IMAGE EVALUATION TEST TARGET {MT-3) t ^ /. ^/ ,<° mp. ' 'R I KTBIt, 7U.'t Ja^ne Street. ^ PEEFACE. The present work was undertaken as far back as the year 1859. But the labor devoted to it at first amounted to little more than tentative efforts to obtain numerical data of sufficient accuracy, and to decide upon a satisfactory method of computing the gfcncral perturbations of the planet. The elements of Neptune employed in the earlier computations were found to deviate too widely from the truth to be used in computing the perturbations of Uranus with the first order of accuracy, and it became necessary to correcit them. This was done during the years 18G-1 and 1865, and the investigation was printed by the Smithsonian Institution in the latter year. It was then found that the adopted elements of Uranus also differed too widely from the truth to serve is the basis of the Avork, and they were provisionally corrected by a scries of heliocentric longitudes derived from observa- tions extending from 1781 to 1861. FinalK' it was found that the adopted method of computing the perturbations, that of the " variation of elements," though not deserving of the disfavor into which it has fallen of late years, was practically inapplicable to the computation of the most difficult terms, namely, those of the second order with respccc to the disturbing forces. Indeed, it appeared to the author that the only method of computing these terms which was at tiic same time general, practicable, and fully developed, W! s that of Hansen. But, were this method adopted, all that had previously been done would have been, useless, even for the purpose of comparison and verification, owing to the expression of the co- ordinates in terms of a disturbed mean anomaly. It appeared to the author that, although this form of theory led to expressions having fewer terms than the other, it was not without its relative disadvantages. Other considerations being equal, he conceived that astronomers generally would gr:^atly prefer to see the perturbations expressed directly in terms of the time, owin^' to the ease with which the results of different investigators could then be compared, and with which corrections to the Hieory may be introduced. Under these circumstances the method deicribed in the first chapter of the present paper was worked out. The question how much it contains that is essen- tially new is one that the author has never clos.'ly examined: it is, however, certain (iii) iv PREFACE. that the mode of con8iilering the subject is well known, being that employed by La Place, Ilerschcl, De rontecoulnnt, Encke, and perhaps others. The method of forming the required derivations of the perturbative function from the analytical development of that quantity, he has not seen elsewhere. With these improved elements and methods the work was recommenced in 1868. The curlier investigations being merely provisional, it has not been deemed neces- sary to present them in the present work. Some of the results, corrected for errors of the older elements, are, however, given for the purpose of comparison. Although this investigation has absorbed the greater part of the author's leisure for more than live years, it is only through the aid of the Smithsonian Institution and Nautical Almanac that he has been enabled to bring it to a conclusion witliin that time. At an early stage of the work Professor Henry responded favorably to a request for aid by the employment of computers; it was, however, not found practicable to use such aid until the perturbations had been completed, and the provisional theory concluded. Then, the comparison of theory and observation, and the construction of the tables, involved a large amount of mechanical compu- tation, and on this part of the work a number of persons have been employed by the Institution at various times, among whom may be mentioned Prof(;ssor F. W. Uardwell, of the University of Kansas, and Dr. C. L. F. Kumpf, late of the Ob- servatory of Leiden. Every part of the work has, however, been done under the author's immediate direction, and, as nearly as possible, in the same way as if he had done it himself, a result which, in one or two cases, has been attained only by the expenditure of an amount of labor approximating that saved by the employ- ment of the computer. In presenting the steps of the investigation, the end has been kept constantly in view to render as easy as possible the detection and correction of any error, or the introduction of any alteration in the elements or other data. It is, of course, impossible to present the steps of the computation with any approach to fulness without far transcending the limits of the printed work : The results given are, therefore, those which it was supposed would be most useful to the futme investi- gator of the same subject. There is reason to believe that the original computa- tions will ultimately become the property of the National Academy of Sciences, so that they may always be referred to for the clearing up of any difficulty in the printed text. The author's acknowledgments arc due to Professor J. II. C. Coffin, Superin- tendent of the Nautical Almanac, and Mr. E. J. Loomis, of the Nautical Almanac Office, for reading the proof sheets of the last twelve tables during the absence of the former abroad. k W'AsnisaTON, July 31, 1813. TABLE OF CONTENTS. Introduction . PAOI 1 CHAPTER I. METHOD OP DETERMININfUTnE rERTrRBATIONS OP THE T.ONOlTrDE, RADIUS VECTOR, AND LATITUDE OF A 1'I.ANET BY DIRECT INTEORATIO.V. Notatfon and general differential formulB) ..... Formation of the re'.iiired derivatives of the pcrturbntive ftmction . Correction of these derivatives for terms of the second order . Integration formula! for perturbations of radius vector Development of functions of rectangular co-ordinates .... Integration of perturbations of radius vector ..... Formnho for perturbations of longitude to terras of the second order Motion of the orbital planes ....... Perturbations of the second order depending on the motion of the orbital planes Reduction of the longitude to the ecliptic ..... E.\pressions for the latitude ....... R 10 12 14 n 23 24 25 27 29 CHAPTER II. APPLICATION OP THE PRECEDINO METHOD TO THE COMPUTATION OP THE PERTURlLVTIONa OF URANUS BY SATURN. Data of computation ..... Numerical expressions for R and its derivatives Perturbations of radius vector Perturbations of longitude .... Perturbations of latitude .... 81 34 44 49 61 I CHAPTER III. PERTURBATIONS OP URANUS PRODUCED BY NEPTUNE AND JUPITER. Adopted elements of Xcptnno ..... Development of li and its derivatives for the action of Neptuno The term of long period between Xcptune and Franus Perturbations of the longitude produced by N'cptnno . Perturbations of tlio radius vector prndneed by Neptune Perturbations of the latitude produced by Neptune . Perturbations produced by Jupiter 53 64 55 58 fiO CI 63 (V) yl TABLE OF CONTENTS. CnAI'TEIl IV. TEBMB OF THE SECOND OKDEIl DUE TU THE AOTIOM OF BATDRN. rrrlimiiiary iiiTCHtipation of the orbit of So,tiirn .... PcrturbatiniiH of Saturn mid Urnnus ...... Formation of llip oxprossioiiH for tlio terms of tlic second order I'erlurbatioMH dopendins on the s(|iiarc of the mass of .Saturn Perlurbatious depending on the product uf the muHava of Jupiter and Saturn Cn AFTER. V. COLLECTION AND TKAN8F0UMATI0N OF THE PRECEDINO PEKTt^IlBATIONB OF URANVB. Terms Independent of the position of the di-sturbiiig planet Secular Tariation.s ..... Au.xiliury expressions on which the perturbations depend Kcduced expressions for the latitude of Uranus Positions of Uranus resulting from the preceding theory Elements III of Uranus .... CHAPTER VI. REDCCTIOS OF THE 0BSEBVATI0N8 OF URANl'9, AND THEIR COMPARISON WITH THE PHECEDINO THEORY. Rcdnction of the ancient observations . Tlieir coniporison with the provisional theory . Discussion of the modern observations Reduction of the results to a uniform system . Adopted positions of fundamental stars Discussion of corrections to reduce the diiferent observations Table of these corrections .... Results of the observations from 1181 to 1830 Observations from 1830 to I8t2 Table to convert errors of right ascension and declination of and latitude .... Tabular summary of results of observations, 1830 to 1872 Corrections to be applied to (i -^ positioi.a of Uranus in the Berlin Jahrbuch and the Nautical Almanac to reduce them to positions from the provisional theory CHAPTER. VII. FORMATION AND SOLUTION OF THE EQUATIONS OF CONDITION RESULTING FROM THE PRECEDINO COMPARISONS. Expressions of the observed corrections to the longitudes of the provisional theory in terms of the corrections to the heliocentric co-ordinntes ..... Expressions of the same quantities in terms of the corrections to the elements of Uranus and the mass of Neptune ...... Table to express errors of heliocentric co-ordinntes as errors of elements Discussions and .solutions of the equations thns formed Concluded corrections to the elements of longitude Corrections to the inclination and node of Uranus to a homogeneous system Uranus into errors of longitude that V5 68 fi9 7G 17 79 80 81 93 98 99 ior> 110 111 111 113 115 120 122 126 127 131 151 158 101 162 105 173 173 TABLK OP CONTEiNTS. CTIAPTEll Vlir. COMPLETION AND ABBANOEMENT OF THE TIIEORV TO FIT IT FOB PERMANENT U8E. Correction of llio covfllcients of tlio loiiff inequality bctwcon UnimiH uikI N'c|ituue for tlio ternia of tiie second order ..... Concluded elements, or cloiuenta IV of Uranus Lon((-pcriod ond secular pcrturbationfl of tlio ulcmoiits Table of these perturbations from A.D. 1000 until A. I). 