40 4.420 7091 315^28 4.539 6620 346.52 4.670 9980 384.39 4.817 5022 431.28 41 .422 6022 3'5-7S •54' 7429 347-09 .673 3064 ^^'•S^ .820 ?2P 45*-'5 42 .424 4982 316.23 •543 '^^7^ 347-67 .675 6191 385.80 .822 6881 433-°3 43 .426 3970 316.71 •545 9149 348.24 .677 9360 386.50 .825 2889 433-91 44 .428 2987 317.19 .548 0061 348.82 .680 2571 387.21 .827 8950 434.80 45 4.430 2037 317.67 4.550 1007 349.40 4.682 5825 387.92 4.830 5065 435-69 40 .432 1108 318.16 .552 1989 349.98 .684 9121 388.63 .833 1234 436.59 47 .434 0212 31S.64 •554 3005 350-56 .687 2460 389-34 .835 7456 ^n-^l 48 •435 9345 3«9-»3 .556 4056 3S'-'5 .689 5842 390.06 .838 373* 438.38 40 j •437 8507 319.61 •558 5'43 3S'-73 .691 9268 390.78 .841 0062 439.29 1 50 4.439 7698 320.10 4.560 6264 351-3* 4-694 2736 .696 6248 391.50 4.843 .846 6446 440.20 51 .441 6919 3^0-59 .562 7i2I .564 8614 352.91 39»^23 2886 441. II 52 .443 6169 321.08 35350 .698 9803 392.96 .848 9380 442.03 53 •445 5449 321.58 .566 9842 354.10 .701 3402 393.68 .85, 59*9 44*-95 f 54 •447 4758 322.07 .569 1 106 354-69 .703 7046 394.42 .854 *533 443-!*7 1 55 4.449 4097 3"-S7 4.571 2405 355-*9 4.706 0733 395^'5 4.856 9193 444-8° 56 •451 3466 323.06 •573 3741 355-89 .708 4464 395.89 .859 5909 ^*)il 57 •453 2865 3*3-56 •575 S"3 356-49 .710 8240 .713 2060 396.63 397-38 .862 2680 446.66 58 •455 "94 324.06 •577 6521 357-10 •!^+ 9508 447.60 50 •457 '753 324.56 •579 7965 357.70 •7'S S9»S 398.12 .867 6392 ■f48-54 60 4.459 1242 325.07 4.581 9445 358.31 4.717 9835 398.87 4.870 3333 449.49 60S bolk- Orl.it. 171° M. 1 1 DUT. 1". 9835 379° 7790 1835 5926 398.87 399.62 400.38 401.14 401.90 0063 4245 X474 2749 7070 1438 5852 0314 4822 9378 3981 8632 3330 8077 . 2872 7715 I 2606 7547 • ^536 1 7574 ( 266a ; 7799 |. 29S6 ) 8222 ) 3509 t 8846 ^ 4^33 t 5 9671 ' ) 5160 I 2 0700 I 4 6291 7 1934 ! 9 762S I 2 3374 i 4 9«7a I 7 502* ! o 0925 2 6881 i 5 2889 7 895° ,0 5065 ;3 1*34 15 7456 ;8 373* [.I 0062 ^3 6446 ^6 2886 1.8 9380 ;i 5929 54 2533 56 9193 59 5909 62 2680 64 9508 67 6392 70 3333 402.66 403.43 404.19 404.96 4°5-74 406.52 407.30 408.08 408.87 409.66 410.45 411.24 412.04 412.84 413.65 414.46 415.27 416.08 416.90 417.72 418.54 1 4'9-37 ] 4.7 0.20 1 421.03 421.86 422.70 4*3-54 4*439 4*5-*4 4-6.09 426.95 427.81 428.67 429.53 430.40 431.28 452.15 433-°3 i 433-9I i 434'^° i 435.69 1 436-59 437-48 438.38 439.29 440.10 441.11 442.03 44*-95 443-87 444-80 446.66 447.60 .H8-54 449-49 TABLE VI. For finding tlit- True Anomaly or llie Time from the Perihelion in a Parabolic Orbit. tf. O' 1 u 3 4 5 6 7 8 9 10 11 12 13 14 13 IG 17 18 19 20 21 22 23 24 25 20 27 28 29 30 31 32 33 34 35 30 37 38 39 40 41 42 43 44 45 48 47 48 49 50 51 52 53 54 55 3G 37 58 59 172^ log M. DilT. 1". 4-870 3333 .873 0331 .875 7386 .878 4499 .881 1668 4.883 8896 .886 6182 .889 3526 .892 0929 •894 8391 4.897 5912 .900 349* .903 1132 .905 8831 .90S 6591 4.911 44" .914 2291 .917 0233 .919 8235 .922 6290 4-9*5 .928 .931 •933 .936 4-939 94* 945 948 951 4-953 .956 •959 .962 .965 4.968 971 974 977 980 4.983 985 991 994 4-997 5.000 .003 .006 .009 5.012 .015 .018 .021 .024 5.027 .031 .034 .037 .040 5.043 44*5 2612 0862 9'74 7549 5987 4489 3053 1682 ■^375 9132 7954 6841 5793 481 1 3S94 3°44 2260 1543 0893 9795 H8 970 8659 8418 8246 8143 Siii 8148 8256 8435 8685 9006 9399 98O4 0.J.02 1013 1697 *454 3285 449.49 450.44 45'-39 45*-35 453^3i 454.28 455*5 456.23 457.20 458.19 459-«7 460.16 461.16 462.16 463.16 464.17 465.18 466.20 467.22 468.25 469.28 47o^3' 471-35 47*-39 473-44 474-49 475-55 476.61 477.68 478.75 479.83 480.91 481.99 483.08 484.18 485.28 486.38 487.49 488.61 489.73 490.85 491.98 493.12 494,26 495.40 496.55 497-71 498.87 500.04 501.21 502.39 503-57 504.76 5°5-95 507.15 508.36 509-57 510.79 512 01 513.24 514-47 173^ los M. Dlff. 1". 043 046 049 052 055 058 061 065 068 071 074 077 080 084 087 090 093 096 100 103 106 109 113 116 119 207 211 214 218 22 1 225 228 232 236 239 *43 3285 4191 5171 6226 7356 8562 9843 1202 2637 I 4149 5738 7406 91 5 1 0976 2879 4862 6924 9067 i 1290 I 3594 i 5980 I 8447 I 0997 I 3629 I 6344 ! 122 9143 126 2026 129 4992 132 8044 136 1181 139 4403 142 7711 146 1106 149 45S8 152 8157 156 1813 159 5558 162 9392 166 3315 169 7328 173 1431 176 5624 179 990S 183 4284 186 8752 190 3312 193 7966 197 2713 200 7554 204 2489 7520 2646 7868 3186 8602 4116 97*7 5437 1247 7156 3165 3» 5 '4-47 5'5-7' 516.96 518.21 5 » 9-47 520.73 522.00 523.28 524.56 525.85 527.14 528.44 5*9-75 531.06 532.38 533-71 535-°4 536.38 537-73 539.08 540.44 541.81 543.18 544-56 545-95 547-34 548.74 550.15 551-57 552.99 554.42 555-86 557-30 558.75 560.21 561.68 563.16 564.64 566.13 567.63 569.13 570.65 572.17 573-70 575-*4 576-78 578-34 579-90 581.47 583-05 584-64 587.84 589-45 591.07 59*-7J 59435 596.00 597-66 599-3* 601.00 6U9 174' lot? M. *43 246 250 *54 257 261 265 268 272 276 279 283 287 291 294 298 302 306 309 313 317 321 3*5 3*9 33* 336 340 344 348 35* 356 360 364 368 37* 376 380 384 388 39* 396 400 404 408 412 416 420 4*5 4*9 433 437 441 445 450 454 458 462 467 47 > 475 3165 9276 5488 1802 8218 4738 1 361 S089 4922 i860 8904 6055 33'3 0680 8154 5738 343* 1*57 9152 7179 5319 3571 1938 04 1 S 9014 7726 6554 5499 4562 3744 3°45 2466 2007 1 67 1 1456 1364 1396 1553 1834 2242 2777 3439 4229 5«49 6199 7379 8692 01 56 1714 34*7 5*74 7258 9378 1636 4032 6568 9*44 2062 5022 8125 480 1373 Diir. 1". 601.00 602. 6(; 604.38 606.08 607.80 609.53 611.26 613.00 64-75 616.52 618.29 620.08 621.87 623.67 6*5-49 627.31 629.15 631.00 632.85 634-72 636.60 638.49 640.39 642.30 644.23 646.16 64S.1: 650.07 652.04 654.02 656,01 658.02 660.04 662.07 664.1 1 666.17 668.24 670.32 672.41 674.52 676.64 678.77 680 92 683.08 685.25 687.44 689.64 691.85 694.08 696.33 698.59 700.86 703.15 705-45 707.77 710.10 712.45 714.81 717.1; 719-59 722.00 175 loK M. 5-480 1373 484 4765 488 8304 493 1989 497 5823 5.501 9806 .506 3939 .5 10 8223 .515 •'.659 .519 7248 DifT. 1". 5-5*4 5*8 533 537 54* 5.546 -55» -555 .560 •565 5.569 -574 -579 .583 .58§ 5-593 •597 .602 .607 .612 5.617 .621 .626 .631 .636 5.641 .646 .651 .656 .661 5.666 .671 .676 .681 .686 5.691 .696 .701 .706 .711 1992 6890 1 946 7158 2529 8060 375' 9605 5621 1S02 8148 4661 1341 8190 5210 2401 9764 7302 5014 2903 0970 9216 7642 625c 5041 4017 3179 2528 2065 1793 1713 1825 21 32 2635 3336 4236 5337 6640 8147 9860 5.-^17 1779 .722 390S .727 6247 -73* 8798 •738 1563 5-743 4544 .748 7742 -754 «'59 -759 4798 -764 8659 5.770 2745 722.00 724.42 726.87 7*9.33 73 1.80 734-3° 736.81 739-33 741.87 744-44 747.02 749.61 752.23 754.86 757-5' 760.18 762.87 765.58 768.31 771.05 773-82 776.61 779-41 782.24 785.08 787-95 790.84 793-75 796.68 799-63 802.60 805.60 808.62 811.66 814.72 817.81 820.92 824.05 827.21 830.39 833.60 836.83 840.08 843.36 846.67 850.00 853-36 856.75 860.16 863.60 867.06 870.56 874.08 877.63 881.21 884.82 888.46 892.13 895.83 899.56 903.31 TABLE VI. For finding the True Anomnly or tlie Tiiiu' from the IVrilielion in a Pariibolic Orbit. 1 " V. 176° ' 177 178° 179° lo(i M. Dim 1". logM. 6.144 62ji9 Diff. 1". 1205.3 logM. DilT. 1". IokM. Diff. 1". 0' ?-770 4745 903.3 6.672 5724 1808.8 7.575 4640 3619 1 •775 705X 907,1 .151 8807 1212.0 .68 3 4709 1824.0 •597 3596 3680 2 .7S1 "599 910.9 •'59 '733 1218.8 .694 4613 1839.5 .619 6295 3744 i a .7S6 6370 914.8 .166 5070 1225.7 -705 5454 1855.3 .642 2868 3809 ' 4 ] .791 "374 918.7 .173 8823 1232.7 .716 7248 1871.3 .665 3452 3877 « 5-797 661Z 922.6 6.181 2997 1239.8 6.728 0010 1887.5 7.688 8192 3948 1 ^. .803 2086 926.6 -'88 7597 1246.9 •739 3758 1904.1 .712 7239 4021 ! 7 .8oii 7798 930.6 .196 2628 I 254. 1 .750 8509 1921.0 •737 0756 4097 8 .814 37S« 934-6 .203 8095 1261.4 .762 4279 1938.2 .761 8913 4176 » .819 9946 938.6 .211 4002 1268.8 -774 '090 '955-6 .787 1889 4257 1 10 5.825 6386 942.7 6.219 0354 1276.3 6.785 8958 '973-4 7.812 9876 4343 11 .83. 3°73 946.8 .226 7158 1283.8 •797 7904 1991.5 .839 3075 443' 12 .837 0008 951.0 .234 4419 1291.5 •809 7946 2010.0 .866 1702 4524 1 Y'i .842 7'95 955-2 .242 2142 1299.2 .821 9106 2,028. g .893 5986 4620 i 14 .848 4634 959-5 .250 0333 1307.1 •834 '404 2048.0 .921 6170 4720 15 5.854 2329 963.7 968.0 6.257 8997 1315.0 6.846 4863 2067.5 7.950 2513 4825 10 .860 0282 .265 8139 1323.0 .85S 9503 20S7.3 7.979 5292 4935 17 .865 8495 972-4 .273 7766 '33'-' .871 5348 2107.6 8.009 4802 5050 18 .871 6970 976.8 .281 7884 '339-4 .884 2422 2128.3 .040 1361 5170 10 .877 5710 981.2 .289 8499 '347-7 .897 0749 2149.4 .071 5309 5296 20 5.883 47J7 985.7 6.297 9617 1356.2 6.910 0353 2170.9 8.103 701 1 5428 21 .889 3993 990.2 .306 1244 1364.7 .913 1 26 1 2192.8 .136 6857 5568 22 .895 3541 994.8 .314 3387 '373^3 ,936 3498 2215.2 .170 5274 5714 23 .901 3365 999.4 .322 6052 1382.1 •949 7093 2238.0 .205 2717 5869 24 1 .907 3465 1004.0 •33= 9247 1391.0 .963 2073 2261.4 .240 9679 ^032 1 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 3298 6580 j 28 •93' 6688 1022.9 .364 7451 1427.11 .018 6437 2359-7 •394 4205 6786 1 20 -937 8213 1027.8 •373 3395 '437-' .032 8796 2385.7 •435 7842 7004 S 30 5-9+4 0030 1032.7 6.381 9910 1446.7 7.047 2729 .061 8271 2412.2 8.478 5044 7238 31 .950 2144 4550 1037.6 .390 7005 1456.4 2439.4 .522 6731 7488 ' 1 32 -95'' 1042.6 .399 46S7 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 i 34 .969 0287 1052.9 .417 1846 1486.4 .106 4921 2524.5 .664 9442 8352 . 35 S-975 3613 1058.0 6.426 1337 1496.7 7.121 7276 2554.2 8.716 0431 .769 2286 8686 , 36 .981 7249 1063.2 •435 '449 1507.0 •137 '434 2584.6 9048 37 .9S8 1198 1068.4 •444 219' 1517.6 .152 7440 2615.8 .824 6779 944' 38 5-994 5464 1073.7 •453 3569 1528.3 .168 5336 2647.6 .882 5925 9870 ; 1 39 6.001 0050 1079.1 •462 5594 1539.2 .184 5171 2680.4 .943 2018 10340 i j 40 6.007 4958 1084.5 6.471 8275 1550.2 7.200 6993 2711.9 2748.3 9.006 7690 10857 ; 41 .014 0192 1089.9 .481 1620 1561.3 .217 0850 •073 5974 11429 42 .020 5756 1095.4 .490 5641 1572.6 .233 6796 2783.5 .144 0401 12064 43 .027 1652 1101.0 .500 0346 158J ; .250 4884 2819.7 .218 5102 12773 44 .033 7885 1106.7 .509 5746 '595^8 .267 5170 2856.8 •297 4963 '357- 45 6.040 4457 1112.4 6.519 1850 1607.7 7.284 7712 2894.8 9.381 5820 14476 40 .047 1372 1118.1 .528 8669 1619.6 .302 2571 2934.1 •47' 47" 15510 47 .053 8634 1123.9 .538 6216 1631.8 .319 9810 2974.2 .568 0247 16704 48 .060 6246 1 129.8 .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 '974' 50 6.074 2535 1141.7 6.568 3320 1669.6 7^374 6475 3101.7 9.909 8535 21715 51 .081 1 21 9 1147.7 .578 3881 1682.A 1695.6 •393 39'8 3146.8 10.047 1256 24127 ! 52 .088 0269 1153-8 .588 5227 .412 4099 3'93-o .200 5829 27144 53 .094 9687 1160.0 .598 7368 1708.9 •43' 7097 3240.7 3289.9 •:74 5584 31023 54 .101 9479 1166.3 .609 0317 1722.6 •45' 2999 •575 3986 36197 55 6.108 9647 1172.6 6.619 4086 1736.4 7.471 1892 3340-3 3392.6 10.812 9421 4345° 50 .1:6 0196 1179.0 .629 8689 '75°^3 •49' 3870 11.103 6719 57 .123 1131 1185.4 .640 4141 1764.5 .511 9029 3446.5 11.478 48S0 58 .130 *4S5 1192.0 .651 0455 I779-0 .532 7472 3502.1 12.006 7617 50 .137 4173 1198.6 .661 7645 1793.8 •553 9305 3559^6 12.909 8516 00 6.144 6289 1205.3 6.672 5724 1808.8 7.575 4640 3618.7 . :j m TABLE VII. For findlnfj tlic True Anonuily in si I'araliolii' Orbit when r is nearly 180°. w .^0 IMIT. to Ao Dim 10 t^ _ —1 Diff. 1 o / f H H / / ti II t 1 II II 153 5 10 15 20 25 3 2 3-°9 19.74 16.43 13.17 9-95 b.T! 3-35 3-3' 3-26 3.22 3.18 3.14 100 5 10 15 20 25 I 6.70 5-33 3-97 2.64 '•33 0.04 1.36 '-33 1.31 1.29 1.26 165 10 20 30 40 50 15.85 14.98 14.16 13.38 12.63 11.91 0.87 0.82 0.78 0.75 1 0.72 0.69 153 30 3 3-63 160 30 58.78 166 11.22 0.65 1 0.62 35 40 0-54 » 57-49 3.09 3.C5 35 40 57-54 56.31 1.24 1.23 10 20 10.57 9-95 45 50 55 54.48 51.51 48.58 3.01 2.97 2.93 2.89 45 50 55 55.11 53-93 52.77 1.20 1.18 1.16 1. 14 30 40 50 9.36 8.80 8.26 0.59 0.56 0.54 0.51 156 5 10 15 2 45.69 42.84 40.03 37.26 2.85 2.81 2.77 161 5 10 15 51.63 SO- 50 49.40 48.32 1. 13 l.IO 1.08 1.06 167 10 20 30 7-75 7.27 6.81 6.37 0.48 0.46 0.44 20 34'53 2.73 20 47.26 40 5.96 0.41 25 31-83 2.70 2.66 25 46.21 1.05 I.02 50 5-57 0.39 0.37 15G 30 2 29.17 2.62 2.58 2.54 2.48 2.44 161 30 45.19 168 5.20 0.36 0.33 0.31 0.30 0.28 0.26 1 35 40 45 50 55 26.55 23.97 21.43 18.92 16.44 35 40 45 50 55 44.18 43.19 42.22 41.26 40.33 1.01 0-99 0.97 0.96 0.93 0.92 10 20 30 40 50 4.84 4.51 4.20 3-90 3.62 157 5 2 14.00 II.59 2.41 162 5 39.41 38.51 37.62 0.90 169 10 3.36 3.11 0.2s 10 9.22 2.37 10 0.89 20 2.88 0.23 15 20 25 6.89 4-58 2.31 2-33 2.31 2.27 2.23 15 20 25 36.75 35-90 35.06 0.87 0.85 0.84 0.82 30 40 50 2.66 2.46 2.27 0.22 0.20 0.19 0.18 157 30 35 a 0.08 I 57.89 2.19 162 30 35 34.24 33-43 0.81 170 10 2.09 1.92 1 0.17 1 0.16 i 40 55-72 2.17 40 . 32.64 0.79 0.78 0.76 0-75 0.73 20 1.76 45 53-57 2.15 45 31.86 30 1.62 0.14 , 50 55 51.46 49-39 2. II 2.07 2.04 50 55 31.10 30-35 40 50 1.48 1-35 0,14 0.13 0.12 158 5 I 47-35 45.34 2.01 163 5 29.62 28.90 0.72 171 10 1.23 1.12 0..1 10 15 43-35 4«-39 1.99 1.96 10 15 28.20 27.51 0.70 0.69 0.68 0.67 0.65 20 30 I.OZ 0.93 O.IC I 0.09 1 20 25 39-47 37-57 1.92 1.90 1.87 20 25 26.83 26.16 40 50 0.84 0.76 0.09 0.08 0.08 ! 138 30 " 35-70 1.83 i.8i 1.78 1.76 163 30 25.51 0.63 0.63 0.61 0.60 172 0.68 35 40 45 50 I3-87 32.06 30.28 28.52 35 40 45 50 24.88 24.25 23.64 23.04 10 20 30 40 0.61 0.55 0.49 0.44 0.07 0.06 0.06 0.05 55 26.80 1.72 55 22.45 0.59 50 0.39 0.05 1.70 0.57 0.04 139 I 25.10 164 21.88 173 0.35 5 23.4- 1.67 1.65 1.62 6 21.31 0.57 10 0.31 0.04 10 21.78 10 20.76 0.55 20 0.27 0.04 15. 20.16 15 20.22 0.54 30 0.24 0.03 20 18.57 1-59 20 19.69 19.18 0.53 40 0.21 0.03 25 17.00 1-57 25 0.51 50 0.19 0.02 '•55 0.51 0.03 1 159 30 1 15.45 164 30 18.67 174 0.16 1 35 '3-94 1.51 35 18.17 0.50 0.48 175 0.07 0.09 1 40 12.44 1.50 40 17.69 17.21 176 0.02 0.05 46 10.97 '•47 45 0.48 177 0.01 O.OI 50 9-53 1.44 50 16.75 0.46 0.46 178 0.00 O.OI 55 8.10 1.43 55 16.29 179 0.00 0.00 1.40 0.44 0.00 160 I 6.70 165 15.85 180 0.00 1 1 611 TABLE VIII. For findiri}; thi; Tinu' from the Pcriliclion in a Parabolic Or])it. log A' o t 30 1 30 2 30 3 ;!0 4 30 3 30 30 j 7 I 30! 8 (I ' 30 i 9 30 10 { 30 11 30 12 30 I 13 30 14 30 ! 15 30 16 30 17 30 18 30 19 30 20 30 21 30 22 30 23 30 24 30 25 30 26 30 27 30 28 30 29 30 5763 i749 5707 5638 5541 54' X 5266 5087 4881 4647 4386 4097 378" 3437 3066 2668 2243 1791 1311 0805 0271 97 u 9124 8510 7869 7201 6507 5786 5039 4266 3466 2641 1789 091 1 0008 9079 8125 7>4S 6140 S109 4054 2973 1868 0738 95S4 8405 7202 5975 0.022 4724 .022 3449 .022 2151 .022 0829 .021 9484 .021 8116 0.021 6726 .021 5312 .021 3876 .021 2418 .021 0938 .020 9436 0.025 .025 .025 .025 .025 .025 0.025 .025 .025 .025 .025 .025 0,025 .025 .025 .025 .025 .025 0.025 .025 .025 .024 .024 .024 0.024 .024 .024 .024 .024 .024 0.024 .024 .024 .024 .024 .023 0.023 .023 .023 .023 .023 .023 0.023 .023 .022 .022 .022 .022 11 30 I O.02O 7913 Diir. V / 30 14 30 4^ 31 69 30 96 32 124 30 152 33 179 ,30 206 34 261 30 35 289 30 316 30 344 30 ! 5^i 37 i 398 30 425 38 452 30 480 506 39 30 1 534 40 ! 560 30 587 41 614 30! 641 1 668 694 42 i 30 43 721 30 747 44 773 30 800 45 825 30 852 46 878 30 f 903 47 1 929 30 954 48 { 9S0 30 IC05 49 1031 30 1055 50 1081 30 1 105 51 1 1 30 30 1 1 1 54 52 ; 1179 .sol ! 1203 53 1 1227 30 1 1251 54 1 1275 30 ! 1298 55 1322 30 \ 5345 56 1368 30 1390 57 1414 30 , 1436 58 145S 30 1480 39 1502 30 1 '5^3 60 K'g -V 0.020 .020 .020 .020 .020 .019 0.019 .019 .019 .019 .019 .018 0.018 .018 .018 .018 .018 .017 7913 6368 4802 3^'; 1607 9979 8330 6662 4974 3267 1540 9795 8030 6248 4448 2629 0794 8941 0.017 707^ .017 5186 .017 3283 .017 1365 .016 9432 .016 7483 0.016 5520 .016 3542 .016 1550 •°i5 9545 .015 7526 •o>5 5495 0.015 3450 .015 1394 .014 9326 .014 7247 .014 5157 .014 3057 0.014 0947 .013 8827 .013 6698 .C13 4561 .013 2416 .013 0263 0.012 8103 .012 5936 .012 3764 .012 1585 .011 9402 .011 7215 0.011 5024 .011 2829 .011 0632 .010 8432 .010 6231 .010 4029 0.010 1827 .009 9625 .009 7424 .0C9 5225 .009 3028 .009 0834 0.008 8 644 Diir. 545 566 587 608 628 649 668 688 737 727 745 765 782 800 819 869 886 903 918 933 949 963 978 992 2005 2019 2031 2045 2056 2068 2079 2090 2100 2110 2120 2129 2137 2145 = 153 2160 2167 2172 2179 2183 21S7 2191 2195 2197 2200 2201 2202 2202 2202 2201 2199 2197 2194 2190 o / 60 30 01 30 62 30 63 30 04 30 65 30 66 30 67 30 68 30 69 30 70 30 71 30 72 30 73 30 74 30 75 30 76 30 77 30 78 30 79 30 80 30 81 30 82 30 83 30 84 30 83 30 86 30 87 30 88 30 89 30 90 loK y 864A 645J 4*77 2103 9934 7774 5621 3477 '343 9220 7108 5008 2922 0849 8792 6750 47^5 2717 0.005 07*9 .004 8760 .004 681 1 .004 4884 .004 2980 ■004 1 1 00 0.003 .003 .003 .003 .003 .003 o.ooS .008 .008 .008 .007 .007 0.007 .007 .007 .006 .006 .006 0.006 .006 .005 .005 .005 .005 0.002 .002 .002 .002 .002 .002 O.OOl .001 .001 .001 .001 .001 0.00 1 .001 .000 .000 .000 .000 0.000 .000 .000 .000 .000 .000 0.000 .000 .000 .000 .000 .000 9245 7416 5613 3839 2094 0380 8698 7049 5433 3854 2311 0806 9341 79'7 6535 5196 3903 2656 1458 0309 9211 8166 7«75 6240 5364 4546 379° 3096 2468 1906 1413 0990 0639 0363 0163 0041 Diff. 1186 2181 2174 2169 2lfio *I53 2J44 a' 34 2123 2112 2100 2086 2073 2057 2042 2025 200X 1988 1969 1949 1927 190^ 188c 1855 1829 1803 '774 '745 1714 1682 1649 1616 '579 '543 1505 1465 142.4 1382 1339 1293 1247 119S "49 1098 1045 991 935 876 818 756 694 628 562 493 423 35' 276 200 ; 122 4' : 0.000 0000 1 •' 1 100 i 3 101 3 102 1 3 1 103 i 3 104 3 105 3 i 106 107 108 109 3 110 3 111 112 3 113 3 114 31 115 116 3 117 3 118 3( 119 ( 3( 120 G12 8644 645S 4*77 2I05 9934 7 .5^^' 7 3477 7 1343 i6 9220 16 7108 16 5008 Dlff. 2186 2181 2174 2169 2160 *>53 2J44 2134 2123 2112 2100 2086 2073 2057 2042 2025 200S 1988 1949 1927 1904 1 8 8c 1855 1829 1803 1774 1745 1714 1682 1649 I6I6 1579 1543 1505 1465 142.4 1382 '339 1293 1247 119S 1149 1098 J 045 99' 935 876 818 756 694 628 562 493 423 35' 276 200 122 41 «0 30 01 30 02 30 03 30 04 30 05 30 00 30 or 30 08 30 00 30 100 30 101 30 102 30 103 30 104 30 105 30 106 30 107 30 108 30 100 30 110 30 111 30 112 30 113 30 114 30 115 30 IIG 30 117 30 118 30 110 30 TABLE VIII. For tintlint; the Time from tin- l'i;rilirli(»ii in :i I'.inibolio Orbit. lug ^V 0.000 0000 9.999 9876 999 9507 •999 ****93 •999 *'039 •999 6944 9.999 •999 •999 •999 .998 .998 9.998 •997 •997 •997 .996 .996 9.996 •995 •995 •994 •994 •993 9-993 .992 .992 .991 .991 .990 5613 4046 2246 0215 7955 5468 2757 9824 6669 3297 9708 5906 1891 7666 S596 3755 8712 3470 8031 2397 6570 0553 4347 9.989 7956 .989 1380 .988 4622 .987 7685 .987 0571 .986 3281 9.985 5819 .984 8186 .984 0385 .983 2418 .982 4288 .981 5996 9.980 7545 •979 «93i* •979 °'77 .978 1264 .977 2202 .976 2993 9.975 3640 •974 4'45 •973 45'° •97a 4739 •97' 4833 .970 4796 9.969 4629 .968 4337 .967 3920 .966 3382 .965 2726 .964 1954 niiT. 120 1 9.963 1069 124 369 614 854 1095 1331 1567 1800 2031 2260 2487 2711 *933 3155 3372 3589 3802 4015 4225 4638 4841 S043 5242 5439 5634 5827 6017 6206 6391 6576 6758 6937 7114 7290 7462 7633 7801 7967 8130 8292 8451 8607 8761 89'3 9062 9209 9353 9495 9635 977' 9906 10037 10167 10292 10417 10538 10656 10772 10885 l* i t 120 ;!() 121 ! :io 1 122 ! 30 i 123 ;!0 124 ."0 125 30 120 30 1 127 30 128 30 120 30 ! 130 '\ 30 ! 131 ! 30 ; 132 30 133 Ol 30 i 134 i 30 135 { 30 ; 130 1 30 137 30 ' 138 i 30 , 130 30 140 1 30 141 30 : 142 1 30 [ 143 30 1 144 30 i 145 i 30 j 140 1 30 147 \ 30 148 30 ' 140 30 1 150 1 log -V 9.963 1069 .962 0074 .960 •959 .958 •957 8971 7764 6454 5046 9.956 3542 •95 5 «945 •954 0^58 .952 8483 .951 6624 .950 4684 9.949 2666 .948 0573 .946 840S •945 6174 •944 3875 •943 1 5' 3 9.941 .940 •939 .938 .936 •935 9-934 ■933 •93' •93° .929 .927 9.926 .925 .924 .922 .921 .920 9.918 .917 .916 .915 .913 .912 99" .909 .908 .907 .906 .904 9.903 .902 .901 .899 .898 .897 9.896 .894 .893 .892 .891 .890 9092 6615 4085 1506 8881 6213 3506 0763 7987 5183 1353 9501 6630 3745 0848 7943 5°35 2126 9220 6321 343 3 0559 7703 4870 2062 9283 6538 383' 1 164 854^ 5969 3449 0985 8582 6243 3972 '774 9652 7610 5652 3782 2004 9.889 0321 j 613 Din. 0995 1 103 1207 I 310 1408 1504 1597 1687 '775 1859 1940 2018 2093 2165 2234 2299 2362 2421 2477 2530 1579 2625 2668 2707 1743 2776 2804 2830 2852 2871 2885 2897 2905 2908 2909 2906 2S99 2888 2874 2856 2833 2808 2779 2745 2707 2667 2622 1573 2520 2464 2403 ^3 39 2271 2198 2122 2042 1958 1870 1778 1683 loK .V o I 50 ' 30 51 30 52 liO 53 30 54 30 55 ' 30 I 50 30 i 57 ' 30 I 58 30 50 30 00 30 61 30 02 30 03 30 04 30 05 30 00 30 67 30 08 30 00 I 30 ! 70 30 71 30 72 30 73 30 74 30 75 30 70 77 30; ' 30 78 30 ; 70 ; 30 80 1) : 9.889 .887 .886 .885 .884 .883 9.882 .881 .880 •879 •877 .876 9.875 ■874 .873 .872 .87, .871 9.870 .869 .868 .867 .866 .865 9.864 .864 .863 .S62 .861 .86i 9.860 .859 .858 .858 .857 .857 9.856 .855 •855 .854 ,854 •853 9.853 •853 .852 .852 .851 .851 9.851 .850 .850 .850 .850 .850 0321 8738 7ii9 5887 4627 3481 2455 '551 0775 0129 9616 9242 9010 8922 8984 9198 9569 0099 0792 1652 2683 3.S86 5266 6827 8570 0500 2620 4932 7439 0145 3053 6164 9482 3010 6750 0704 4875 9266 3878 8714 3775 9065 4584 0335 6319 2538 8994 5687 2620 9794 7209 4868 2770 0917 9.849 9309 •849 7948 .849 6833 .849 5966 •849 5346 .849 4974 9.849 4850 Diir. 11583 "479 11372 1 1 260 1 I 1 46 I 1026 10903 10777 10646 I 05 I 3 10374 10232 10088 9938 9786 9629 9470 9307 , 9140 8969 8797 8620 8439 ' 8257 I 8070 7880 7688 7493 7294 7092 6889 6682 6472 6260 6046 5829 5609 5388 5164 4939 4710 4481 4249 4016 j 3781 3 544 i 33°7 ■ 3067 2826 i 2585 '■ 234' : 2098 I '853 ' 1608 1361 '"5 S67 620 372 124 TABLE IX. For fiiuling iho True Aiimiiiilv or tlie Tiiiiu liuiu iliu I'oriliclion in Orlatx of grcnt eccentricity. X .1 o // () 0.00 1 0.00 'Z 0.0 1 .1 O.OJ 4 0.12 5 0.23 0.39 7 0.62 H 0.93 « '•33 10 1.82 11 2.42 la 3.14 Vl 3 99 14 4.99 15 6.13 k; 7-43 17 8.90 IH 10.55 10 12.40 'M 14-45 ai 16.70 'M 19.18 •Z3 21.89 'Zl 24.83 25 28.03 2ft 31.48 27 35.20 28 39.19 29 4347 »0 48.04 :u 52.91 i.1 58.09 •M 63.59 34 69.42 35 75'57 36 82.07 37 88.92 38 96.12 39 103.68 40 in.6i 41 119.92 42 128.62 43 137.70 44 147.18 i:» '57-05 40 i;'7.34 47 178.14 48 189.16 49 200.71 50 212.69 51 225.10 52 i37->5 53 251. 