2200 Mean elements of Uranus ..... Expressions for tbo concluded theory of Uranus CIlAl'TEll IX. OENEBAL TABLES OF UBANU8. Enumeration of the qnantitics contained in the several tables Precepts for the use of the tables .... K.xainplea of the use of the tables .... Tables of Uranus ...... Subsidiary tables ...... vU PAOI 178 181 183 184 IS4 186 190 195 198 20« 279 ERRATA. Pages 100 to 105. In computing the latitude from the provisional theory the values of the secular terms of iij and tk ou page 97 have been iatcruhaugcd. The provisioual latitude, therefore, requires the correction — 0"..')3 r Bin V + 0".63 T cos v where \—g+ 12° 45' -|-2<; sin g. This correction is not applied in the subsequent investigation. Its circot would have boon to change the value of b deduced on p. 170 l)y probably 0".2 or 0".3. Tlic effect on the other tlcuienta of latitude would have been much smaller, and therefore unimportant. Page 122, line 1,5. Add : the corrections in the si.xth column being omittoil. Pago 151. Add foot-note: In forming these comparisons the corri-'ctions to the heliocentric longitude in the si.xth column of the provisional cphemeris, pages 100 lo 105, are not applied. Page 159. Equation 7. In this equation the coelTicienta of ix and ip have been multiplied by ^, instead of f , the factor of il. The effect of this error enters into all the subsequent results, but in the compari.sons of theory and observation it is corrected. Page 184. The element hero represented by > (kappa) is the same which, in the preceding chapters, has by mistake been represented by k, and which is defined on p. 24. The k of Chapter VII 1 is, therefore, not the same with that of preceding chapters. ON TIIK ORBIT OF URANUS. INTRODUCTION. The connection of (lie [jlunot Uranus witli the most brilliunt astronomicnl acliicvenjcnt of tlic century lends u peculiar interest to its theory. The researches of Adams and Le Vcrrier showed that the observed motions of that planet were represented, at least approximately, by the action of i theoretical planet havinj^ the lonj^itude of Neptune. I'eirce showed that the ih tion of Neptune itself aecoiuited for tiiese motions within the limits of possible error of the observations nsi'd by Lc Verrier. It remains to be seen wlicth'-'- the n^'reement between theory and observation still subsists wiien the comparati' '; few observations used by those investi.; itors are reduced with tlie more relined data now at our disposal, and when the great mass of additional observations made both befun; and since the date of Lc \'errier's researches are included. The circumstances connected with the discovery of Neptune have been so exhaustively recounted by a number of authors that it would be difficult to add anything not already familiar to astronomers without transcending our present limits. I shall therefore confine myself to such nn account of previous researches on the theory of Uranus as may give an idea of their nature and extent, and facili- tate their comparison with the methods and results of the present investigation. The perturbations used by Bouvard in his tables are those of the ^lecanique Celeste. Although not affected with any striking error, the numerical methods adopted in their computation are necessarily too rough to allow of much interest attaching to their comparison with the results of the more recent researches. It is essential to a clear understanding of subsequent researches that we classify the methods which have been or may be adopted in the computation of the general perturbations of the planets. This computation comprises two distinct operations: (1) the development of the disturbing forces, or some quantities of which these forces are fmictions ; (2) the integration of the equations of motion inider the influence of these forces. In each of these operations three methods have been employed. In developing the perturbativc function, we have first the purely analytic method used by the great geometers of the last century. In this metliod this function is developed in powers of the eccentricities and mutual inclination of the orbits of the two planets, and the numerical coefficients are found by substituting the values of the elements in these expressions. A is only applicable when the eccentricities 1 Uaroh.lSTS. ( ^ ) :i\ THE ORBIT OF URANUS and mutual inclination are small, and has for that reason fallen, of lute, into a certain disrepute. The extended tables publislied Ly Le Verrier' liave, however, added so much to its facility for use that it is not wholly unworthy of attention. At the other extreme stands the purely mechanical method, in which spccifil values of the disturbing force are computed for many combinations of the mean anomalies of the two phniets, and tlic values of the coefficients in the general expression for the force thence deduced. Between these two stixnds what 1 conceive we may designate as the Cauchy- Hansen method, in which the development is made mechanically with respect to the one ])lanct, but the eccentric anomaly of the other is retained as an undeter- mined quantity. The germ of this mctliod is found in several papers, by Cauchy, in the earlier volumes of tlie Comiitca Rendiis of the French Academy, which have since been combined into a liomogeneous memoir by Puiseux.^ Tlie object had in view by these authors is only the computation of inequalities of long period. But Hansen has taken up the essential principle of the metliod, first, in his piize memoir on the perturbations of comets, crowned by the French Academy of Sciences, about 1848, and afterwards in his " Auseinaiiderscfzuiiff ei'ner zwcckmiimgcn ilethode znr JJcrecIinung tier Storiiugcn (hr klcinen Planetcn"^ and applied it to the general development of perturbations Among the three metliods of integration, the first in point of analytical elegance and generality, but the last in order of convenience in use, is that of the variation of elements, a method with which the name of La Grange is inseparably associated. In the second the direct integration of the differential equations wliich express the perturbations of longitude, latitude, and radius vector is effected by special devices. In the first of these methods the problem is presented in this form : The equations of motion being completely integrated for the action of the principal forces only, how must the arbitrary constants of integration vary in order that the same expres- sions may represent the motion of the planet under the influence of the disturbing forces? In the second method, the same thing being presupposed, the question is, what expressions must be added to the integrals of undisturbed motion in order that the sum may represent the integrals of the disturbed motion X The third is Hansen's method, in which the co-ordinates are partly expressed in terms of a certain function of the time known as th? disturbed mean anomaly, determined by the condition that the true longitude in the disturbed orbit shall be the same function of the disturbed time that the longitude in tlie elliptic orbit is of the simple time. Although the last two metliods have a great advantage over the first in the com- putation of the periodic perturbations, I conceive the first to be best adapted to the computation of the secular variations, and perhaps, of terms of very long period in the mean longitude and the elements of the orbit. •*■ • Annales de V Ohservatoire Imperial de. Paris. Tome I. ' AiinnleK de roharrvaloire Jmpi'rinl de Paris. Tome A'll. • AbhamUitnrjen der KOniglich, Sdchsischcn Oescllscha/t der Wisscnscha/lcn. Band V. VI, VII. •I* THE ORBIT OP URANUS. 3 In his researches on the motion of Uranus, the first thing done by Le Verricr was to reconii»ute the perturbations by Jupiter and Saturn. It will sufficiently describe his method of doing this to say that in the developments he used the purely mechanical method for the action of Saturn, and the algebraic development of tlie perturbative function for the action of Jupiter, while in the integration he used the method of the variation of elements. After completing the perturbations of the first order he made the earliest attempt at a complete determination of those of the second order. Beginning with the terms of tliis order which arise from the secular variations of the elements, he determines them by recomputing the terms of the first order for the epoch 2300, and assuming that the general term will then be given by interpolating between the two terms thus found, supposing them to increase uniformly with the time. This proceeding has the sanction of such high authority that it is worth while to call attention to its want of rigor. The dif- ferential coefficient of each element being given in the form (la J J, , = k cos Ot. (It k being a function of the elements, the perturbation of the first order will be (a = , sua lit. b When we take into account (he variation of /•, and suppose it of the form h^ -}- Ic't, the process is equivalent to supposing that in this case la = ' — sni bty whereas it really contains the additional term, J j cos bt, which appears to be neglected in tlic process in question. It will be seen that the neglected coefficient is equal to the secular variation of the term during the time that its argument requires to increase by an amount equal to the unit radius. It is tlierefore the more important the longer the period of the inequality. To obtain the periodic terms of tlie second order Le Verrier begins by determin- ing the ten principal terms of the perturbations of tiic elements of Saturn produced by Jupiter. Next he takes up the terms in the mean longitude of Uranus which depend on the square of the mass of Saturn. The only sensible terms he finds are — T'.n sin {^— 30 — 0".35 cos (^' — S^) + 0".43 sin {i," — ^' + 40 — 0".2l cos (^' — 4^' + 40, ^, <^', and ^" being the mean anomalies of Uranus, Saturn, and Jupiter, respectively. The terms depending on the product of the masses of Jtq)iter and Saturn are then taken up. lifteen arguments are found the coefficients of wliich vary from a small fraction of a second to one or two seconds, while a single one of long period amounts to 32". When the method of variation of elements is used, it is necessary not only to determine these variations to quantities of tlie second order, but, in the tiansforma- THE ORBIT OP URANUS. tion of the perturbations of the elements into perturbations of the co-ortlinates, to carry tliis transformation to terms of the second order also. This Le Verrier avoids by showing that the terms of the lowest order with respect to the eccentricities thus introduced are destroyed by certain terms in the perturbations of the elements, so that it is only necessary to omit both classes of terms. These terms are of that fictitious class which disappear of themselves by a simple change of elements. When, instead of the eccentricity and longitude of the perihelion, we take h and /■•, which represent the products of the eccentricity into the sine and cosine of this longitude respectively, these terms disappear of themselves both from the perturba- tions of the elements and of the co-ordinates. It is not likely that any of the neglected terms of tliis class exceed 0".l. As soon as the elements of Neptune were known, the nature of its general action on Uranus became of interest. This subject was taken up by Prof. Peirce, whose results are found in the Proceedings of the American Acatlemy of Arts and Sciences, Vol. T, pp. 334-337. This paper is accompanied with a comparison of his theory of Uranus with observations, to wliich simihir comparisons of the theories of Adams and Le Verrier are added. Tliis comparative exhibit is of sufficient interest to be given here. The numbers given arc probably excesses of computed over observed longitudes. Residual Differknceb between the Theoreticai, anp Observed Longitudes of Uranus, | riio.M the Tueouies of 1'eirce, Le A'ekrieb, a.nd Adams. From he Ver- From Le Ver- From Adams's From I'eiroe's theory of Neptune adopting for its rier'a test or- rier's original original the- mass ory with his from tlie mo- hi!) best orbit se(!0iid hypo- That of Strnve Tliat deduced That deduced dern ob^^erva- of hypotheti- tlietical planet from his own by Peirce from by Peirce from Year. tions without cal planet, of of which mass observations Bond's & Las- Bond's obser- any external which the is of the satellite sel'a observa- vations of Las- planet. mass IS tions combined sel's satellite jjVi- zi^-^- Tr'tT^' riHi- •n\n- ft // // // // // 1000 + 289.0 — 19.9 + 50.0 — 124.7 + 13.0 -f 0.8 1715 -1- 279 + 5.5 — 0.6 — 99.6 4- lO.O + 8,7 1750 + 230 9 — 40 — 4.0 — 102.4 — 12.7 + 4,0 1709 -Y 123 3 + 3.7 + 1.8 — 67.0 — 10.0 — 0,0 1782 + 20 5 + 23 0.0 — 18.3 — 5.6 — 3.0 1787 + 2,0 — 12 — 0,2 — 4.7 — 12 — 0,5 1792 — 7.8 + 3 — 1.1 + 1.0 -f 5 + 0,3 1797 — 6.7 — 1.0 — 0,;') -f 3 3 -f 0.8 + 0,3 1803 — 34 + 0.8 + Ifi + 3.2 4- 1.2 + 0,8 1808 + 3.8 + 0.8 0.0 — 1.3 — 0.6 — 0,4 1813 + 4.5 — 09 — 1.0 — 2.3 + M — 0,3 1819 + 3.8 + 0.4 — 22 + 0.9 + 0.7 + 1.0 1824 — 7.0 — 54 + 1.1 — 1 6 — 1.9 — 2.0 1829 — 7.8 — 2.2 + 2.0 + 25 + 1.3 + 0.8 1835 — 45 — 0,8 — 1.2 -f 3.9 + 24 + 2.0 1840 + 07 + 22 + 1.3 - 1.3 — 1,3 — 1.1 1845 + 6.5 — 0,3 — 2.8 — 1 2 — 0.9 ■4 i In this paper Professor Peirce presents the results of a complete computation of tlie general perturbations of Uranus by Neptune in longitude and radius vector, THE ORBIT OF URANUS. I r but witliout any details v/hatcvcr of the investigation, or any statement of the methods employed. Tlie minuteness of the residuals in the last column of the preceding table shows that employing these perturbations by Neptune, and those of Le Verrier by Jupiter and Saturn, we had a theory of Uranus from which quite accurate tables might have been constructed. But this never seems to have been done. The cphemeris of Uranus in the American Nautical Almanac was intended to be founded on this theory, but the proper definitive elements do not seem to have been adopted in the computations, as the cphemeris does not correspond with the theory. Although twenty-five years have elapsed since the epoch of these researches, I am not aware of any published work of importance on the theory of Uranus during the interval. Mr. T. H. Safford has, however, made a very extended investigation of the subject, but has published nothing more than a brief general description of his work, which may be ^ound in the Monthly Notices of the Royal Astronomical Society, Vol. 22. Like Professor Teirce, he took I^e Vcrrier's perturbations by Jupiter and Saturn, but, instead of using general perturbations by Neptune, he computed the effect of the action of this planet by mechanical quadratures for the whole period of the observations of Uranus, and thus corrected the elements and the mass of Neptune from modern observations alone. The mass in question deduced was 1 20039 Mr. Safford does not give the representation of the modern observations, but pre- sents the following comparison of the ancient ones, alongside wliicli wo place for comparison the corresponding numbers of Pcirce's theory and those of the present investigation. Excess op Observe Date. No. of obs. Safford. ifino 1 + 5".0 , 1-15 3 — 4.2 ltr>o 2 — 1 2 It 53 1 _ 0.2 1"5G 1 — 0.0 17G4 1 + 0.4 nco 8 + 4.5 Peirce. Newcomb — 0".8 -u" — 8.T — 8 — 4.0 + CO + 2.0 — 1.4 6 THE OllBlT OF UKANUS. CHAPTER I. METHOD OP DETERMINING THE I'EUTURIJATIONS OF LONGITUDE, RADIUS VECTOR, AND LATITUDE OF A I'LANET BY DIRECT INTEGRATION. Let us conceive a plane determined by the condition tluit it shall pass through the sun and contain the tangent to the' orbit of a planet at any moment. If the planet were acted on by the sun alone, the position of this plane would be invariable, but, under the influence of the disturbing forces of the other planets, it is subject, at each instant, to a motion of rotation around the radius vector of tlie planet. We may regard this as the instantaneous plane of tlic plaiu>t's orbit. Tlie disturbing and the disturbed planet will each have its own instantaneous plane. Let us now put : — V, the longitiule of a planet counted from a determinate point in the instantaneous plane of its orbit. V, its distance from the node of intersection of its own orbit with that of another planet, the mutual inclination of the two orbits, sin I y. the radius vector of the planet, its logarithm. fi, the attractive force of the stui upon unit of matter at unit distance. a, the mean distance corresponding to the observed mean motion of the planet, determined by tlie condition »r m and n being as usual the mass and mean motion. a„, the value of