15 54 265.01 55 279.-. I 5ft 293.88 57 309.02 58 324.6s 59 340.70 00 357.26 iiiir. 0.00 0.01 0.04 0.07 o. n 0.16 0,23 0.31 0.40 0.49 0.60 0.77. 0.85 1.00 1. 14 1.30 1-47 1.65 1.85 2.05 2.25 2.48 2.71 2.94 3.20 3-45 3-7* 3-99 4.28 4-57 4.87 5.18 5-50 5.83 6.15 6.50 6.85 7.20 7.56 7-93 8.3, 8.70 9.08 9.48 9.87 10.29 10.70 11.12 11.55 11.98 12.41 12.85 13-30 13.76 14.20 14.67 15.14 t5.6o ib.c'^ 16.56 Hill. 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.00 1 0.001 0.002 0.002 0.003 0.004 0.005 0.006 0.008 0.010 0.01 2 0.014 0.017 0.010 0.025 0.030 0.035 0.041 0.047 0.055 0.064 0.073 0.084 0.096 0.109 0.123 0.139 0.156 0.175 0.196 0.218 0.243 0.269 0.298 0.328 0.361 0.397 0.436 0-477 0.521 0.567 0.617 0.671 0.727 0.787 0.851 0.919 .001 .000 .001 .COl .001 .001 .00s .002 .002 .002 .003 .003 .005 .005 .005 .006 .006 .008 .009 .009 .011 .012 .013 .014 .016 .017 .019 .021 .022 .025 .026 .029 .030 ■°Ti .036 .039 .041 .044 .046 .050 ■054 .056 .060 .064 .068 bU 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0,000 0.000 0.000 0.000 0.000 0.000 0,000 0.000 0.000 0.000 0.000 0,000 0,000 0,000 0,000 0,000 0.000 0.000 0.000 0.000 0,000 0.000 0,000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 o.ooo 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 O.OOI 0.001 O.OOI O.OOI 0.002 o.ooz 0,002 0.002 0.003 11' Dill. c 1 It : 0.000 0.000 ' 0.000 0.000 0.000 o.oco 0.000 0,000 0,000 0.000 0.000 0.000 0,000 0.000 c.ooo 0.000 0,000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 O.OOI 0.000 O.COI 0.000 O.OOI .000 0.000 0.002 .001 0.000 0.002 .000 0.000 0.003 .001 .001 0.000 0.004 0.005 .001 0.000 0.000 0.006 .001 0.000 0.008 .002 0.000 0.010 .002 .002 0.000 0.012 0.000 0.014 .002 0.000 0.017 .003 0.000 0.020 .003 0.000 0.024 .004 0.000 004 0.028 0.000 0.033 .005 .006 .006 0.000 0.039 0.045 o.o-^o 0.000 0.052 .007 .008 0.000 0.060 .008 0.000 0.068 0.000 0.078 .010 0.000 0.088 .010 0.000 O.I 00 .012 .013 0.000 O.I 13 0.127 0.000 .014 0.000 0.142 .015 0.000 0.159 0.177 .017 .018 .020 o.coo 0.000 0.197 o.coo 0.219 .022 0.000 0.242 0.267 0.294 .023 .025 .027 .029 0.000 0.000 0.000 0.323 0.354 0.388 0.424 0.462 0.000 .031 .034 .036 .038 0.000 o.coo o.coo o.coo .040 0.502 0.000 0.546 .044 .046 0.001 0.592 O.OOI 0.641 .049 O.COI 0.693 .052 .056 O.OOI 0.749 0.002 cat ccccntiii'ity. o.ooo o.ooo o.oco o.ooo o.ooo o.ooo 0.000 o.ooo o.ooo o.ooo o.ooo o.ooo 0.000 o.ooo o.ooo 0.000 )00 0.000 >0I 0.000 300 0.000 )0I 0.000 501 0.000 301 0.000 301 0.000 302 0.000 302 0.000 302 0.000 DO 2 0.000 30 3 0.000 JO 3 0.000 004 0.000 304 0.000 30 5 0.000 300 O.O-'O 306 0.000 '°l 0.000 D08 0.000 308 0.000 3IO 0.000 DIO 0.000 312 0.000 31-? 0.000 314 0.000 3«5 0.000 "7 o.ooo 3l8 0.000 320 0.000 322 0.000 323 o.oco D25 0.000 327 0.000 329 0.000 331 0.000 =34 o.ooo '^^ 0.000 35S 0.000 340 0.000 H4 0.00 1 34b 0.00 1 349 O.COI 552 O.COI 3Sb 0.002 TABLE IX. Kor liiidinK tlie Triio .Vnonmly or tlie Tiiiu! t'roni the I'tiilielioii in Orhitnof urcnt epcpntricify. »l 0'^ m 111 (»r (M ou 70 ri 7a 73 74 75 70 77 78 71) 80 81 S'Z 83 81 8U iX) 91 1W »3 01 05 UU 97 98 99 100 U 101 (I ;!() 102 (I 30 103 30 101 30 103 30 lOG 30 i 107 30 i 108 30 109 Dlff. 357.26 409.x 6 42X.38 44;40 4(16.92 4S6.96 507.51 52S.58 550-«7 572.29 5^4 94 61S.12 64. .85 666.13 690.96 716.34 742.29 768.81 795.90 823.57 851.84 880.70 910.16 940. -.3 970.92 1002.24 5034,20 1066.81 1 100.08 1 134.02 1168.64 1203.95 1239.97 1276.72 1314.21 '3 52-45 1391.46 1431.27 1471.88 1492.50 >5'333 •534-3» «555'64 1577.12 1598.82 1620,75 1642.91 1665.30 1687.93 1710.80 1733.92 1757.28 1780.90 1804.77 1828.90 1853.30 1877.97 17.04 >7-54 18.02 18.52 19.02 19.52 20.04 20.55 21.07 21.59 22.12 22.65 23.18 a3'73 24.28 24.83 *5-3X aS-95 26.52 27.09 27.67 28.27 28.86 29.46 30.07 30.69 3'-32 31.96 32.61 33-i7 33-94 34.62 35-3' 36.02 36-75 37-49 38.24 39.01 39.81 40.61 20.62 20.83 21.05 21.26 21. 4S 21.70 21.93 22.16 22.39 22.63 22.87 23.12 23.36 23.62 23.87 24.13 24.40 24.67 0.919 0,990 1.066 1. 145 1.230 1. 318 1.411 1. 510 1.61 3 1.721 1.8,5 '■954 2.078 2.209 2-345 2.4S8 2.637 2-793 2.95 b 3.125 3.302 3.4X6 3.678 3.878 4.087 4.303 4.529 4.764 5.008 5.262 5 527 5.801 6.087 6.385 6.694 7.016 7-350 7.698 8.060 8-437 8.829 9.032 9.238 9-449 9.664 9.883 10.108 '°-337 10.570 10.809 11.053 11. 302 "-557 11.X17 12,083 12.354 12.632 12.916 1 3.207 IMH. 071 076 079 0X5 08X 093 099 103 108 "4 119 '24 '3' 136 «43 '49 '56 163 169 177 1X4 192 2 00 209 216 226 235 244 254 265 274 286 298 309 334 34i* 362 377 392 203 206 211 219 225 229 233 239 244 249 255 260 266 271 278 284 291 0.003 0.003 0.003 0.004 0.004 0.004 0.005 0.005 0.006 0.006 0.007 0.007 0.008 0.009 0.009 o.oio 0.01 1 0.012 0.0! 3 O.OI4 0.0 1 5 0.016 0.017 0.018 0.020 0.021 0.023 0.024 0.026 0.02S 0.030 0.032 0.034 0.036 0.038 0.041 0.044 0.047 0.050 0.053 0.058 0.060 0.062 0.064 0.066 0.068 0,070 0.072 0.074 0.077 0.079 0.082 0.084 0.087 0.090 0.093 o 096 0.099 //' nin. 0.749 0.807 0.869 0.935 1.004 1.077 I.I 54 '■235 1-321 1. 41 1 1.505 1.605 1.709 1.819 •-934 2.05s 2.1X1 2.314 2-453 2.599 2.752 2.912 3.079 3-255 3-439 3.631 3-«33 4.044 4.266 4.498 4-74' 4.996 5.263 5.83S 6.147 6.471 6.812 7.171 7-549 7.946 X.I 52 8.364 8.582 8.805 9.035 9.271 9-5 '3 9.761 10.017 10.280 10.550 10.828 II. 1 14 1 1.408 11.711 12.022 12.343 12.673 ! -058 .061 I .066 .069 I •°73 •°77 .0X1 .0X6 .090 .094 .100 .104 .110 .115 ; .121 .126 i .133 •'39 .146 : .153 i .160 .16- .1- .1.S4 I .192 ' .102 ' .211 .222 .232 I -243 1 -255 I -267 I .281 i .294 ■309 -324 •34' i -359 .378 i -397 I .206 I .212 .218 .223 ! .230 .236 I .242 .248 .256 .263 .270 .278 .286 .294 •3°3 .311 .321 .330 f," O.002 0.002 0.002 0.002 0.002 0.003 o 003 0.003 0.004 0.004 0.004 0.005 0.005 0.00b 0.006 0.007 0.007 0.008 0.008 0.009 O.OIO O.OII O.OIl I 0.013 0.014 0.015 0.016 0.018 0.019 0.021 0.023 0.025 0.027 0.029 0.032 0.035 0.038 0.041 0.045 0.049 0.053 0.055 0.05X 0.060 0.063 0.066 0.069 0.072 0.075 0.078 0.0X2 0.0X5 0.089 0.093 0.095 O.I 02 0.107 0.1 12 0.117 615 my. TABLE IX. For finding tha True Anojnaly or .lie Time from the Perilielion in Orbits of great eccentricity. X o , 109 .•'-It 110 .3(1 111 II ;ju US () A Diir. H'~ 113 II .'ill 114 II :iO 113 :!0 lie ;io 117 :jo 118 :!0 110 ;iO 120 1) .iO 121 (I :;o 122 n :!() 123 II :u) 124 fl ;m) 125 nn 120 30 127 l> 30 128 ;^o 129 30 130 20 40 131 20 40 132 20 '10 133 ' 20 40 134 20 40 135 20 40 130 1877.97 1902.91 I92X.I3 1953.64 1979.4+ 2005.54 2031.94 ^058. 64 2085.66 21 13.00 2140.66 2168.66 2197.00 2225.69 2254.73 2284.13 2313.91 2344.06 2374.60 2405.54 2436.88 2468.64 2500.83 253345 2566.51 2600.03 2634.02 2668.49 2703.46 2738.93 2774.91 2811.43 2848.50 2886.13 2924.33 2963.12 3003.53 3042.56 3083.23 3124.57 3166.59 3209.31 3252.76 3282.13 33H-85 334190 3372.31 3403.09 3434-23 3465-74 3497.03 35--;.9i 3562.60 3595-69 3629.20 3663.13 3697.50 3732.31 3767.58 3803,31 3^39-52 24-94 25.22 25.51 25.8c 26.10 26.40 26.70 27.02 27-34 27.(11 28.00 28.34 28.69 29.04 29.40 29.78 30.15 30-54 30.94 3'-U 31.76 3' '9 3i..02 33.06 33-52 33-99 34-47 34-97 35-47 35.98 36.52 37.07 37-63 38.20 38.79 39-4« 40.03 40.67 41.34 42.02 42.72 43-45 29-37 29.72 30.05 30.41 j<->4 31-5' 31.89 32.28 32.69 35.09 33-51 33-93 34-37 34.81 35-27 35-73 36.21 3.207 3.504 3.808 4.119 4.438 4.764 5.097 5-439 5.789 6.148 6.515 6.892 7.278 7.674 8.080 8.496 8.924 9-363 19.813 20.276 20.751 21.240 21.742 22.258 22.789 23.336 23.X98 24-477 25,073 25.687 26.320 26.973 27.646 28.341 29.057 29-797 30.562 31-351 32.167 33.011 33-885 34.789 35-725 36-367 37.025 37.699 38.389 39-097 39,822 40.564 41.326 42.108 42.910 43-733 44-576 45-442 46.33> 47-245 riilT. .297 .304 .311 .319 .326 -333 .342 •350 -359 -367 •377 .386 .396 .406 .416 .428 •439 .450 .463 -47 5 .489 .502 .516 -53' •547 .562 -579 -596 .614 •633 -653 -673 .695 .716 .740 -765 -:'S9 .816 ,844 .874 .904 .930 .642 .658 -674 .690 .708 .725 .742 .762 .782 .802 .823 -843 .866 .889 .914 DilT. 48.183 1 •'-*? 49.«47 ' '"^ 50-' 38 964 .991 0.099 0.102 0.106 0.109 0.1 13 0.116 0.110 o. 24 0.-28 O.! 32 o-'.)7 0.142 0.147 0.152 0.157 162 168 •74 180 1S6 0.J93 0.200 0.207 0.214 0.222 0.230 0.239 0.248 0.258 0.268 0.278 0.289 0.300 0.312 0-325 0.338 0.352 0.367 0.382 0.398 0.415 0-433 0.452 0.465 0.479 0493 0.508 0.523 0-539 °-555 0.572 0.590 c.6og 0.629 0.649 0.669 0.691 0.714 0.738 0763 0.788 .003 .004 .003 .004 .003 .004 .004 .004 .004 .005 .005 .005 .005 .005 .005 .00(1 .006 .006 .006 .007 .007 .007 .007 .008 .008 .009 .009 .010 .010 .010 .011 .011 .012 .013 .013 .014 .015 .015 .oi6 .017 .018 .019 .013 .014 .014 ! -015 .015 .016 .01 6 .017 3T8 .Olr i -024 Diff. 12.673 13.013 «3-363 13.724 14.095 14.478 14.874 15.282 15.702 16.135 16.583 17.045 17.522 18.015 18.524 19.050 •9-594 20.156 20.738 21.339 21.962 22.606 23.273 23.964 24.680 25.422 26.191 26.988 27.815 28.673 29.564 30.489 3 '-450 32,448 33-4*<5 34-563 35.685 36.852 38.067 39-331 40.649 42.022 43.452 44-439 45-455 46.500 47-5:'5 48.68?. 49,820 50.992 52.199 53-442 54-723 56.042 57.401 58.302. 60.247 61.736 63-273 64.857 66.491 340 35^' 361 371 3*«3 396 408 420 433 448 462 477 493 509 526 544 562 582 601 623 644 667 691 716 742 769 ',/7 827 858 891 925 961 998 03; 078 122 167 ■■'»5 264 318 373 430 016 045 075 107 138 172 207 243 281 319 359 401 445 489 537 584 634 C bill. 0.II7 0.122 0.128 - '34 0.141 0.148 0.155 0.162 0.170 0.178 0.187 o 196 0.206 0.216 0.227 0-239 0.251 0.264 0.277 0.291 0.306 0.322 0-339 0-357 0.376 0.396 0.417 0.439 0.463 0.4 88 0.515 0.544 0-574 0.606 0.640 0.676 0.715 0.757 0.800 o.!(46 o 896 0-9-19 1.005 1.045 1.087 1. 130 1.175 1.221 1.8 1.273 •-325 1.379 i-n6 1.495 '-55« 1.623 1.692 1.764 39 1.917 2.000 2.087 .005 .006 .006 .007 .007 .007 .007 .008 .008 .C09 .009 .010 .010 .011 .012 .012 .013 .013 .014 .015 .016 .017 .018 .019 .020 .021 .022 .024 .025 .027 .029 .030 .032 034 .036 •039 .042 .043 .046 .o;o -053 i -040 I 'J42 I -043 i -c+i I -04-^ .054 .057 .059 .063 .065 .061} .072 .075 .078 -oi'3 :,87 For 13 13 13f 13! 14C 141 142 143 144 145 ' 4 140 3 4 5 147 1 1 1 21 ,SI 41 ' 51 118 1 11 2( .•Ji 4! 50 149 »1« ; eccentricity. TABLE IX. For finding the True Anomaly or tlie Time from the Perihelion in Orbits of great eccentricity 136 n 2(1 4(1 137 20 ■ill 138 U 20 40 130 20 40 140 20 40 I 141 J 20 I 40 142 j 20 ;;o 40 50 143 10 20 oO 411 ,')0 144 10 20 ,'.0 4'i aO 145 10 20 .•!0 ' 40 0(1 146 10 20 30 4(1 ■ 50 147 V) 20 30 40 SO 148 10 20 .'iO 40 50 149 01 DifT. I 387(1.21 3913.41 3951.12 39*'9-3; 4.028. n 4067.42 4107.28 4147.72 4188.75 4230.38 4272.63 4315.52 4359.06 4^05.26 4448.15 4493-73 4540.03 4C87.07 4610.88 4634.88 4659.07 4683-;' 4708.05 4732.84 4757-84 4783.05 4808.46 4834.10 4859.95 4S86.02 4912.31 4938.83 4^05.58 499^ S 5019.78 i;o4;-23 107+-93 5102.88 5131.08 5159-53 5188.24 5217. 2T 5i4''-45 5^75-95 5305-73 533)-79 5366.13 5396.76 C427.67 5458.88 j49°-39 5^22.20 5554-33 5586.77 5619.52 5652.60 5686.01 5719-75 5753 ^i 5788.26 6 ! 36.69 37.20 37-71 38.23 38.76 39-31 39.86 40.44 41.03 ,1.63 42.25 42.89 43-54 44.20 44.89 45.58 46.30 47.04 23.81 24.00 2- .9 24.39 14-59 24.79 25.00 25.21 25.41 25.64 25.X5 26.07 26.29 26.52 26.75 26.98 27.22 27.45 27.70 27.95 28.20 28.45 28.71 28.97 29.24 29.50 29.78 30.06 3°-34 3"-f'3 30.91 31.21 31.51 31.81 3?--<3 32.44 3^-75 33.08 33-41 33-74 34.08 34-43 B 50.138 51.156 52.203 53.280 54.388 55.528 56.702 57.910 59-154 60.436 6'-757 63.119 64.523 65.971 67.465 69.007 70.599 72.243 73-941 74.811 75-695 76.595 77-509 78.439 79-385 80.347 81.325 82.321 83-333 84.363 85.411 86.478 87.564 88.668 89-793 90.938 92.103 93.290 94.498 95-729 96.982 98.259 99-559 100.884 102.234 103.610 105.012 IC6.441 107.897 109.382 1 10.896 112.439 1 14.013 1 15.619 117.256 1 18.926 120.631 122.370 x 24. 1 44 I ■5-955 127.804 I Din. 1.018 1.047 1.077 1.108 1.140 1.174 1.20S 1.244 1.282 1.321 1.362 1.404 1.448 1.494 1.542 1.592 1.0+4 1.698 0.87c 0.884 0.900 0.914 0.9 3D 0.9+6 0.962 0.978 0.996 1.012 1.030 I.C48 1.067 1.086 1 . 1 0.1^ 1.125 1. 145 1. 165 187 20S 231 ^5 3 277 300 1.325 , '-350 1.376 I 1.402 I J -4-9 I I •45'' ; -485 1.514 '•543 , 1-574 I 1.606 I «-637 i 1.670 1.705 ' 1-739 I '-774 1. 811 1.849 Diff. 0.788 0.815 I 0.843 ! 0.873 ' 0.904 1 0-936 1 0.969 1 1.004 ! 1.041 I 1.079 1119 I 1.161 1.205 1.251 : 1.299 j 1.350 j 1.404 : 1.460 1.518 1.549 1.580 1.612 1.645 '.679 i 1.714 J-749 I 1.786 ' 1.823 1.862 1.901 j 1.942 I 1.984 i 2.026 i 2.070 j ?.. 116' 2.162 I ;..2io ! 2.259 I 2.309 2.361 2.414 : 2..^69 j 2.516 j 2.584 2.643 i 2.704 , 2.767 2.833 2.900 2.969 3.040 3-"3 3.188 3.266 3 •34'^ 3.428 3-513 3.601 3.691 3--'84 3.881 oir i)iff. .027 .028 .030 .031 .032 .033 -035 .037 .038 .040 .042 .044 .046 .048 .051 .054 .056 .058 .031 .031 .032 .033 ■034 -035 -Oj5 •037 .037 •039 .039 .041 .042 .042 .044 .046 .046 .049 .05c .052 •053 .055 •057 .058 .059 .061 .063 .066 .067 .069 .071 .073 .075 .078 .080 .082 .085 .o?8 .090 -093 .097 66.491 68.178 69.920 71.718 73-575 75-493 77-475 79-523 81.641 83.830 86.094 88.436 00.860 93.369 95-9''7 98.657 01-443 04.331 07.324 08.861 10.427 12. 022 13.646 15.301 16.986 i?704 20.452 22.233 24.049 25.899 27-/85 29.707 31.666 33.663 35.698 37-774 39.890 42.048 44.249 46.494 48.784 51.120 53-533 5 5-934 58.415 60.947 <'3-53« 66.168 68.860 71.608 74-414 77.280 80.206 83.194 86.246 89.364 92 549 95 804 99-130 02.528 206.002 1.687 1.742 1.798 1.857 1.918 ..9S2 2.048 2.1-8 2.189 2. 264 2.342 ; 2-424 2.509 2.598 2.690 ' 2.786 ' 2.888 2-993 1-537 i.^ee 1-595 1.62^ 1.655 1.685 1.718 1.748 :.78i 1.816 1.850 1.886 1.922 1.959 1.997 2.035 2.076 2.116 2.158 2.201 2.245 2.290 ■ 2.336 2.383 ' 2.431 2.481 2.532 ^ 2.584 2.637 2.692 2.748 2.806 2.866 2.926 2.988 i 3-052 '3.118 3.185 : 3-255 ; 3.326 3-398 i 3-474 Diir. 2.087 2.178 2.274 , 2-375 I 2.480 I 2-59' I 2.708 2.831 2.960 3.096 3.239 3-390 3-549 3-7'7 3.893 4.080 4-277 4.484 4.704 4.819 4.936 5-057 5.181 5-309 5-440 5-575 5-715 5-858 6.005 6.157 6.313 6.473 6.639 6.809 6.984 7.165 7-35> 7-543 7-740 7-943 8-'53 8.369 8.592 8.822 9.060 9-3 ' 9-Si'3 o.;.i5 1 .083 10.359 10.645 10.940 11.244 11.558 11.883 12.218 12.564 12.921 13.291 '3-673 14.067 .091 .096 .101 .105 .1 1 1 .117 .123 .129 -136 •'43 .151 .159 .168 .176 .187 .197 .207 .220 .115 .117 ! .121 .124 .128 .131 -135 .140 .143 .147 .152 .156 .160 .166 .170 I -'75 I .181 ; .186 1 .192 : -'97 I .203 I .210 , .216 ' .223 [ .230 .238 .244 .251 .260 J .268 .276 .286 -295 .304 .314 -325 ' -33 5 -346 -357 .370 .382 -394 i TABLE X. 1 For finding the True A»onialy or the Time i'roni tlie Perihelion in Elliptic and Hyperbolic Orhits. A Gllipge. ll.viiorlM la. 1 1 log li Dill. log a logl.Diff. los; ball 11. 1)1 ir. log Jl DIIT. log C '"Sl-'^'"^ Lalfn^WIT 0.000 0.000 ! 0.00 0000 4 37 53 68 0.000 0000 4.23990 1.778 0000 7 23 37 5' 66 0.000 0000 4.23982,, 1 77« 1 .01 0007 .001 7432 .24286 .783 0007 9.998 2688 .23686 767 .02 0030 0067 .003 49X5 .24583 .788 0030 .9<)6 5493 .23392 762 .03 .005 2659 .24885 •794 0067 •994 8414 .23098 758 , .04 0120 .007 0457 .25190 •799 0118 •993 »45o .22807 753 j 0.05 0188 84 99 114 130 «47 0.008 8381 4.25497 1.805 0184 81 9.991 4599 4.22 5 1 8„ 1 748 i .06 0272 .010 C432 .25806 .81 • 0265 94 109 123 137 .989 7859 .22230 743 ! .07 0371 .012 4613 .idl 16 .816 0359 .988 1231 .21943 739 .08 0485 .014 2924 .26427 .821 0468 .986 4711 .21659 734 ' .09 0615 .016 1367 .26741 .827 0591 .984 8298 .21376 730 O.IO 0762 162 °-°i7 9945 4.27057 1.833 0728 152 165 178 193 206 9.983 1992 4.2io94„ 1 725 .11 0924 T78 .019 8659 .27376 .839 0880 .981 5791 .20815 720 1 .12 1102 >94 21 1 .021 7511 .27697 .845 1045 •979 9694 •20537 716 1 •«3 1296 .023 6503 .28020 .85, 1223 .978 3699 .20260 711 i .14 1507 227 .025 5637 .28344 .857 1416 .976 7805 .19986 706 0.1s 1734 7,61 0.027 49 '6 4.28670 1.863 1622 220 9.975 2011 4.1 971 2„ I 700 .16 J977 .029 4340 .28999 .869 1842 233 2a6 •973 6316 .19440 695 : •'7 2238 177 29J 311 .031 3913 .29331 iP 2075 ■972 0719 .19170 690 ; .18 2515 .033 3636 .29665 .882 2321 260 .970 5218 .18901 685 .19 2 8 09 •°35 35" .30001 .888 2581 Z73 .968 9813 .18633 679 0.20 3120 3*8 345 363 381 398 0.037 3542 4^3°339 1.895 2854 286 9.967 4502 4.18367.. 1 672 .21 3448 .039 3730 .30679 .901 3 HO 299 3439 ,,^ .965 9285 .18102 666 : .22 3793 .041 4077 .31022 .908 .964 4159 .17840 661 ' •*3 4156 •043 4585 .31368 .915 375' 325 338 .962 9124 •«7579 b55 ; -4 45 37 .045 5259 .31716 .922 4076 .961 4180 .17319 649 1 i 0.25 4935 416 434 452 471 488 0.047 6099 4.32066 1.929 44'4 1 C I 9.959 9324 417061, 1 643 ; .2b 5351 .049 7109 .32418 .936 VAl 363 5'28 376 5504 389 .958 4556 .16803 637 ■-1 5785 .051 8290 ■32773 •943 .956 9875 .16547 t\3' i •*«! 6237 .053 9646 .33131 .951 •955 5281 .16292 625 1 -'^ 6708 .056 1179 .33492 .958 •954 0771 .16038 618 ; °^3o| 7196 0.058 2893 4.33856 1.966 6294 9.952 6346 4^1 5785,, J 6.3 i TAP T.T! Y Par+ T T , 1 T 0.00 .01 .02 .03 .04 0.05 .06 i ■°l 1 .08 .09 O.IO .11 1 ''2 i •>3 1 •H 1 0.15 .16 1 -'7 ! .18 ; ..9 1 0.20 Ellipse. Ilypcrbolii. T Ellipse. Ilypeihola. ! A Diir. A Dlff. A Diir. A Difl'. 0.00000 .0099a .01969 .02930 .03877 0.04808 .05726 .06630 .07521 .08398 0.09263 ..0I16 .10956 .11783 .12599 0.13404 .14198 .14981 •>5753 .16515 0.17266 992 977 961 947 931 918 904 891 877 865 ^53 840 827 816 805 794 783 772 762 7S» 0.00000 .01008 .02033 .03074 .04132 0.05209 .06303 .07417 .08550 .09702 0.10875 .12069 .13285 .14522 .15782 0,17067 .18375 .19709 .21068 .22454 0.23867 _ 1008 1025 1041 1058 1077 1094 1114 1133 1152 1173 1194 1216 1237 1260 1285 1308 •334 1359 1386 1413 0.20 .21 .22 •23 •24 0.25 .26 •27 .28 .29 0.30 •3' •32 •33 •34 •39 0.40 0.17266 .18008 .18740 .19462 .20174 0.20878 .21573 .22258 •22935 .23604 0.24265 .24917 .25561 .26198 .26826 0.27447 .2S061 .28668 .29268 .29860 0.30446 742 732 722 ;;: tn 677 669 661 652 644 637 628 621 614 607 600 III 0.23867 .25309 .26779 .28280 .29813 0.31377 1442 1470 150I 1564 i 1 1 i 618 rboHc Orbits. "~^ ^ 1 UifT. , log mlfU.Diff. 82,, 1.771 86 .767 J92 .762 398 ■758 io7 •753 qi8„ ..748 130 1 •743 HI •739 659 •734 37C .730 o94„ 1.725 8.,- .720 s-;7 .716 200 .711 986 .706 I712n 1.700 440 .695 (lyo .690 (901 .685 5633 .679 ^367,. 1.672 .666 7840 .661 7579 ■655 7319 .649 706 1 „ 1.643 6803 .637 6S47 .631 6292 •5-3 6038 .618 5785n 1.613 eiliola. Diff. 1442 1470 I 501 1533 1564 TABLE XL For tlie Motion in n Parabolic Obit. V I"i?ft I)ilT. 1 0.000 0.000 0000 0.060 .001 .000 0000 .061 .002 .000 0001 I .062 .003 .000 0002 I .063 .004 .000 0003 I I .064 0.005 0.000 0004 0.065 .006 .000 0006 2 .066 .007 .000 0009 3 .067 .008 .000 0012 3 .068 .009 .000 0015 3 3 .069 0.010 0.000 0018 0.070 .01 1 .000 0022 4 .071 .012 .000 ooi6 4 .072 .013 .000 0031 S .073 .0,4 .000 0035 t .074. 0.015 0.000 0041 0.075 .016 .000 0046 I 7 8 8 8 8 9 .076 .017 .000 0052 .077 .018 .000 0059 .078 .019 .000 0065 .079 0.020 0.000 0072 0.080 .021 .000 0080 .081 .022 .000 0088 .082 .023 .000 0096 .083 .024 .000 0104 .084 ; 0.025 0.000 01 1 3 0.085 1 .026 .000 0122 9 .086 ! .027 .000 0132 10 .087 i .028 .000 0142 10 .088 i •°^9 .000 0152 lO II .089 0.030 0.000 0163 II II 12 12 '3 0.090 i •°3i .000 0174 .091 i •°'i- .000 0185 .092 •033 .000 0197 •093 .034 .000 02og .094 0.035 0.000 0222 0.095 .036 .000 0235 '3 13 .096 .037 .000 0248 .097 .038 .000 0262 '4 '3 'S 14 16 •5 16 16 .098 .039 .000 0275 .099 0.040 0.000 0290 O.IOO .041 .000 0304 .101 .042 .000 0320 .102 .043 .000 0335 .103 .044 ,000 0351 .104 0.045 0.000 0367 16 «7 >7 18 18 0.105 .046 .000 0383 .106 .047 .000 0400 .107 .048 .000 0417 .108 .049 .000 0435 .109 1 0.050 0.000 0453 18 «9 19 19 20 0.1 10 .051 ,000 0471 .III .052 .053 .000 0490 .000 0509 .112 .113 1 •°54 1 ,000 01,28 .114 0.055 0.000 0548 0.1 15 .056 .000 0568 20 .116 I -057 .000 0589 2! 21 .117 .058 .000 0610 .118 .059 .000 0631 21 21 .119 0.060 0.000 0652 0.120 lot; /It 065a 0674 0697 0719 0742 0766 0790 0814 0838 0863 0888 0914 0940 0966 0993 1020 1047 1075 1103 II32 1161 1 190 1219 1249 1280 I311 1342 1373 1405 1437 1470 1502 1536 1569 1603 1638 1673 1708 "743 1779 1815 1852 1889 1926 1964 2002 2040 2079 21l8 2158 2198 2238 2279 2320 2361 2403 1445 2487 2530 2S73 0.000 2617 0.000 .000 .000 .000 .000 0.000 .000 .000 .000 .000 0.000 .oco .000 .000 .000 0.000 .000 .000 .000 .000 0.000 .000 .000. .000 .000 0.000 .000 .000 .000 .000 0.000 .000 .000 .000 .000 0.000 .000 .000 .000 .000 0.000 .000 .000 .000 .000 0.000 .000 .000 .000 .000 0.000 .000 .000 .000 .000 0.000 .000 .000 .000 .000 Ditl. 1 1 0.120 22 .121 1 23 .122 i 22 .123 1 *3 .124 *4 ^ 1 0.125 ; H .126 24 .127 24 .128 2.S 25 .129 i 26 0.130 ! 