a corrected for the constants introduced by the perturbations, so that, as in the elliptic motion, we have p = logrt+/(/, p, ct), we shall have in the disturbed motion p = log CTo +/(^ e, ct) -|- periodic terms only. «!, the mean distance of an outer planet, whetlier it be a disturbing or disturbed planet. », the logarithm of a. a, the ratio of two mean distances, taken less than tmity. li, the perturbative function. o, r, P %« Mt«9 THE ORBIT OF UIl ANUS. J h, the coefficient of any term of -i- R, so that we have B = Z COS N «1 m' being here the mass of the disturbing planet. X, the mean distance of the planet from the node, or the mean vahic of v. w, tlie distance of the perihelion from the node. g, the mean anomaly. I, the mean longitude, or the mean value of v. 4/, the angle of eccentricity so that c = sin t^^. r„ the radius of the planet in the undisturbed ellipse, r,, the quotient of r„ divided by the mean distance, wliich is a function of the eccentricity and mean anomaly only. T, the time after the epoch 1M50, Jan, 0, Greenwich mean noon, counted in Julian centuries. V, the integrating factors of the periodic terms, or the ratio , ^ being the change of the angle in iniit of time. «, the eccentric anomaly, and, in tlie tables, the argument of latitude. We have for the value oi li Mt»9 It m m r (cos v cos v'-)-sin V sin v' cos y) V 1^ — 2)-r'(cosvcosv'-|-sin v sin v' cos y)-\-r''' i or, if we suppose r replaced by its value in p, namely we shall have With this value of R it is well known that the differential equations for the longi- tude and radius vector of a planet arc dh- dr^ " (!+»')_ ^J^ de d(' ^ r fPv de + 2, dr dv dt dt ~~ (^li = F cJp' (1) If we multiply the first of those equations by 2 ' P and the second by 2 J^ and dt dl add them together, putting, for brevity, f*^^ (/p 1^ r/? dv jy „ cJp f/<+ ov dt -^^''^- (2) and then integrate, we shall have ';:+y';:-'^'-^'+'"'='>,(c+fir.n,o d( 8 THE ORBIT OF URANUS. ^~d^^ ~=t^{2c+2fiy,m+f) c3) Lot us now represent by ,, that elliptic value of . which satfsfies the equation Subtracting this equation from the last we have ^^t::zt^:js::^:::tz:'h'''': -' '-'^ -''- ^^^ ^^-^o- the logarithm of the radh ^ ." he .t ,7 " "" 'T'f ''^'^""^'''^'^ "^ ^'"^'"S for the latter its value '""''''' '"^'^'''^ °^ '^ "^'-'If' ^^c substitute for the latter its value and put Then ^P = P — po. ^('•'-0 = »v^p-f,-„^y^etc. Substituting these valiip ^ Hilt.. " dt^ + & ( V5p) = fc ( 2Sjyjidt + ^\_<^ (ro'V) , f^ ¥ rx. • ^ ^ ^ dp) dC' I- -9,. ' (*) an equation which gives the perturbations of radius vector we siLi b4in by ^:::!:ir;^i:^^:::zz:t:7 "^ ^"'' '-'-' ''- '"--^ of 5v, K ^p, ap', ^y, and that from the e we wi h " ^r""''''? ^ '" *'" "•'"'''^ We must first carrv this innrnv,-.. . . . ^ ' *" "" '"''"""'^ approximation, ber of (4). To effLuhi^rr; 7 '", ''V' '""^'""^ "^ ^ '" ^'^ ^^""^ "-- of the elements and , time w mtT "' '™'" '" ^--'"P--^* of ^ i" terms to the quantities which or 'iJo ^^ ^Zu^'^'^t 't^'^'' ''''' ^^^"^^^ and r, is, in its developed form, a futtbn of^T ' '> '""?'"" "5 ^''Z ' P' P'' development being effected bv s nbsfih f / u L"' "' ^' "' "' "' ""'* ^ '^'^ values in terms of the second '"''''''*'"? ^"'^ ^''^ «-* set of quantities their 01 me second. Ihe substitution is as follows: v = ;i + /},, ^-^'+^y, (5) P = » + «/'5' , P' = »'+ =i.J".-.v and J','> being, as usual, the coefficient of cos i^ in the development of (1— 2acos-fa=)-' in multiples of cos ^, and » — 1 the sum of the exponents of the eccentricities in E. It would have been much more convenient if in effecting this development the derivatives of fe'j' had been taken with respect to » instead of a. In fact the derivative ^ ^ when expressed in terms of the derivatives with respect to » is of da '^ the form «"--•=".--.:-+«. -^„;- + etc. + ».,.-^ <9a" 6to Therefore, when expressed in terms of the derivatives with respect to », A will be of the form a'-' ( (0)' t> ^a» = 71 oo ^» consequently wc have for the derivatives of A from formuloe (8) 6A_ = (0) a -^t + (1) ( a -^ + a» -^^,' ) + (2) ( 2a' -^^-,i + a' ^j)+ etc. 5a ~r - ^„, / -r v-y v-* "^oT ^ ""* ~5a»' + "' ^ -)+ etc. The derivatives of A being formed in this way, those of h are immediately deduced from the equations 6h 6 A oVi d^A dri' ~ ^^ dra' Wlien 8 is equal to |, ^ is of the form a I (0') i'l' + (!)'« :^'^ + (2)' a' ^^''l + etc. I I <9a ^a" 3 W The quantity within parentheses is of the same form with A, in the case of s = |. If we represent it by A' we sliall liave 6A / dA' A SA' , . d^A dri ,3 =^ + 2a-^-+a~^^ dA , oM' A being the same form with A, tlie derivatives ^'"' and 5» (9ir will be of the form (9), substituting | for the index |, and (0)', (1)', etc., for (0), (1), etc. In the case of s = | the derivatives are obtained in the same way, which is too simple to need elucidation. We have now to pass from the derivatives of h to those of * , the coefficients of the perturbative function. The form of these derivatives will depend not on whether the planet is disturbing or disturbed, but on whether it is an outer or flq^ 12 THE ORBIT OF URANUS. inner one. Let us then suppose for the present, that a and « refer to the inner planet, and put », for the logarithm of the mean distance of the outer cue. Wo then have for the derivatives relatively to » 5- A a, 1 d'h 5»» a, dvf n » and for the first derivative relatively to », using the symbolic notation, ^A ^ = 1(Z)., -1)7». 5»j «i The symbols in the second member being distributive, vire have by successive differentiation The quantity A is a function of o, the ratio of the mean distances or oi C'—'i, being the neperian base. Hence D.y* = — D,h, which substituted in the last equation gives h 6" a, (-1)" dK (Z>, + 1)»A. (10) This formula gives for the first two derivatives 5»i d«A Substituting in the general formula; (7) these expressions for the derivatives relatively to v and », we have expressions for the derivatives of E relatively to V, V, p, p', it being understood, however, that all the quantities are expressed in functions of the elements of elliptic motion. In order to compute the perturbations of the second order we must carry E and such of its derivatives as enter into the difierential equations (1) to quantities of the first order with respect to the perturbations. Let us then represent by Vo, Vo', Po7 p'o» yo. t^e elliptic values of v, V, p, p', and y, which we have assumed in the Erst approximation to the perturbations, and by 5v, 6v', etc., the quantities to be » * THE Oil BIT OF URANUS. ta added to v„, v,,', etc., to make the true vnluos of v, V, etc., whether perturbations or corrections of the elements. \Vc shall then have <9v„» dm. h\ + dRo d'R„ ■ W + JR^ d'R, ^ oiVg^v'o ^ C^VoOlpo S'R, d'Ro ^po^Vo ^ ^pu<5v'« ^ df\ .3p + - ip + 5p + 5p'o S'Ro ^V„c)p'o ^po^p'o c'Voo'yo Sfodyo (11) The value of DftR may be found either by cqiuition (2), or by differentiating with respect to the time as introduced by the co-ordinates of the disturbed planet. When quantities of the first order only are considered the latter operation is very simple, but it is different when terms of the second order come in, because the true longitude of the planet is then expressed in terms not only of its own mean longi- tude, but also of the mean longitude of all the disturbing planets. The result can still bo obtained in the same way by separating all the mean longitudes introduced by the co-ordinates of the disturbed planet from those introduced by the co-ordinates of the other until after the differentiation relatively to <'. Let us now resume the equation (4), representing its second member by // Q, so that it becomes where (It Vq (12) By the operations already given Q has become a known function of the time. It is well known that the integration of (12) may be effected by finding two values of rj^hp which satisfy this equation when the second member is neglected, or, in other words, by finding two variables x and y which satisfy the equations / , /'(I +»»)„_ when the required integral is '5p = ^ Ji/ dx ill yji I yjxQdt — xjyQdtX. The above differential equations are satisfied by the rectangular co-ordinates of the planet in its assumed elliptic orbit. The position of the axes of co-ordinates being arbitrary we shall take the line of apsides for the axis of X, the perihelion being on the positive side. If we put Co = sin 1^, we have dy dx , :r — - ■ yj/=A«(i + ' X dt dt M (1 +770008 4- |cos^= ^„ • cos V ^a (cos n — c), y = r sin v = a cos ^ sin m, ^ = cos n — e, ri ■=-. sin M, As ^ and >? are to be expressed in the form i=--\^Pi cos hj, the finite integrals extending to all values of i from —oc to +00, we shall deduce general expressions from 'p^ and q^ arranged according to the power of the eccen- tricity. Since « = jf -|- e sin tt, •we have by Lagrange's theorem cos V, = cos or g — esm'^fl ■'- ^ — etc.; '' '' 2\ eg 3! fV 'sm using the notation We then have C09 u^ — 2 -i — -., n + l. n! = 1.2.3 n= r(H, + 1). 0! = 1! = 1. THE ORIHT OF URANUS. 16 Siibstitutiiif,' ill tlie gciicnil term of the above scries for sin g its value in imaginary cxpuuuutial functions 2 sin flf = yiTl' (o-»*^- » — cO^-\) we find by the binomial theorem, using the notation of combinations, «,_w(» — 1) (» — /,4-l) _ „! l.'i.JiVr.a' "«!(»» — «)! I n+l il4 '1+1 j Differentiating n — \ times with respect to g, and putting together the first and last terms, the one after the first, and that before the last, and so on, we find _ 2"+' ^Tl'^^^'^^^'j} = (M -(- 1 )»-i(„('.+i)(^'-r ^ c-<"+"<' ' -0 — C(n — 1)"-' (c'— ""• -' + c-'"-')" '-"') + etc. n+l 1^1 Substituting for the exponentials their values in circular functions, and dividing by 2"+' we have ^"-' sin"+V ~ Og"-' - L { (» + l;"-' cos (« + l)g- b{n - 1)"-' cos (n - \)g * I n+l + C (« — 3)"-' cos (« — 3) j7 — etc. I n+l ) the series terminating at the lust positive coeflficicnt of g. Substituting this last value in the general term of the series which gives cos «, we have cosM="F ~^^^ |(«+l)-'cos(n+l)flr-r'(„_l)''-icos(«-l)^+ctc. I Let us now substitute for n another variable /, putting in the first term of the last factor t = n -f 1, in the second / = ?i — 1, in the third t = »i — 3, etc. The limits of finite integration with respect to t will then be in the first term, -f-l to -\-'+'_| : (7e'+* — etc. > cos lo ''°'"-raci-l)!2'-'' -(l+l)!2'+»];/ +(i+3)!2'+»„* j "^ - 1. (i-l) ! 2^» I 2^ i(t+l) <-.!. + 2«i(t+l) (t-l-2) (i+3) *+« i We have, therefore, for all values of i different from zero i'-«e'-» jl h ^^ _J_^- —J^^' -ctclrU) '''=^-'=(i-l)!2--M <+^"2^i^+lT"^*4^2*(i+l)(i+2)(i4-^ •; '^ To obtain the value of p„ we remark that the only constant term in cos u arises from the term — e sin^ jr ; its value is therefore — i e. The constant term in ^ = cos u — e is therefore | e, whence l)o = -3e. (15) The values of (ji may be obtained in a similar way by developing sin « by La Grange's theorem. But the development is rather more complex, and it is easier to derive them from 2>(. Let us take up the equations g ^ cos tt — e ri = sin u u — e sin u = g Considering m, like ^ and >?, as a function of the independent variables e and «/, we have by differentiation 6u _ d (e sin w) _ _ 6e Se ~ du _ d (cy;) _ sin M ' de ' 6'e 1 — e cos u 1 Comparing («) and (I>) du _ _______ dg 1 — c cos u du . Su d^du 6^ de dg dudg dg dg de (a) (i) Putting in this equation for g and >? thcii developed values this equation becomes 2 ipt sin ig =2 - \f^ - sin ig ^ de which gives by equating the coefficients of sin ig (16) *T THE OliniT OF URANUS n The following are special values of i\ and g',, developed to the sixth power of the eccentricities, as derived from the preceding formula; : 2)o = — 3e 11)2 Pt 2^ 16807 46080' , 1 8' ^192 9216 1 ^2' 3 1 c» + 48 ,_ 27 , 243 , ^8*-" 128'' +5120*' 24-..' 1.-/ 125 ^^"=384' 27 . 3125 9216 . e" 2" 80 16807 '46080' (16)' Having the developed ^ and yi in terms of time, let us resume the equation (13). As only purely linear operations arc performed on Q in this equation, it follows that if we represent its several parts by Q^, (^., etc., and by (^p„ ^p^, etc., the values ^p obtained by putting Q=z Q^^ Q = Q^^ etc., we shall have ^p = ^p, + (^p2 + f tC. We have, therefore, only to find the separate values of ro'5p corresponding to the different terms of Q, and to take their sum. Let us then represent, as before, by m any one term of R. 3 April, 1873. " h cos (iX + a +/u' +JL}) 18 THE OUBIT OF URANUS. We then have, considering only terms of the first order with respect to the dis- tiirbing forces, UtE = sm Jv, (17) fu,R = m'ihv cos N: •where we put for brevity. -n ^ — cli , a, ,, 1'= -.;-,-,-— Let us represent by Q^ the terms in Q which are of the first order with respect to the disturbing forces, so that we have The general term in B, will tlien give rise in Q^ to the term m d a, cosN. a, • dx> In the case of the action of an outer on an inner planet this expression becomes ai V ' d-of while in the contrary case it is — ( 2tj7t — Ji — *^ M cos N, both derivatives being taken with respect to the logarithm of the mear distance of the inner planet. In the integration it will be more convenient to substitute for ^' and X the mean longitudes counted from the perihelion of the disturbed planet. If we put the angle N will become, Since corresponding to each set of values of i' and / there are several values of/' and y, it will be convenient in the numerical computation to combine these different terms into a single one, because after forming the derivatives of R there is no need that u, ' + {i! + i +i) u] cos [tT + ig\ a, _ — k sin [iV + (i' + i +j) w] sin [t7' + ig-\ «i If we put h = 2 i cos [ j'J + (»•' + i +y) o], k.=Xlc sin [j'o' + (*' +i+j) u], the sign 2 being extended so as to include all values of J and f which correspond to tlie given values of i and i', we shall have for the general terms of Qa ^ I h cos (*Y + ii/) + /.-. sin (lY + if/) I , or, when we represent the angle i'l' -\- i-^^'^X I i'+j-v+,\ 1/h.cos [N,-\-{i+j)g-\+k,sin [Ar,+(i+y)^]J + } r+t-r-j \ \ A-oCos [A\+{i-j)g]-\-k, si" [iV.+('-i)i/] \ + 1 .'+,->'_, \ \ h cos [iV,-(i-y)«/]+A-.sin [iV;_(/-j)»/] \ -\-\v.,—v_j\ lAvcos[iVi-(i+y);7]+A-.sin[Ar,_(t+y)i/]| The sign 2 of finite integration here includes the separate combination of every value of i with every value of j, except those combinations which make the ♦ Tlio imlloes ? and j, in tlipsn eqnatidns, arc not to be ponfoundcd with the coefficients of x and u In the general terms of R and <^. We need not use the latter at present. ta!^ i 20 THE ORBIT OP URANUS. coefficient of tfie time under the sign mn or cos vanish, and so render the corre- sponding value of V infinite. These cases have to be treated separately. To find, from the expression, the coefficient of the sine or cosine of cosine Ni-\-ug in r,^Sp, we put, in the four lines of this equation, e : follows: In the first, i-\-j=u.'.j= u — i; " second, i — J=:u.'.j= i — u; " third, — i-\-j=u.\j= u-\-i; " fourth, — i —j = u .•. j= — u — i. In the above expressions i and j being independent, and including all values from — oc to -foe, »■ and u will also be independent, and include the same range of values. Substituting for J its value in it the coefficient of VI a r becomes 16a,(l-f-m) [k, cos ( A^i + itg) + k, sin {N, + ug)} + i>i ?-(«+() (''-( — »'(«+.■))• Since q^ = — q^j this expression reduces immediately to " \ + J'. '?.■+»)(»'("+.)— »'-i )' or, substituting t — u for i in the second line 22 (j',?(,-„) + i',(-.)?() (''( — »'«-()• Hence, writing N instead of Ni, • 8ai(14-w») This expression fails for the particular case iV= vg, where the value of j'_„ will be infinite. If we take each term of Q of the ibrm m (Ai"> COS vg + A;*"' sin vg). and substitute in the general expression (13) it will be found that the terms in r^-fip which have the infinite values of v as a factor are to be omitted, and replaced by ^ *a,(l + wi)" (20) The two parts of i-j'Sp thus found include all the terms of the first order with respect to the disturbing forces. But when terms of the second order are taken into account, we shall find terms in Q proceeding from secular variation in wliich the time appears as a factor, outside the signs sin and cos. Let us represent such of these terms as depend on any angle N by Q = V}^± (k, cos N-\-k, sin N) * THE ORBIT OF URANUS. 31 ; and use the symbol j'j, as before, to represent the ratio of the mean motion of the phinet to the coefficient of t in the angle ^-|- i reduces to 1 «»'« is a,(l+J/0 ^ V 2 f i'.?(u-o (C..-0- c. ) cos (iV+ «^) -! i),7„_„ («<- «,.-„) sin (iV+ w^/) ) "'"' li'i'Z(a4.)(c,„,. )-"-<) cos (.V+«j/) -ri'*,.H)(-'~V+o)s»n(^V+«i/) i or, putting t — u for i in the last line, 1 m'arT^ ^ 2„^, 1 i'/Au-o -i'(,.-,,7, ! S (c>„-.)- '■.) cos ( A^+ vy) + («. - «„_.) sin (A^+ wjr) ( ; HI 2'i THE ORUIT OP URAXUS. to which expression is to be added, in lieu of tlie terms which will have iufinite values of p as a factor. 