26 .131 : 26 .132 ; 27 27 •'33 i •>34 27 28 0.135 ; .136 28 29 29 .139 1 ; 29 0.140 j .141 29 30 3' 31 .142 ■«43 .144 0.145 31 3J .146 •«47 32 32 33 .148 .149 0.150 32 .151 34 .152 •'53 •154 33 34 35 0.155 ' 35 35 36 .156 .158 .159 0.160 37 .161 37 .162 37 3« 38 .164 38 0.165 .166 39 .167 39 .168 40 40 .169 ' 0.170 40 .171 4« .172 4« 4' 42 •173 .174 0.175 42 42 .176 •177 43 .178 1 43 .179 44 0.180 In- 0.000 2617 .000 2661 .000 2705 .000 2750 .000 ^79 5 0.000 2841 .000 2886 .000 2933 .000 29^9 .000 3026 3074 3121 3169 3218 3267 3316 3365 34'5 3466 3516 3567 3619 3671 3723 3775 3828 3S82 39 35 39S9 4044 0.000 4099 .000 4154 .C ;.-. V .0 ■' :; .001^ .1 0.000 .000 .000 .000 .000 0.000 .000 .000 .000 .000 0.000 .000 .000 .000 .000 0.000 .000 .000 .000 .000 437S 4435 4493 4551 4609 4668 4726 4786 4846 4906 4966 5027 5088 5150 5212 0.000 5274 .000 5337 .000 5400 .000 5464 .000 5528 0.000 3592 .000 5657 .000 5722 .000 5787 .000 5853 0.000 5919 0.000 .000 .000 .000 .000 0.000 .000 .000 .000 .000 0.000 .000 .000 .000 .000 Dlfl'. 44 44 45 46 45 47 46 47 48 48 49 49 49 49 50 5' 50 51 52 52 52 52 53 54 53 54 55 55 55 55 56 56 5'^ 5^ 58 59 58 60 60 60 60 61 61 62 62 62 64 64 64 65 65 65 66 66 OIU TABLE XI. For the Motion in a I'anibolic Orbit. V log ii Diff. V ipg/i Diff. >? lOgfl 1 Diff. 0.180 0.000 5919 67 67 67 68 68 0.240 0.001 0603 no ° 300 O.OOI 6733 .181 .000 5986 .241 .001 0693 90 301 .001 6848 "5 .182 .000 6053 .242 .001 0784 9' 91 9» 92 302 .001 6963 "5 116 1 116 i 117 ; .1X3 .000 6120 .243 .001 0875 303 .001 7079 .,84 .000 6188 .244 .001 0966 304 .001 7195 0.185 0.000 6256 69 68 0.245 O.OOI 1058 n» ° 305 O.OOI 7312 1 .186 .000 6325 .246 .001 1 1 50 92 92 306 .001 7429 117 117 i 118 i .187 .000 6393 70 69 70 .247 .001 1242 307 .001 7546 .188 .000 6463 .248 .001 1335 93 308 .001 7664 .189 .000 6532 .249 .001 1429 94 93 309 .001 7783 119 118 ' 0.190 0.000 6602 0.250 O.OOI 1522 ° 310 O.OOI 7901 .191 1 -191 .193 .000 6673 .000 6744 .000 6815 71 7' 71 ■ill .253 .001 1617 .001 1711 .001 1806 93 94 95 3«« 312 313 .001 8020 .001 8140 .001 8260 119 120 ; lao ; .194 .000 6887 72 71 .254 .001 1901 95 96 3'4 .001 838: 121 1 121 [ 0.195 0.000 6959 71 73 0.255 O.OOI 1997 96 97 315 O.OOI 8502 121 1 122 .196 .197 .000 7031 .000 7104 .256 .257 .001 2093 .001 2190 316 317 .001 8623 .001 8745 .198 .000 7177 73 .258 .001 2287 97 3.8 .001 8867 122 .199 .000 7250 73 74 .259 .001 2384 11 319 .001 8989 1 :.^2 ! 124 i 0.200 0.000 7324 0.260 0.001 2482 93 ° 99 320 O.OOI 9113 ^ 1 .201 .202 .000 7399 .000 74/3 75 74 .261 .262 .001 2580 .001 2679 321 322 .001 9236 .001 9360 123 1 124 j .203 .000 7548 75 76 76 .263 .001 2778 99 323 .001 9484 124 ' .204 .000 7624 .264 .001 2877 99 100 3M .001 g' 39 125 125 1 0.205 0.000 7700 76 77 77 77 78 0.265 O.OOI 2977 100 lOI lOI 102 lOI 315 0.001 97 34 126 i 126 ; 127 1 127 127 .206 .207 .208 .209 .000 7776 .oco 7853 .000 7930 .000 8007 .266 .267 .268 .269 .001 3077 .001 3178 .001 3279 .001 33S1 326 327 328 329 .001 9860 .00: 9986 .001 01 1 3 .002 0240 1 0.210 .211 0.000 8085 .000 8163 78 79 79 79 80 0.270 .271 O.OOI 3482 .001 3^85 103 ° 103 103 33° 331 0.002 0367 .002 0495 128 i .212 .213 .oco 8242 .000 8321 .272 .273 .001 3688 .001 3791 331 333 .002 0624 .002 0752 1 29 i 128 .214 .000 8400 .274 .001 3894 104 334 .002 0882 130 ; 129 0.215 0.000 8480 80 81 81 81 82 0.27s O.OOI 3998 105 ° 104 106 in " 335 0.002 lOII .216 .000 8560 .276 .001 4103 336 .002 1141 1 30 1 .217 .000 8641 •177 .001 4207 337 .002 1272 131 i 131 ! 131 132 ■ .218 .000 8722 .278 .001 4313 338 .002 1403 .219 .000 8803 .279 .001 4418 1O3 106 339 .002 1534 0.220 0.000 8885 82 82 0.280 O.OOI 4524 in-, ° 340 0.002 1666 .221 .222 .000 8967 .000 9050 .281 .282 .001 4631 .001 4738 107 107 341 342 .002 1799 .002 1931 133 132 : .223 .000 9132 .283 .001 4845 107 108 108 U3 .002 2065 '34 .224 .000 9216 .284 .001 4953 344 .002 2198 133 »35 0.225 0.000 9300 84 85 86 0.285 0.001 5061 108 345 0.002 2333 1 .226 .000 9384 .286 .001 i^i69 346 .002 2467 '34 1 .227 .000 9468 .287 .001 5278 1 09 IIO log lli 347 .002 2602 "35 136 136 136 137 .228 .000 9551 .000 9638 .288 .001 5388 348 .002 2738 .229 .289 .001 5497 349 .002 2874 0.230 .231 0.000 9724 .000 9810 86 87 87 87 88 0.290 .291 O.OOI 5608 .001 15718 no ° III 112 112 112 350 351 0.002 3010 .002 3147 .232 •133 .000 9897 .000 9984 .292 .293 .001 5829 .001 5941 35^ 353 .002 3284 .002 3422 '38 '38 139 .234 .001 0071 .294 .001 6053 354 .002 3560 0.235 o.ooi 0159 88 88 89 90 0.295 O.OOI 6165 ..3 ° 113 114 114 114 355 0.002 ■' 99 .236 .001 0247 .296 .001 6278 356 .002 3838 139 139 .237 .001 0335 .297 .001 6391 357 .002 3977 .238 .001 0424 .298 .001 6505 358 .002 4 I 17 140 '4' «4' 1 .239 .001 0513 .299 .001 6619 359 .002 4258 0.240 O.OOI 0603 0.300 O.OOI 6733 360 o.oci 4399 1 02U DifT. I 4 3 )i 10 ^o )0 !: Da '3 V5 57 89 13 36 60 84 39 34 60 86 13 40 67 95 2+ 5 2 182 III 4J 72 03 34 66 99 3> &■; |3 3 I IS 116 I 116 117 117 117 118 \\l 119 120 120 121 121 121 122 122 X'.\2 124 123 124 124 125 125 126 126 127 127 127 128 129 130 129 130 131 131 131 132 133 132 134 I 133 : '35 134 135 136 130 136 137 i 1 1 137 *4 ! ,38 )0 .38 139 >9 n 139 139 140 . 141 ; >4» ! 19 1 TABLE XI. For the Motion in a Puraholic Orbit. l0K>i I 0.360 .361 .362 •363 .364 0.36:; .366 ■367 .368 .369 0.370 •371 ; •371 •373 •374 I °-375 • •376 i •377 I •37^1 •379 I 0.3S0 I .38, .382 .383 .384 0.385 .386 .387 .3i,.- .389 0.390 .391 .392 •393 •3'>+ 0.395 .396 •397 .398 •399 0.400 .401 .402 .403 .404 0.405 .406 .407 .408 .409 0.410 .411 .412 { .413 .414 0.415 .416 •4»7 .418 .419 0.002 4399 .002 4540 .002 4682 .002 4824 .002 4967 0.002 5110 .002 5254 .002 5398 .002 5543 .002 5688 0.002 5834 .002 5980 .002 6126 .002 6273 ,002 6421 0.002 6568 .002 6717 .002 6866 .002 7015 .002 7165 I)ilT. 7315 i 7466 j 7617 I 7769 I 7921 j 8073 I 8226 8380 8534 8689 I 8844 I 8999 I 9155 ; 93>' ! 9468 9626 9784 994^ ! oioi j 0260 I 0.002 .002 .002 .002 .002 0.002 .002 .002 .002 .002 0.002 .002 .002 .002 .002 0.002 .002 .002 .003 .003 0.003 °420 I .003 0580 I .003 0741 I ,003 0903 .003 1064 0.003 .003 .003 .003 .003 0.003 .003 .003 .003 .003 0.003 .003 .003 .003 .003 122; 1389 1553 1716 I88I i°45 221 1 2376 2543 2709 2877 3044 3213 3381 355° 4« 41 42 43 43 44 44 45 46 46 46 47 48 47 49 49 49 5° 5° 5« SI 5^ 52 52 53 54 54 55 55 55 56 56 57 S» S« s» 59 59 60 60 61 62 61 63 62 64 64 66 65 67 66 68 67 69 68 69 70 log/* DIfl'. 0.420 ; 0.003 3720 0.120 ,-•.2 I .^22 4^3 424 J.25 4.^6 427 428 429 430 431 43i 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 45 3 454 455 456 457 45S 459 460 46, 462 463 464 ill 467 468 469 470 471 472 173 474 475 476 477 478 479 480 0.003 .003 .003 .003 .003 0.003 .003 .003 .003 .003 0.003 .003 .003 .003 .003 0.003 .003 .003 .003 .003 3720 3890 4061 4232 4404 4576 4749 4V2 3 509b 5271 5445 5621 , 5797 ' 5973 ' 6150 I 6327 I 6505 j 6683 I 6862 7042 0.003 7222 .003 7402 .003 7583 .003 7765 .003 7947 0.003 8130 .003 8313 .003 8496 .003 8680 .003 88(-5 0.003 9050 .003 9236 .003 9422 .003 9609 .003 9797 0.003 9984 .004 0173 .004 0362 .004 0551 .004 0741 0.004 0932 .004 1 1 23 .004 1315 .004 1507 .004 1700 0.004 1893 .004 2087 .004 2281 .004 2476 .004 2672 0.004 .004 .004 .004 .004 0.004 .004 .004 .004 .004 2868 3064 3261 3459 3657 35*56 4°55 4^55 4456 4657 0.004 4858 70 71 71 72 72 73 74 73 75 74 76 76 76 77 77 78 78 79 80 80 80 81 «2 82 83 83 83 84 85 85 86 86 87 88 87 89 89 89 90 91 91 92 92 93 93 94 94 95 96 96 96 97 98 98 99 199 200 201 201 201 login Diff. 0.480 .481 .482 .483 •484 0.485 .486 •487 .488 •489 0.490 .491 .492 ■493 •494 0.495 .496 •497 .498 •499 0.500 •51 .52 •53 •54 0-55 .56 •57 •58 •59 0.60 .61 .62 .64 0.65 .66 .67 .68 .69 0.70 •71 .72 •73 •74 0.75 .76 •77 •78 •79 0.80 .81 .82 i^ .84 0.85 .86 ■87 .88 .89 0.90 0.004 .004 .004 .004 .004 0.004 .004 .004 .004 .004 0.004 .004 .004 .004 .004 0.004 .004 .004 .004 .004 0.004 .005 .005 .005 .005 0.006 .006 .006 .006 .007 0.007 .007 .007 .008 .008 0.008 .009 .009 .001; .010 4858 5061 5263 5467 5670 5875 6080 6285 6492 6698 6906 7113 7322 7531 774° 7951 8161 8373 8585 8797 9010 1173 3397 5681 8029 0441 2919 54fH 8079 0765 35^5 6361 9^74 2268 5345 8508 1759 5103 8542 2081 I o.oio 5723 .010 9473 .on 3336 .0. ! 7316 .c- . 1419 0.012 5652 .013 0022 .013 4536 .013 9202 .014 4031 0.014 9033 .015 4219 .015 9603 .016 5202 .017 1033 0.017 7120 .018 3486 .019 0165 •019 7195 .020 4629 0.021 2519 203 202 204 j 203 205 205 207 206 208 207 209 209 209 21 I 210 212 212 212 213 2163 2224 1284 2348 2412 2478 2545 2615 2686 2760 2836 2913 2994 3077 3163 3251 3 344 3439 3539 3642 3750 3863 3980 4103 4233 4370 4514 4666 4829 5002 5186 5384 5599 5831 6087 6366 6679 7030 7434 7900 621 TABLE XII. '< log Jll] log m-i ^1' 1 z 3 1 m ill. / mi m I m ' )"a I o / / / > 1 / ' 1 - 1 — 00 0.0000 90 90 180 180 180 o| 1 4.2976 9.9999 -» 13 90 20 90 20 178 40 178 40 '79 359 359 5' i) 1 s 3-395° 9.9996 4 46 90 40 90 40 177 20 '77 20 178 358 358 9 •] i.8675 9.9992 7 8 91 91 176 176 '77 357 357 '4 4 2.493» 9.9986 9 3* 91 20 91 20 '74 40 '74 40 176 356 356 18 m 2.2044 9.9978 11 55 91 41 91 41 '73 '9 173 .9 I7S 355 355 23 1.96S6 9.9968 '4 '9 92 t 92 I '7' 59 '7' 59 174 354 354 :^8 7 1.7698 9-9957 16 4i 92 •...'. 92 22 170 38 170 38 172 59 353 ' 353 32 H 1.5981 9-9943 '9 7 92 42 92 42 169 18 169 18 171 59 352 I 352 37 9 '•4473 9.9928 21 3* 93 3 93 3 167 57 167 57 170 58 35' 2 35« 42 10 1.3130 9.991 1 23 57 93 *S 93 as 166 35 166 35 169 57 350 3 35° 47 11 1. 1922 9.9892 26 ^3 93 46 93 46 165 14 165 14 168 55 349 4 349 52 12 1.0824 9.9871 28 50 94 8 94 8 163 52 '63 5* 167 54 348 6 348 56 13 0.9821 9.9848 3' 17 94 3« 94 3' 162 29 162 29 166 S' 347 8 348 I 14 0.8898 9.9823 33 46 94 53 94 S3 161 7 161 7 165 48 346 II 347 6 15 0.8045 9.9796 36 '5 95 17 95 '7 '59 43 '59 43 164 44 345 '4 346 II 1(( 0.7254 9.9767 38 46 95 40 95 40 158 20 ,58 20 163 40 344 '7 345 '6 17 0.6518 9.9736 4' 18 96 5 96 5 '56 55 156 55 162 34 343 =' 344 -I IS 0.5830 9.9702 43 5' 96 30 96 30 1 55 30 '55 30 161 27 342 27 343 27 10 0.5185 9.9667 46 26 96 56 96 56 '54 4 '54 4 160 '9 34' r- 342 32 20 0.45S1 9.9629 49 ^ 97 23 97 i3 '5* 37 15a 37 '59 9 340 38 34' 37 21 0.4013 9.9588 5' 41 97 5° 97 5° 151 10 '5' 10 '57 58 339 45 34° 43 1 o«> 0.3479 9.9545 54 22 98 19 98 19 149 41 '49 4' 156 45 338 53 339 49 1 23 0.2976 9.9499 57 5 98 49 98 49 148 II 148 II '55 29 338 338 54 24 0.2501 9-945' 59 5' 99 20 99 20 146 40 146 40 '54 II 337 9 338 25 0.2053 9.9400 62 40 99 53 99 53 '45 7 145 7 152 50 336 19 337 6 2(> 0.1631 9-9345 65 33 100 28 100 28 '43 32 '43 r- '5' 25 335 18 336 '3; V. 0.1232 9.9287 68 30 101 5 loi 5 '4' 55 141 55 '49 56 334 38 335 »9 2S 0.0857 9.9226 7' 33 loi 45 loi 45 140 15 140 ■5 148 22 333 49 334 25: 20 0.0503 9.9161 74 4' 102 27 102 27 '38 33 '38 33 146 42 333 ' 333 32 30 0.0170 9.9092 77 58 103 13 103 13 136 46 136 46 '44 55 332 12 332 39 31 9.9857 9.9019 81 23 104 4 104 4 134 56 134 56 142 59 33' 24 33' 46, 32 9.9^65 9.8940 85 105 I 105 1 132 59 132 59 140 5' 330 37 330 54! 33 9.9292 9.8856 88 54 106 6 106 6 '3° 54 130 54 '38 27 329 49 330 2| 34 9.9040 9.8765 93 11 107 22 107 22 128 38 128 38 '35 39 329 2 329 10 35 9.S808 9.8665 98 7 108 58 108 58 126 2 126 2 132 '3 328 14 328 19 30 9.8600 9-8555 104 20 III 13 III 13 122 47 122 47 127 29 r-7 27 327 28 !— 3« o2.2 1 9.8443 9.8443 116 34 116 34 116 34 "6 34 116 34 116 34 326 45 326 45 This tabic exhibits tlie limits of the roots of the equation sin (s' — C) = vio sin* z', when tlierc are four real roots. The quantities irii and ma are the limiting values of m^, and the values of 2/, z^', z^', and z^', corresponding to each of these, give the limits of the four real roots of the equation. 622 : 1 i i / O ' ' o o o; o 3^9 5' TABLE XII. 3 5« 9 357 '4 356 iS 355 23 354 ^S 353 3^ 35i 37 35« 41 35° 47 349 5- 348 56 r 8 I 348 I ill 347 6 ; Hi 346 •! ^ »7j 345 '6 5 21 I 344 -I 1 27 343 -7 I 32 1 34a 31 38 45 53 o 9 34» 37 340 43 339 49 338 54 b 19 5 i« 4 38 3 49 3 ' 2 12 I 24 1° 37 9 49 338 337 336 6 >3 335 19 334 15; 333 J" 33* 33> 39 46 330 54' i 330 •* r 9 i 329 10 8 14 328 19 - 27 327 i8 6 45 326 45 C log m. log m.j i 1 z J r »' \ \ "a n " "'. nig ™, "'1 '"i m^ » ' 1 1 1 ' / ' / + 00 00 0.0000 90 90 180 180 180 1 4.2976 9.9999 I I 20 I 20 89 40 89 40 177 37 180 55 181 2 3-3950 9.9996 a 2 40 2 40 89 20 89 20 175 «4 181 51 182 Ol » 2.8675 9.9992 3 4 4 89 89 172 52 1S2 46 183 4 2.4938 9.9986 4 5 20 520 88 40 88 40 170 28 183 42 184 A 2.2044 9.9978 S ° 6 41 6 41 88 19 88 19 168 s 184 37 185 1 t* 1.9686 9.9968 6 8 I 8 I 87 59 87 59 165 41 185 32 186 1 7 1.7698 9-9957 7 I 9 22 9 22 87 38 87 38 163 18 186 28 .86 59 H 1. 5981 9-9943 8 I 10 42 10 42 87 18 87 18 160 S3 1S7 23 187 59 9 '•4473 9.9928 9 a 12 3 12 3 86 57 86 57 158 28 188 18 188 58 10 1. 3130 9.9911 10 3 >3 25 13 2; 86 35 86 35 156 3 189 13 189 57 11 i.:922 9.9892 J« 5 J4 46 14 46 86 14 86 14 «53 3/ 190 8 190 56 12 1.0824 9-987« 12 6 16 8 i6 8 85 52 85 52 151 10 .-91 4 191 54 13 0.9821 9.9848 13 9 17 31 17 3' 85 29 8s 29 148 43 9' 59 192 52 14 0.8898 9.9823 14 12 18 53 18 53 85 7 85 7 146 14 192 54 '93 49 15 0.8045 9.9796 15 16 20 "7 20 17 84 43 84 43 '43 45 '93 49 '94 46 IC 0.7254 9.9767 16 20 21 40 21 40 84 20 84 20 141 14 1:4 44 195 43 17 0.6518 9.9736 17 26 23 5 23 5 83 55 83 SS 138 42 '95 39 -.6 39' IS 0.5830 9.9702 18 33 44 30 •H 30 83 30 83 30 136 9 196 33 197 33 lU 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 23 27 23 82 37 82 37 130 58 198 23 '99 22 21 0.4013 9-9588 22 2 28 SO 28 50 82 10 82 10 128 19 i9c^ 17 200 1 5 , 22 0.3479 9-9545 a3 15 30 >9 30 19 81 41 81 41 125 38 200 II 701 07 23 0.2976 9-9499 24 31 3» 49 3' 49 81 II 81 II 122 55 201 6 2-02 Oi 24 0.2501 9-945 « 25 49 33 20 33 20 80 40 80 40 120 9 202 202 5 I \ 25 0.2053 9.9400 27 10 34 53 34 53 80 7 80 7 117 20 202 54 203 41; 20 0.1631 9-9345 a8 35 36 28 36 28 79 32 79 32 114 27 203 47 204 32 27 0.1232 9.9287 30 4 38 5 38 5 78 55 78 55 III 30 204 41 205 22 2.S 0.0857 9.9226 3' 38 39 45 39 45 78 IS 78 15 108 27 205 35 206 II 20 0.0503 9.9161 33 18 41 27 41 27 77 33 77 33 los 19 206 28 206 59 30 0.0170 9.909a 35 5 43 13 43 J3 76 47 76 4; 102 3 207 21 207 48 31 9.9857 9.9019 37 » 45 4 45 4 75 56 75 56 98 37 208 14 208 36 32 9-9565 9.8940 39 9 47 I 47 > 74 59 74 59 95 209 06 209 23 33 9.9292 9.8856 41 33 49 6 49 6 73 54 73 54 91 6 209 58 210 II 34 9.9040 9-8765 44 21 5' 22 51 22 72 38 72 38 86 49 7 10 50 210 58 35 9.8808 9.8665 47 47 53 58 S3 58 71 2 71 2 81 53 211 41 211 46 3(t 9.8600 9-8555 52 31 57 13 57 13 68 47 68 47 75 40 212 32 212 33 -f-36 52.2 9.8443 9-8443 63 26 63 26 63 26 63 26 63 26 63 26 213 15 213 «5, This table exhibits the limits of the roots of the equation sin (3' — C) = «!o sin* z', ^he liiniting I when there are four real roots. The quantities r\\ and ?% are the limiting to each of I values of m^, and the values of s/, Sj', s,', and r/, coi'responding to each of these, give the limits of the four real roots of the equation. 623 TABLE XIII. For finding tlie Ratio of the Sector to the Trianjcle. 1 lot! s"- o.oooo 0.000 0000 .OOOI ,000 0965 .0002 .000 1930; .0003 .000 "■I'H .0004 .000 3858 0.0005 0.000 4821 i .0006 .000 ';7X4 .0007 .000 6747 .0008 .000 7710 .0009 .000 8672 O.OOIO 0.000 9634 .0011 .001 0595 .0012 .001 • 556 .0013 .001 2517 .0014 .001 347« 0.0015 O.OOI 4438 .0016 .001 5398 .0017 .001 6357 .0018 .001 7316 .0019 .001 8275 0.0020 O.OOI 9234 .0021 .002 0192 .0022 .002 1150 .0023 .002 2107 .0024 .002 3064 0.0025 0.002 4021 .0026 .002 4977 .0027 .002 5933 .0028 .002 6889 .0029 .002 7845 0.0030 0.002 8800 .0031 .002 9755 .0032 .003 0709 .0033 .003 1663 .0034 .003 2617 0.0035 0.003 3570 .0036 .003 4523 .0037 .003 5476 .0038 .003 6428 .0039 .003 7380 0.0040 0.003 8332 .0041 .003 9284 .0042 .004 °^J? .0043 .004 ii86 .0044 .004 2136 0.0045 0.004 3086 .0046 .004 4036 .0047 .004 4985 .0048 .004 5934 6883 .0049 .004 0,0050 0.004 iH^ .0051 .004 8780 .0052 .004 9728 .0053 .005 0675 .0054 .005 1622 0.0055 0.005 2569 .0056 .005 35'5 .0057 .005 4461 .0058 .005 54°7 .0059 .005 6353 0.0060 0.005 7298 965 965 964 964 963 963 963 963 962 962 g6i 961 961 961 960 960 959 959 959 959 958 958 957 957 957 956 956 956 956 955 955 954 954 954 953 95 3 953 952 952 952 952 951 95> 950 950 95° 949 949 949 949 948 948 947 947 947 946 946 946 946 945 logs3 I DitT. 0.0060 .0061 : .0062' .0063 .0064 o.ocfi5 .0066 .0067 .0068 .0069 0.0070 .0071 .0072 .0073 .0074 0.0075 .0076 .0077 .0078 .0079 0.0080 .0081 .0082 .0083 .00S4 0.0085 .0086 .0087 .0088 .0089 0.0090 .0091 .0092 .0093 .0094 0.0095 .0096 .0097 .0098 .0099 O.OIOO .0101 .0102 .0103 .0104 0.0105 .0106 .0107 .0108 .0109 o.oi 10 .0111 .0112 .0113 .0114 0.0115 .0116 .0117 .0118 .0119 0.005 7298 .005 8243 .005 9 1 87 .006 0131 .006 1075 0.006 .006 .006 .006 .006 0.006 .006 .006 .006 .007 0.007 .007 .007 .007 .007 0.007 .007 .007 .007 .007 0.008 .008 .008 .008 .008 0.008 .008 .008 .008 .008 0.009 .009 .009 .009 .009 0.009 .009 .009 .009 .009 2019 2962 3905 4X47 5790 6732 7673 8614 9555 0496 1436 2376 3316 4^55 5 1 94 6133 7071 8009 8947 9884 0821 1758 2694 3630 4566 5502 6437 7372 8306 9240 0174 1108 2041 2974 3906 4838 5770 6702 7633 8564 0.009 9495 .010 0425 .010 1355 .010 2285 .010 3215 o.oio 4144 .010 5073 .010 6001 .010 6929 .010 7857 O.OIO 8785 .oio 9712 .Oil 0639 .oil .oil 1565 2491 3417 945 944 944 944 944 943 943 942 943 942 941 94' 941 94' 940 940 940 939 939 939 938 938 938 937 937 937 936 936 936 936 935 935 934 934 934 934 93 3 933 932 932 932 932 931 931 931 930 93° 93° 930 929 929 928 928 928 928 927 927 926 926 926 log «2 ma: 0.0120 0.0 .0121c .0 .0122' .0 .0123J .0 .01241 .0 ' 1 .0 0.0125; °-° .0126! .0 .OI27J .0 .01281 .0 .0129' .0 0.0130^ 0.0 .0131, .0 .0132 .0 .0133! .0 •01341 -o 0.0135! 0.0 .0136 .0 .0137 .0138 .0139 0.01401 0.0 .0141! .0 .OI42J ,0 .0143 ■° .0144 0.0145 .0146 .0147 .0148 .0149 0.0150 .0151 .0152 •0'53 .0154 0.0155 .0156 .0157 .0158 .0159 0.0160 .0161] .01621 .0163' .0164S 0.0165; .oi66| .0167; .oi68i .0169 0.0170 .0171 .0172 .0173 .0174 0.0175 .0176 .0177 .0178 .0179 0.0180 I 0.0 .0 .0 .0 .0 0.0 .0 .0 .0 .0 0.0 ,0 .0 .0 .0 0.0 .0 .0 .0 .0 0.0 .0 .0 .0 .0 0.0 .0 .0 .0 .0 0.0 .0 .0 .0 .0 0.016 3417 4343 5268 6193 7118 8043 8967 9890 0814 '737 2660 3583 4505 5417 6348 7269 8190 9111 0032 0952 1871 2791 3710 4629 5547 6465 7383 8301 9218 0135 1052 ! 1968 { 2884 1 3800 1 4716 1 5631 6546 j 7460 8374 9288 5 0202 5 5 5 5 5 5 5 5 1115 2028 2941 I 3854: 4766 56781 6589 I 7500' 841 1 ! 9322 0232 1142 2052 2961 3870 4779 5688 6596 7504 8412 926 925 925 925 925 924 923 924 923 923 923 922 922 921 921 121 i;2l 921 920 919 920 919 919 918 918 918 918 917 917 917 916 916 916 916 9' 5 9'5 914 914 914 914 913 913 913 913 912 912 911 911 911 911 910 910 910 909 909 909 909 908 908 908 C24 TABLE XIII. For finding the Ratio of the Sector to the Triangle. log .a 923 922 922 921 921 121 1 >t2I i 921 ' 920 919 920 919 919 918 918 918 918 9'7 i 9>7 1 917 916 916 916 916 9«5 915 914 914 914 914 913 913 913 913 912 912 911 911 911 911 910 910 910 909 909 909 Tol 908 908 0.0180 .0181 .0182 .0183 .0184 0.0185 .0186 .0187 .0188 .0189 0.0190 .0191 .0192 .0193 .0194 0.0195 .0196 .0197 .0198 .0199 0.0200 .0201 .0202 .0203 .0204 0.0205 .0206 .0207 .0208 .0209 0.0210 .0211 .0212 .0213 .0214 0.0215 .0216 .0217 .0218 .0219 0.0220 .0221 .0222 .0223 .0224 0.0225 .0226 .0227 .0228 .0229 0.0230 3231 .0232 .0233 .0234 0.0235 .0236 .0237 .0238 .0239 0.0240 0.016 8412 .016 9319 .017 0226 .017 1133 .017 2039 0.017 *94S .017 3851 .017 4757 .017 5662 .017 6567 0,017 .017 .017 .018 .018 0.018 .018 .018 .018 .018 0.018 .018 .018 .018 .019 747 » 8376 9280 0183 1087 1990 2893 3790 4698 5600 6501 7403 8304 9205 0105 0.019 1005 .019 1905 .019 2805 •019 3704 .019 4603 0.019 55°^ .019 6401 .019 7299 .019 8197 .019 9094 0.019 9992 .020 0889 .020 1785 .020 2682 .020 3578 0.020 .020 .020 .020 .020 4474 5369 6264 7159 8054 8948 9842 0736 1630 2523 3416 4309 5201 6093 6985 7876 8768 9659 0549 1440 0.022 2330 0.020 .020 .021 .021 .021 0.021 .021 .021 .02 1 .021 0.021 .021 .021 .022 .022 niff. 907 907 907 906 906 906 906 905 905 904 905 904 903 904 903 903 903 902 902 901 902 901 901 900 900 900 900 899 899 899 899. 898 898 897 898 897 896 897 896 896 895 895 895 895 89+ 894 894 894 893 893 893 892 892 892 891 892 891 890 891 890 logj5 0.0240 .0241 .0242 •0*43 .0244 0.0245 .0246 .0247 .0248 .0249 0.0250 .0151 .0252 .0253 .0254 0.0255 .0256 .0257 .0258 .0259 0.0260 .0261 .0262 .0263 .0264 0.0265 .0266 .0267 .0268 .0269 0.0270 .0271 .0272 .0273 .0274 0.0275 .0276 .0277 .0278 .0279 0.0280 .0281 .0282 .0283 .0284 0.0285 .0286 .0287 .0288 .0289 0.0290 .0291 .0292 .0293 .0294 0.0295 .0296 .0297 .02981 .0299! 0.0300' o 0.022 ,022 ,022 022 ,022 022 022 022 022 023 023 023 023 023 023 023 023 023 023 023 024 024 024 024 024 024 024 024 024 024 024 024 025 025 025 025 025 025 025 025 025 025 025 026 026 026 026 026 026 026 026 026 026 026 026 027 027 027 027 027 027 2330 3220 4109 4998 5887 6776 7664 8552 9440 0328 1215 2102 2988 387s 4761 5647 6532 74'7 8302 9187 0071 0956 1839 2723 3606 4489 5372 625A 7136 8018 8900 9781 0662 •543 2423 3303 4183 5063 5941 6821 7700 «S79 9457 0335 1213 2090 2967 3«44 4721 5597 6473 7349 8224 9099 9974 0849 1713 2597 347 « 4345 5218 Dicr. 890 8§9 889 889 889 888 888 888 888 887 887 886 887 886 886 885 885 885 885 884 885 883 884 883 883 883 882 882 882 882 881 881 881 880 S80 880 880 879 879 879 879 878 878 878 877 877 877 877 876 876 876 875 875 875 875 874 874 874 874 873 Io«»5 0.0300' .0301 .0302 .0303' .0304, 0.0305' .03061 •03071 .0308, .0309 0.0310 .031 .0312 •03>3 .0314 0.0315 .0316 .0317 .0318 .0319 0.0320 .0321 .0322 .0323 .0324 0.0325 .0326 .0327 .0328 .0329 0.0330 .0331 .0332 .0333 .0334 0.0335 .0336 •0337 •0338 .0339 0.0340 .0341 .0342 •°343 •0344 0.0345 .0346 .0347 .0348 .0349 0.0350 .0351 .0352 •0353 .0354 0.0355 .0356 •0357 .0358 .0359 0.0360 0,027 .027 .027 .027 .027 0.027 .028 .028 .028 .028 0.028 .028 .028 .028 .028 0.028 .028 .029 .029 .029 0.029 .029 .029 .029 .029 0.029 .029 .029 .029 .030 0.030 .030 .030 .030 .030 0.030 .030 .030 .030 .030 0.030 .031 .031 .031 .031 0.031 .031 .031 .031 .031 0.031 .031 .032 .032 .032 0.032 .032 .032 •032 .032 0.035 7120 DifT. 5218 6091 6964 7836 8708 9580 0452 >3i3 2194 3065 3936 4806 5676 6546 74«5 8284 9«53 0022 0890 1758 2626 3494 4361 5228 6095 6961 7827 8693 ! 