1 m'ar-hi-e ^aJi^^^^-^-^^^^^n (22) for' "o^ ^''' ^""^ *^'*' ^''''''''' °^ «. "' ''°' "^ *""* a] "' ''" "^ ^" *''^ expression The formulae 19, 20, 21, and 22 give the complete expressions for the perturba- tions of the logarithm of radius vector by successively substituting in it all tho terms of Q. Perturbations of Longilade, We now pass to the perturbations of longitude. In the Mecanique Celeste (Premiere Partie, Liv. ii. Chap, vi.), Laplace gives an equation (Y) by which the perturbations of longitude, which are of the first order, may be derived from those of the radius vector without the formation of any other derivatives of i? than those which enter into Q. But tlie formula does not seem easily adapted to the case in which the perturbations of the second order are taken into account, we shall therefore derive all the perturbations of longitude from the second of equations (1) liy integration this equation gives do ft { rSE , ) dt=r"-\J6y<^(+(^\ IS being the arbitrary constant of the integral, llepresenting, as before, by sub- scnj.t zeros tlie values of the co-ordinates corresponding to the ellipse to which the orbit IS supposed to reduce itself when the disturbing forces vanish, we have fA'o n"-n cos i// fiO because the constant to which the integral must reduce itself in the elliptic motion a^>i cos T^ o w i- - ~- Subtracting the last equation from the preceding, and putting V — I'o = Sv, we find dhv _ ,1 rdRj /I 1 V 1 Developing -^.^ to terms of the second order with respect to the disturbing force -^ = ,\(l-2' in formula (11) will then contain only periodic terms. In computing the terms of the first order we determine the value of 5p from the equations (19) and (20), using the value of ^o in il»). '-I'l^en those of tv are obtained by integrating the equation lit = l + m«^ ^v'^'-2"^°«^r.'- (24) Having found tlie values of hv and Sp for both planets, they are to be substituted in (11), to obtain 5^, 5-^^and h ^!^. But, rigorously, Sv and ^v' arc not the 6v op same with So and M, owing to the movement of the orbits of the planets, and the corrections for Sy are also to be added. Considering, for the present, only the perturbations of the second order, which depend on Iv, ^v', Sp, and 5p', we may use the following equation for hE, and similar ones for its derivatives: s« ='"*»+ '5 w+ vR m ^p+'^-^-.^p', 6\' ^"'^ RcU + 8 ^^-^^ and dq, lot us represent by 0„ 0,j, etc, the lon- gitudes of the ascending nodes of the several orbits of tlic disturbing planets on that of the disturbed planet. We shall then have dq_^ dl dr, dt ScosO, "'" — 2 sinO, dl dp — „ dh\ , „ . ^ dr, ,,^=scoso<^;+vsmo,^y;. (28) These equations completely define tlic instantaneous motion of the orbital plane. They cannot, however, be rigorously integrated in their present form because p and q as integrals have no completely defined signification. To do this it is neces- sary to express the difttrential rotations dp, dq, etc., in terms of the differentials of any elements wo may select to define tlic position of tbe orbital plane, and then to integrate the equations thus formed. But, for the purpose of constructing tables of the planets we may consider ^), q, etc., to represent small rotations of the planes of which the powers and products may be neglected, and the integration is then quite simple. Peiiitrhadons of the second order depending on the motion of the orhital planes. R being a function of the five quantities of r, r\ v, v', and y, the motion of tlie orbital planes introduces tcrnis of tiic second order by clianging the values of v, v', and y. These terms we liavc hitherto neglected. To investigate them let us refer the rotations of both planes as given by (28) to the node of the disturbing on the disturbed planet as the principal axis. If wo represent by dy;,dK;d/;\ and (//.;' the rotations corresponding to this axis, and designate by the subscript 1, the quantities which refer to the disturbing planet whose action we are considering, and by 2, 3, etc., the other planets, the equations (28) will be replaced by these S' ---(«.-".) *■• •».+v, !„(„,_„,)*., the summation commencing with / =: 2. By formuhc of tbe same kind wo arc to find the difl"erential rotations dy;' and dk' of the orbit of the disturbing planet, produced by the action of all the planets. 4 April, 1873. Iw ^ 96 THE ORBIT OF URANUS, These rotations will be nroimd tlio snmo principal axis with the rotations dn and *//.-, but around a secondary axis in the piano of the disturbing orbit, and therefore making an angle y with the secondary axis of the disturbed orbit. A geometrical construction will now sliow (p.ite simply that the infinitesimal rotations hn, hk, bn\ and hic' will produce the following changes in v, v', and y. ^v = cot ybk — cosec yhK hV = cosec yU- — cot yhk ' (Op) hy = hri — A»7 If wo substitute these values in the general fornndic (11) the terms of the second order added to hR will be (6R . dR SD /o'^ . SR \ \6v ^°^ ^ + <5v' ^°^^^ ^'r^ -(^^cosccy+ ,^cotypk^ (30) ^k The first two terms of this expression may be put into the form {i(2f + ^:;)(— y+eoty)-j('^f__;^;?_)(._^_,,,^)| - { K 7! + t^) (^"^^'^ y + cot y) + I ( f^ - I'JL) (eosec y - cot y) I hk. But, cosec y + cot y = cot i y = ^^^ ^ y, a cosec y — cot y = tan ^y= " cos i y and in the general term of R, by (7) rR m'h Ox ~ a, 0'+y)sin-Ar Making these substitutions, and putting, as before, the above value of SR reduces to m'h , w/cosjy clh . , . (31) ,WJ THE OllUIT OF URANUS. 9t 1! Tlio currcsnoiiiliiig tLriim of h and h ', < uiul may ho ulitaiiicd in tliv satiie way liy Miibntituting ^, uud '[ tur A' iu (:3()) uud coutiuuing the correspuuding sub- stitutions of the general terms of tlie derivatives of It as given on page 9. Tlic equation (31), besides being of tlie second order witli respect to the disturb- ing forces, is also of tlie second order with respect to tlie mutual inclinations. For Ms hkt hr„ and hy; arc of the first order with respect to both quantities, and, when- ever ( is not zero, h is a quantity of the second order, containing a^ as a factor. It is, therefore, only in exceptional cases that the terms of th(! second order depend- ing on the motion of the orbital planes can become sensible. JiediwIuM of (lie loiigUiiJe in the or/til to lo tjitude on (he ecl!j)lic. The integration of (23) gives a value of ftv, which, added to the longitude iu orbit corresponding to tlie pure elliptic motion gives the longitude in the disturbed orbit, counted from a fixed point in tiie moving plane of that orbit. The position of this fixed point is comi)letely determined by the condition that the instanta- neous rotation of the plane in question around tiie axis perpendicular to itself is always zero, so that the motion of-the point of reference is always perpendicular to the direction of the plane. IJut, altliougli tliis instantaneous rotation is zero, the integrated rotation is not rigorously zero when W(.> consider tin; terms of the second order. It follows that the value of r, the longituoe in orbit, and the position of the piano of the orbit do not rigorously determine the position of the planet: we must also know how the fixed point of reference has changed its position iu con- fiequence of the motions which the plane has undergone. Let us consider the relative positions of this plane at two epochs. If the fixed point were equally distant from the common node of the two planes, the integrated rotation of the plane around its own axis would be zero. But, these distances not being equal, tiieir difference is a correction to be ajiplied to the longitude of the planet in its orbit. Suppose, now, that at the end of any time the inclination of the actual orbit to tlie primitive orbit is <^, and the distance of its ascending node from the present position of the moving axis of a; is 0. A rotaticm around tiu' line of nodes will not change the quantity sought. But, if we represent the infinitesimal rota- tion around an axis perpendicular to it by dr we shall have cos dp — sin dif =; di\ dq and dk being the instantaniMins rotations around th(> resp(>ctive axes of a* and y. By this rotation it is easy to see that the relative distance of any two fixed points, one on each plane, from the node, will be altered by the quantity, dr (cosec ^ — cot ^) = dr tan \ <^, the relative longitude of the fixed jioint on the moving plane being increased by this amount. Tiio correction to the longitude in orbit from tliis cause is, therefore, (// — dr tan J f/(/). (32) 1 -f- cos tp The approximate value of the integrated correction is therefore 5^ =2 /('/'(/'— My)- For every pair of periodic terms in jt and + tan i <^ S cos {hh + W) — sin ((V + 'V) \ (3fi) In the case of Uranus, tan <^ is so small that this equation will be sufficient for a long tmie before and after our epoch. In the application of the method to other planets the mode of operation must depend on the circumstances of each particular case. The differential equations (34) between 0, t, and <^ are rigorous, and their integrals may be approximated to in various ways, out of which that best applicable to the particular case must be selected. Expressions for the latitude. If the position of the orbital plane and of the ecliptic were each determined by tlie i)receding formuUr, there would be no perturbations of the latitude, the lati- tude itself being given rigorously by the equation sin [i = sin ^ sin (« — t). = sin (J) cos T sin v — sin ^ sin t cos v. But the instantaneous values of ^ and t, or of sin ^ cos t and sin <^ sin t, are troublesome to tabulate ; it will therefore, in practice, be found mor" convenient to use only their mean values, and to consider their changes from this mean as per- turbations of the latitude. Representing by the sign h the deviations from tiie mean values, which are of course arbitrary, we have cos /35/i = cos <^ sin {v — t) 5(^ — sin <^ cos (v — t) hr. Let us substitute for S<^ and ^t their values given by the integration of (34) to lJ 80 THE ORBIT OF URAXUS. quantities of the first order, in which case and t may be assumed equal. Tliese vahies are h