9559 ; 04241 1290 : 2154 1 3019 ! 38831 4747 56m| 6475 7338 8201 9064 9926 0788 ; 1650 ; 2512 ' 3373 . 4234 1 5095 5956 6816 7676 8536- 9396 0255 ; 1114 1973 1 2831 ! 3689! 4547 5405 6262 873 873 87a 872 S72 872 871 871 871 871 870 870 870 869 869 869 869 868 868 868 868 867 867 867 866 866 866 866 865 866 864 865 864 864 864 864 863 863 863 862 862 862 862 861 861 861 861 860 860 860 860 859 859 85? 858 858 858 858 857 858 40 U25 TABLE XIII. For findinp tlie Kiitio of ilie tScctor to the TrinnRlc. logja 0.0360 .0361 .0362 .0363 .0364 0.03651 .0366 .0367 .0368 .0369 0.0370 .0371 .0371 •0373 .0374 0.0375 .0376 .0377 .0378 .0379 0.0380 .0381 .0382 .0383 .0384 0.0385 .0386 .0387 .0388 .0389 0.0390 .0391 .0392 .0393 .0394 0.0395 .0396 .0397 .0398 .0399 0.040 .041 .o.;2 .043 .044 1 0.045 .046 .047 .048 .049 0.050 .051 .052 .053 .054 0.05s .056 .057 .058 .059 0.060 0.032 .032 032 033 033 °33 033 033 033 033 °33 033 033 033 034 034 034 034 034 034 °34 03+ 034 034 034 03 s 035 035 035 035 03 s 035 035 PifT. 7120 7976 i<^33 9689 0546 1401 j **57 I 31 12 ; 3967 i 48221 5677 I 6531 73«5i 8239 9092 9946 0799 1651 2504 3356 420S 5059 6762 7613 8464 93«4 0164 1014 1864 2713 3562 441 1 5^59 6108 03s 6956 035 7804 035 8651 035 9499 036 0346 036 1192 036 9646 037 8075 I 038 6478 039 4856 040 3209 041 1537 041 9041 042 8121 ° 043 6376 p^l 044 4607 ■ 856 857 856 857 855 856 855 855 85s 855 854 854 854 853 854 852 853 852 852 85, 852 85, 850 850 850 850 849 849 849 848 849 848 848 847 848 847 846 8454 8429 8403 8378 8353 8328 8304 8280 045 281J 8207 046 0997 \l\ll 046 9»57' '^° 047 7294 048 5407 ^'37 !ii3 8089 049 3496 '^ 052 5626 I 0.060 .061 .062 .063 .064 0.065 .066 .067 .068 .069 0.070 .071 .072 •073 .074 0.075 .076 .077 .078 .079 0.080 .081 .082 .083 .084 0.085 .086 .087 .088 .089 0.090 .091 .092 .093 .094 0.095 .096 .097 .098 .099 0.1 00 .101 .102 .103 .104 0.105 .108 .109 O.IIO .114 0.1 1 5 .116 .117 .118 .119 logs! I Dinr. 0.052 .053 .054 .054 .055 0.056 .057 .058 .058 .059 0.060 .061 .061 .062 .063 0.064 .065 .065 .066 ,067 0.068 .068 .069 .070 .071 0.071 .072 .073 .074 .074 0.075 .076 .077 .077 .078 5626 I , 3602 ''^76 1556 7954 9488 7932 7107 7909 7397 ^ijgg 0994 7»44 88.7 7823 66,8|78oi 7780 4398 I 2157 ^^^59 989 |7738 76 1 2 7717 53°« life 2984 , 0639 765s 8274:7635 5888:''6'4 3482,7594 ^^ 7575 1057 8612 6146 3661 II 7575 7555 75 34 7515 )6l '3*3 ■57 ■^■^'^A ^' 7476 8633 6000 !7457 3527 m 0945 ,7+'8 8345 ^T ^^■' 7380 5725 3087 0430 77S4 5060 0.079 2348 .079 9617 7362 7343 7324 7306 7288 7269 80 6868 7=51 .081 4101 .082 1316 0.082 8513 .083 5693 .084 2854 .084 9999 .085 7125 0.086 4235 .087 1327 .087 8401 .088 5459 .089 2500 0.089 9523 .090 6530 .091 3520 .092 0494 .092 7451 0.093 4391 .094 1315 .094 8223 .095 5114 .096 1990 0.096 8841 7233 7215 7197 7180 7161 7145 7126 7110 7092 7074 7058 7041 7023 7007 6990 6974 6957 6940 6924 6908 6891 6876 6859 0.120 121 ,122 123 124 »25 126 127 128 129 130 131 132 133 134 135 136 '37 138 139 140 141 142 143 144 146 '47 148 149 150 «Si 152 153 154 "55 156 157 158 159 i6c 161 162 163 164 165 166 167 168 169 170 '71 172 '73 '74 175 176 177 178 179 logi« 0.096 ,097 098 098 099 100 100 lOI 102 102 103 104 104 105 106 106 107 108 108 109 110 I 10 III 112 112 113 114 114 116 116 "7 118 118 119 Diff. 8849 5692 2520 933« 6127 2907 9672 ; 6421 I 3'54; 9873! 6576 i 326A; 9936 6594 3237 9865 647 8 9660 6229 2783 9323 5849 2360 8§S7 5340 1809 8264 4704 1131 7S44 3943 0329 6701 3059 6843 6828 6S11 6796 6780 6765 '■'749 6733 6719 6703 6688 6672 6658 6643 6628 6613 6598 6584 6569 I 655t 6540 6526 6511 6497 6483 6469 645s 6440 6427 6413 6399 6386 6372 6358 6345 33' 119 9404 , !!? 5735 e^Jg 6304 6292 6278 6265 6*252 121 2053 121 8357 122 4649 123 0927 123 7192 124 3A44 124 9082 125 5908 126 2121 126 8321 ttt!6238 6226 '3 62 I 62 6187 127 4508 V 128 0683 I ^'75 128 6845 '"''^ 129 2994 129 913' i 6 «30 5255 I 6 131 1367 6165 149 6137 124 131 7466 132 3553 i 132 9628 I 133 5690, 134 1740 60Q 608 6075 6062 6050 6038 347778:^^30 135 3804 we TABLE Xm. For fincling the Riitio of tla- Sector to the Triangle. 1"K »« Did'. 6^.6 65.. 1 6497 r 6483 6469 6455! 6440 , 6427 6413 6399 6386 1 I 6371 : 163581 ! 6345 j 63311 6318 6304 6292 i 6278 1 6265 j 6252 1 6238 I 6226 16213! i 6200 i 6187 I 6175 1 6162 1 6149 1 6137 6124 6112 6099 6087 6075 6062 6050 6038 6026 0.180 .l8i .182 .i8j .184 0.181; .186 .187 .188 .189 0.190 .191 .192 .193 .194 0.195 .196 .197 .198 .199 0.200 .201 .202 .203 .204 0.205 .206 .207 .208 .209 o.iio .211 .212 .213 .214 0.215 .216 .217 .218 .219 O.220 .221 .222 .223 .224 0.225 .226 .227 .228 .229 0.230 .231 .232 .234 0.23s .236 •237 .238 .239 0.240 0.135 .136 •«37 •«37 0.138 .138 .139 .140 .140 0.141 .141 .142 .143 •«43 0.144 •'44 .145 .146 .146 0.147 .147 .148 .148 .149 0.150 .150 .151 .151 .152 0.152 •«53 •'54 •'54 •'55 0.155 .156 .156 •157 •'57 0.158 •'59 •'59 .160 .160 0.161 .161 .162 .162 .163 0.164 .164 .165 .165 .166 0.166 .167 .167 .168 .168 3^i 98 1 8 5821 1811 7789 3755 9710 5653 158;; 75°4 3412 9309 5'94 1068 6931 2782 8622 4450 0268 6074 1869 7653 34*7 9189 4940 068 1 641 1 2130 7838 3535 9222 4899 0565 6220 1865 7499 3113 »737 4340 9933 5516 1089 6652 2204 7747 3J79 8802 43'5 9817 S3'o 0793 6267 1730 7184 2628 8063 8903 4309 9705 0.169 5092 6014 6003 5990 5978 5966 5955 5943 5';32 5')i9 5908 5897 5885 5874 I585' 15840 5828 5818 5806 5795 5784 15774 575« 1574' 5730 57'9 15708 5697 15687 '5677 5666 5655 5645 5634 5624 5614 5603 5593 5583 5573 5563 15552 5543 5532 5523 5513 5502 5493 5483 5474 5463 5454 5444 5435 5425 5415 5406 5396 5387 1 0.240 0. .241 .242 •*43 .244 0.245 0. .2+6 . .247 .248 .249 0.250 0. .251 .252 •*5 3 .254 0.155 0. .256 •*57 . .258 .259 0.260 0. .261 .262 . .263 .264 0.265 0. .266 .267 .268 .269 0.270 0. .271 • .272 •*73 , .274 0.275 0. .276 • .277 . .278 .279 0.280 0. .281 .282 .283 .284 0.285 0. .286 .287 .288 .289 0.290 0. .291 .292 .293 .294 0.295 0. .296 .297 .298 .299 0.300 0. U>g «3 .169 .170 .170 •17' 171 172 172 «73 173 '74 •'74 •'75 •'75 .176 .176 •'77 178 .178 '79 •'79 180 .180 .181 .181 .182 1S2 184 184 '!« '85 186 186 187 187 188 18S 189 189 190 190 191 191 192 192 '93 93 194 194 •'95 .195 .196 .196 •'97 •'97 .198 .198 '99 •'99 5092 0470 5838 1197 6547 1887 7218 2540 7853 3156 8451 3736 9013 4280 9538 4788 0029 5261 04S4 5698 0903 6100 1288 6467 1638 6800 '953 7098 2235 7363 2483 7594 2696 779' 2877 7955 3024 8085 3138 ^183 3220 8249 3269 8281 3286 8282 3271 8251 3224 8188 I 3'45 ! 8094; 7968 : "94 i 78"! 2721 I 7624 I 2518 I 7406 200 2285 Ditr. 5378 5368 359 350 340 331 322 3'3 303 295 28 5 277 267 258 250 241 232 223 214 205 '97 188 '79 '7' 162 '53 '45 137 128 120 1 11 102 095 086 078 069 061 °53 045 037 029 020 012 005 4996 (4989 '4980 4973 4964 '4957 4949 494' 4933 4926 4917 4910 4903 4894 4888 4879 I logi« 0.300 I 0.200 301 ' .200 ,302 .201 303 .201 .304 .202 505 0.202 306 .203 307 .203 308 .204 1050 309 .204 1285 7'57 2021 6878 1727 6569 1403 6230 310 311 312 3'3 , 3'4| 3'5| 316 I 3'7 , 318, 3 '9 1 0.205 .205 .206 .206 .206 0.207 .207 .208 .208 .209 I 320 I 0.209 321 t .210 322 i .210 323 j .211 3-4 ! ^2" 325 326 I 3*7 i 328' 329 330 I 33' ! 332 i 333 I 3 34 ! 336 337 338 339 34° 341 342 343 344 0.212 .212 .213 •2'3 .214 0.214 .214 .215 .215 .216 0.216 .217 .217 .218 .218 0.219 .219 .220 .220 .220 345 I o-"' 346 I .221 347 ! •2*2 348 ! .222 349 i ^223 35° 35' 352 353 354 356 357 358 359 360 0.223 .224 .224 .225 .225 0.225 .226 .226 .227 .227 0.228 5862 0667 5464 0254 5037 9813 4581 9342 4096 8843 3582 83.5 3040 7759 2470 7'74 1871 6562 1245 5921 0591 5253 9909 4558 9210 3835 8464 3085 7700 2308 6910 1505 6093 0675 5250 9818 4380 8935 3483 8025 2561 7090 1613 61 30 0640 5143 9640 4131 8615 3093 7565 2031 Ditr. 4872 • 4864 14857 1 4849 I 4842 J4834 ; 4820 ' 4812 J4805 j 4797 I 4790 478J 4776 4768 4761 i 4754 4747 4739 4733 4725 47 '9 47" 4704 4697 4691 4683 4676 4670 4662 I 4656 ■ 4649 j 4642 14635 I 4629 '' 4621 4615 4608 4602 ' 4595 4588 .4582 4575 4568 j 4562 '' 4555 4548 4542 4536 4529 4523 45 '7 45'° 45°3 4497 4491 4484 I 4478 4466 027 TABLE XIII. For finding the Ratio of llie Swtor to tlu> Triangle, logd" Din. 0.560 0.228 .361 .22X .362 •363 .364 0.365 .366 .367 .36S .369 0.370 •371 •37» •373 •374 o-375 •376 •377 .37S •379 0.380 .38. .382 ■383 .384 0.38s .386 .387 .388 .389 0.390 .391 .392 •393 ^'•395 V)6 •397 •398 •399 0.400 .401 .402 .403 .404 0.405 .406 .407 .408 .409 0.410 .411 .412 .413 .229 .229 .229 0.230 .230 .231 .231 .232 0.232 •*3 3 •*33 .233 .234 0.234 .235 .235 .236 .236 0.237 .237 .237 .238 .23S 0.239 .239 .240 .240 .240 0.241 .241 .242 .242 .243 0.243 .243 •24+ .244 .245 0.24s .245 .246 .246 .247 2031 6490 0943 5390 9831 A265 5694 3116 753* 1942 6346 0743 5135 9521 3900 8274 2642 7003 '359 5709 0053 439' X723 3050 7370 1685 5993 0296 4594 8885 3171 7451 1725 5994 0257 45'4 8766 3012 7252 1487 5716 9940 4158 8371 2578 0.247 6779 .248 0975 .248 5166 .248 9351 •249 3531 0.249 7705 .250 1874 .250 6038 .251 0196 4459 4453 i4447 4441 4434 '4429 4422 '4416 I4410 ,4404 14397 '43/2 43^6 143/ 9 :4374 14368 4361 4356 4350 4344 4338 4332 4327 4320 43«5 4308 43°3 4298 4291 4286 4280 4274 4269 4263 4257 4252 4246 4240 4235 4229 4224 4218 4213 4207 4201 c 4»96 4191 J4185 4180 i4»74 I4169 :4>64 ;4i58 14153 ,4>47 252 2^38 ;4;42 •252 6775 .414 I .251 4349 1.415 o.;;5i 8496 .416 .252 20 .417 .418 .419 1.420 .253 090 •253 5°32 0.253 9'53 4'37 6'4'3 4126 4121 0.420 .421 .422 •423 •424 425 426 427 428 429 430 43' 43?. 433 434 ■t55 43' 43-' 438 439 440 441 442 443 444 445 446 447 448 449 450 45« 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 lo«,VJ 0.253 •254 .254 •255 .255 0.255 .256 .256 •257 •257 0.258 .258 .,.58 •259 .259 0.260 .260 .260 .261 .261 0.262 .262 .262 .263 .263 0.264 .264 .264 .265 .265 0.266 .266 .266 .267 .267 0.268 .268 .268 .269 .269 0.269 .270 .270 .271 .271 9' 5 3 3269 7 379 1484 5584 9679 3769 7853 1932 6006 0075 4»39 8198 2252 6300 0344 4382 8415 2444 6467 0486 4499 8507 6509 0503 4492 8475 2454 6428 0397 4362 8321 2276 6226 0171 4111 8046 1977 5903 9824 3741 7652 '559 5462 Diir. 4116 4110 4«o5 4100 !4°95 1 4 090 4084 14079 i4074 4069 4064 4059 I4054 14048 4°44 4038 4033 4029 4023 4019 4013 4008 4004 3998 3994 3989 3983 3979 3974 3969 3965 3959 3955 395° 3945 3940 3935 393' 3926 3921 I 39«7 391 1 3907 3903 3898 3893 0.271 9360 1 3884 3879 3874 3870 3865 3861 3856 3852 3847 3842 3838 3834 .272 7141 .273 1025 •273 49041 0.273 8778 1 .274 2648 ! •274 6513 •275 0374 .275 4230 0.275 8082 .276 1929 .276 5771 .276 9609 ■277 3443 3^8^5 0.277 7272 628 lug «' - 480 0.277 7272 481 .278 1096 482 .278 49 1 6 483 .278 8732 484 .279 2543 485 0.279 6349 486 .2X0 0151 487 .280 394'» 488 .280 7743 489 .281 1532 490 0.281 5316 491 .281 9096 492 .282 2872 493 .282 6644 494 .283 041 1 495 0.283 4'73 496 .28^ 7932 497 .284 1686 498 .284 5436 9181 499 .284 500 0.285 2923 501 .285 6660 502 .286 0392 503 .286 4121 5°4 .286 7845 505 0.287 1565 506 .287 5281 5°7 .287 8992 508 .288 2700 509 .288 6403 510 0.289 0102 51' .289 3797 512 .289 7487 513 .290 "74 4856 514 .290 S'5 0.290 8535 516 .291 2209 5'Z .291 5879 518 .291 9545 519 •292 3207 520 0.292 6864 051! 521 •293 522 •293 4168 523 •293 7813 524 •294 '455 525 ' 0.294 5092 1 526 .294 8726 527 1 -295 2355 528 .295 5981 529 195 96c 530 I 0.296 3220 .296 6833 53' 532 533 534 535 536 537 538 539 540 •297 0443 •297 4049 .297 7650 0.298 1248 .298 4842 3.594 .298 8432 359 .299 201 8^35 .299 5600 0.299 9*78 3582 3578 1)1 ir. 3824 3820 3816 38,1 3806 3802 3798 3794 378<; 3784 3780 3776 3772 3767 3762 3759 3754 3750 3745 3742 I 3737 I 3732 i 3729 3724 3720 3716 37" , 3708 ; 3703 : 3699 i 3695 i 3690 ! 3687 3682 3679 3674 3670 3666 3662 3657 j 3654! 365° i 3645 , 3642 I I 3637 3634 : 3629 3626 ! 3621 i 36,8 j 3613 i 3610 3606 3601 3598 ntff. 3824 3806 j 3802 , 379« 3794 378<; ■ 3734 3 i 378° , i 3776 -i 3772 \ 3767 376a 3I 2 6 16 li '3 30 I JJ] ZI 45 §sl 9* I 'oo ! r°3 35 09' 79 45 •07 ', 68 i 13: •55 1 192 1 26 55 .81 02 20 33 43 '49 3759 3754 ! 3750 ' 3745 374* ) 3737 373* 3729 3724 3720 1 37»6 37'' 3708 3703 3699 3695 3690 3687 3682 3679 I 3674' 3670 3666 3662 ; 3657 ; 3654 ! 3650 1 3645 , 364* I' I 3637 j 3634 ' 3621 1 3618 3613 3610 3606 3601 ,50 48 }32i 18 ': 00] 78 i 359» 3594 359° 3586 3578 TABLE XIII. For fmilint^ the I{;itio of i\w Sector to tlu- TriiiiiKlc. 1 luK-.l 0.540 0.499 9'7'' 54' .300 2752 54» .300 6323 .300 9890 543 544 .301 3452 545 0.301 7011 546 .302 0566 547 .302 4117 54« ,302 7664 549 .303 J208 550 0.303 A748 : .303 828J. .304 1816 551 55* 553 •3°4 5344 554 .304 8869 555 C.305 2390 55b •305 5907 557 .305 9420 55« .306 2930 559 .306 6436 560 0.306 9938 , I HIT. 3574 357' 3 5<'7 3562 3559 3555 355' 3547 3544 3540 3536 35 3* 3518 3515 35*' 35'7 35'3 3510 3506 3502 i'>H.i^ 0.560 .^61 .562 .563 .564 0.565 .566 .567 .568 .569 o.;70 •57' ■57» •573 •574 0.575 .576 •577 .578 •579 0.580 0.306 .307 .307 .308 .308 0.308 .309 .309 .309 .310 0.310 .310 •3" •3" ,311 0.312 .312 .312 •3'3 ■3'3 0.313 9215 9938 3437 69 3 • 0422 3910 7394 0874 4350 7823 1292 4758 8220 1678 5'33 8584 2031 5475 8915 1351 5785 niff. 3499 3494 349" 3488 3484 3480 3476 3473 3469 3466 3462 3458 345 5 345' 3447 3444 3440 3437 343 3 3430 0.5S0 .581 .582 •583 .584 0.585 .586 ■587 .588 .589 0.590 •59' .592 •593 •594 0.595 .596 •597 .598 •599 Q.600 log i« i Piff. 3'3 3'4 3'4 3'4 •'5 3'5 3'5 3.6 316 316 3'7 3'7 318 318 318 3'9 319 3'9 320 320 9115 26a I 6064 9483 2898 6310 97 '9 6525 9913 3318 6709 0096 3480 6861 0238 3612 6983 0350 37'4 320 7074 3426 34M 34" 9 34'5 3412 3409 340^ 3401 3398 3395 3391 3387 3384 338, 3377 3 374 3367 3364 3360 TABLE XIV. For finding the Kalio of the Sector to the Triangle. Elliiiac. ! 0.000 0.000 0000 .001 .000 0001 .002 .000 0002 .003 .000 0005 .004 .000 0009 0.005 0.000 0014 .006 .000 0021 .007 .000 0028 .008 .000 0037 .009 .000 0047 O.OIO 0.000 0058 .oil .000 0070 .012 .000 0083 .013 .000 0097 .014 .000 0113 0.015 0.000 0130 .016 .000 oia8 0167 .017 .000 .018 .000 0187 .019 .000 0209 0.020 0.000 0231 .02 1 .000 0255 .022 .000 0280 .023 .000 0306 .024 .000 0334 0.025 0.000 0362 .026 .000 0392 .027 .000 0423 .028 .000 0455 .029 .000 0489 3.030 0.000 0523 Diff. Iljporboln. I I 3 4 S 7 7 9 10 II 12 '3 14 16 '7 18 19 20 22 22 *4 *5 26 28 28 30 3' 3» 34 34 Diff. 0.000 0000 .000 0001 .000 0002 j .000 0005 .000 0009 0.000 0014 ; .000 0020 i .000 0028 ' .000 0036 I .000 0046 i 0.000 0057 ; .000 0069 I .000 0082 ' .000 0096 1 .000 01 II 0.000 .000 .000 .000 .000 0.000 .000 .000 .000 .000 0127 0145 0164 0183 0204 0226 0249 0273 : 0298 i 0325 ! 0.000 0352 .000 0381 I .000 0410 ; .000 0441 '' .000 0473 ! 0.000 0506 ' 10 II 12 '3 '4 '5 16 18 19 '9 21 22 23 S4 15 *7 27 29 29 3' 3» 33 0.030 .031 .032 .033 .034 0.035 .036 .037 .038 .039 0.040 .041 .042 .043 .044 0.045 .046 .047 .048 .049 0.050 .051 .052 .053 .054 0.055 .056 .057 .058 .059 0.060 t s EIII118P. 0.000 0523 .000 0559 .000 0596 .000 0634 .000 0674 0714 0756 0799 0844 0889 0.000 .000 .000 .000 .000 0.000 .000 .000 .000 .000 0.000 .000 .000 .000 .000 0936 098.1 1033 1084 "35 1188 1242 1298 '354 1412 0.000 1471 .000 1532 .000 1593 .000 1656 .000 1720 0.000 1785 .000 1852 .000 1920 .000 1989 .000 2060 0.000 21 31 Difr. 36 38 40 40 4a 43 45 45 47 48 49 5' 5' 53 ^t ^f, ^t 58 59 61 61 64 65 67 68 69 71 7' Iljiiorliola. 0.000 .000 .000 .000 .000 0.000 .000 .000 .000 .000 0.000 .000 .000 .000 .000 0.000 .000 .000 .000 .000 0.000 .000 .000 .000 .000 0.000 .000 .000 .000 .000 0.000 0506 °539 0575 061 1 0648 0686 0726 0766 0807 0850 0894 0938 0984 1031 1079 1128 1178 , 1229 1281 I '334 i 1389! '444! 1500 '558! 1616 I 1675 1736 1798 i860 1924 1988 DifT. 36 38 40 40 4' 43 44 tt ii 49 50 5' 5» S3 55 '! Is 59 61 62 62 64 64 62U TABLE XIV. For (ii)(Iing Uie ]\;itii) of the Sei'tor ici tlie Triangle. ^ KlliiMi!. Diff. "I,vi«'iI)oItt. I Din. 0.060 .061 .062 .063 .064 0.065 .066 .067 .068 .069 0.070 .071 .072 .073 .074 0.075 .076 .077 .078 .079 0.080 .081 .0S2 .0S3 .084 0.085 .086 .087 .088 .089 0.090 .091 .092 •093 .094 o.cg5 I 0.000 .096 I .coo 0.000 .oco .000 .000 .000 0.000 .000 .000 .coo .000 0.000 .000 .000 I .000 i .000 I 0.000 .oco .000 .000 .coo 0.000 .000 .000 .000 .000 0.000 .000 .coo .000 .000 0.000 .coo .000 .000 .coo .098 .099 000 CJO oco o.ioo I 0.000 .101 : .ooo .102 [ .000 .103 I .coc .104 .000 0.105 .106 .107 .108 .109 o.iio .II I .112 •"3 114 o.ii; .116 .117 .118 ,H9 0.000 .000 .000 .000 .000 coco .000 .000 .oco .oco 0.000 .000 .000 .000 .000 2131 2204 2278 2354 2431 2509 2588 2669 -751 2834 2918 3004 3091 3180 3269 3360 3453 3546 3^*4' 3738 3«35 3934 4034 4136 4139 4343 4448 4555 4663 4773 4884 4996 5109 S^H , 5341 5458 5577 5697 5819 5941 6066 6192 <^3'9 6448 6578 6709 6842 6976 7111 7248 7386 7526 7667 7809 7953 8098 8245 8393 8542 8693 73 7-, 76 77 78 79 81 82 84 86 87 l9 89 91 93 93 95 97 97 99 CO 02 °3 «4 I 05 I 07 I 08 I 10 I I O.I 20 I O.OOC 8845 12 13 15 17 J7 '9 20 23 24 26 2 7 19 30 31 33 34 35 37 38 40 41 42 44 45 47 48 49 S« 51 0.000 .oco .000 .000 .000 0.000 .coo .oco .000 .000 0.000 .000 .000 .000 .coo 0.000 .000 .000 .000 .000 0.000 .000 .coo .000 .000 c.coo .000 .000 .000 .000 0.000 ,000 .000 .000 .000 0.000 .000 .000 .000 .000 0.000 .000 .000 .000 .000 0.000 .000 .000 .000 .000 0.000 .000 .000 .000 .000 0.000 .000 •coo .000 .000 0.0 DO 1988 1 2054 ; 2I2I 2189 : 2257 1 2327 j 2398 2470 1 2543 2617 2691 2767 2844 2922 3001 :,o8i 3162 3244 3327 34" 3496 1 3582 j 3669 ! 3757 : 3846; 393*' 4027 j 4119 4212 - 4306 j 4401 449(1 i 4n93 4691 4790 4890 ' 4991 5092 5 "95 5299 , 5403 ' 55'^9 5616 5723 I 5852 I 594» 6052 6163 6275 6389 0503 ■ 6618 6734 6851 6969 7088 ■ 7208 ' 7 3 '^9 745 > 7574 7<''9^' 66 67 68 68 70 71 72 73 74 74 "6 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 9? 95 97 98 99 100 lOI 10 1 103 104 104 106 107 107 109 109 III II I 112 114 114 115 116 H7 118 119 120 1 2 1 122 123 15.4 Klli|.sc. I Diff. lI.v|)(rbolu. Diff. 20 j 0.000 2 I 24 25 26 I 27 I 28 I 29 ] .000 i .000 I I .000 I .oco } 0.000 .000 i .000 .00 .00 30 31 32 53 .34 I 0.00 .00 .00 .00 .00 35 I 0.00 36 I .00 37 ' .00 38 I .00 39 i -oo 40 I 0.00 41 ! .00 .00 .00 .00 42 43 44 45 o.co 46 .00 47 I .00 48 ; .00 49 : .00 50 o.co 51 : .00 52 \ .00 53 I -O" 54 , -co 0.00 .00 .00 .00 .00 5'^ : 56 58! 59 60 i 0.00 61 62 63 64 65 66 67 68 69 7= 71 7'-' 73 74 75 76 77 78 79 1.^0 .00 .00 .00 0.00 .00 .00 .00 .00 0.00 .00 .00 .oc .00 o.co .00 .00 .002 .002 0.002 8845 8999 9»54 93 1 1 9469 9628 9789 9951 CI 15 0280 0447 061 5 0784 0955 1128 1301 '477 1654 1832 2012 2193 2376 2560 i 2745 2933 I 3121 I 33«« I 3';o3 i 3696 1 389' j 4087 j 4285 4484 4684 4886 I 5090 j 5295 1 5502 - 5710 5920 ; 6131 6344 i 6559 '>775 6992 721 ( ; 7432 7654 : -S78 8103 ; 8330 I 8558 I 8788 I 9020 I 9253 { 9487 9724 9961 02Cl 0442 '^f'85 »54 155 '57 158 '59 161 162 164 165 167 168 169 171 173 '73 176 '77 178 180 181 •83 184 185 188 188 190 192 '93 195 196 198 199 200 202 20,). 205 207 io8 210 211 i'3 2'5 216 217 Z19 221 22X 224 225 227 228 230 232 " iS 234 237 '37 240 241 243 7698 7822 794!< 8074, 8202 I 8330' 8459 ; 8590 ; 8721 i 8853 j 8986 I 9120 j 9255 I 9390 9527 j 9665 i 9803 i 9943 ] 0083 ! 0224 I 0366 I 0509 I 0653 [ 0798 I 0944 j 1 09 1 i 1238 i 1387' 1536 ' 1686 ,838 j 1990 2'43 : 2296 I 2451 I 2607 I 2763 ! 292T : 3C79 ! 3238 , 3398 j 3 i-f' 176 I TABLE XIV. For finding tlic Iviitio of tla- Sector to the Triangle. >.ooi 678a i Kllipse. Diff. Ilyperbolii. Diir. 178 179 180 0.180 0.002 0685 O.OOI 6782 .iSi .182 ' .1X3 ,002 .002 .002 0929 i>75 1422 244 246 247 .001 .001 .001 (i960 7139 7319 .184 .002 1671 249 .001 7500 iSl ' 0.18; 0.002 1922 0.001 7681 .186 .187 .002 .002 2174 2428 252 »54 .001 ,001 7864 8047 1^3 '83 ; .188 .1S9 .002 .002 2(183 2941 255 258 258 .001 .001 Si 3 1 8416 iS4 185 :86 , 0.190 0.002 3 '99 261 262 263 266 O.OOI 8602 187 187 189 1S9 190 ' .191 .192 .002 .002 3460 3722 .001 ,001 8789 8976 .193 .002 3985 .001 9165 ! ''94 .002 4251 267 .001 9354 0.195 0.002 451S 268 270 O.OOI 9544 191 191 .196 .197 .002 .002 4786 5056 .COI .001 9735 9926 .198 .002 5328 272 .002 0119 0312 '93 .199 .002 5602 274 275 .002 193 J95 0.200 .201 0,002 .002 5877 6154 177 0.002 .002 0507 0702 «9; .202 .203 .204 .002 .002 .002 6433 6713 6995 279 280 282 283 .002 .002 .002 0897 1094 1292 '95 '97 ,98 19S . 0.205 .206 .207 .208 0.002 .002 .002 .002 7278 7564 7S5, 8139 286 287 288 C.O02 .002 .002 .002 1889 2090 199 ?oo 201 .209 .002 8429 290 2(;j .002 2:91 201 203 0.210 .211 .212 0.002 .002 .002 8722 9...T5 9311 207 296 0.002 .002 .002 2494 2(197 2901 203 204 .213 .002 9608 9907 297 .002 3106 331' 205 .214 .002 299 300 .002 205 207 0.215 ■ .216 I -217 .218 .219 1^.003 .003 .003 .003 .003 0207 0509 OS; t 1 1 19 1427 302 30s 305 308 3C9 0.002 .002 .002 .002 .002 3518 3725 3931 4142 435i 207 207 210 2IO 210 0.220 .221 .222 0.003 .003 .003 1736 2047 ^359 311 312 316 318 0.002 .002 ,002 4774 4986 211 212 113 *'5 1 ••■^^3 .224 .003 .003 2674 2990 .002 .002 5«99 54«i 0.2?,:; .226 .227 0.003 .003 .003 3308 3(127 3949 3-0 0.001 .CCi2 .00?. 5627 5842 (1058 216 .228 .229 .003 ,003 4272 4597 3^3 3*5 327 .002 .002 6275 6493 217 21s 218 0.230 .231 .232 •■233 0.003 .003 .003 .00 ? 4924 5252 5582 59'4 328 330 332 0.002 .002 ,002 .002 6711 6931 71 ;i 7371 220 220 220 .234 .003 6248 334 336 .002 7593 2 22 223 0'23.5 .236 .237 .238 •'•39 0.003 .003 .003 .003 .C03 6584 6921 7260 7601 7944 337 339 34: 343 345 0.002 .002 .002 .002 .002 7816 8039 X263 8487 87,3 223 224 224 2 2fi 22(1 0.240 0.003 8289 0.002 8939 c Ellipse. 'I 0,240 ] 0.003 8289 .241 j .003 8635 .242 ; .003 {',983 .243 ' .003 9333 .244 I .003 9(185 C.245 .24(1 •247 .248 ,249 0.250 •15' : .252 •253 •154 0.155 I .256 1 ■257 ': .258 i •259 j 0.260 I .261 i .262 ; .263 I .264 I 0.265 ' .266 I .267 ' .268 j .269 I 0.270 .271 •272 •273 •274 : 0.275 .276 .277 .2-8 .279 0.280 .281 .282 .283 .284 0.285 .286 .287 .288 .289 0.290 .291 .292 .193 .294 C'-295 .296 .297 .298 •299 0.004 .004 .004 .004 .004 0.004 .004 .004 .004 .004 0.004 .004 .004 .004 .004 0.004 .004 .004 .004 .004 0.004 .004 .0C4 .004 .004 c.004 .004 .005 .005 .005 0.005 .005 .005 .005 .005 0.005 .005 .005 .005 .005 0.005 .005 .005 .005 .005 0.005 .005 .005 .005 .005 0.006 .006 .006 .006 .006 Diff. 1 IIjpi.-rlMilii. Diff. 0039 0394 0752 11 11 I '472 j '835 I 2199 2566 i 2934 33°5 3677 4051 4427 4804 5184 5566 5949 '1334 i 6711 7111 7502 I 7894 1 S289 ; 8686 I 9085 I 9485 I 9888 I 0292 I 0699 ; 1107 j '5'7 I 1930 2344 2760 3178 359S 4020 4444 4870 5298 5728 6160' 65941 7010, 7468 ! 7908 i 8350 8795 9241 ' 9(189 0139: o5<)i : 1045 1 502 i960 : C.3OO D.Ocd 2421 346 348 350 352 354 355 35'^ 359 361 363 364 367 368 37' 372 374 376 377 380 382 387 390 391 392 395 397 399 4c o 403 404 407 408 410 4'3 414 416 418 420 422 424 J 26 428 430 432 434 436 43S 440 442 445 44'' 448 450 452 454 457 4 = 8 461 0.002 8939 .002 .002 9166 9394 217 228 .002 .002 9(123 9852 229 229 231 0.003 0083 .003 03 '4 231 233 .003 054^ .003 077S .003 101 1 233 234 0.003 .003 .003 1245 1480 1716 235 236 .003 1952 .003 2189 237 238 0.003 2427 .003 1666 239 .003 2905 239 .003 3146 241 .003 3387 241 241 0.003 3628 .003 3^*71 243 ■>°3 .003 4114 435« 2-^3 2^4 .003 4603 -4l 245 0.003 .003 4848 5094 246 24: 248 .003 .003 534^ 55^9 .003 5838 249 249 0.C03 6087 .003 f'337 250 .003 (1587 .003 .003 6839 7091 252 252 253 0.003 7 344 2'>J .003 759'' .003 7852 2s4 .003 .003 8107 8363 256 257 0.003 .003 8620 8877 =^V .003 9135 259 260 260 1 -003 .003 9394 9^' 54 ! 0.003 9914 -61 1 .004 0175 26-' ' .C04 0437 .63 263 264 .004 .004 0700 0963 0.004 .004 1227 H91 264 266 .004 1757 266 .004 2023 267 .004 2290 267 c.004 .004 .004 2826 3095 269 169 269 271 271 .004 3 3<'4 .004 3<'35 0.004 39°^ (i.'il I : w TABLE XV. For Elliptic Orbits of j;reat eccentricity. TABLE XVI. For Hyperbolic Orbits. « or« li.g«oi>rlon/V Diff. log N Diff. 7 21 36 49 64 t i.r fi IdgfloW'OK^V Diff. 9 II II li log N Diff. 436 1 450 464 47'; 493 o 1 2 3 o.ooo oooo .ooo oooo .ooo coco .ooo oooo 0.000 oooo .000 0007 .000 0028 .COG 0064 30 31 32 33 0,000 0066 .000 0075 .000 0086 .000 0097 0.000 6400 .000 6836 .ooo 7286 .000 775c 4 .ooo oooo .000 01 1 3 34 .000 0109 n .000 8229 5 o.ooo ocoo 0.000 0177 78 92 107 120 »3S 35 O.OOO 0122 \l 0.000 8722 508 5^3 537 ' 567 6 .ooo oooo .000 0255 30 .000 0137 .000 9230 7 .000 oooo .coo 0347 37 .000 O153 t8 .000 9753 8 1 9 1 .000 oooo .ooo ooo I I .000 0454 .000 0574 38 39 .000 0171 .000 0190 «9 20 .001 0290 .001 0842 I 10 1 11 18 o.ooo ooo I .000 ooo I .ooo 0002 I 0.000 0709 .000 0858 .000 1 02 1 149 163 178 191 206 40 41 42 0.000 02 10 .000 0232 .000 0255 22 0.00 1 1409 .001 1990 .001 2586 581 ! 596 ■ 611 i 13 .000 0OO2 .000 1 199 43 .000 0281 27 29 .001 3197 626 1 14 ,000 0003 1 .000 J 390 44 .000 0308 .001 3823 640 1 1 »•"' 0.000 0004 I 0.000 1596 220 45 0.000 0337 3' 33 36 38 40 0.00 1 4463 6.55 i 670 685 , 700 71S i 16 1 17 1 18 .000 0005 .000 0007 .000 0009 2 2 2 .000 1816 .000 2051 .oco 2299 235 248 263 277 4a 47 48 .000 0368 .000 04CI .000 0437 .001 5118 .001 5788 .001 6473 ! 10 .000 001 I 2 .000 2562 49 .000 0475 .001 7173 20 ! 21 0.000 001 3 .000 0016 3 3 4 4 s 0.000 2839 .000 31 31 292 50 51 0.000 0515 ,000 0558 43 48 SI 0.001 7888 .001 8618 730 744 760 775 790 1 22 .000 0019 .000 3437 320 334 349 52 .000 0604 .001 9362 23 24 .000 0023 .000 0027 .000 3757 .000 4091 53 54 .000 0652 .000 0703 .002 0122 .002 0897 25 0.000 0032 5 6 0.000 4440 363 ;02 55 0.000 0757 60 0.002 1687 806 2G .000 0037 .000 4803 50 .000 0815 .002 2493 820 27 .oco 0043 7 .000 51 81 57 .000 0875 6,1 .002 3313 836 851 866 ; 18 .000 C050 7 9 .000 5573 407 420 58 .000 0939 68 .002 4149 2» .00c 0057 .000 5980 i»« .000 1007 71 .002 5000 1 30 0.000 0066 0.000 6400 00 0.000 1078 0.0C2 5?66 ' morn log Q or log Q' log I. Diff. loglmlfll.Diff. m or n 0.10 log Q or log C log I. Diff. loglmlfll.Diff. 0.00 0.000 OOCO 2.1 149,, 9.998 7021 3.41256,, 2.1046„ .01 9.999 9870 i.4>597.. 2,1146,, .11 .998 4308 3.45326,. 2.1025,. ^ .02 •999 9479 2.71675,, 2.1 142,, .12 .998 1342 3.49028,, 2.1003,, •03 .999 8828 2.89259,, 2. II 37,, •13 •997 8»23 3.52423.1 2.o978„ .04 •999 79«7 3-oi74«n 2.II30n .14 .997 4654 3-55547" 2.0952, 0.05 9,999 6746 3.ii4«i.i 2.1121,, 0.15 9.997 0936 3-58453" 2.0923^ .06 .999 5316 3.19290,, 2.11I0„ .16 .996 6971 3.61 154" 2.0892™ .07 .999 3628 3.25940,, 2.IC97,, •17 .996 2760 3.63679,, 3.66048,, 2.0860,. .08 .999 1082 3.31687,, 2.1082,, .18 •995 8305 2.0826, .09 .998 9479 3-3674S" 2.1065,, .19 •995 3608 3.68276,, 2.0790, O.IO 9.998 7021 3.41256,. 2.1046,, 0.20 9.994 8671 3^70378« 2.0752, J 0.0035 .0036 .0037 .0038 .0039 0.0040 .0041 .0042 .0043 .0044 0,0045 .0046 .0047 .0048 .0049 0.0050 .0051 .0052 .0053 .0054 0.0055 .0056 .0057 .0058 .0 -0 .cob . 032 TABLE XVII. For special Perturbations. 9. 9', '{' o.oooo .0001 .0001 .0003 .0004 0.0005 .ocoh .0007 .0008 .0009 O.OOIO .001 I .0012 .0013 .0014 0.0015 .0016 .0017 .0018 .0019 0.0020 .0021 .0022 .0023 .0024 0.0025 .0026 .0027 .0028 .0029 0.0030 .0031 .0032 .0033 .0034 0.0035 .0036 .0037 .0038 .0031} 0.0040 .0041 .0042 .0043 .0044 0.0045 .0046 .0047 .0048 .0049 0.0050 .0051 .0052 ■0053 .0054 0.0055 .0056 .0057 .0758 .0 -o .00b ' For positive values of tlio Argument. log/ BiiT. log/', log/" DilT. 0.477 •477 .476 .476 476 1213 0127 9042 7957 6872 5787 4''02 3618 *534 1450 0367 9284 8201 7118 6035 476 476 476 476 476 476 475 475 475 475 475 4953 475 3^7' 475 2789 475 1707 475 0626 474 9545 474 8464 474 7383 474 6303 474 5223 474 4143 474 3063 474 "983 474 0904 473 9815 473 8746 473 7667 473 6589 473 55" 473 4433 473 3355 473 2178 473 I20I 473 o«24 472 9047 472 7970 472 6894 472 5818 472 4742 472 3666 47^ 472 472 471 471 47' 471 471 471 47 « 471 47 « 470 470 470 470 2591 1516 0441 9366 8292 7218 6144 5070 3996 2923 1850 0777 97-^4 8632 7500 (14S8 086 085 0S5 085 085 085 084 084 084 083 083 083 083 083 082 082 082 082 08 1 081 081 081 080 080 080 080 080 079 079 079 079 078 078 078 078 077 077 077 077 077 076 076 076 076 075 075 075 075 074 074 074 074 074 073 073 073 073 072 072 072 0.301 0300 .300 9431 .300 8563 .300 7695 .300 6827 5959 5092 4224 3357 2490 1623 0756 9889 9023 8157 0.300 .300 .300 .300 .300 0.300 .300 .299 .299 .299 0.299 .299 .299 .299 .299 0.299 .299 .299 .299 .298 0.298 .298 .298 .298 .298 0.298 .298 .298 .29S .298 0.298 .297 .297 ,297 .297 0.297 .297 .297 .297 .297 0.297 .297 .296 .296 .296 0.296 .296 .296 .296 .296 0.296 .296 .296 .296 •295 .295 7291 6425 5559 4693 3828 2963 20q8 »233 0368 9504 8639 7775 69 1 1 6047 5184 4320 3457 2594 1731 0868 0005 9143 8280 7418 6556 5695 4833 3972 3H0 2249 1388 0528 9667 8807 7946 7086 6226 5367 4507 3648 2788 1929 1070 0212 935 3 8495 : 869 868 868 868 868 867 868 S67 867 867 1867 867 , 866 866 I 866 ' 866 ! 866 I 866 i865 ; 865 1 865 865 \ 865 1S64 1 865 ' 864 ' 864 864 ' 863 864 863 863 1 863 863 : 863 ■ 862 863 862 862 861 862 86 1 862 861 861 860 861 860 861 860 860 859 860 859 860 859 859 858 859 858 For negative values of tlie Argument. log/ 0-477 •477 ■477 •477 •477 0.477 •477 •477 •+77 ■478 0.478 .478 .478 .478 .478 0.478 .478 .478 •479 •479 0.479 •479 •479 •479 •479 0.479 •479 .4S0 .480 .480 0.480 .480 .480 .480 .480 0.480 .481 .481 .481 .481 0.481 .481 .481 .431 .481 0.482 .482 .482 .482 .482 .482 .4S2 .482 .483 0.483 .483 .483 .483 .483 •483 1213 2299 3385 447 « 5558 6645 7732 8819 9906 0994 2082 3170 4*59 5348 6437 7526 8615 97"5 0795 1885 2975 4065 5156 6247 7338 8430 9522 0614 1706 2798 '3891 4984 6077 7170 8264 9358 0452 1547 2641 3736 4831 5926 7022 8118 9214 0310 1407 2504 3601 4698 5796 6894 7992 9090 0188 1287 2386 3485 4584 5684 6784 riff. log/', log/" 086 086 086 087 087 087 087 0S7 088 088 088 0S9 0S9 089 089 089 090 090 090 090 090 091 091 091 092 092 092 092 092 093 093 093 j 093 094 I 094 j 094 I C95 I 094 095 095 095 096 096 096 096 097 I 097 ' 097 : 097 \ 098 - 098 I 098 I 098 I 098 j 099 I 099 099 099 100 100 0.301 .301 .301 .301 .301 0.301 .301 .301 .301 .301 0.301 .301 .302 .302 .302 0.302 .302 .302 .302 .302 0.302 .302 .302 .303 •303 0.303 .303 .303 •303 .303 ©•303 •303 .303 •303 .303 0.304 .304 .304 .304 •304 0.304 •304 .304 .304 .304 0.304 .305 .305 •305 .305 0.305 .305 .305 .305 .305 0.305 .305 .306 .306 .306 .306 Diff. 0300 1 169 2037 2906 3776 4645 55»5 6384 7254 8124 8995 9865 0736 1606 2477 3348 4220 5091 5963 6835 7707 8579 945' 0324 1 196 2069 2942 3815 4689 5562 6436 I 7310 8184 i 9058 i 9933 I I 0807 1682 \ 2557 , 3432 4308 1 5183 6059 6935 7811 8687 9563 0440 1317 2194 3071 3948 4825 5703 6581 7459 8337 9215 0094 0973 1851 2730 869 868 869 870 869 ; 870 869 870 87c 871 870 871 870 871 871 872 871 872 872 872 872 872 873 872 873 873 873 874 874 874 874 874 875 874 875 875 875 876 875 876 876 876 876 876 877 877 877 877 877 877 878 878 878 878 878 879 879 878 879 oyy TABLE XVII. For special Perturbations. q. q', q" 0.0060 .0061 .006a .0063" .0064 0,0065 .0066 .0067 .0068 .0069 o 0070 ."lo-ri .0-072 o<.73 .coy4 0.0075 .0076 .0077 .0078 .0079 0.00?0 .0081 .0082 .0083 .0084 0.0085 .00S6 .0087 .00S8 .0089 0.0090 .0091 .0092 .0093 .0094 0.0095 .0096 .0097 .0098 .0099 O.OIOO .0101 .0102 .0103 .0104 0.0105 .0106 .0107 .0108 .0109 O.OIIO .oil I .0112 .01 13 .0114 0.0II5 .0116 .0117 .0118 .0119 .0120 For iiositive values of tliu Argument. log/ Diff. log/', log/" Diff. 0.470 6488 .470 5416 •470 4545 - .470 3274 .470 2103 0.470 1 1 32 .470 0062 .469 8992 .469 7922 .469 68 5 2 0.469 .469 .469 .469 .469 0.469 .4')8 .4(8 .408 .468 0.468 .468 .468 .468 .468 0.467 .467 .467 .467 .467 0.467 .467 .467 .467 .467 0.466 .466 .466 .466 .466 0.466 .466 .466 .466 .465 0.465 .465 .465 .465 .465 0.465 .465 .465 .465 ,464 0.464 .464 .464 .464 .464 .464 5782 4713 3644 ^575 1506 0437 9369 8301 7233 6165 5098 4031 2964 1897 0831 976? 8699 7633 6567 5502 4437 3372 2307 1243 0179 9115 8051 69S8 5925 4862 3799 2736 1674 0612 955° 8488 7427 6366 53°5 4244 3'83 2123 1063 0003 8943 7884 6825 5766 4707 3648 2590 1072 1071 1071 1071 1071 1070 1070 1070 1070 1070 1069 1069 1069 1069 1069 1068 1068 1068 1068 1067 1067 1067 1067 1066 1066 1066 1066 1066 1065 1065 1065 1065 1064 1064 1064 1064 1063 1063 1063 1063 1063 1062 1062 1062 1062 1061 1 06 1 1 06 1 1061 106 1 1060 1060 1060 1060 1059 1059 1059 «°59 1059 1058 0.295 8495 .295 7637 .295 6779 .295 5921 .295 5063 0.295 .29 T .295 .295 .295 0.294 .294 .294 .294 .294 0.294 .294 .294 .294 .294 0.294 .294 •293 .293 .293 0.293 .293 .293 .293 .293 0.293 •293 .293 •293 .292 0.292 .292 .292 .292 .292 0.292 .292 .292 .292 .292 0.292 .291 .291 .291 .291 0.291 .291 .291 .291 .291 0.291 .291 .290 .290 .290 .290 4205 3348 349 « 1634 0777 9920 9064 8208 7351 6495 5640 4784 3928 3073 2218 1363 0508 9653 8799 7945 7090 6236 53*^3 4529 3675 2822 1969 1 1 16 0263 941 1 8558 7706 6854 6002 5150 4298 3447 2595 «744 0893 0043 9192 «34i 749 « 6641 5791 4941 4092 3242 2393 1544 0695 9846 8997 8149 7300 858 858 858 858 J858 ^857 857 :857 I 857 i857 i856 856 857 856 855 i 856 I 856 ^55 855 I 855 ^855 ;855 854 ; -^54 I 855 i854 853 854 854 1853 :853 853 853 852 1853 '852 852 852 852 851 852 851 851 850 851 85, 850 850 850 850 849 850 849 849 849 849 849 848 849 For negative values of the Argmnont. log/ 0.483 .483 •483 .484 .484 0.484 .484 .484 .484 .484 0.484 .484 •485 .485 •485 0.485 •485 .485 .485 .485 0.485 .485 .486 .486 .486 0.486 .486 .486 .486 .486 0.486 .487 •487 .487 .487 0.487 .487 .487 .487 .487 0.488 .488 .488 .488 .488 0.488 .488 .488 .488 .489 0.480 •4'9 .489 •489 .489 0.489 .489 ■489 .490 .490 .490 Diff. 6784 7884 8984 0085 I186 2287 3388 4490 5592 6694 7796 8898 0001 1 1 04 2207 33" 4415 5519 6623 7728 8833 9938 1043 2149 3255 4361 5467 6573 7680 8787 9894 lOOI 2109 3217 4325 5433 6542 7651 8760 9869 0979 2089 3199 43°9 5420 653' 7642 8753 9865 0977 2089 3201 4314 5427 6540 7653 8767 9881 °995 ! 2109 I 3223 lOT 100 lOI lOI lOI lOI 102 102 102 102 102 103 103 103 104 104 104 104 105 105 105 105 106 106 106 106 06 107 107 107 107 108 108 108 108 109 109 109 109 no no no no ni m 1 1 1 ni n2 n2 1 12 1 12 113 "3 »i3 113 "4 «i4 'H i'4 114 log/', log/" 0.306 .306 .306 .306 .306 0.306 .306 .306 .306 .307 0.307 • 3°7 .307 .307 .307 0.307 .307 .307 .307 .307 0.308 .308 .308 .308 .308 0.308 .308 .308 .308 .308 0.308 .309 .309 .309 •3°9 0.309 .309 .309 .309 .309 0.309 •309 •309 .310 .310 2730 3610 4489 5369 6248 7128 8009 8889 9769 0650 1531 2412 3293 4J74 5056 5938 6820 7702 8584 9466 0349 1232 2n5 2998 3881 4765 5648 6532 7416 8301 9185 0070 0954 1S39 2725 3610 4495 5381 6267 7153 8039 8926 981a 0699 1586 q. q'. 0.310 2473 .310 3360 .310 4248 .3IC 5136 .310 6023 0.310 6911 .310 7800 .310 8688 .310 9577 •3J' 0465 0.3M •3" •3" •3»« •3" •3«" «354 2243 3'33 4022 49 '2 5802 Diff. 880 ; 879 880 8-^9 880 881 880 880 881 881 88i 881 881 882 882 882 882 882 882 883 883 883 8S3 883 884 883 884 884 885 884 885 ' 8,84 885 886 885 885 886 886 886 886 887 SS6 8S7 S87 887 887 888 888 887 888 889 888 889 888 S89 889 890 889 890 890 634 TABLE XVII. For special Pertiirl)ation.s. 3nt. tf Diff. 88o i o 9 879 1 880 ! I 8-5 880 8 881 y 880 9 8S0 y 881 881 I 881 881 !> 881 t 882 882 8 882 .0 882 )2 4 )6 8S2 882 883 , 1-9 !2 883 i 881 i 883 i 885 ' 884 885 ' 884 )4 ' 885 (9 1 886 ^S ! 885 10 885 886 886 '/ 8S6 >J 886 !9 i6 SS7 SS6 SSy S87 ■ 887 )0 [8 ■3 887 888 888 887 , 888 1 I 889 !8 '7 i 888 1 : 889 i 888 'i 889 14 ■3 13 ,2 i 889 1 890 ! 889 2 )2 1 890 ; 890 0.0 .0 .0 .0 .0 0,0 .0 .0 .0 .0 0.0 .0 .0 .0 .0 0.0 .0 .0 .0 .0 0.0 .0 .0 .0 .0 0.0 .0 .0 .0 .0 0.0 .r .0 .0 .0 0.0 .0 .0 .0 .0 0.0 .0 .0 .0 .0 0.0 .0 .0 .0 .0 I o o .0 .0 For positive viiliiea of the Argument. 20 21 22 23 24 ^5 26 27 28 29 30 3' 3^ 33 34 35 36 37 38 39 40 4' 42 43 44 45 46 ; +7 i 48 49 CO I 51 52 53 54 55 56 57 58 59 60 61 62 63 , 64 65 66 67 68 69 70 71 71 .0 73 .0 74 0.0 75 .0 76 .0 77 .0 78 .0 179 .0 180 log/ Dlff. I log/', Ing/" I DIIT. 0.464 .464 .464 .463 .463 0.463 .463 .463 •463 .463 0.463 .463 .462 .462 .462 0.462 .46 ^ .46-... .462 .462 0.462 .462 .461 .46, .461 0.461 .46, .461 .46: ,461 0.46 1 .460 .460 .460 .460 0.460 .460 .460 .460 .460 0.460 •45.' •459 •459 •459 0.459 •459 ■459 •459 •459 0.458 .458 .458 •458 .458 0.458 .458 .458 .458 •458 •457 2590 1532 °474 9416 8359 7302 6245 5188 4132 3076 2020 0964 9908 8853 779« 6743 ';688 4633 3579 1471 0417 9364 8311 7258 6205 5«53 4101 3049 »997 0945 9894 8843 779i , 6741 5690 4640 3590 2540 1490 0441 9392 8343 7194 6245 1 5197 I 4149 3101 2053 I 1006 j 9959 i 8912 7865 i 6818 577i I 4726 ' 3680 j 2634 1589 0544 9499 058 058 058 057 057 057 056 056 056 056 055 055 055 055 °55 054 -154 054 "54 053 °53 °5 3 053 052 052 052 052 052 051 051 051 051 051 050 050 5<: 050 050 049 049 049 049 049 048 048 048 048 047 047 047 047 046 046 046 0415 045 045 °4S 0.290 .290 .290 .290 .290 0.290 .290 .290 .290 .289 0.289 .289 .289 .289 .289 0.289 .2S9 .289 .289 .289 0.289 .288 .288 .288 .288 0.288 .288 .288 .288 .288 0.288 .288 .288 .287 .287 0.287 .287 .287 .287 .287 0.287 .287 .287 .287 .287 0.286 .286 .286 .286 .286 0.286 .286 .286 .286 .286 0.286 .286 .285 .285 .285 .285 7300 6452 5604 4756 3909 3061 2214 1367 0520 9673 8826 7980 7134 6287 544> 4596 3750 2904 2059 1214 0369 95^4 8679 7835 6990 6146 5302 4458 3615 "•77' 1928 1085 0241 9399 8556 7713 6871 6029 5187 4345 3503 2661 1820 0979 0138 9297 8+56 7615 6775 S93S 5095 4155 3415 ^575 1736 0896 0057 9218 8380 754« 6702 848 848 848 847 848 '847 847 847 847 847 ' 846 846 847 ' 846 ■845 ' 846 846 845 ,845 '845 845 , S45 844 845 844 ■844 : 844 i843 844 1843 843 844 842 843 1843 842 842 ,842 842 842 ,842 ' 841 84. 84. 841 84. 841 840 , 840 1 840 840 840 840 839 1 840 ;839 839 838 839 839 For nog-itive values of the Argument. los / Dlff. 0.490 •49° .490 ■49° .490 490 490 49' 49' 491 491 491 491 49' 491 49' 492 492 492 492 492 492 492 492 493 493 493 493 495 493 493 493 493 494 494 494 494 494 494 494 494 494 495 405 495 495 495 495 495 495 495 496 496 496 496 496 496 496 496 496 497 3223 4338 545 3 6568 7684 8800 9916 1032 2149 3266 4383 5500 6618 7736 8854 9972 1 09 1 2210 3 3^9 4448 5567 6687 7807 8927 0047 1168 22>;9 3410 453- 5<^54 6776 7898 9021 0144 1267 2390 35'4 4638 5762 6886 8010 913s 0260 1385 2510 3616 4762 588S 7015 8142 9269 0396 1524 2652 3780 4908 6037 7166 8295 9424 055+ I log/', log/" j I'iff- I1I5 1115 1115 11 16 III6 11,6 ° 1116 1117 M17 H17 1 1117 1 1118 ! 1118 1118 1118 1119 1119 1119 1 1 19 ! MI9 1120 1120 1120 1120 II2I iiai 1121 1122 1122 1122 1122 1123 1123 1123 1123 1124 1124 1124 1124 1124 1125 1125 1125 1125 1126 .126 '= 1126 1127 1127 1127 II27 1128 1128 . 1128 1128 1129 1129 1129 31 1129 1130 I 5802 1 6692 1 7582 1 8472 1 9363 2 0254 2 I 145 2 2036 2 2927 2 3S19 2 4710 2 5602 2 6494 2 73S7 2 8279 3 9172 3 0064 3 °957 3 '850 3 2744 3637 453' 54^5 6319 7213 13 8108 13 9002 13 9897 14 0792 14 1687 14 2583 '4 3478 '4 4374 '4 5*7° 14 6166 7062 7959 8855 97 5 2 0649 5 1546 5 i444 5 334' 5 4239 5 5' 37 5 6035 5 6934 5 7832 5 873' 5 9630 0529 1428 2327 3227 4127 5027 5927 6827 7728 8629 9530 ; 890 S90 890 891 891 891 891 891 891 , 891 892 892 893 892 i893 \ 892 ,893 893 894 ,893 894 '894 894 894 ;89S 1894 1 895 '895 .895 896 895 : 896 896 896 i 896 897 . 896 897 1897 897 , 89S 897 898 ,898 898 1899 ■■ 898 899 ,899 ■ 899 899 ,899 900 900 I 900 I 900 i 900 goj . 901 ' 901 I 635 si i n TABLE XVII. For special Perturbations. 1. 9'. 7" 0.0180 .0181 .01X2 .0183 .0184 0.0185 .0186 .0187 .0188 .0189 0.0190 .0191 .0191 .0193 .0194 1 0.0195 i .0196 i -oi'jy .0198 .0199 1 0.0200 ! .0201 ] .0202 I .0203 .0204 0.0205 .0206 .02Q7 .0208 .0209 0.0210 ' .02II .0212 .0213 .0214 0.0215 .0216 .0217 .0218 .0219 0.022n .0221 .0222 .0223 .0224 0.0225 .0226 .0227 .0228 .0229 0.0230 .0231 .0232 .0233 .0234 0.0235 .0236 .0237 .0238 .0239 .0240 For positive values of the Argument. log/ 0-457 9499 •+57 »454 •457 7409 •457 6365 ■457 S3J> 457 457 457 457 457 456 456 456 456 456 456 456 456 456 455 455 455 455 455 455 455 455 455 455 454 454 454 454 454 454 454 454 454 453 453 453 453 45 3 453 453 453 453 453 452 452 452 452 452 452 452 452 45* 452 45 « 451 4SI 4177 3*33 2189 1 146 0103 9060 801/ 6975 593 J 4891 3849 2808 1767 0726 9685 8644 7604 6564 5524 4484 3444 2405 1366 0327 9288 8249 7211 6173 5135 4097 3060 20 -.'.3 0986 9949 8912 7876 6840 5804 4768 3733 2698 1663 0628 9593 8558 7514 6490 5456 4422 3389 2356 1323 0290 9258 8226 7»94 Diff. 1045 1045 1044 1044 1044 1044 1044 1043 1043 1043 '°43 1042 1042 1042 1042 1 041 1 041 1041 1 041 1041 1040 1040 1040 1040 1040 1039 1039 1039 1039 1039 1038 1038 1038 1038 1037 1037 1037 1037 1037 1036 1036 1036 1036 i"35 1035 1035 1035 1035 1035 1034 1034 1034 1034 1033 »033 1033 1033 1032 1032 1032 iog.r, log/" 0.285 .285 .285 .285 .285 0.285 .285 .285 .285 .284 0.284 .284 .284 .284 .284 0.284 .284 .284 .284 .284 0.283 .283 .283 .283 .283 0.283 .283 .283 .283 .283 0.283 .283 .2S2 .282 .282 0.282 .282 .282 .282 .282 0.282 .282 .282 .282 .281 0.281 .281 .281 .281 .281 0.281 .281 .281 .281 .281 0.281 .281 .280 .280 .280 .280 6702 586A 5026 4188 3350 2512 1675 0838 0000 9163 8326 7490 6653 5817 4981 4«i5 3309 2473 1637 0802 9967 9132 8297 7462 6627 5793 4958 4124 3290 2456 1623 0789 9956 9123 8290 7457 6624 5792 4959 4127 3295 2463 1631 0800 9968 9137 8306 7475 6644 5814 4983 4«53 3323 2493 1663 0S33 0004 9174 8345 7516 6687 Diff. 838 838 838 838 838 1837 i837 1838 837 I 837 : 836 ^37 S36 ; 836 I 836 ' 836 836 835 '835 :835 ;835 1835 i834 :^35 ^54 833 833 833 833 833 833 832 !" 832 832 832 831 832 831 831 831 831 830 831 830 830 830 830 830 829 830 829 829 829 636 For negiilivc viiluos of the Argiiincnt. log/ 0.497 •497 •497 •497 •497 0.497 •497 •497 •497 .498 0.498 .498 •498 .498 •498 0.498 .498 .498 •499 •499 0.499 •499 •499 •499 •499 0.499 .500 .500 .500 .500 0.500 .500 .500 .500 .500 0554 1684 2814 3944 5°75 6206 7337 8468 9600 0732 i86j. 2996 4129 5262 6395 7528 86b2 9796 0930 2064 3199 4334 5469 6604 7740 8876 0012 1 149 2286 3423 4560 5697 6835 7973 9111 Diff. 0.501 0250 .501 1389 .501 2528 .501 3667 .501 4807 5947 70S7 8227 9368 ; 0509 i 1650 2791 : 3933 5°75 6217 I 7360 ; 8503 ; 9646 0789 1932 3076 4220 5364 6508 7653 i 8798 0.501 .501 .501 .501 .502 0.502 .502 .502 .502 .502 0.502 .502 .502 .503 .503 0.503 .503 .503 .503 .503 .503 log/', log/" 130 I 30 130 >3> 131 131 131 >3^' 132 132 132 133 133 133 ' 133 ! >34 ! 134 I '34 i 134 I '35 i '35 I '35 ! '35 ! 