' and f.q' being omitted because, I'cing purely secular, they may be included in the mean values of <^ and t. Substituting in tlie expres- sion for S^ cos ;3V/J a«i):6| a'Z)'6'J 2.14447 0.33969 0.5878 1.081 3.44 13.6 1 0.55207 .68314 .4990 i.in 3.40 13.8 2 0.20836 .47198 .7396 1.152 3.59 13.9 3 0.08687 .28491 .7123 1.463 3.68 14.5 4 0.03793 .16270 .5632 1.596 4.30 15.1 5 0.01702 .09010 .3998 1.485 4.87 16.9 6 0.00777 .04896 .2653 1.231 4.98 19.1 7 0.00359 .02624 .1682 0.940 4.60 20.5 8 0.00168 .01392 .1022 0.675 3.91 20.4 9 0.00079 0.00733 0.0615 0.463 3.11 18.8 Dciivadrcs with re-fjtcct to (log '({ = ») o/a »/>:?/;>. i n = 0* 1 ^2 3 4 DJ>*> IM aDah^f) Dx (oi^Z)^/,';') D. WI^'J.'I') A,(a*Z);i'» 0.33969 0.9275 2.257 6.68 27.4 1 68314 1.1821 2.175 6.93 27.4 2 47198 1. 2116 2.631 7.05 28.3 3 28191 0.9972 2.888 8.07 29.2 * 16270 0. 7259 2.722 9.09 32.3 $ 09010 0.4899 2.285 9.33 36.4 6 04896 0.3143 1.762 8.67 39.0 7 02624 0.1944 1.276 7.4i 38.9 1 01392 0.1161 0.879 5.93 36.0 ft 0.00733 0.0688 0.586 4.50 31.2 & coiid (Jcrivativcs. f Dii'? Z>;(a2)o6"?) Diwi^f'V 2)-(a'^;6'|>) © 0.9275 3.184 11.19 47.4 :t . 1.1821 3.357 11.28 48.2 t 1.2116 3.843 12.31 49.4 $ 0.9972 3.885 13.85 53.4 ,* 0.7259 3.448 14.53 59.6 t 0.4899 2.775 13.90 64.4 t 0.3143 2.076 12.19 65.0 Vo lues of a"+^Dlb"> « ^ a'Djr^ .Wl'''^ «'/'i6| $ 1.865 2.674 8.104 30.8 ^ 1, 1.267 2.844 7.77 30.8 f 0.761 2.412 7.63 29.9 . t'^ 0.433 1.790 6.92 28.7 . t 0.240 1.224 5.73 26.8 ft 0.130 0.792 441 23.5 g^ 0.070 0.493 3.2U 19.6 k April. 1873. 8a "f- 34 THE ORBIT OF URANUS. Derivatives with rcs2>ect to {log a = ».) 1 V„{a'Dabp =fi<»,7i 4.539 13.452 65.1 1 4.111 13.46 54.1 2 3.173 12.45 52.8 3 2.223 10.50 49.5 4 1.464 8.18 44.0 5 0.922 5.99 36.7 6 0.563 4.19 29.2 Second derieatii'ca. i a^DvB^» a,DvH a ,i5 a,DvE'2 UJKE'V 1.267 4.111 13.46 54.1 4.111 17.57 81.0 1 1.313 3.856 12.95 54.0 3.856 16.80 79.8 2 0.850 3.167 11.98 51.8 3.167 15.15 75.8 3 0.500 2. 318 10.31 48.4 2.0I8 12.63 69.0 4 0.281 1. 573 8.24 43.1 1.573 9.82 59.6 5 0.155 1.014 6.18 36.6 1.014 7.20 49.0 The notation B'l^ and Ej; is that of liC Vcrvier in liis devolopment in the first volume of '■^Anwdea de V Ohwrvnioirc Inq)erii.d dv Paris." Numeric d exjircssion of R and its dciivatircs. We next proceed to the computation of the coefficients h and their derivatives. As an example of the most convenient form of computation we present in full that of the coefficient of — cos (jX' — (t— 1)X — w) in the expression of R for the action "1 of Saturn on Uranus. In this computation I use the tahles given by Le Verrier in his "Annales de VOhsermtoire" tome i, ])ajTes 358-383, comparing the develop ment with that of Professor Tcircc in the Astronomical Journal, vol, i, as a control. (•aw THE ORBIT OP URANUS. 86 y=o; y=i- • -3 —2 —1 +1 2 3 4 6 6 i> +4 3 2 1 —1 —2 —3 —4 6 (0)Xt<,'> +0.5212+0.8334]+ 1-1041 —1. 10414 —0.83344 -0.6212 —0.3034 — 0.1702'-0.0929J 0)y.alJ^bf —0.2849 —0.4720— 0.6831 —0.3397 —0.68314 —0.47198 -0.2849 —0.1627 —0.0901 -0.049(1 Ao, (50)(0 a, (50)(«) -16.1775 +O.2303 +0.3614—15.7565 —0.3397 —0.78728 —1.30542 —0.8061 —0.4601 —0.2603 —0.1419 (0)6«j" (i)x«/>- —13.55 —11.25 — 5.52 0.00 0.00 + 2.92 + 5.73 + 6.83 + 6.46 + 8.54 + 7.7S + 4.78 +0.51 0.00 + 1.18 + 2.56 + 3.25 + 3.06 (2)X«««. + 1.43 + 0.74 — 0.00 —0.59 — 1.00 — 2.22 — 2.65 — 2,82 — 2.40 (3)X«»fi'« — 0.73 — 0.5S — 0.59 —0.54 — 0.59 — 0.68 — 0.73 — 0.80 — 0.74 Aa,(51) (1,(51) + 48.53 + 4.71 — 4.31 — 3.31 + 47.20 —0 62 — 1.69 + 1.30 + 6.46 + 6.38 (0)X6'/' mX'D,. —18.70 —13.33 — 4.42 0.00 + 4.42 +13.33 +18.76 + 19.41 +17.01 + 13.1o| +10.38; + 4.10' —0.68 — 1.37 + 2.83 + 6.27 + 7.48 + 7.02 (2)X.'/>'- + 1.43 o.oo; — 1.00 —2.35 — 3.00 — 6.92 — 7.13 — 6.70 — 5.60 (3)X«'i^« — 1.46 — 1.15 — 1.18 —1.08 — 1.18 — 1.15 — 1.46 — 1.60 — 1.48 ao,(D2) 1,(52) + 32.36 — 5.69 — 4.10 + 29.86' —4.11 1 1 — 1.13 + 9.09 +16.44 + 18.53 +16.96 (0)X"^,(') —3.00 — 3.40 — 2.63 0.00 + 2.63 + 3 40 + 3.00 + 2.25 + 1.55 (1)X<',£,(') +2.32 + 3.17 + 3.8(1 +4.11 + 3.86 + 3.17 + 2.32 + 1.57 + 1.01 Aii,(tiO) 0,(60) + 16.18 + 17.41 +4.11 + 6.67 —0.68 — 0.23 + 6.49 + 6.32 + 3.82 + 2.60 1 j-2i representing the several values of j in the different terms which correspond to one and the same set of values of i and t". jl THE ORIJIT OF UK ANUS 87 To obtain the dcrivativo with respect to y wo notice that all tlie apprcciahlc terms in the difleicnt values of /<, whicli depend upon the mutual inclination, are of the form li = ff- A, where ) + (") u-j-w +2m> K h. i i' h /, h /( /. —3, + 4 — 2.31 4- f..39 — 1.01 — 0.20 —2, 3 — 3.42 4- !).!t7 —0.95 + .02 — 0.51 — 9.40 —1. 2 — 3.54 —439.59 —.03 + .02 — 93.30 + 430.35 0, 1 +287.30 — 9., 55 +0.13 — .02 —.04 4-. 09 + 285.20 + 9.19 + 1- + 58.38 — 50.05 —0.02 — .05 —.03 + 48.16 + 48.90 2,-1 — 3!).7fiO — 3(!.:W8 —0.074 — .071 —.016 — 47.138 + 35.531 3—2 4- 35.13 — 2-2. 179 —0.105 — .313 —.007 + 30.f,3 + 21.44 4,-3 4- 20.3!) — 12.(!0 —0, 1 1 + -n + 17.90 + 12.55 ft,— 4 4- 11.25 — CMC, —0.10 4- .23 + 9.98 + 7.00 r.,— 5 4- ('..00 — 3.f. + .22 + 5.38 + 3.78 88 THE ORBIT OP URANUS. Valuks of h. P — 2u — W — U — 2«' P" h -2(«) + 2« h. /,. i i' h h h —1, 3 +0.04 — 0.27 42.24 +0.11 +38.72 + 17.03 0, 2 +0.00 +31.05 — 0.04 +0.18 + 0.58 —31.15 + 1. 1 —0.74 — 0.83 — 1.08 —0.01 — 0.03 + 2.15 2, +3.00 — 5.19 + 1.89 +0.18 + 0.32 + 4.14 3,-1 + 1.2(i8 — 0.200 + 2.082 +0.111 — 1.912 + 5.216 4,-2 +4.1(i — 5.458 + 1.707 +0.001 + 1.44 + 4.570 5,-3 +3.13 — 4.03 + 1.20 +0.04 + 1.13 + 3.30 (1,-4 +2.11 — 2.74 + 0.87 +0.02 + 0.75 + 2.31 7.-5 + 1.35 — 1.70 + 0.55 +0.01 + 0.49 1 + 1.40 P — 3« — u — 2u .Jui u — 3„' t Id P' -(-) — 2M — :!(w) t)U u 2« h. h. i i h h h /(, /) h + 1.+2 —.12 —.11 —.14 +.02 +.17 + .00 2, 1 +.01 —.07 + .03 —.02 —.02 + .01 + .04 3, +.18 -.40 + .34 —.08 —.02 . + .04 — 15 + .213 4,-1 + .222 —.700 +.520 —.117 —.010 + .020 —.325 + .410 5,-2 +.424 -.847 +.,558 —.122 —.012 + .021 —.170 + .400 (!,-3 +.:i08 —.773 + .408 —.107 —.008 + .014 —.145 + .400 :,— 4 + .327 —.027 + .305 —.080 —.005 +.009 —.107 + .370 8,—;-. _-t-Al __: —.47 +.20 —.06 —.083 + .280 A — 4w 1 — u' Itio — 2«' — 2u ow u — ij P i — («) — 2(o) — 3(..) — 4(") ;.. /'. i i' h h h h h 4, —.04 +.04 —.02 —.04 5,-1 +.02 —.08 +.08 —.04 + .01 —.03 + 003 <■),— 2 + .030 —.105 + .100 —.047 + .008 —.045 + .028 7—3 + .043 —.114 + .112 —.048 + .008 —.049 + .031 «,— 4 + .042 —.100 + .103 —.044 + .007 1 —.044 + .032 »,— 5 +.0;$.') —.000 + .080 —.030 +.000 —.037 + .029 Vai.les of DJi. P lo b) F («) i i D„h />../l Dvho D«h, -4. +4 + 3.18 —3, 3 + 3.24 —2, 2 + 2.39 —1, 1 + 0.40 0, + 171.03 — 2.075 + 171.51 + 1—1 +8740.50 — 2.09 + 8749.12 — 3.02 2, 2 + 407.72 + 20.83 + 472.40 + 18.05 3,-3 + 277.00 + 2.87 + 278.25 — 0.36 4,-4 + 153.87 + 4.02 + 155.46 + 1.41 5,-5 + 82.12 + 4.00 + 83,54 + 1.80 THE OK HIT OP URANUS. 30 *- Vai.i Ka (IF 7Vi. 1 F — u t — W u— 2u' «' — 2- +u' u F -(") — 2(«) + (") Dvh 2w Dvh, Dvh. i %• Vvh Dvh Dvli Dvh Dvh —3, +4 — 9.28 + 19.4 — 5.32 — 19.00 -2, 3 — 9.74 + 18.5 + 2.05 — .06 —.12 +.08 — 7.97 — 17.43 -1, 3 — 4.fi3 +907.62 + .05 — .08 —.13 + .07 + 1K0.44 —888.65 — ", 1 — 5.5(i.40 — 26.17 — .10 — .09 — 14 —.12 —561.82 + 25.60 1, + 30.16 — 71.56 + .04 — .19 —.11 +.02 + 15..37 + 69.80 2,-1 + 2(1,-). 17 — 86.47 — .19 — .26 —.07 +247.58 + 84., 30 3,-2 + 83.10 — 74.60 — .42 + .63 —.04 + 68.34 + 73.49 4,-3 + 68.34 — 54.9 — .,55 + .29 —.02 + 5V.70 + 53.9 6, 4 + 49.04 — ;!6.9 — .58 + .68 —.01 + 42.19 + 36.7 (>,— 5 + 32.1 — 23.2 _ .53 + .90 + 28.1 + 23.4 P — 2u — w — w — 2w' F -(") Dvh _2M + 2« Dvh, Dvh. i { i>o/l Dvh Dvh -l,+3 + 0.16 — 0.66 + 86.02 +0.47 — 7S.76 —34.19 0, 2 + 0.32 —64.55 + 0.15 +0.60 — 12.92 + (;2.54 + 1. 1 + 3.27 — 2.70 + 4.69 +3.02 — 1.22 — 2.20 2, + 2.,')6 — 8.78 + 5.00 +0.60 — 3.74 + 6.00 3, 1 4-13.745 —15.870 + 7.232 +0.463 + 3.933 + 12.153 4,-2 -flO.2,') —IS. 93 + 7.85 -1 0.32 — 0.77 + 15.07 ,5,_3 + 10.76 — IS. 07 + 6.99 +0.21 + 0.69 + 14.68 (1,-4 + ',1.39 — 15.0 + 5.59 +0.13 + 1.22 + 12.4 T,— 5 + 7.3 —11.5 + 4.21 +0.09 + 1.2 + 9.0 P — 3« _ _2« — u' , — « — 2w' •1 ' — u — u P i i — (") 1 — 2(") -3(„) ^u — u' +2u Dvh', Dvh. Dvh Doh Dvh Dvh Dvh 1 Dvh + 1. + 2 +0.03 +0.18 + 0.05 +0.22 + .22 1 +.08 —0.31 —0.17 2, I + 0.12 —0.26 +o.:i3 — 0.09 —.08 + .17 —0.08 —0.09 3, +0.21 — 0.S9 + 0.98 —0.31 —.09 + .13 —0.59 +0.13 4,-1 +0.S9 —2, ((2 + 1.S9 —0.54 —.09 + .13 —0.84 + 0.69 ;'>,— 2 + 1.0S —3.00 +2.53 —0.67 -.08 + .11 —1.38 + 1.30 (•),— 3 + 1.40 —3.47 +2.73 —0.69 —.07 +.09 — 1.33 + 1.68 1,-4 + 1.46 —3.44 + 2.60 —0.