136 136 136 '37 '37 '37 '37 I .37 '.III 138 I '39 ! '39 '39 '39 140 I 140 140 140 141 1 '41 I .41 i 141 ! 142 142 I 142 '43 143 '43 '43 '43 '44 '44 '44 '44 145 145 0.3 •3 3 3 6 9530 7 043' 7 1332 7 2234 7 3'35 7 4037 7 4939 7 584' 7 6744 7 7646 Diff. 7 8549 7 9452 8 0355 8 1259 8 2162 8 3066 8 3970 8 4874 8 5778 8 5683 8 7588 8 8492 8 9398 9 0303 9 1208 9 2114 9 3020 9 3926 9 4832 9 5738 9 6645 9 7552 9 8459 9 9366 320 0273 320 ii8i 320 20S8 320 2996 320 3904 320 4813 320 320 320 320 320 321 32' 321 321 321 321 321 321 321 32' 5721 6630 7539 8448 9357 0266 1 176 2086 2996 3906 4816 5727 6637 7548 8460 321 9371 322 0282 322 1194 322 2106 322 3018 322 3930 901 901 902 901 902 902 902 903 902 903 903 903 904 903 904 904 904 904 905 905 904 906 905 905 906 906 906 906 906 907 907 907 907 907 908 907 908 908 909 908 909 909 909 909 909 910 910 910 910 910 9" 910 9" 912 911 911 912 912 912 912 TABLE XVII. For special I'erliirbutions. ?. '/'. 7" DilT. > 1)01 901 Q02 0.0240 .0241 .0242 .0243 .0244 0.0245 .0246 .0247 .0248 .0249 0.0250 .0251 .0252 .0253 .0254 0.0255 .0256 .0257 .0258 .0259 0.0260 .0261 .0262 .0263 .0264 0.0265 .0266 .0267 .0268 .0269 0.0270 .0271 .0272 .0273 .0274 O.C275 .0276 .0277 .0278 .0279 0.0280 .0281 .0282 .0283 .0284 0.0285 .0286 .02S7 .0288 .0289 0.0290 .0291 .0292 .0293 .0294 0.0295 .0296 .0297 .0298 .0299 .OJOO For i)OsitiTO vnliiea of the Argument. log/ 0.451 7194 .451 6162 45' S'30 451 4099 451 3068 Diff. log/', log/" Diir. 451 451 450 450 450 450 450 45° 45° 450 45° 450 449 449 449 449 449 449 449 449 449 449 448 448 448 448 448 448 448 448 448 448 447 447 447 447 447 447 447 447 447 446 446 446 446 446 446 446 446 446 446 445 445 445 445 445 2037 1006 9975 8945 79«5 6885 5855 4825 3796 2767 1738 0709 9681 8653 7625 6597 5569 4542 3515 2488 1461 °435 9409 8383 7357 6331 5305 I 4280 I 3255 ' 2230 i 1205 I 0181 I 9'57 8133 : 7109 I 6085 5062 4°39 : 3016 2993 j 0970 1 9948 I 8926 I 7904 ' 6882 581:1 4840 3819 2798 >777 0756 9736 8716 7696 6676 5657 1032 1032 1031 1031 1031 1031 1031 1030 1030 1030 1030 1030 1029 1029 1029 1029 1028 1028 1028 1028 1028 1027 1027 1027 1027 1026 1026 1026 1026 1026 1026 1025 1025 1025 1025 1024 1024 1024 1024 1024 1023 1023 1023 1023 1023 1022 1022 1022 1022 1021 102 1 1021 I I02I ; 1021 1 02 1 1020 1020 I 1020 j 1020 1019 ; 0.280 .280 .280 .280 .280 0.280 .280 .280 .280 .279 0.279 .279 .279 .279 .279 0.279 4273 .279 3446 .279 2620 .279 1794 .279 0968 6687 5858 5030 4201 3373 2545 I7>7 0889 0062 9^34 8407 7580 6753 5926 5099 0.279 .278 .278 .278 .278 0.278 .278 .278 .278 .278 0.278 .278 .278 •277 .277 0.277 .277 .277 •277 .277 0.277 .277 .277 .277 .277 0.276 .276 .276 .276 .276 0143 9317 8492 7666 6841 6016 5191 4367 3542 2718 1894 1070 0246 9422 8599 7775 6952 6129 5306 4483 3661 2838 2016 1194 0372 9550 8728 7907 7086 6264 o-27<' 5443 .276 4622 .276 3802 .276 2981 .276 2161 0.276 1340 .276 0520 .275 9700 .275 8880 .275 8061 .275 7241 829 828 829 828 82S 828 82S 827 828 827 827 827 827 827 826 827 826 826 826 825 826 825 826 825 ,825 i 825 824 825 824 1824 I 824 824 'S24 823 824 823 823 823 ;823 j 822 ! 823 ' 822 822 ] S22 j 822 822 : 821 i 821 822 821 821 I 820 1 821 820 I 821 ! 820 820 820 819 820 For nugativo vuluos of tiio Argitiiioiit. \osf DifT. log/', log/" Pi IT. 0.503 .503 .504 .504 .504 0.504 .504 .504 •504 .504 0.505 .505 .505 .505 .505 0.505 .505 •5°5 .505 .506 0.506 .506 .506 .506 .506 0.506 .506 .506 .507 .507 0.507 .507 .507 .507 .507 0.507 .508 .508 .508 .50S 0.508 .508 .508 .508 .508 0.509 .509 .509 .509 .509 0.509 .509 .509 .509 .510 8798 9943 1089 2235 3381 4527 5674 6821 7968 91 15 0263 1411 2559 3707 4856 6005 7154 8303 9453 0603 •753 2903 ; 4054 6356 ■ 7508 ] 8660 ! 9813 : 0965 ; 2117 i 3^7° : 4423 5577 , 673' I 7885 ! 9039 I 0194 ' '349 2504 3659 , 4814 i 5970 7126 8282 9439 : 0596 ' 1753 : 2910 I 4068 ; 5226 j 6384 ; 7543 I 8702 ' 9861 '' 1020 0.510 2179 .510 3339 .510 4499 .510 5659 .510 6819 .510 79S0 45 46 46 46 46 47 47 47 47 48 48 48 48 49 49 49 49 50 50 5° 50 51 5' 5' 52 53 52 52 I 53 53 54 54 54 54 I 55 ' 55 , 55 55 55 ' II 56 57 57 ; 57 58 1 58 i 59 59 1 59 ; 59 60 60 60 60 6i 0.322 •322 .322 .322 .322 322 322 323 323 323 323 323 323 323 323 323 323 323 324 324 324 324 324 324 324 324 324 324 324 325 325 325 325 325 325 325 325 325 325 325 326 326 326 326 326 326 326 326 32fi 326 326 327 327 327 327 3930 4843 5756 6668 7581 8495 9408 0322 1236 2150 3064 3978 4893 5808 6723 7638 8553 9469 0384 I 300 2217 3133 4049 4966 5883 6800 7717 S635 9553 0470 1389 2507 3225 4144 5063 5982 6901 7821 8740 9660 05 So 1500 2421 3341 4262 5183 6104 7026 7947 S869 9791 0713 '635 2558 3481 327 4404 327 5327 327 6250 327 7'74 327 8097 327 9021 920 920 920 921 920 921 921 921 922 921 922 922 922 922 923 923 923 923 923 924 923 924 637 TABLE XVIII. Elements of the Orbits of Comets wliicli have been observed. ] 3 a. S 5 'H -H ^ =* ^ J3 c5 . b4 t4 U . b4 ►-• ..s O 0) >i; -2 '1 to-i k5 '«S ■ 'i.2 a; .2 -d 'u So'tc ^ ii-m Nill S'^M K * « • = - Hi M l-H »-H xt^i::^ SasJ K^^K^ oxzi^-i V V a! « 6 TS -o T3 -B "O rs .2 S S 2 2 2 £ 2 g *: tc ^ .. - &•> t* iP ^ ^ ^ ^ ■*J 50 .^ .^ -*-! tl3^ ^ - - j^ £' 1 £ S a Qj c O b^ O I- K >- o ■K k- t -f i« s -W E & & ^ aJ ^ a a -r, t- (S 5 « s « a ~l s« Ss; s X 5 ^00 00 in o *■ 1 tl 1 e< t-^VO «3 r> H to in o ^ O ^ ^ o -^ so OO *♦■ so to O tl ' m o o 00 N. so VO tJ-00 \o toso to r^ 00 to w ^ O vO t^ ^r\ o r^ OO O m to X to tl ^ tl 00 tc 00 ^~ o t^vD Cn »n r^ m .- ^ t^ t^oo so to H OO •- SO oc t-- *-• « 1^00 so to f- T*- fO .^ OS O. tJ- in r-, t*^ ir\ •>l-oe o ^ ■>*• in mso vo r^ OS O -« OO SO OO (^\0 r*^ vj-i so 1^00 SO OC «- O SO >1- 1^ vo 00 u-1 inoo O ONOO 1^ t^ 1-^00 OOO OS Th in OS OS t^ oc U-* r~- o r-- oc 00 m t^ t^ m r~ r-- tl r^ j C\ Ov 0\ 0\ Ov CT^ ^ ^ ^ ^ OS as OS OS o Os OS OS ON OS Os O^ 0^ ^ On Os Os OS OS OS Os ON Os Os cK ' i M OS to u to SO 1 oc Os c 0. q q q q q q q q q q q q q q q q q q q q q q q M )•« M H M q q OS q >"■ •-■ •-■ d -" q q q q ov 1 M M M M ' t O o o o o O O o O O rt m ■<*■ m 1 ^ O o o rl w CT\ to O ^n O in m t^oo so m n so OS in t~- 1 o t^ Th tl OS OS to m TO ^ m in to in H m M in NN m to tl 11 11 tl to ^rr$2.2 rj-vo « « On r^ t^oo fo-o moo O so t-- tl OS r^ * « in t^ n tJ- w in m t^ tl 00 O to in **- ' 'l-vo >- r^ M w tl 1-^ toso •>*• M m t- ■♦ n •* tl to r^ t^sc 5; O 00 O O O O O O O O O O o 1 O 'J- m 1 fl tl tl V O I. vO t-^ ^ ro in o o o O 00 M t-^ w OS m tl m C 5 00 OssO so O 1-^ J a •i- ^r^ O fO h- »n to ro in rr to to '^ to T^ to M to tJ» ir .-o NN tl to tl M rt OVOO 00 •<»-00 O so O ^ m in t^ to t^ t^ to f» 1^ OO 00 tl in OS Os in tl in Os m « 00 ir^ "^ O- rt 0\ m 00 t< ti o M mo OS >- ■<*• so to ^so O 00 so to -^ -. OS t^ to tl « - - t< - D to « t« » ►. tl tl « tl tl H - N 1-. to i O O O O O O O O O O O O O O » "^ t^ On t^ to to in 00 00 O M t^ tl to to Q t^ tl i <*■ to - .^ O m Os tl OS t: to to ^ to in e) to tJ- to tl tl to ■♦ • t^ tl tl i m r» f» m to w to tl H tl f< M to « M tl « tl to to n tl to ~ rl rl to w IN ; » O O O CT\ •* >« "^ O OS O • O O t^ m w to O •♦ in inso Os 1 ■+ - tl tOTj- tl tl m Tf- * -. in to ^ tl Tt* to to g O 00 tOVD O SO tl M Os-« OS T»- OS OS to so m tl to w m -^ •* so M ■+ m « m to tl ■* m M ■+ -. M to m in X tl rl •+ « in rt in vo vo ■+ to « t-~ to t- n 00 so tl OS m « M m OS T^ x O tl tl to E^ M rt m M M H » tl n M « n n rl " •^ 05 ©"o o 00 t- «o — o (N r-H lO r-( © 5^ d lO CO 00 to ■^ 00 l-» 00 f f CO lO 03 "* Ol © to «: i-l (M r-l (M IT) CO I-l 1-1 Oq CO 11 CO n tl r-1 «^ 5-1 Ul r^ 11 01 n Ol 01 Jan. March Nov. Oct. July Aug. April .June March Dec. Sept. April Feb. Sept. Jan. July March June Oct. Nov. Oct. Nov. June Oct. Feb. Dec. .Jan. Sept. Aug. Oct. CO r-( O Ci lO 00 ■»f © 1^ 'H 05 «0 ■M t^ ^ -i< 05 1^ to 00 lO CO to 00 Ol O n to d 01 CO to CC 1- o O "t -rf" M to «C 1^ f^ CO -o 00 -^ O C5 CO to©co-o l~ 00 CO 10 to I~ c; o: o CO CO CO lO lO 1- X rH (M IC UO lO lO 1- CC 05 o; o o © c-i IM C-l CO CO CO CO If tl -f< It* ■^ -*! lO IC o lO >C iC lO 10 f— 1 T-1 d i—t f-H 1-1 1-H 1-1 ^1 ^^ '"*'"''"' ^" iH»ieo^»9 «et>.x8i© ■F^ ^IM ^ i» ^r-oCCi© -"»1 W •* 19 «e 1- GT St © iM 91 OS •»»< »9 ^. TH ^NrHFHnr^ ^H ^ rH ^N 91 »l »1 ®1 91 91 91 91 91 91 99 W 50 ©• CO so 088 TABLE XVIII. Elements of the Orbits of Comets which have been observed. ^ ■-■£, S' -r' o =: v "S 3 so r^ o o n fi rl Tt- rl 00 » o ^ ■+ un r--^ r-^ fi_ r^ o\ o^ d^ o^ o I- O O O CT^ » »' ><' d o o o ^ rt o^ a\ t^ M M r1 f*^ t*^ u-t ^ I t^ t^>o o o CN r^ ^*^ f* ■" O O O O O O ^r\ 0\ t\ On ^ « ^ ^ fl f~- ■<}- ON OX .. r- rl rl O fi rt r»i M 1- u-) O »nvo O^ rl ^ d r^ ►. r) rt •* ». O M H r" r) « cl - rt Cl 1—1 (M • • a CU = o ,-- 1 cc ?o CC 1- ® : CO ic- ic 1^ * r lo ic »c 1^ »* 1 .i „ r-l .-( rl -- J ,-< (Jl M i»»< »« 9 MMOeOSM 1 3 c S C D'Arre-^t. Peters and Sawitsch. Hind. La Caille. Hind. 1 Bessel. 1 Pingre. Bessel. Halley. Meehain. Lindelof. Halley. Henderson. Halley. 'C jj _• o - J- Ji Vogel. Burckhardt. ILUlev. La Caille. ] i 1 c! oj oS na T3 T3 2 2 2 £ j; 2 g 2 s''' 1 i 1 s'l ,^3 ^ ^ oS 1 «5< tc vo I10O O rl m m r*^ tJ- o ». r^ -c o\ r- VO On ■«*■ O fO rl m in *t- 'n rl t-~ C7. t-^ ^ d CTsOO CT\ 00 O ^ O "H U^ W-i *♦■ Tn CTn O O CC vO ^ -^ O rl ^ r^ t^ u-» owo 0^ 0^ ^ Cn C^ ON OS NO rl »1- O t-- t^ On to ON •>»- O O 1^ On tooo "• rl ON •+ ■<*• q 0_ tooo + On On ON On <-i oo 00 ON 1*^ ^ ^ ON i-i 'noo ^ ro On ON *^ O '"^ OC ro «ri fo rl vn o^-O tJ-oc 0^ t;- r-;- t^ CTv O t^ 0^ ON cS CO 00 O NO t^ 00 CI w NO »n n r^NO On *. 11 r^ rl to r^ in rl ONOO 00 of>oo' c^ rfi O M OO OO 00 ON 0\^ o t-. t-- rl fo On t\ O '^ o r-^ t-i fo ro "i* t^OO NO On CN ON o* o» to "J- t-- O 00 t-. «nNO rl 00 1 *♦• I •- ON O CO j CnO t-cc CO ; o» o •+• to rl 1 a\^_ to c^00 j On d C^ O^N ^ V « q q q q q 00 o NO ON O o d «■ -.' «■ «■ q q q q q « o ^ « O »^ O 1^ t^oo r» ^ ON On Ov Cl o ON r- to f* ON*^ OO NO o> o\ ON q d d d d w q q q q q q q q q q ro to o «n o q q q q •- t CO rl •rf- o o in4 « ^ t^ m ONOO 00 tJ- rl in m . O NO ON t^ >« ° NO rl oo m t^ O "• O m •H to in rl 00 M oo M M »< rl t^ w t-^ On ro M rl to t^ to rl O w 00 u-i t^ n ro X rl i-i vo t^ to On rl t^ rl 00 1^ NO irivo o M ^ ^ ^ rl O (-~ ro m NO M 00 NO Q O M m o NO o rl •* rl ►. <7n rl M O to m rl w NO ^ ^ o vo O^ ^ ^NO 00 m rl M CO M if *'^oo w rt* >^oo CO 00 00 in m .*■ « « rj. ^ m •+ rj rl ~ T» O t^oo On m m t^ )i to in 1 5 m u-iNO On -.in .*• V 00 •« NO «n •- ■+ ro •« rl On t^ in Tt- o o rl r*^NO NO r*^ oo o O q mTi- o Th * rl •+ "^ lo 00 ro moo ^ rt- On r~.00 OO rl rl O O m to « in rl NO On -1 rl ro t1- w 00 to t^NO 00 rl On On to rt « rl rl ONOO oo t\ " ■* O On "" r~. m rl M « « to rl M rooo NO i^ m i^NO ■1 rl » rl O mt in rf- n to Ti- m o ^ *^ CO cl 4 .*• O O NO t^ M in On — NO rl CO c< cl CO O ro «A On w rl ro H i«* On »- NO >i-i 00 OC ro f J t^ On 00 ** u^ rl t- O to Th tJ*oo rl to ir, w to rl in rl •4- O \D to 1^ i M M rl rl O to rl -< rl t: t 00 NO r< o *n M H w ro ^ moo t^ On tJ- t-i in M in NO 00 i^no rl H >« rl O " 00 »< n * 00 »noo NO to rl « « "-« oo rooo m >- n - to ro M m O "n 11 to to to ^ On On t-- to in in ro O i-i O NO t^ to t^ ^ ^ to t-~. m 1^ m to CO n t-~ On m 11 rl ^ ^ in to m rl rl M vo oo rl NO 00 to to rl to rl O m*0 to 11 - - - tJ- in to t^ ^ O rl t-.NO NO t^ « rl n H ro rl u-> u-i N- vo vo •*• On ^ "i- to m ro rooo tl On w ro ro t-^ 1^ tl m rl C> O to rl to .^ rl ti inNO 00 in CO in to CO rl O m rl c< ^ cl rl NO O CO CO rl « ^ » rl •+ r^ 00 n ^ m S » 00 ON OnOO in r*N r*^ m * ON r« w 00 w in O rl in M On M n >*• t- tooo >n w w M rl m o NO o O NO NO NO 00 00 to 11 t4- to to n moo 00 O M 11 O ON CO in O in w »l-NO Tl- m t-. to tJ- rl ". t^ to On r-N o n M M M 0-000 ■>*-00 00 CO to tJ- m rt t}- tJ- I^NO 00 O r» O ^ M ro (^ ^ U-, ro lo to -^ On -a- ^ ro w H rl C> (-- O rl •* ejN « ON w rl CO inNO oo 1^ M M «D 00 00 00 >0 ^ O a, 3 S e^i lo o SO '-o 00 00 OS 05 OS IC lO lO lO lO «D i>."oo''e-f to" e-) r-( 1-1 CI • ti > !> _• !■- oo 00 (N r-( O "-I i-H lO ■» CO XI 50 <0 «D Ttl ■^ ■* 1-1 CO ■^ lo 00 e^ t^ so «o o 1^ r- CO 50 O ® O 00 t^ •* CO 00 .-1 rH tH rt 00 O (M CO 'f t^ 00 00 QO 00 CO O 'O -O CO CO oTo: CO CO rH S<1 i-( i-c CD Ct lO OC C5 00 00 05 05 oa CO CO » -O 50 I> CC O rH Tt* rl i-< CC r-t r-( -fcj b c ^ e .-1 (M CO I^ Ctt OOOOn t^ t- 1^ 1~ t^ 1^ CO O 00 l^ s>i i-i :c n ^ c s c c C = CS = S j cc I-: ►-5 i-s >-j j CO 0-. t- 1- o» 1 (?1 Ol CO rt CO l--l^l~l-l^ ^^„^^ _„„__ _„___ _„___ „_„„_ _ ,-_„ _,_„„ ,, 1 6 »H ®1 M 1^ >9 19 IIS »5 lis »a «stN.xes© us iS 19 >9 tt »-l«lM.*>9 «Sr»ar Si© 1 099 TABLE XVIII. Elemontfl of the Orbits of (.'onicts which liavc been observetl. '2 1 3 4-t *^ tZ J 4J -2 ^ M -^ "2 .5;'2 d % 2 = -ci 3 n 2 -r >^ 'i; '3 o,2 « = fe ^ If 6 «^s - 2 '1 " y= waJwl :| ^ I'- ' ■ • ■o On OO 00 ONOO rt- tl M NO o i-~ ^-NO On -4 t - OO r^ to ro r^ cr\^ n ^\D 00 c^sO *** m r'l I •, <- ■ to O ONOO tl tn to tl NO to to tl On tl tl ONVJ O tJ- lo c 00 rl <-• Tt- Tj- rl r^ fl rnsO O OO O CV ' • O 00 tl tl to On lo iri »o 00 -H 00 00 NC •I- »0 to tl Cn 00 ON 1^ t^ m Cv ^ "^ t*^ 1^ O^ O^ O \D t~^ t^NO O 00 1^ On On O "- 00 On I-^OO on >. 00 O NO to lO ^CTsCT>CnO C^CT^^^O^ C^ iTv ON CN On On On On On On On O O ^ OO On ON On On On On On >e M s ;i o 00 « m O tn O ON o On O to NO •.*■ M •+ to tl OO NO "4 On ON Tt- to NO to to rh tl ^ to to 1^ » 4 ■+ OnnO ON * tl OO On On 00 ^ ^ NO ONOO O tl O tl On to •+ OOOOO OO0OC\ O O ON o OO ON r^ O O t^ O O O On O O >o O O O 00 o «--->-■" mmmmO ». M « O - w O O ■" >" O HI •" "4 ►- « 1-1 -; «• -• d -; 1 t w vo O « i^ r^oo f< On O to O •-* O y^ Q m IT) ONOO M NO t^ tl O NO •+ ■+ H tl tJ- to M>^i-i^ri- »^HH w^ tl ^ to to Wl M n tl to M to tl tl Hi* N- »o to 1 **» V *^ 00 t^^ VO r^ CNVO to tl OO »*• to w lo **■ »^ ut to tn O to to to tl to On Tj- 1- NO 00 1 M c*^ tl un M r*^ ^ to to un »0 tJ- to tl M w tl to tl Tl- - U-, t«4 to to Ul ° o ■■l-Tht^oovo-vOM On ■>*■ "1 t) tl O 00 " "i M r^ M to tl ■4- tl 1- t~- .+ to f-oo tl -^ 11 O t^ to o t^ OO t-v wo tJ- ^ to M M NO 00 to to c^oo w to t ult^O\Tt-f> vOO\OOt-^ ^r\ Tt- trioo to O O ON Th tO06 O "i- 00 oo to o 11 1/1 -O -1 *^vrjf»ui^ Mrlin r» w tl to to ITN to to « to to tl TO m M to to tl tl -- CO to ^»j-wLot--oo rtoofiOO O O to t< t1- « to On tl « lOOO •* 'I- til M Tj- It w O tl to On tl tooo to c; cnr^w-rt-M »r^ Mv/^u^ ■<1- tt to « « « Ul tJ- •+ tl -t -i- -« to tl wit^^ir^r^ rimT^-oro On OnOO no •* •>1- ■/-, 1-1 OO ■<}• t^ t- to o r^ t^ « to to tl to r^ ly^ NO •+ ■!• Tt- to 000^ ^^ mm-^min to t^ T^ ITN tl •ri to tl 00 tl tl tJ-00 r-^ tri to NO NO to On 1- w rl H tl w to to .H rt M n M tl « « M IH tl to M !•«*■•> t^oo m O O 00 On O to t^oo H- NO to ui tl u-i i- to N*' tl t/^ tl tl t^ to to ^ roi-^x-ic^e) Ot-~ooooO 00 to tl 1-^ ■+ to to l-« NO tl tooo r^ t^ ■>!• l/N tl 1-1 to to "l- -4 ONOO 00 *!- tr, tl to to 13 r^w-iMH. Tj-ir^t^M to U-l "1 M « M « tl « tl « to ti-, M r^ r» t^ r-- r^ i^oo ri t^ m to ^ ^ tJ- uo to NN t*-nO OO ^ ty-i 'tr ^O o •+ to I-- t^ NO NO OnnO O O On I~~nO 00 O'-'OntJ-oi-^ Mi^r»v£30 wi to o 00 <-i « r^ - 00 >i- t)- to ji to OO O On to li-l t« tl w M tl t< -4 tl m w ti^ 1-t tl 11 to tl M H4 to tl tl tl ►. tl w 11 •» tJ-\0 t^ONON ^w^CTs^ 00 M tl 00 vo to M NO O On O O « to O On t^ O NO to « M ' iHi-iwr*^iH tl r*^r^^tn LO Ln N^ U-l 1-4 m tl to 1- r^ i/N tl tl to ir^ tl li-l tl rh >o to »« g « O ►« i" 1-^ "100 »1- t^ ■* to 00 m tJ- tl M to tOOO On vO to M to T^ to N* w to o f-oo OO t-~ r^ ttttt-iu-iui H«^>^«i-t to ^ U-) T^ ^ ^ to to to Tl- to M tl to to to -1- 4^ to to to ' ■C'^O'+t^O CT\>"I^fnm H 00 O to 00 to ^ O to iri ly^ ^ Ov tl tl O Tl- M " t}- t---oo o to h tl « M t> m tl « tl w tl tl «4 >H tl -1 OOOOrtCO COODr-rHlM t^ CO oo"^ iM t^ CO t^ t 5-1 05 CO U5 lO '1' O 00 t^ OS CS -1 I-- i» O CO 1 rH (M Ul i-H (M rH 1-H W rH IM .-H --1 M 1-1 ffl t-H rH 1— t C<5 5-1 51 rH 5-1 5Nl CO .j:.^ _ ■ -d , l-H 1 , o a J: o "3 -O "S ^ M > U 0*0 s O fc- *^ o -^ 1^^ ■g -a -g, M c -^^4 a: -*!'-! £< o -= O O c£ ;^ 1-5 fe; t^ ClCCCO'l't^ OOQOI^OCOJ 05 C5 C-l CO -t< C0 50C500 i-H oi eo "* C5 O O I-- r- J(5 -f l.'^ lo CO CO ! f-f-ti-fM* -ti-fioiraic lO lO tr --D -o CO CO O 1^ 1^ t- 1^ i^ i~ i^ CO ;» cc^ CO cx) OC CO CC CC CO I^I-l^l-l- 1^ t^ [^ t^ t^ 1, 1^ I- i^ 1^ 1- l^ t^ I- t^ (^ i^ t^ t^ i- t^ t^ t~ w t^ ',:; i:; !:; !^ S 1 iHoios^iis tfi>>ces%o tHINOS-"*** «0 (N. X s» © 1-1 91 as ■<* 1* «I>.XOt© -H (51 M -^ »« O t'«l>»t»l'>I>> t«t^l»l»(X> CC(X o O ^ o »o o Wi OO 00 o 6\ 6 6\ 6^ 6 M so so ? 00 »rs u^ ro t^ OS T»- -o t-,vO M M lr^ ITS s^ O "« 00 r^ OS M OssO M OS o' cfs 0> o' so "h - OS O OS "soo r-^ ti to NN Th fOOO ^00 « vrs M tl OS f» OO OS 1^ M t^ SD 00 OS OS OS Os t^ to '•f ^ OO tooo so in •i- "*-oo o 00 •t- in 1^ Os OS tl Os M ro tl Os r-. Tt- q q OS Os Os o' o' OO o 00 so t1- Os in - TJ-OO 11 M O vO .* to CTs tl I^ "t- o lO in fo M CTs in Os 00 m Os Os o" (7s Os OO Os m O m t~- in CTs 1-^ in M to o OO to M rl rt- to Os in Os O 00 OO M Os OS 'T ? '^. ". '*'. OS OS OS OS rt OS r-- M o On O m tl M so so 1 m csoo f- tl T^ Tt- ro in in •+00 00 OO so OO O O so OO Os C . Os « M IN to O OS 00 rt 00 00 00 00 O O O OS M tl 00 00 invo tl r^ t^ O 00 00 M t- -1 Tt- so m in Tt- Tt- ►. Os CO r^ O OS o O OO tooo tl to o 11 OSM -t- O (^ m m tl Tt- Osoo m O Os Os OS OO so Os M H M CA O m m p^ m M M M M O O M o M M M O O M M O M M *«* i » c Th i-~ M ^ w t^ tJ-OO M p) C^ tJ. U-l n 00 H ■+ OVO t-N t~- ■>*• M Tt- « M u-t Ti- tl ro ro OS OS w vO ro ^so sn to T*. t) * to to to ^ rt Th o Tf*^ ■* un Tj- •+ M to ^ O to r» « so in ■+ rh r^ Tt- r- o tl Tt- Tt- tl mo O 00 tl tl M t^ M t^Si) in r^ tl in m O Tt- tooo r^ to to to tl toso tl to to to M Tt- M m toso ^ Cs m M M to in M tl M 00 M N l^ 1-^ MM M m OS M to M tl toso t^ to t^ to tl Ir o T^ to M m, tl tl to tl OS m tl M tl ~ M Tt- to O tl 00 t1- ^ tl a V i- \o ■+ O 00 vo ^ f) rl t^ O i'^ m f^vo H OS ^ H - H •+ OS ro o rooo rt ro osoo w n •« H so t^ M O tl « to rt to OS « m rt to w n to ■>*• r~. OS M OS to « tl H Ti- to to W t< tooo osoo M to m to o OO tn t-~ t^ to tl ►« Tt- OsSO Tt- so Os rt ■♦ M r- tO to M O OssO M so M M M M to O SO to t^oo tl •« tl Tl- m Tt- >i tl so tl to m f 1 so N to H to tl to m tl Tt- tl M M tJ- in M M Tf M M M H tl t1- o to to tl M Tt- Os m to M tl Tl- m Tt-so O tl M to in 00 00 Tl- Tl- ■+ Tt- tl M tl to to O so Tt-00 tl m to tl h 5 OS !-» 'I- "1 r- u^ .!i- ro V "J-oo cys ^ ■* r~. On n 00 w OS P* *n w M t^ rt wt c^ rt ro CO to o n H * Th t< lO •<}■ ITS toso tnoo M r- to to ti t-^ O tooo r~ r^ tl M in f* •- ■+ t^so I-. "l- •* tl tl so tl OS in ^ m OS -^ O to mm m so so O Tl- M m -f m 00 tl OS Os Tt- tO tl Tt- ^ to to tl 00 Qsoo ro Tl- M M to M T*- m Th tl t-~ tl tl tl Tt- m t^oo to Tt- tl Tt- N m M so Os r^ o Os in o OS r-.so MM tl O tl Tt- t^ T*- m Tt- to tl tJ- 00 Tt- o 1-^00 to tJ- tl m tl H m 1^ tl in in t-^ Tt- OS 00 00 so to o in to so to M m r^ m «^ to Os t^ Os r^ m so OS Tl-sO OS 1 M tl Eh •9 OssO 0» OS OS g OO U-l I'^OO so ■< OS t^ t-~00 t^ m M »!• u-i OS O 11 so O 1^ H so tJ- m tJ- w u^ t« t-~ •'I m M M ri SO to m t-v >- 00 t^ to ^ to in N« Os tl « to mm MM SO SO »n in to Tt- m O to to toso in tl H moo to M •^ M M tl M o n Os Tt- in M m Os M (JS to tl m Tt- m M t^ tl t~ tl M tl tl M d 00 M H t^ M in to to M to O to o so M tl M Tl- ro Tt- tl so to t^ tl O t^ tl M t1- OO M OO OO m to ^ M in tl tooo M M tl M 00050 00 rH l-H (M r-l C X cTj cnci O 5-1 (M CO CO O Od O Cd O) 1^ 1^ t- t^ t- lO to t^ CO 00 Ci C". 05 C5 ov I- l^ I^ t^ t^ t-Tisoo oseo OS 05 --I 5-1 Tf OOiOOO t-.. i~ CO 00 cc f— t i-H r-i rH r-! i-i 1— 1 00 00 e-i 52;nftMS lO --0 tt 1^ 00 O O o o o CO cc CO CO OO 1-1 1— 1 1— 1 r-^ f-H IM CT5 C-l O lO tH (M ?— 1 rl 1-1 CO O ^ n ?! O rt -H n ^ as CO 00 X -JO »-1 1— 1 1-1 r1 1— I Tt 05 10 rH t^ 1-1 ?l cc ^? >c -xj CO : X 3 X 00 X 1 ©©e©»^ rH 1-1 1-^ 1-( 1-H « t^ « a* © 1^ fH ^a tH 91 r- ©ISO ^ »e (51 (N ©1 IN 91 «Dt>.X©© 91 91 91 91 «« ^ 91 M ■«*< »S CC €0 w w w C550 W ^ "^ , 41 641 TABLE XVIII. Elt'tnentrt of the Orbits of C'ometn whieli have been observed, \ i" • .a C i I a c mil c o c 'C s 1. .1 1- f, S >r< £.= £ l^ ■v -H t. i; a ■3 ? " t ■= a 1 i 1 1 « CM n t s £ ■£ o .is OK 1 J'L 1 1 III 0^ .a * aS .b wa . 1 r, to it- ■< O l-~NO Ov , so rl to tovo 1 ■- — 00 00 o to if NO rH rl I-. 4nO - ~ O ONvO I-- to rn to tnxi ^'^ OC 00 tr. rl 00 r^ ri rl rooo C> ^/^ "^00 00 r'^^ fl O rl 1^00 m tr^ CT^ fn to *C ON 'n O ■^ O rl rl On I- »/^vO O t*^ rl On CN r~- »n ;7s ■OVO 1^ OvvO O rl VO « fo r~00 'l-vo 5C O 1- rl CN 00 'o - -. oo to lo i~. rl *t* O r*^ l-- O Ov ^oo rt « rl -^ SO 00 00 ^ tn CN u-> O ON "^ ^ r'l >o H t ~ rl - O I*- rl «o,00 O On il-V) to NO ^ •+ O On •+ r- 1^ to rl O O rl On -t-,1 , On r»oo rl rl «o to to it- rl 00 O VO 00 On to to it- ONOO On 00 ij- ON to to to NO On f*^ M to Ov O to O 0^ ON CT^ ^ ONOQ Ch ^ ^ Ov Ov Ov ON ON ON On Onoo On on On Ov ON Ov Ov O ON Ov On 1 %> •n N« fn -+ 00 o\ 00 t^ oo t'^ -^ 1 00 •+ VO O 00 M w to rl »»• O «n ro t^ VO n ON O ON O O 00 00 N- o CO tl o "^ 00 »i-^£) On T** mo OC 'f ON -^ 00 Ov l^ O O to O fl «" to it- If l~- rl 11 rl VO it- Ov if- tn ON •+ it- Onoo O 00 O 00 r- to to t^ « « •" d - « o' - O - N- ~ « Ml O O w O H H n O « "< 1 i 00 O ■*■ r^oo ^ to Ln u^ ^ VO t^ •+ r-. •!• m m w O 1-^ On OvvO n ly^ to r^ *t- to •*• "4- ri- tl in u-1 fl rl 00 On to to ro It- rl it- t^ Ov it- to rl to 1 -N OO 1^ rl rl j If it- H wi If 1 •** Tj- * t^ rl tooo n to rl to to if O VO to rl rl H it- rl c«; H 00 vo rl ' M M M to „ Ov M r^ O ° 00 VO « 00 « C^ m ro ro r-- I^ to n ro rl vO 'f -<-vO to r^ to to VO On m tn ro On to to ON r^ to rl oo t^ ^ it" to n it- to to n H if n ro to iv, to ON ^ r- »< rl « •+ "t t^vo •<*■ O ro ir\ iri fl rl O Ov OnoO if to O M ON ^ M to rl M M tooo H ONOO -> tl r. •+ rl H rl vo 1^ Ov n to to On ON h> H l^ if if M to to 1 ct ro to to w n •<1- rl rl to 1*- ro On to^O O t^ ro r-. o • On O On O ro t-^ rl to rl r* c< tJ- in H4 i^ ON ro fcN m On « to rl fl i-i •* to If 00 it- li- tooo H rl H to On it-vO t^ •* 1*- to O to to M to rl to to n oo rovo 00 r^ if rl rl VO rl ro rl 00 »n to *f-00 Tj- rl •"j- »}• r)- H* ro ro U-, M e\ (^ ^ n ro to rj- w ro it- ONvD rl n rl to H it- rl tooo Ov w M to to n VO if ro ON If ; to rl rl if tl j t: V lo l-» OS «n t1- to ,* 00 ON t^ -> "1 « rl -t- 1 to it-vO n to ij* to w to to 00 00 ON O 1 it- it- tl to to r^ t-^ n Ov - to H M to rl if NO H ro On 1 to to rl ro 00 >noo t-^ M M 11 n rl r^ C\ rl t^ ov VO ro ov t-o « rl « w rl » •+ If to 1^ t^vo 1^ rl n M « O t--00 <7\ t-- M »n t-i o •-* M to M m to r^ to ro t^ to to r^ to On to H r^ rl O 1^ to to M H to rl M M ro n t^ Ov if vO VO rl O rl r-. O ; rl •" rl rl r< <^ ON r4 ro Ovoo ^ to ~ ■+ OS rl rl fO CO to tJ- On ON 00 On rl rl ij* to to u-iOO 00 O*.^ to if. to it- to H n to H it- t^ O 00 VO lj- TO to lt- oo if O "< H to if ro, rt to So O 00 r< VO M u^ ro ro rl iJ-vO to U-i to fO CO 00 ONOO ro^O rl to n r< f*1 to t^ to f^ to ij- rl NO - 1-. f^ « to to if H l~~ it- to * to to it- to rl 00 vO On to 0\ to to rl to rl 1 rl n r< ti-i rl ro ro w w rl oe n M to r--vO VO to H M H rl O rl On rl O rl rl rl VO t^vo to ro H H MM rl On ON tovo 1 t< CI M S fci X >. t^ O rt cS rt -S al'lll a. -J o 0) a •OO CO 05 cv o ?-H 1— « 1— 1 1— 1 ^H CO CC 00 GO 00 r-t n T— 1 i-H rH O pH iM Cl CI rH IM (M W (M C/1 CO CO CC 00 1— ' rH t-H »-H »-H CI ^r •* fl lo CI CI C) CI CI CZ) OC 00 CO CO r-t t— i f— * rH 1-1 »0 »f^ »C to 'X) (M ?M -ri 'M (TI CC 00 oc^ yj 'JO f— 1 1— 1 ^) 1— • 1— 1 «0 !0 o i-~ t^ CI CI d CI CI CC X CC CO CC r-l ^^ f—t 7—i r^ t^ CS © O CI CI CI CO ?5 M CO CC CC CC CO C4 Cl CO •* lO CO CO CO CC CO 00 00 X CO CC i rH tH ^> rH rH ) 1 'M (51 M ■<** 1* SCtvXCS© setvacot© iH 51 ee •* >« WtN.QC«© »H (51*5 ^19 i r>» !-• IN. (,» t-. -^"""" '1 642 TABLE XVIII. ElementH of tlic OrhitH of C'onietH wliicit Imvc been ob(ierve 01 - t- i 5ft M ■^ «n •+ «" ^ r~^o o» ri «' fCvO ■N >-• cc 05 O « rl ■M B4 3 OvvO 1-- rl o* CTv ef\ C7N 0' o 00 u-1 ■* o M *^ *^ •vo 1^ rl rl Th ■>*• iri ■+ „ 00 vO rl •-" « u-^ m f^ r~ tn On «*■ " o Ov r- rl 1-^ ■+ •* M ^ -X 00 00 »»t ti « rl "*• »/N rl 00 r^vo 00 l^ -J- rl rl yr\ rl m rl v^ ^ ro OS ^ m rl n 'l- rl -UrO » m On 1 tn in rl t^ Ov •+VD NO rl n rl o rl M rl rl n 00 •+ O M Ln •&■ r^. ■" m v£! OS »ri ON r*^ ro M U1 rl rl o\ Ov LTlVO ** ■^ tS'S ©"(M-O" (MCI r— 1 c; — (- c- -, ^— ;i, rt s '^H •J. -< IM (M CO •* O CO CO CC CC CO 00 CO X v: cc 1-t «1 M •* >» i> r» »>• t^ r» >> 6 > A (/) id 3 1 .. i. • , C ^ c •3 V 1-^ c * 1 ■2 c ~ j- M W O"— — -^1 •j; ?; K :- ;3 s ul — W "^ 1- ^ 2-: J.'c'^ w ' « oS aJ 45 o ^ c! b 4! -3 'O ■3 -B 73 t3 -TS 1 s 2 2 s u i. 2 2 2 .2 ' a ^ 6lj^ t(j - sP~ -• tc^ tf:.^ tc-: tc 1 ♦- -c ^ V ' S g 2g £ .5 '3.- * a! ^ 2' 1 = ^11 ft «ft £ J! 2 ji £ b 2' ' S 2 ' ai S^'l Mft = s s vu Mft Mft M G^ ft OS 'i* un r-- m 00 o M tri in ve r» 00 ^^ rl O oo NO in to o r-N t^ n •♦• t^ rl *♦■ ^00 00 00 Os« - ^ T»- l-^Vf) O NO in f.^ m o oo On rl — tl o. m o rl ■■ In in r^ T«-i w^ — O M O O tn Ov O r^NO I-. rosO (*^^.£l M NO "N rl O NO O OO 0*5 O On 00 NO o »o o — On T*^ I - r- w «-«-i sO f n 30 *n>a '^ (n ir. *f rl -4 On 1- in 00 "■ •+ 1 - On ON tl tl 00 r- tn t - "^ rl rl >5 nO y= r-.'x o - so •+ O !-■ rl r" 00 OC "t rl O NO ->5 ro On O tl rl — 00 *t- •1 v5 On in r^- -^ 00 i-^ "i- I-- r^ ^ r O NO in M ONOO OO »- -1 00 ONNO rn rl •■ so *f^ iv^ "^ ^ O^ CN OS O On On t-~ O O O On OS On ON CA O On On ON On Onoo On On On ON in M TO M m i ^ O *r\ "^ NO •!- I~- OO l^ OS TO O rl rl NO O « in ON r- ON On 00 NO ' »/% O r^ >^ m rl O tn ON ON »0 30 to •1-nO rl NO 00 On 00 oo n rl t- *1r ri 00 d ^ t^ OO »n On w I-- Os m O ^*^ r- m O 00 OO to tn -. I 00 rosO TO fi "1 -^ rl oo oo r-- c^ "N CO nC -.O TO 00 "I- ■+ nO TO O TO On tn ^ lA, -t- »n « 00 u-i I-- »r\ O I--- On "*• ON NO -♦• On C iri 1^ js O ON t-- rl SO to rl « On m On OO 1- rl 1- •i-^ '*• o as »rN h^ O O OO •+ On m ONNO rl 00 On On O* On 1 ^ 00 CTvM c^ O ONVO O O "-NO 0\ o ONOO ON r-- 1- On I-- On On On ON ON ON 00 O "-« o « O - « « >- - O O O 0^0 O O w d " -« d t vn U100 fi O rl r'NNO r^ On NO TO O M t^ <7nvO on i1-no m TO rl ^ ^ t^ m r- in •t" inso Tl NO M M %r( H in rl rl r^N m *f f^ 1^ **• m m in m in m I** rl rl ^ »n m w rl tl in rl to w in M u-i ro >« t^ ro H- rl tJ-sO 00 ^ rl ui to to O tn « r-.NO tJ* m mso «J- 00 {?N ONIO NO 00 00 00 00 •t* f» ^ M M in in rl m ^ m rl »f rl to in to M M rn rn iH rl (O >n rl ^ f*i t^ r*^ ro CT\ *J M M M u^ ut On r-- r*^ m in rl « tl oo in NO NO 00 m r-. M O u^ r^ O On c^oo o m tl On ^ •*• r*^ t^ vn M r- ro »n - Tj- ■* tJ- m * - rl- M tooo i'^ to M •>*• *• ^ 0^ OS Os On ^NO OO ON'''0 tJ- O tooo 00 rl 00 n m\o tl 00 ON tJ- rl ' a r^ to u^ tJ- in m T^ 11 « rl Tt TO M in -^ to NK tl m rn *i- rl ^ -+ ^ ^ to tl j Th u^ ^ CN^ NO oo «i- t^ - t^ ON to M 00 NO r-. i-^ ■+ 1 lo r^ r- -N o 1 ■* - moo vO OS - 't- t O TO W-i r^ M m oo ■>♦■ f1 o tr. O NO TO — m *i- m m M 'i- O I>.nO so vO rl t^ tn t^ On — to 1 m TO M tl »H rl tn rl 11 tl M m m to m n tl -H i-i tl rl w m ro NN fi CO 1 :; OS rl T*- 1-^ O t^ rn t^ to t» t^OO rl OnnO « rl ^ in O lo O m r< •*■ X "4 sO r^NO 00 t« CO tn M rl t)- rl » » t)- Ln in m M H w m rl rl in NH rl to "i- X rl tn w ^ in rn TO w t^ w 00 rl "< onno o On TO i^ f 1 O m rl ■+ NO rl 00 r-- On t^ m t^ rl M in rl tl rj- t^ ; 1: •" rt to t» N« W >« m H M «*■ rl to TO TO rl m ^ tl tl to tt- ^ ■* rl M M ro <4- 1 t^ "*• t^ tl o ^ rl 1^ r^oo M On tl OnnO n rl rl r- On Osso O tl rl oo NO t^NO t ON ■+ 1^ w^ lO OS 00 M rl m rl r- 00 ^ '^ t-N On On OnvD moo O ^ ONOO -f NO On t^ m '^ rl r^ r'. ■* in M TO M M r*i 11 m rl tl TO M cl « rl « w -t tl x rl » rl tl to M *9 rt w 00 H Cl t^ in ON On On to^o NO in tl On t^ »♦• O M n m ^ in n tl OnOO in f*N m t-. <7n 0> ^ m rl m tJ- « On TO to ^" to rl in m to ^ TO « ^ tl to BN in in *m to to ■^ w 1 S 0\ rl t-~ r'lvD m t^NO r^ ^ O to tl in w 00 I*- On « m in ^ w O 00 in Th in On ON 1 r^ -, SO *♦• 1 •- TO rl ^ 1 r*! r^ M w rj- m rl rl mm tl TO n w M w in rj- tn m »!- m •COO rl m w m o n ON M ro M 00 so m NO m rl rl oo CO On in «- « NO 0N05 NO to On fN tl E^ M M rl MX r< M M M M n ^ ^ >« rl eOUiOi'^ N u e h 0.0 S-S O 1-5 -»! Hi ■< 1-5 lO iC oo o © © © N M CO CO CO -t -f 'Jf lO iO iO lO CO «o ;o «c © o © © r^ t^ t^ t- t- 1 - CO CO Tj< -)( -t< •^ -*< •!<'♦<•*-»<-»< -11 -t< -f I*! -f *tl -f -* ■'t' -1" •^ -»< f -*< -fi -tl -f -t -r -t* CC 00 00 X CO 00 C» CO OO 00 CC re OD CC ZC CC CC CC CC :c CC CC cC' CC CC CC CC CC CC CC 00 CO X X CO ,— ,,-^,-,^^^,-,^r-,.-^T^r-,^^^^ rr . ^ ,j »r>.x«t© 1-4 (51 M ^ »« «eiN-GC«tO 1-1 (51 M 1* »(5 »r>.xet© -Hi5lOS'«i(S (»r-X©© 1 o r> ■>» [<•!>• QfD (X x an OD (K 00 X X X «S cscsct««t ctoseseto ©«©©© (51 (51 91 ei (51 ©©©©-- (51 (51 (51 (51 81 643 TABLE XVIII. Element of tlie Orl)its of Comets whicli have been observetl. >> 1 1 a. e S . § • 3 5 d •v ^ S r : - ^, yi p., ^ ^ ilisl ^<:^1^ O-^"^-^ BKi^al c2-2;2s;5 H^'^Sx'W S ■" " .• »^ £ K ■»■:„• (jj q3 . ^ ^ -*^ 00 ■*-! iCU bC^ SO _ *J u CC^ 0=: >vO=^=^5: t) 0=:u Co u == Sj i ^- jU _____ __ Co'-" OtH OJ-c^ ijS^ijS "^ "^^^ e^a f« pp:5 Q W Ci «Q ""wo ;! 3 O O O Ov -« r» r*- ^ ^ rr. 5 rl Tt- e*-;^ « ^ w, i-v O M M t-> U1 f1 M v^ rl tr-i^xi tJ- 00 r-- r-oo ^^ •4- H vD r* i-^ i^vO <:#- u- T^ rh »/*. i-l O O 00 o^ r^o 00 ;-• 00 (.. - rt r-. ri H r* r- o « ii^OC *^ f^ m rl •-'^ ft 00 >/^ rf t---vo t^ t~- t^ fi rl 00 f^ rj- r» Oh "> -n o M ^ ^ « _ r. ■rh OC OC « rj- « r» y; 1 /-. -H m I--. «. C^ t-- r-^OC DC r* fl "^-OO fO OnOO to tl •* - rr M O O-' m ri oo 'i-'^ ry. on t- I--. i-i — Pi fi •-< !-• t^ ('- ri "^ f*^ r*iDC ^'^ ' o 00 SO "-> c^ »r, "" ^ C7\ i/~i f) ■>D »''. c^ t'l »/■■ 00 '■" "-^ rj- "^ t*". •*■ 'n H c^^o (>vc t-- v^ sT) m oc l •—1 CN O ON O 1- t» O t> « »J^ Ca Tl- ri ri rt- crs as « f'^ r-- »n q 00 t-- ».i^oo r- q q q 00 \ r^ d a\ d c^ d O' C^ C>. O^ ds dv d d ds <:> (^ 6 ds '^ (^ 6 6 c^ <> 6 c\ 6s <> 6\ 6s d d d d^ m <^ ^ T»-^0 vo rt-'O rt- -+• »/100 -n "O i »C - O i^- w: ri ^ 1/- ^c el ■^- X) rh ■^ "-r r-^ 1/1 ^ '^ 00 >■■< r^ M CN n rO j 00 r^. lO vo 'sD v^ r-. \C I-' *- 'H Or! r) 3C C7S00 i-r^ -« ■> cTi t-^- »- c^ V DO 00 OO O cl >r: 'O »>-.>=crs-^c^^rl m-fJ-r-CNt^ r» rt M r*^ OOC M 1 r-. r--Dc -+■ O^' 1- rl V/IDO ^ (^ »-< c^'C i-n ',--: t-. r-- C7N ri 00 ^O ".D rj- CJv J\ ON iri m Cn tJ- m ui — 0>oo rl Ooc -o •+■ Os 0^00 Cn O- '^1 tl ^n o o o o o 1*1 v^ O oo ^- c^' CTN ON i-- a> it^ c^oc CN 000 c> On OS C^^ oc r-. ; " d c d «■ o d d -■ d « d d d d d « «' -.' w d d d d d d d d d d d d d d' d s mo Ov -*- CI C< OO f1 t- «-o CTC\00f"1 wrf^r^N- 0000N''-*'O C> r'l m M *J* M 1/-. M ^'^ M U-^ W f. ir^ rr\ (-*-, (Ai M cl »-< ■'j* ■^ "^ Cl c^ »n t-( vr >^ M vo ui c( rt '-n M ui '" .. f» t^ Ln >H 00 ►^ 'O c^.oo t^ M ,n i.^ r^ C7S OS r» 00 «!t-cc -i- ^00 MVDoc r^vO rooovC rj-r-- ir, M r* \r\ in ^mir. N. r^ -^ •' OC vo VO kO ^ M I-* ro 1-- « '^w^'Svi»o^'sOoor^ ^«vnrtw McoNu^f*^ mc^^MioM s "" r-. o CO M \D ^* m « M hs \rioo \0 oc rt t^ fo o> vn f* f< fi ►H 1/1 rn ►•. (;> i-~- 00 r^. m p [ c? i-t rn m i4-i »y-i n rt Tj- rt f» rl ui PH f. m c^ rl •;*• rl ■«♦••-. rl t*i '^■'i'^" «nMfi f^t ^n fl O f» »'•) a^OC r*! -f -*- r^i-n^jDCso oor--inr-. rf-oooso-^ M r*i M m ^ Ovoo Cs f*"- "< O r^ ir> i-»Tj-';fsD'*- '^ tl ti ^ ^ ti <^0C ^ m •^♦■^OrtO mitO'^vOm n ti M r) Id cl rn CI N to »-t t* ti t*i (■n m el w c» tn (^ M rt M M Cl M 1 ;; vo vnoo in I^ O iJ-> OO tl CO vO fi c> r- o^ »■'> c^ ri-^o f»o >-n m-*.o n rtOMM OoOO^/^Tt- v^- •^ W, Tt T) »^-H- c^ r» O oo t^ fjo CTi r OS inHOOt^ON'^'^t-.Orl oli-«OTfr Ooe vOr^iro MMOrtw •- M n "^ « M (4 M f1 M M M .it MM t-l M rt rt rt »H ► •tr.Kr.K. -.^.r.. •«•, •^r^•>.■kK tx oT-c^oo cc'oT T-.rcr?co"'t" Ci (M S^l CO Oi -H CCM T 'M r^ i(:> LO Oi ^H if^rr-Too'i-Tco' o'cToo'co'Tr »- TI (>3 i-H <;■» cc> — t — Tl rH Tl .-. CJ Tl (M rH CI CM Tl Tl t-H 0 i^ IT >^ »0 1 cc X '? ci'»/i'i* ■r^slM'H«>r^ «i>.arrtO "-siM-t*!* i o ' 1.1N f"^ 1«-1 *«« ?-* _< _m iM f*.] SI SI CI SI SI SlSlSl'flCS «f"^WW «*5!<5M'* -ti-^-^-i.'^ i 1 ''" 1 rjj »» ^1 (Jj <« SI SI 51*1 '51 SI SI SI SI SI SI SI 01 SI «1 SI SI 1*1 SI SI SI «J SI SI SI SI SI SI SI SI i 1 1 - r o cs 1 1 1 .1 6U TABLE XVIII. Elements of the Orbits of Comets whicli li /^OO 1 ^ C^ r*^ f^ w O O O O O c>OC O •- a\ o »^' f» *'"' OS Cv^ 30 t-- d d d d d •*■ ti-i 11 oe H t-. fl ^ I-. ro C 1 rl to 1 M n '^^ ^ I-' It •- tl « OO ^ CO P-1 0\ »r> ■<}■ t-s to ro M Ov K/-) -O o ■± r*> 1/^ *n TO .-< C* rh r« ». M >o l-o ro H M h>< Ov rl w M M »' o'croo'co'cr t- t^ I - CC CO lO ifti 1" '' i'^ X oo t: t: » 1- 91 «•♦<>'? ^ •* -t -* •*! -3 ?"' •3 i -5 1 ^ s^ ^ ■ u 11 v ^ 1 c . c «-; .— . at >^» — ^ 1 O p^.ij . ^ to ■11. i •I i.1 .1 ~i? S^ ! Ef^SwS ^ C X i^ W X Z^ ^ ^ PH>H '5^-/^ J Q 6 0 00 ^ in Tt-\0 NO -< T»- Th rl lO I-. •+ o NO cl ir^ r- M NO r- oc on CI '^ -. O t^ - r- .- th n *iJ [^ CO o 't'OO f*^ 00 t^ 00 C.1 VO >3 fl ^ "H t4- rl ti t^ in »n n o 1X3 to >J -f ■* cl lO H. to •+ O 00 cl H, CO CO W r-- C^ "^ r» NO OC in On to rl »r> rl r» cn Cl "^ in ro n 00 Oi t-.oo rl 00 CNvO t- 'r) On invO o oo r^vD ro rl vn o 1-- HI ^ l^li? — fl CO ONOO O O rl c^ -f oo n rl u-ivo oc *1- « 00 rl CN in *" O t- ^^ »^ H HI 30 q '♦• On Cl -. q d 6\ d\ 6\Q d <> d d <> CTl C^ CTi CTi C^ (ji oi c> cfi d On On Oi OS Jx C?v CTn d 00 On d d d \o Cl •i- 1 fl LTi « f< •*■ oo O Oi M fl 00 rl HI I ' CO N* \D CO CO c\ O t-. ON t-^ r<^ ONOO in <7v O CO h. cl ■4- CO t-. rl ON NO t-- .«■ m vo vo r-~ t-. CO in \o - On ^ m r- ON O »*• "■ «■ d d d - -■ i' >-■ d «■ d d -■ d „■ d -.' d "' d -' -«■ '^' d d d -■ t 0^ rt CTv *^ w r-. in u-i Thoo O HI 00 CO lO to tOlO CO •* « 10 t~- ON ^ rl ^ ►■• « M <^ in r^ in ' tJ- Tj- ^ « M CO ^ in CO CO cl fl f 1 to in in .^ -^ ti rl r* to 1 ■- ^ O K M rh c) \0 M »n CO i^ ^ mic r-- in <1- m rl 1 rl 00 inoo On 1 tl 00 r^ t-- fl 00 rl H t^ to rt M ^ cl in in H H rl rl rl rl HI tr^ M vn t<\ M rt M 1 - rt CO CO M " C< 00 VO HI HI •H CO a\f>Q CN 00 Oi in N- CO r^ 'O rl in t--- inoo ^ CO in ^ 00 t^ r-N 1^ M 00 NO 1 1 « 00 r^ rt- t- cl r--oo **• « VO ^OO vO 00 r--i£J 00 10 t^ TJ- NN (50 ^ ^ oo CO Tf- f 1 CO ^ in (> ■+ el in fl ClOO fo rooo to in f 1 lO 10 -^lO t^vo fl fl tr> CO fo in — m M CO ^ in in in in m in CO to rl m fl fl d c* « ft « in ^ CO ^ H oo OsM O m o I'OiO ca -j- in r^O O rl sD ■+ in in CO rl n* in n On ON CO HI - ^ ly-i r^ Tf r*i NO M in in a, ^ xr,^ t> ■4- rj- M m ^ ^ in in to in rl **■ in - ^ « un rl .f CO to O •<*• <0 rt- ON CN r- *oo ■* •+ !7-00 in ■+ NO r^ iniO M ON r-~ *1- in •*• in HI to to - OnOO 00 t^ C< VO CO o in in d 00 O cl r^ >♦■ CO cl to in M in tl- ON t-. ON CO NO in to t^NO I-. CO « to c) ** CO CO M Cl fc^ CO to M CO ^ rl fl to »N « fl - n fl fl M 5 ^ in M o ^ tJ- CO CO, u-» 00 I HI m H t^iO l^iO CO in in NO f^OO 00 tj- d f< « NO t^ 00 in m r^ HI CO rl HI fl CO HI rl n * •+ rl «. rl M in in ro CO CO th m M M oo ^ fl ^-. Ml CO o in\o M ONfl rj- O n 0' rl m n to ^- t^ « HI 00 fl to in 00 NO fl 1: " ir; « UN to - N ■>*- « CO m rl CO rl ■+ rl - CO -f rl ro « „ Tf fl HI fl in in Th u-i-sO VC t^ Cs it* in CO O H. n CO CN cooo Oi ^ in n h^ rl if rl Oi •+ in ^ ro in Tf On HI rl « ON m rl C. ^ rl CO tr, vt- r-- r-^ in^ « ^ If t-^ in OMO 00 00 in fl NO rf NO ^ t^NO ! 1- H ~ "^ M HI rl fl M HI fl to -4 -■ fl CO n n ** 1 ft> t^ CN « ro p^ lO 00 O invc O t^-OO Oi Oi rl oo ^ CO r^vc ^4- in On H. 4 '»• cl .+ »f « to t'-OO in in CO CO ir-, CO CO T^ vo to rl in CO » Tf in CO fl -* HI in in CO rl rl •+ ■+ - rl i? c) moo « ^ 'O »n Cn rl O o «" (^ t~. t~~ to to to t-- rl On in ON NO ■- t^NO fl On ON t- •*• •>♦• cl rl r« « cl Tj- fl -, - 1^ in ro i4- .f to to N- in 'f ■+ ~ - M fl ro M « t-^ ^•. cooo M O inia t-~ vO CTi fl 'f •♦ O H. 00 -< - - 00 I*- ON CO 0\ M t^OO CO f« M rl M rt cl - - M M MM rl rl \ &1 ififfcTyrM i-rc-T'S'rT'S x'c^T— "t-^o" !i\'i^\:cn-t o-jTr-ro-ri-.- 1^ i^ii rlni ■"-"-r'rTi^." Cl Hi r-c — ( I. -• £; 2 Cf, ;u ^, 1-5 ^ c = o ^ X l-I 1-5 C 1^ < /; ft r:; bi:_; c C - < ^ HIHIO >-5 J--J»t- CO 00 oc X 00 X C5 O O 3 o Hi Hi ,— ri iri -M •M C^ CO S « 2 2 3 -f -f -t" -r IC •p X t;. 1- lO i.t> iC "0 >o iC! ir? -c -x -^ ?o ;c -^c «£> XI ■£> (£> -w ■£ ■■£> :c --s -c X X * X X X oo X X a: 00 X' X X X X X X X X X X X •/■/ X X ■X X X X X HH T^ HI X X X X X HI HI HH H- 1 Hi f' HH HI HI HI H" HI Hi Hi rH Hi HI n HI HI HI ! «e ! - /■ St o '- 91 r? >* >!? •£ 1 . r « s — 91W *'9 :e 1^ :r Si e -^9l«-t? ir >* "9 ;£ ^ ^ « ^ '<^ :c -^ ;£ tt 1- 1^. In. In. In. |, 1.1-. I'. I-. X ©111 '51*1 'it 91 91 91 SI 91 91 9191 91 91 '51'^I'M^I^I 9I9I9I919I 9I919I919I 91 91 91 91 91 615 TABLE XIX. Elements of the Orbits of the Minor Planets. 2 mil coSoH H .2 v* orj V, to « 5 .s . .IS t;-T!~.a c g.s.sss "■' a a •- (iaspar . Hind. j Gaspar 1 Ga.spar Hind. Hind. 03 i »—:/:- t-i Ci CO I- <■, 7. P3 Q — 'U ■^ '~ '■"' c/:, oj ^/? ^ -vj g S O P. ft. ^, < O < < t^ 1^ I- X Ci 'Ti' *t 't -r -r Vj cc CC CO CO S V. !l^ S 1-5 o = -s — ^ • O O '(^ .o o CO 'y, xj CO CO . T c-l jr. rt r- N f I I— s < 4 < -i; Cl CI C\ C-1 CI i* to '--J >r: -o CO CO 'X X CO ■3 -J t-t (« 'i • • B o .-3 ^ '^ ;:i. c; ^ a a ■^- rt C S 5 C iJ •n to .O ..-5 O : o o c s. CI CI 01 ^^ « .- .o ..-: ,,-. .o CO CO rx CO CO h^ -^ h5 r^ H-l . r- C^ lO CO I J3 ja >^ > J- U .^ <^ /^ ^, r^ 1-3 r: :■■:-* -f f ic uT .-1 .o .n r/j 'X V:* ■/) VD c a ?; -5 a »; = C 4J tt-r i gJ3 • . _< .4 ra — - rJ; C O -<'■< O -+ ('I N a^ »-^ M O vO O -t- r» rt ►-. -rt- -t- r< r- ►- T*- rt- -t «T+ 6 6 6 6 6 ^ i-< ONOC vo t^ ro to c«^ ^f d d 6 d 6 CC oo *t- •+ ON O ro »n r*^ O u^ H- O ^ rl O^-oc « rl H oo %c ^ i-t ri m »*! -t- Tt- ^ d d 6 6 d O O c^ »/^ rt *^ f 1 O CT^^ Os m a\ >ri o\ vy, m O r^ •-• \£) C^sC OO oc rf- r*^ rn ro r*^ d d d d d CC 00 l-^ Pi O t^ ^ ro ly^O »n C^ irv r*^ rO O "^ Cn t-. O OC \0 -" O^0O r*^ -^ -i- •+ «^ d d d d c On r-. 0\ T^ ^r\ ^ O ro^£3 O I-" f* r4 "i-oc rl (?v -t* O r^ r~. ro ri :^ -+ O t^ rt- C^ rj- tJ- c*^ d d d d d O oc « \0 00 oc -+• rl ^ O yo rl t-. c^ t~^ rj- -+ -i- rt- rt- 6 6 6 6 6 00 VO u-1 O 11 VCJ oo 00 Pi r» O O c^ O O oc O VO o ►^ a\ 4* r^ i^- I--, t-^oc o^oo •i-t O CO 00 o rl sT; Os CNOo rl o O oo I-" oo oc ro r^ i-t O VO ro m rn ON pi -sO pi ■4- r^ rl C^ 'o »o pl o CNtc oo oo ro 00 p^^o oc r-- t-^ O u-1 ^oc oo '^o \o ■-< r^ »o t^ O '-' M i^ c> pi d d oc O ^ rl ro ^- oc r*"OC r^ -rt* pi ^ p^X; ro to VO N- o ^ pi lo, lo -I r-oo ro "4- t*"\d t*^ ro « po po lo O t^oc vi c> oc 00 OC -^-oo •sO -+• PI rj- ro -t- o n -+ ^ oc o^ rl p*^ r-^ >o Cn ■■ c*^ pi CN-o ^ a^ >n I- oc so '^ t~^ oc c^ r-oc c^ 00 o <:> r*^ -' O M3 'O «o *i- »o r~--oo wioo oo »oo oc t^ CO pi 1^ lo d ro to ro O 00 so CC r--oo ^ oc »ooo POM ooHP^^'1- MCNMr-»-pi p*^popipit^ sot^oocc^-^ poHv'^t^f*^ p*^ r^*.D w r-. 5; rl r^ »o\o O M rl p<^ CO tJ- to rt- to O ^ O VO f 1 VO PO VO to LO i-t O LO rf- ■+ -^ f^ PO Pi to «+■ -i- Pi t*^ rl oc o^ t*^ t^ rt" to to to ■^ poosr-«io lopi-t-^o O P» O O t M Pl CO »■ t^ o ■«*■ r- ^n ■<4* pl po i-i r^ r^ Pl ^00 TJ-OC 00 pl 00 ^ (-. to to CNC'»'*'PiPo O'jcr^ Ti-'^o T*- po r^ ON Pl MtOpl-+Tj- U^, POP-.I-. -t"'i"*+ Pl o 1^ toso ■^ to CTsoo -^t" t~-. p* •+ c>vo pi ooootooo O00«*ip» -o Pl • to Tj- to tooo O O VO oc POsO Cv ■^- Pl >o po -i- \0 t^OO VO ■+ ►1 ■•4- to W r- ro o r-- ^- to rl to rf- Pl Pl OOO VO VO •^\D r--oo »o to ON f-vo VO -^ O" VO t-' TJ* P0\0 ON pi »^ to to r^ Ov r^vo pi VO -+ P4 »o to VO to tooc ■+ Tt- -^ -I- to _ O •+ to t^ VO rf* to to to to rJ-OC OOv*-" rOLoO"^0 t^ 0*vO OvvO ^ Pt Pl 'J- ^Pl CvvO Pl q q c^ qvvq VO to OvOO t^ r- CNvo VO Pl pl to Pl cs to tooc vonoo o»t^cor^io ^opl^^'-•^-- pioopi'.oto t--. tosc Pl 00 to to rl 11 O to Cv ► rl O Pl O to PH OC r- r^ O '+• r^ 'ooo L*^ 1-* w »o 4* PO ON r- r-- »-* po to ^ w to tJ- OO Ov O 00 vC to to M VO OO 11 pl N- tl t-^ M to I-I rl ^, *-« VO to ■»!• fl Ti" lo pl ►I r-. -t- to f 1 Pl to H< ^ lo to to to\o po Pl ro -i-00 O 11 rl Pl -:t- O O 00 to t^ rl- Ov ONO •1 ^Pl 1 to PI VO rl tJ- VO Pl O n "^ to — O ^ n Pl t-- ov r-- Pl r^ to rj- w 1^ VO O Pl LO rl (o ^ .« 00 VO VO to M to to ON Pl pl ON li On LO ON Os to CN >OOC to ■•i-VO LO to to (O t 00 VO n 00 OO w rorl -^h OO Pl ■^ On to ■+ Pl VO -1- PO PO tJ- to O Pl VO « rj- On to Pl Pl w to to LO q O O Pl 00 r^ r^ « -4- "-^ <1 n to to to ^sO -+ to r^ On pi Pl LO Pl 'i-OO M O Os LO TJ- -i- to to, to -t-sO oo rJ* C^ 't- t*^ O n to to '^-nvOtoH- OsOvOOOOv t^ O OvOO VO to ^00 LO pi to o Pl to to NN n cl toOvor^O wOstoplLo vOn to O oo n to ON oc VO »- VO to to Tt" rl LO to VO ^00 Os r--oo to o Pl tl »o Pl -i- O f 00 VOOO •- CO •+ ^ to H O OV ■+ LO CO rl 00 »o Pl .-.to so "-; t-- rl O VO N VO ON ON VO to vosO 00 Pl to to VO pl VO l^ Pl VO n tooo Pi VO, ro P) M M 1 ^ rl PO >0 Ov PO -+ rl ro tl O VO O H to W LO Pl -^ Pl r-. w ONVO tJ- CO M pi M PO to to n rl o o o o o t^ r^ **• ■^* CO :?« & 1 G ^ P •-: "-s /; t-5 W2 •lO tC »0 * ^■'i Pl NM .1 o o o o o o o i-^ o c^i CO « (M a a G GXi 3 rt «J 3 « l-5H5^t-3&M «c o -y: CO -^ to to "^ tO «o 00 ;/: OD c/: CO to On 1 ON Ml M to rf- to ^ Pl *f *^ LO M to to ONVO O 't* H to^o Pl PO Pl Pl rl ►-< n w o o o 1 - o* cj od o C^ CI c^ « 3 S O S to f- -^ -*■ '^ eo '-T '.c to '1^ c/i CC en ca c lO «0 '-O «o «o CO CO 'X CC CO LO H t-^ Pl LO LO to pl to Pl to t^OO M N Pl po M O* PO r^-\Q >0 rf- O O O to M to »M to o o = o o 5 ri rt ?3 3 M 'O (M «0 'O L.O to to to '-O CO CO CO 00 QC -i- VO rl to pl •*• po 00 ■+ O to c> n Pl -4- 1-~ •+• flS ce ^4> = 3 t-3>-2 Ht, < ^ t^ ifi CO '■'^ :C CO '■-: tc to -^ CO CO CO oo x to c4 ^=3 a "-. i: c! *" » 4 a r = ^ X di J n 1.-5 >o >^ O^ -X CA) C*v ri r-- c> t^ 6 6 6 6 ^ ri 1^ >-r. O -i oc r--oo ^ M -1- ir. -t- >/^ ^ rri t^ a^ r» II o o a> c>^o ^ Vi-i C^ i^OC >>> -i- "i"^ "^ ■r^ M >/^ ('^ f^ ^ 00 1^,00 ^ ro+ ■4-"-, ►« O O + "^ ro rl oo >'^ vO t~. « O tl « - I r»T+ M O i-i ON O 'I- "^ O - CO NH ej ~b b f^- T "^ N -1- CO 00 -i- O CO ;> o + O r c) + I - ■ <^ ir> ^^ o c* r- e-l CI . . ■ ^v" Kj w. ^ < >-; ..T tC O -^ — '-"^ '^ S '-i <« CO 'li i S o o .2 4* •- ?» f? "* '/} t3 . . '-J -a -va —5 ~3 -3 a S|a a a a a 3 c . a •gcESg •ggcf ^ c i: ~ "3 J) s-S to U O = OT w: O C y, ;/j C O A y. 3 S'S § S2 ^ siuS^ ^• "^ O S J5 J= = O 3 o o o o 3 = s a< Ci^oua ^H fl^ o^ r« ^ fi,»JCOfi( hJ'J! a s = a ■9. c u ° S. — •A K ^ C 3 X 2 5 w 5 '" 2 "^ 3 3 .a « "3 ;5 J J ^' Cn C i :3 01 *oriO r»r»t^r»^ -ri- o ^D O \0 ^ y^ i~^ O :>■ Qfc H- t^ Cl 1/^ N r^ C--: ■+ ^ M c> r) ■X) t'l T«-i f4 t*^ ■:}• io 'hoc 'irzr^ K*^ O »«". i^'Xi r» '^•^^'+t*^ '^'T^T^'i' '^'l'*^^^ ddodd 6 6 6 6 d 6 6 6 6 6 O VO '*• O H sO w" iri -"t-i^ r^ « ^ ri O f^-^ ^ -*- rf- d d 6 d d -t- O ►- ^D O C" « •-. v£> »r^ tl rl -h -+• C> Th -rf- Tf rt- CO d d d d d It ys c> r^ «^ CN -f r> M O •1- «y" c^y. -h r- O t-- r) rn ■H- *1- c^ "1* v^ 6 6 6 6 6 o o u-i rl O O oc -t - « O ^ r O -^ OO 1-^ iri H* t^ dvc O r» 'H »" O oc t*~ r» to C\ t-' n CN ri O O fi O O toy^ to t-. r^ OO ^^ ri o r^ r» C^ r) »o o ■OvC -C Cv -f O "t^ to rooci CN "i* tovT^ 'T* r^ r) 00 r^ to \o toxi ▼i- o^ ro 1/^, t-^ \r\ r\ oc rl ■^- >/^ rt OO ^v^£) ^ OO tn o 'sO -t -t t-^ <-n CO ON 1^ CnsO 00 t^ r--- oc rl ON tooc ■+ 1 1 ^ O^ 'J*! OC >o t^ r- o oc O t^oc O « — i/~ fi o oc H- i'^ O ^ ri -t-^ ON ^'^ ■o ^ o^oo >n oc o^ r-'-c oc r» « t-^ r» vC t^vO 00 On c» i^ t-~- ON to u-» u^ ^ to 1 to «^ .' ^ -:■• Pi M H to ►* O ' 00 to CO t-« OO i tJ- tJ i^ ly^ t}- iri r» OO »n to CvOO ^ •1 O m*o O ^ vr, to f^ O r< to rt 'f 't- t^ ro5C O r» d -4- ■+ ^-C^ t-s CN ^ ^ CN ■+ T- -1- n « W-t ^ f » tooo -h ><-. f» '^ CO to to to CN -r On -i- ir^0O VD tn tJ* tn CO tv.'sO to W '4- CaO 00 to ir> ^D Oc\d '5C t^ "i- tr-, -J- CO ON '^ ^ COvC fOvr.^M00 toincl^OO OiCNt^ t--.^ On O O O O c» CO O r-. »J^ iH CO Tt* w^ r* ci 00 OC ^ 00 to c* ■4* ^ 00 q o ctnoc H 1-^ ON *-) CTn in ■«#- to -t- ■+ C*\0 tt »n ^^ C^ t^OO Cl U-1 CT^ »+ t^ * Tt* u-i r< « vr^ ro rt « t--- O ONOO t^ «^ -t-'sD ^ CO \D »rn CO O OO »n t^o ri CTN ft M M tn N4 M t}- M C4 1 in "I- o^ O to to M to **• ON WOO 00 in "n H CO 00 fO\o O ^ *noo CO co^c n ^j^^ <*^ d c^ i t^ oc >n ti-^oo to OO c» u-^ M jro to in i-^oo « a\ ^ ON inoo V 4- !> Tt- ►^ oc M H CO M in — ct t^M m -• w t» ri c^ -«f\0 O t^ •+ ON C» I - 00 d CO ci c^ in ^t- -i- CO •sO O t~~* '■O'.O CO CO O* -t -+■ M OO f % ^ to ^ M N M M <- t--^ in CT* 4-00 CNM 4 <«» ^ t^ CO I- ^ in O 00 C4 M CO rl vo •+■ '» in -i- o> r^ J* to q q -j-'O t^ mvo oc O O •-' CO to r-~ » T*-in 1^ rt O »f '^^ O CO 00 On t-- O^ O ■+ ^ 'f to ^ -f+ H O CO O "4' fo in On 1-^ c» sD OO d 4 •1- •*- o CO T--.00 d v.* (^ CO i-i lr^oo ON M CO CO ri t^ m 3N O c» -♦• O « O r» On O nD r-- ON M rl •-« N-* M O ci 00 in »+ >n N OO t^ 4 ro t-v Os »o O fOOC CO > OC ON f*N 4 ct to to O '/-.'JO 00 m -«♦• ►-* in\£) o -noc 00 ~N NT n 4 c4 CI »y- u-1 to O* r- •-« 00 CO rt *J- 1-« to CO cooo OO oc ON moo "-< On NO C* X ^ -I ■^-^ 00 to ri to CO 1^ CO in e^ O • OC On ON ; Cl C^ Tt- ■ « OO to M It -f On ^ to*c 1^ O 1 to ci tO'SC t\ m r-- o -i- mNO in to to •H »f OO C» ■+ t^OO O -t-oc -1- tl it O ON tl -t- to d NO OO y'r. 6 in r) to >n00 -t-NO to C4 •i-\jD lo On O ■«+• o -t o o t«~ to hH ro N O -t 'TN '-n (3 O m t^ 'T CO to . «♦-« M iH O ^ ^ » o o >" ■»•; sj ei M o y I/-1 O M It H ^ 'n »n ^ Tf- «t c» rj- cl in ft N in On ■^ f On moo 00 00 w N w rl ___ 'O o o o o NO o r-- (J 00 r*- rj- H COi« It O « O 1^ tt fi ^M rl N \0 'O A 00 ^o -4 N O - :o"« «■ tC I- --i -* noo f rl to CO rl - o o o « 7> 1^ 1-^ 1^ 9 to -^ ft ^i; « f-, Cl CONO » in -J- ' r^NO to -t y: 00 ON On 'n O o o c c: — I- I- o r: -/ -c ^ ^^.'«'5^ = ^ c ;:* ^sg'si I'ltjil iiiii liiii to o to M ^ .:i ■< jr W '^^•i'¥ yllt -li^^- i^ip i-s^H S Luther. Peters. Tuttle. TempeL Peters. D'Arre.st. Peters. j Luther. ! Watson. Pogson. Tempel. 1 Luther. (rasparis. Luther. Peters. Tietjen. Pogson. Peters. Stephan. Luther. Stephan. O o So TT 05 t- 0> M ■— C-1 i.-5 r)< e-) r-iW C^W C-lrHrHr-l 1-M 1-t (M CJ N CI (M i>V M -J* 00 cc CO CO CO c/; !» CO r/; CO CC C^ C^ CS r-( I-* rH •+< -^ lO lO .ri t3 CO «5 o o o CO CO o) go xi CO c/) CO CO <» CO 2 «^ rh »/-iV£) 00 ro -+ i^ foo ^ »/-) fi -•;- ri H f^ w OQ ^ ddcJdd 6 6 6 6 6 rt -t-r* 00 T*- rOfltot^r^ ONf'^ONwfl O lOOO ■+1-- NOVO OnO^ oo o to ri oo o "^ >n Ht i-o <^ »n o »^ to to o to rJ NO ctn r^ «ri T^oo r-.fi On'+'+OOnO •**TfcOtO-+ r^ti^^,4.-rf-*^ ddddd dddddd o 1 1 o i O O CO vo u^ ri vxs c^ CN M O O r^-X! O <^ -i- -i- «*! tn 1/-) O t/^ ri oo \h 6 -^^ ri On ri >n -rfs CTn r-^ o oo tr^oo fcooo 00 o CO rf- C» O 00 O IHNO e)MM^M ooooNOm >noo O t^ ri- »n t^ CN '1-1 »v-i\0 to O lAc^r^t^d ficodNtifit^ to t^ to t-- fl m •1-NO r^ fO'C t^ t^ ON OnOO no W-i t^OO NO 00 ^ K4 0'itri>«t-< 0't*>J*>«^ t trivodcsci fid4*<^»rj tmi oo oo 11 »nNO r-' p» r^ r4 •^ toM ^-tovotiiAfifir^ co^i-tocotri cor»u-tc< t» cocoOnOnO ONONONt^O-ti- n Tfco rj-coritom clrO'+tOM H. -f 0\0^ f< \n '^ *^ " " '" 1 ?S i rl i/~i irwo r* OO ^O On •'^ f^ OOOflOOO NOOwOMO V 1 o 1 5- w-t-as-^-rJ- OvO O^O iH *-• r» H »/^ rl »^ r^ ^^ O <•.■-• NO r» -** H cc oo »/i >/-) ^MMMi-4 T^ritntONH ivT-iNrtto t^M'H-rtr^ON in to Nio tj-tntMcoj-i t^ H vn ON M ri- tr^NO r< c< 1 t. t^ CT^ £> d^ r*" *i t^oo ri tri ^ C^ -f -foe vo oo &Mi^ N w _^or>.(^t^o\ HM f*^^ 00 Owo ON»ON* roOM ^OiTMiinto NOOoovnO -^ wiMON^to dN(>t^t^*i-d iTN to -+ vn in M HNO't-rltl incort-wOONi cointortto li-iciTt-co m ti NO r-- 1^ to f-NO i^ w n « HHr*o oot^i-^wr^M CO r4 c» to M t— 1 M pa pq ' b oovnOHO mO*^»-«u^ On t^ On t-" On 0^\0 «nOO to m -V00«0^ OON t^30 ri „ M t^ ON I^ ^- C*^00 M tJ- m "MO to r^ ONi^H'+u-i c^ c> d d 00 cox' d d^ t^ CO w H f< -+ to N Cl to5o OO rt f » On M w^ O tONO COmMmCI COflli-iCO M CO w 00 ON H 00 r^oo On ■+ in ^ tooo tort HtoO-^ONr- « w CO to to CO c^ rl •O :^ \o •-. oo »-" oo M vo ly-i r-- r- »MO^0 rnOs NOCO»ht> N -i- Eh ^ t~^ i-Ti I/-1 H O NO vO '^' '-'I ^ O t^OO t*^0\ w ^o O O w »-i w »H M m to f«n H CO w _ m ft »n O rn u-ioo vO m O Oootori'+ro u-\rt«o »n 00 rJ On »n O •^* ri w CN On tn w tJ »n ^4* rt M m nOcow^OOn tn«i»u^ ^NO vC H to CO tJ- f< ^ ♦ to rt t^ t1->^ 00 ^\0 On f< NO r^tOM^un i^incomro CO to rl to t*s ro Epoch and Mean Equinox. Berlin Mean Time. ooooo ooooo CO O -^ O M 1^ O -^ rH t-i i-jHjOt-^&H t-5 »-5 O •-: O ^ eo -*< «c 'i* PT (C lO "^ 'j^ COCCOOCOOO cococococo o o o =: o o lO «o =: cr -^^ <©^00 a^ tn t^roO t^doc H O <« rl O rt O '"00 t~> r^ »n o c^^o O « DO r>. >n 1.^ 1*^ -t- r» r*> rl r" O O 8 so ^ C^^O o o o >- o o o o o o o o o o o o o o q 6 6 6 6 6 + 1 I + + 88 o o d d I I CO t^NO O »-« *'^ •+ rt O O CO h- r» O ONvO OnOOO Ono inr-.0^ ^ooo t^roHi wi^ ■-+• in^o NO tooo NO NO OO O O f* ONTj-vnTf-o Noooqqqq 66666666 vn rONO in w 00 -^ d N »^ CO ++ 111+ t ^00 ^ O t*> rt r» CO (N q t^ VD M »noo It W 1 CO t^ M 00 O^ND t^ in M cl tJ- ^ o t^ to M w rt O ^ M On w H O ^ to w 00 NO (■» O* Ni f CO r* to vn II II II rt ON i d M M M ^ 00 -i- »r^ ON .^ f"00 O so VO On t^ OO 00 i .* ON" ri O ^ On .+ ONSO t-« r» rh M r< 11 <^ + 1 + + -t- + + H wOnOnOOO ►h CO ^ t^ CO »n*o -t" O^NO O "^ CO m «n to ti w lO M CO O tOOO CN "-• t^ M 'i- to r» t^ - „ '+■00 a> rt t^ o^ (^ to o l^rt^ro-^XNO^- fiOHmot^qoN to CO d t'^ tosO f O30 ri> M m rl ri to O to ^ f ) m O OO i'^ to t^rt « H ^ ^ -I O -+• rt t*^ »^ "^ '■O "• QnO —' (^t-^fN I 5 a I 0^ a IS SB ^ • . !• c 5 o i: s ^ h 0.5 $ 9- fc > W S -s II 64S oo •-< o i 3 r^ r-. r*-! tn ^-1 ri oc tJ O m U1 OnOnO ! M CO t H CO 3 O ?s 3 O D o « -t-rt ro O 88 O o 3 d d ■f + 1 1 w in ^ r» t-~ M O O>vo fl l-l tH ^ «+ trvx ^ VO 00 0\ rj-in •+ O O O O O o V^KO »n M fl li-) CO 1 1 1 + r» rot-N r-^ H4 00 W M M O w OM-. M ON rt "n DO >0 CO c» 1 1 1 1 1 1 1 1 tn ON ■* tooo 1-^ »n M r^ tn c^ \0 ^3 C^ t-^ OOOO M fl o f» wt- o • + -H- H- \£) VO 00 •-< CO iy-1 CO c^ w CO COOO C^ '-' t^ \ r\ M o*' t^ f*^ lo co^ ro3c ^ f J lio O oo *^ '^c< t-< rt ^ I -+ f» »r^ t^ *r» f \0 ►- c*^ 1-- t*^ I M M *i« CO .!-i>« TABLE XXL Constants, &c. Base of Naperian logaritlinis Alodulus of the common logarithms RadiiiH of a Circle in seconds " " II II minutes . " " " II degrees Circumference of a Circle in seconds II II II when ?■ = 1 Sine of 1 second . c = 2.71828183 ^0 =^= 0.43429448 . J- = 2002(i4.80() r — 3437.7468 . r :::=. 57.29578 1290000 . ■"■ = 3.14159205 log 0.43429448 9.03778431 — 10 5.31442513 3.53027388 1.75812203 0.11200500 0.49714987 0.000004848137 4.08557487 Equatorial horizontal parallax of the sun, according to Encke .... 8'^5711C 0.9330390 Length of the sidereal year, according to Hansen and T ?/"^''!'. ■ '■ , 365.2503582 days 2.50259778 l^ength ot the tropical year, according to Hansen and *^-^'"''*'^" 365.2422008 // 2.50258095 - o''4(j()0OOO24 ^''' ^'"^'^ ""^ '^' *'''*'''"' ^''"' '' '^"' ^^^^'^' '^^^" '"'"""' '■"""''"" '' Time occupied by the pa-ssage of light over a distance equal to the mean distance of the earth from the sun, according to Struve 497.»827 Attractive force of the sun, according to Gauss . /,• :.- 0.017202099 " " " " " " '/ in se- *=°"^«°f^'"^ 3548.18761 3.55000057 2.0970785 8.23558144 — 10 Constant of Aberration, according to Struve 20" 4451 " " Nutation, // n peters " 9' '.2231 Mean Obliquity of the ecliptic for 1750 + t, according to Uessel .... 23° 28' 18".00 — 0".48308<- 0".00000272295<» Moan Obliquity of the ecliptic for 1800 + t, according to Struve and I'etors . . 23° 27' 54".22 — 0".4738< — 0".0000014C^ General Precession for the year 1750 + t, according to Bosscl " " " " II II Struve 50".21129 + 0".00024429G6i 50".22980 + 0".000226< Masses of the Planeto, the Mass of tue Sun being THE UNIT. Mercury , . wi— - 1 Veni" Fvtth Mars . • • • • 4805751 1 390000* 1 354936 1 2680637' Jupiter Saturn Uranus m — - 1 1047.879 Neptune . Oitf 24905 1 18780* EXPLANATION OF THE TABLES. Table I. contains the values of the amjlr of the vertical and of the logarithm of the eai-th'« radius, Avith the geographical latitude as the argument. The adopted elements are those derived by Bossel De- noting by I, the radius of the earth, by ^ the geographieal latitude, and by f' the geocentric latitude, we have / = ^ — 1 r 30".65 sin 2

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 formuUe for the parallax in right ascension and 111 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. _ Tabt,e III. gives the intervals of mean time corresponding to given natervals of sidereal time. It is required for the conversion of sidereal into mean solar time. Table IV. furnishes the numbers required in converting hours minutes, and seconds into decimals of a dav. Thus, to convert Vih 19m 43.5s into the decimal of a day, we find from the Talkie 13/i =0.-5416667 19m ==0.0131944 43s =0.0004977 O.Sa = 0.0000058 Therefore 13/i 19m 43.5^ = 0.5553646 651 652 THEORETICAL ASTRONOMY. y//^ ^< Tho decimal corresponding to 0.5s is found from tliat for os by changing the place of the decimal point. Tarle V. serves to find, for any instant, the number of days from the beginning of the year. Tims, for 1863 Sept. 14, 15/i 53//i 37.2.*?, we have Sept. 0.0 = 243.00000 days from the beginning of the year. Ud 15h 53m 37.28:^ 14 .66224 Required number of days = 257.66224 Tarle VI. contains the values of JI/=75 tan ^v + 25 tan'' ^v for values of v at intervals of one minute from 0° to 180°. For an ex- phmation of its construction and use, see Articles 22, 27, 29, 41, and 72. In the case of parabolic motion the formulae are m = 15- 3 M=m{t — T), wherein log Cu = 9.9601277. From these, by means of the Table, v may be found when t — jT is given, ov t — T when v is known. From I, =. 30° to V =^ 180° the Table contains the values of log M. Tarle 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 smttJ •''/200 w + \ t — T-- 200 sin' ^v w being taken in the second quadrant. The Table gives the values of A^ with ?y as the argument. As an example, let it be required to find the true anomaly corresponding to the values t — T=22.5 days and log q = 7.902720. From these we derive log il/= 4.4582302. Table VI. gives for this value of log M, taking into account the second difierenccs, V = 168° 59' 32".49 ; but, using Table VII., we have w-=168°59'29".ll, \ = 3".37, EXPLANATION OP THE TABLES. 653 and Jicnce v = w-\-A,=: 168° 59' 32".48, tlie two results agreeing completely. Table VIII. serves to find the time from the perihelion in the case of parabohe motion. For an explanation of its constrnction and use, see Articles 24, 69, and 72. Table IX. is used in the determination of the true anomaly or he tune from the perihelion in the case of orbits of great eccen- Art. 41 '"°'*''"^*^'^" i« f»"y ^'^Pjained in Art. 28, and its use in Table X. serves to find the value of . or of < - ^ in the case of e hptic or hyperbolic orbits. The construction of this Table is ex- planied .n Art. 29. The first part gives the values of log B and l!t f ,'' t^ie argument, for the ellipse and the hyperbola. Li the case of og C there are given also logl.Diff. and log half II. Diff., expressed in units of the seventh decimal place, by means of which the interpolation is facilitated. Thus, if we denote by log (C) he va ue which the Table gives directly for the argument next less than the given value of .1, and by aA the difference between this argument and the given value of .1, expressed in units of the second decimal place, we have, for the required value, log C== log (C) + A^ X I. Diff. + A^^ X half II. DifT. For example, let it be required to find the value of log (7 correspond- ing to ^ = 0.02497944, and the process will be :- ^ Arg. 0.02, (1) (2) log (C) = 0.0034986 logl.Diff. =4.24.585 log half II.Diff. = 1.778 (1)— 8770.6 logA^l =9.69718 2IogA4 —9304 A^= 0.497944, (2)= ^^8 gio^S J^o log C= 0.0043771 The second part of the Table gives the values of ^ correspondinff to given values of r. x & Table XI. serves to determine the chord of the orbit when the extreme radu-vectores and the time of describing the parabolic arc are given. For an explanation of the construction and use of this iable, see Articles 68, 72, and 117. 654 T1IE(»11KTI('AF. ASTIJONOMY. Tablk XII. exhibits the limits of tlio real roots of the equation sin (s' — Z) =^ vig sin* /. The construction and use of this table arc fully explained in Articles 84 and 93. Tahles XIII. and XIV. arc used in iluding the ratio of the sector included hy two radii-vectorcs to the trianj^h^ included by the same radii-vectorcs and the chord joininj^ their extremities. For an explanation of the consti'uction and use of these Tables, sec Articles 88, 81), {);], and 101. Table XV. is used in the determination of the chord of the part of the orbit described in a <>;iven time in the case of vcrv 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. Sec Articles 118 and 119 for an explanation of tin; use of the Table, and also the explanation of Table X. for an illustration of the use of the columns lieadcd log 1. Diff. and log half II. DiiK Table XVII. is used in the computation of special perturbations when the terms depending on tlu! 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); /T, the longitude of the perihelion; Q,, the longitude of the ascending node; /, the inclina- tion of the orbit to the plane of the eclijitic; e, the eccentricity of the orbit; and g, the perihelion distance. The longitudes for Xos. 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 arc KX PLAN AT ION OF THE TAHLKS. 066 comets [f peri- of the Incliiiii- oftlie 2-215, 1)1-267, ^quinox 0, 172, lies tire moasurod from the mean equinox of the beginninf^ of the next year. The longitudes for Xos. 19 and 27 are measnred from the mean c<[iiinox of LSoO.O; for Xo. 18(1, from the mean ecjninox of .luly .'5; for No. 187, from the mean cc^uinox of Nov. 9; for No. 200, from the mean equinox of July 1; for No. 202, from the mean ecjtiinox of Oct. 1 ; for No. 20G, from the mean equinox of Oct. 7; fiir No. 211, from the mean e(iuinox of 1848.0; for No. 216, from the mean ecjui- nox of Feh. 20; for No. 220, from the mean ecjuinox of April 1 ; for No. 250, fntm tiie mean equinox of Oct. 1; and for No. 276, from the mean equinox of 1865 Oct. 4.0. No.s. 1, 2, 11, 12, 20, 23, 29, 41, 53, 80, and 177 give the elements for the successive appearances of Ilalley's comet; Nos. 104, 116, 126, 143, 149, 157, 167, 170, 176, 178, 183, 194, 210, 220, 235, 249, and 260, tliose for Encke's comet, the longitude- 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 aji- pearances of liiela's comet; Nos, 187, 216, 250, and 276, those for Faye's comet; Nos. 197 and 238, those for Rrorsen'.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 re[)reseut 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 I-tcpcrtor'min der Cometcn- Astronomic (Munich, 1864), or Galle's Comctcn-Vcrzcichniss ap])ended to the latest edition of Olbers's Mdhodc die Jkihn eines Cometcn zu berechnen. Table XIX. contains the elements of the orbits of the minor planets, derived chiefly from the Berliner Asironomisches Jahrlmch fur 1S6S. The epoch is gi.'on hi Berlin mean time; J/ denotes the mean anomaly, (p the angle of ec'.v>ntricity, /z the mean daily motion, and a the semi-transverse ax'^. The elements of Vesta, Iris, Flora, Metis, Victoria, Eunomia, Meljiomene, Lutetia, Proserpina, and Pomona arc mean elements; the others are osculating for the epoch. The date of the discovery of the planet, and the name of the dis- coverer, arc also added. w'-<^ IMAGE EVALUATION TEST TARGET (MT-3) & // ^/ ^>- ..•^ / « ^ % 1.0 I.I 11.25 ■ Ml ■^" ■■■ - Hi Ik U IIIII16 V] % vl ^% ^j^.^ -^^ •^ 7 n. / Photographic Sdences Corporation 23 WEST MAIN STREET WEBSTER, NY. MS80 (71«)«72-4S03 i V ^/q \ ^^ i\ \ ^.^ ,.<> 7x ^ ^ 656 THf:OUETICAL ASTRONOMY. Table XX. contains the mean elements of the orbits of the major planets, together with the amount of their variations during a period of one hundretl years. The epoch is expressed in Greenwich mean time, and L denotes the mean longitude of the planet. Table XXI. gives the values of the masses of the major planets, and also various constants which are usetl in astronomical calcula- tions. APPENDIX. M=. 171° 36' 10" + 39".7& (t ~ 1750), -^ = 50".2113 + 0".0002443 (t - 1750) + (0" 4889 - 0".00000614 (t - 1750)) cos (A - M) tan ,., (i) = - (0 ■.4889 - 0-.00000614 (,, _ 1750)) ,i„ (; _ 3/) by .., we have, according ,0 I^,, ''""'' "' "'"' '""" ^•«'+ ^. -^^- = 0".17926 - 0".0005320786 r, -^^ = 50".37572 - 0".000243589 r, = 23° 28' 18".0 + 0".0000098423 r», and if we put ai dt ^' dt sim,~± = n, dtuS w c;!"""""' '"^'"'^ '" "•^"' -"- w -^ 41 (fa -^ = m -\- nt&n d aia a, dd dt = n cos o, (2) 067 658 TIIEORfmCAL ASTRONOMY. and the numerical values of m and n are, for the instant f, TO = 46".02824 + 0".000r,086450 (« — 1750), n = 20",06442 — 0".0000}>70204 (t — 1750). To determine the precession during the interval t' — <, we compute the aniual variation for the instant J (/'+ t) and this variation mul- tiplied hy t'— t furnishes the required result. Ji. Nutation. — The expressions for the equation of the efpiinoxes and for the nutation of the obliquity of the ecliptic arc, acconling to Peters, A?. = — 17".2405 sin ft + 0".2073 sin 2^ — 0".2041 m\ 2C + 0".0677 mi ( C — T') — l".2(;i)4 sin 20 + 0".1279 sin (© — r) — 0".02i:jHin(O4-r), Ae r= -f 9".2231 cos ^ — 0".nS97 cos 2^^ + 0".0S86 cos 2C + 0".o510 cos 20 + 0".0093 cos (© + r), for the year 1800, and AA = — 17".2577 sin SI -} 0".2073sin 2Q, ~ 0".2041 sin 2C + 0".0677 sin (C — T') — 1 ".2t)9o sin 20 + O".127o sin (Q — r) — 0".0213 «in (O + r), Ar = + 9".2240 cos Ji — 0".0896 cos 2$J (f'.OSSS COS 2C + 0".5507 cos 20 + 0".0092 cos (Q + r), for the year 1900. In these equations SI denotes the longitude of the ascending node of the moon's orbit, referred to the mean cfjuinox, C tlie true longitude of the moon, the true longitude of the sun, F the true longitude of the sun's perigee, and P the true longitude of the moon's perigee. The values of these quantities may be derived from the solar and lunar tables, and thus the required values of a^ and AS may be found. The equations give the corrections for the reduction from the mean equinox to the true equinox. To find the nutation in right ascension and in declination, if wo consider only the terms of the first order, we have (4) The values of aA and Ae are found from the preceding equations, and for the differential coefficients we have Ao=: da aA -t- da de AC, a3 = aA f ds (is Ae. APPENDIX. 659 !0f vcd the wo (4) and -J— = cos e + 8111 £ tan o sill o, = — cos a tan d, ^1 — !r= COS & Sin £, ---- sin o. (5) The tonns of the second order arc of sensible niagiiitiide only when the body i.s very near the pole, and in this esisc by eoiiijuitiiij; the secoru differential eoeffieients the eoinplete values may be iouiid. Ii the reduction of the place of a ])lanet or comet from the mean eqninox of one date t to the trne equinox of another date /', the determination of the correction for precession and of that for mitatiop niav be efl'eetcd simnltancoiislv. Thns, let r denote the interval t' — / cxju'csscd in parts of a ycijr, and the snni of the corrections for f recession and nntation j^ivcs Ao = mr -\- aA cos £ -f~ ('i" + ^^- ^h» £) sin a tan <* — A£ cos a tan ^ cos e — cos 3 sin e) — 20".44r)l sin © cos a sin S -j- 0".3429 cos /' (sin o sin 3 cos e — cos d sin e) — 0".3429gin/'cososin<5, ao) in the case of the right ascension and declination. In these formula) /^denotes the longitude of the sun's perigee, and they give tlic cor- rections for the reduction from the true place to the apj)arfcat place. 1). Intensity of LUjht. — If we denote by r the distance of a planet or comet from the sun, by J its distance from the earth, and by C a constant quantity depending on the magnitude of the body and on ita capacity for reflecting the light, the intensity of the light of the body as seen from the earth will be /= r»J« (11) "When the constant C is unknown, we may determine the relative brilliancy of the comet at different times by means of the formulii J5 = (12) APPENDIX. 661 In the ca.sc of ti.e plnnct.s wc adopt as tlie unit of tho intensity of lif^l.t the vahie of /when the planet is in opposition nngarithms or of two numbei*s from left to right (which will be effected easily and certainly after a little jmictice), the sum or difference to be used as the argument in the tables may be retained in the memory, luid thus the required number or arc may be written down directly. The luunber of the decimal figures of the logarithms to be used will depend on the character of the data as well as on the accuracy sought to be obtainetl, and the nse of ai)proximate formula) will ha governed by the same considerations. Whenever the formulas furnish checks or tests of the accuracy of the numerical process, they should be applied; and whenever th'^se are not provided, the use of differences for the same purpose should not be overlooked. By proper attention to these suggestions, much time and labor will be saved. The agreement of the several proofs will beget confidence, relieve the mind from much anxiety, and thus greatly facilitate the progress of the work. THE END.