(12 , —.05 + .06 —1.20 + 1.78 8,— >) + 1.32 —3.00 + 2.20 —0.51 1 ! +.04 —0.99 + 1.62 P — 4« ■ ,iui W — 2a— 2« — u — ow -w 1 P i i' — (u.) — 2(u,) _3(„) Dvh 4(«) Dvh, Dvh. Dvh Dvh Dvh 1 ■)vh 4, + .02 —.09 ' +.13 -.08 + .02 —.06 —.01 fi,— 1 + .0.5 —.22 , +.30 —.18 + .04 —.14 —.02 (;,— 2 + .10 —.38 ' +.48 —.26 + .05 —.24 + .01 ■J,— 3 + .15 —.52 +.62 —.32 + .00 —.30 + .04 «, -t + .18 —.59 +.66 —.33 + .06 —.32 + .09 n,— 5 + .19 — .59 +.65 —.31 + .05 —.31 + .11 The values of I)vfi, needed in compntins; the perturbations of the second order with respect to the masses being obtained in the same way, by the simple substitu- tion of the second derivatives of the functions h,aDuh, etc., for those functions themselves in the expressions for /*, it is not necessary to present the details of the computation. After obtaining h and its derivatives., ic will be found convenient to change the arrangement of tlie terms. Ilitlurto we have kept in one series those in which the sum of the indices are a constant. Now, we shall put together all those in which 40 THE OKBIT OP UIIANUS. he index of the disturbing planet has the same vnhie, nrranfriug the individual terms ot each scries according to the index of the disturbed phmet. Thus the index of the product of any term, as h cos N, by any multiple of the mean anomaly of tlic disturbed planet, as^y, "ill be found in the same series witli that of .V itself and J lines above and below. ' The next process will be the formations of the required functions of the mean anomaly of Uranus, |^J, '^J^, "', , bg r. Their values are as follows :- rr= = Po = » = 1 1.001 l();i +.0()().55()7 "''' "'^'~ +.()9;3!);J3 ros g -.04(i8SS«) cos g +.0468889 sin g +.0938294 cos g Iniw. '"'!^ -.001(i494 cos -V +.0032988 sin 2y +.0055012 cos 2^ tZtt '" ? -;;;|"<)^^-^ -« 3y +.0002190 sin 3^ +.0003357 cos 3^ +.000020 cos 4y -.0000035 cos Ag +.0000142 sin 4g +.000020G cos Ag Considering only those terms wuich are of the first order, the value of D'^R may be found in two >yays, tlie agreement of whicli will aft'ord a check upon the entire dcvvelopment of the perturbative function, and upon the computations of 7? and These are (1) by direct differentiation, witli respect to the time as con. ov tamed in the mean anomaly of a single planet, whereby each term in R of the will produce in D\R tlie term R =■ /* cos N a. D\R = — "* inh siniV and (2) by forming the expression D\R = "'^ '•'■» 6'v (It GR dp, "^ Op lit ' As several "mechanical multiplications," like those indicated in tliis hst expression, are to be performed, the following example of the fom o^ com- putation IS presente.1. It exhibits the formation of the product of tliose terms of -.: in wliich t' = _ 1 by ih - ' -^^ (sin) m S" X .04G915 X .002750 -'^ -10 +1 2 3 4 .5 + .02 + l.fi.'i _2S7.42 +:!4,S1.14 + r,4.42 + C.988 +1.18 +0.10 I " -i- OH — l;!.48 +10,1.32 + 2 5r)3 4-0 ■?f^i^t»ism''i"iimii&imm THE ORBIT OF URANUS. 41 The Tinltiplicr^ on the left arc each one-half the cocfflcinnt cos j«iio''' to the corresponding terms of ', 111 "' "« in' d^ Perturhalions of radiu6 vector. Let us now resume equation (19), and put for brevity (44) ■«.(l + »0 If we give to u the successive values 0, +1, —1, -f2, —2, -fS, —3, we have + i S< (iVA(-i) +i>u-i)V() ('—''(i-o) 1 ''".cos (.V+ g) -\-k. sin(iV+i/) | r,=,^p=i/'X ] +|2.(i>, I A'o cos iV -|- /i", sin iV ( . \hcQs{N+g)+k.smiN+g)\ ■ ; Z-„ cos (iV - flf) + A-. sin (JNT - i/) ( •f i ^ • + (Pili + Pill) (''3 — v-i) I 4- etc. etc. 1 {l;cos{N+2g)-\-l:s{n(N-\-2g)l + \M + i^r {2Wi (»'j — J'-j) ■) + ( Piqs +i'3?.) (»^, - »'_3) \ J /.•„ cos ( A^- 2g) + Z-, sin (AT - 2g) \ 4- etc. etc. ) + p»q3 (»'» — n) + (pi74+r.?i)(»'« — »'-i) + (7'i72— i'//i)(»'2 — J-i) -|- etc. etc. j /.-. cos (iV+ 3sr) + 1; sin (JV + 3g) \ m TUE ORBIT OP UBANUS. i6 + i^ + (/"74+ivy.)(". -»■-)! I /,„ cos (iyr- 3^7; + /.-. sin (iyr_3j7) I + {Pi'h — iV/i) ("-1 — »'-2) I ' -|- etc, etc. I A law of the factois of /.-, cos (iV+ «//) + A-. sin (^+ ««/) wliicli will be noticed in the above expression, is this: lloprcsenting this i:tctor by A'„, wo have the index i representing the coefficient of (j in iV, so that only half the valnes of it' need be separately computed. As the computation of r,«<^p from these formulie .an be arranged in such a way as to be very simple, the computation of the termv i.i which the index i' is —1 is Here presented quite fully. The logarithms only are omitted, being used only in the cases in which they are more convenient than a table of products. In prac- tice I find it convenient to write them in red ink immediately under the numbers which tliey represent. via First, to find M, it will be noticed that in the expression .,"V' y the a in the a,(l -j- VI) numerator represents the moan ( and q^ from (U?)' we find, for Uranus, i M mi — — •i -iVC/v/i— 7'//i) — I- 0.0784 10,044 3.343;> 0.0028 0.2;5.')8 O.llh 0.008 0.005 0.()();{ In units of the sixth place of decinuds. In the computation the first three lines are copied from previous pages. 46 THE OllBIT OF URAxNUS. i = — 1 • -4 —3 —2 —1 +1 2 3 4 5 V -0.1460 -0.1700 -o.soeio -0.35000 -0.85062 -0.68004 -1608.18 -1.17868 +6.76040 +0.87118 —1.10 1- 0.466a 40.03 k„ +0.08 + 1.38 +276.62 + 20.85 - 70.56 +142.56 •l— •-! -0.08 -1.28 + 84.70 — .28034 + 6.7U -286.60 + 104.17 +2.04476 + 1.70 —6.2938 + 0.18 —0.6536 — .0887 — .14452 — .82301 +7.299:14 + 0.0784 .,-._, — .2047 — .3690 — .9675 +7.0190 +1.2218 + 1.0O85 + 1.4913 — 6.51b6 — 10.044 'i~'o — .0.'),'?5 — .0910 — .1893 — .63361 +7.9330 —5.8883 —0.4056 + 3.343:t 'i—'l — .1797 — .3338 — .9140 + 7.1100 +1.4110 + 1.6392 —6.4417 -«- 0.0028 '..-». — .39 —1.00 +6.96 + 1.13 +0.82 +0.86 + 1.41 — 0.23B8 »»— • V .144 — .280 — .823 +7.299 +2.045 —6.29 J + 0.118 •a— '-1 — .3(;9 — .9i;8 +7.019 + 1.222 + 1.005 + 1.-190 0.008 — .33 — .91 + 1.41 +11.82 +7.93 +1.64 +O.Mi —5.89 + 0.005 + 0.0003 1.00 +6.97 — .19 —20.60 •j-'-l — .09 142.56 X{'|-'-,) —12.35 — 39.9(;5 —117.327 + 1040.59 +291.49 —897.2 —78.8 0.0784 X(',— -,) — 0.02 — 0.03 — 0.075 +0.550 +0.10 +0.08 +0.1 — 05 — 10044 A'o -12.87 -80.68 +0.914 -40.010 + 1.901 -110.777 + 1040.00 +201.57 +B0.I4 -807.1 -70.8 X (•,-•„) +0..'i32 + 6.364 —79.68 + 4.1 + 3.:tJ:l3 Xf»i-»-,) — O.OCl -1.12ii —3.056 +23.770 + 4.72 + 6.48 -21.B + o.ot>a: X(«j-»-j) —0.001 -0,003 + 0.020 + 0.003 A', — 0.20 —1.186 +80.187 - 1.135 —1.721 — 74.»« +64.62 —17.4 — .2:i.')a A-:. Xfvj-O —0.07 +o.oi;« —0.205 + 194 +80.14 —0.48 -74.1I6 + 1.48 +64.6 +0.034 + .118 X('j-»-,) A'. —0.04;; —0.114 +0.!-2S + 1.022 -0.00!' +0.144 — 1.577 +0.1-2 -0.86 + 0.18 + 1.66 + 1.58 .008 A-*. -0.048 + 1.03 —.013 + .003 .(H)5 X (•,-•„) X {•,-»,) + .007 + .035 -.011 + .005 + .004 -.007 -.003 + 176117 +.004 ^.000 +.(M2 + 21698 h'„ h — 1 — 2il —11076 —23197 + 991 2 K, h —314 -45152 —15113 —51 •■•. + 19 K , h — 5i! + 1712 -t-628 + 5964 -71 Kj /■, n + 2,-2 -1-2378 —8 -132 — 3 + 72 + 21 + 126 K'k, + 8 A',/-„ + 3 + 1 r,i»,(.Oi.) - 1 - —0848 f 170557 -17508 -82458 -4151 -116 X.0-. •• i -43S -l->2.-3 —-21 — lii.',;t -1!)5 —438 + 8283 —821 —1053 —195 — 5 X. 002751 —26 +:!(> +485 -48 -62 —26 +485 — 11 — 4H II —62 X.OOOIfix — 3 - 4 — 1 —2 -4-30 — 3 cos ^1^ (co.-») + 8 + 84 + 26 -1112 -1393 + 175235 —676 -10315 -22086 — 2982.'i4 +56614 -5227 — 1606 -875 — 10 + 1 A, <. — 39 + 175 ! +21483 + 12548 —31 A' ,t. — 7 —6 — »(;37 —14556 +116 -2 A, i. + 36 —9 + 104 + 322 (1 n —293 —306 — 1 A'f,i, — 11 II I) - I r,«», («in) + 1 + 8 -1602 -0058 -812600 J 1*8204 + 11044 4 281 X. 0409 15 { — 79 —453 — 79 — Ilt.ilii -453 +3i;(i'.i -146116 + 518 + 3lil>!l + 13 + 51H II — 5 -27 —860 + 215 X. 002751 — 27 — HIIO + 215 + 3M x.oooiim .^ — 53 4- 13 -1-2 '1 - 3 COs*l^(:.ill) - » - 151 - 21tlt3 341M6 -800351) 4 64020 f 18864 4!>«1 it" • . THE O K B 1 T OF URANUS. 47 ' In forming the next ten lines, it will be noticed that the value of r„ correspond- ing to any vertical column is found u columns to the right. It is therefore necessary to extend the line v two columns at cath end. The extension on the right is, however, omitted for want of space. In performing the subtractions it will be convenient to copy the j^'s again on the lower edge of a horizontal strip of paper, and, in forming tlie differences r„ — r„' to lay the; strip above the line of r's, and u — u' colunnis to the riglit. On the left of each line of differences is written the factor by which that line is to be multiplied. The mode of formation of the A"s is evident from the formula. It will be seen that the same computation which gives 7C gives also K_„, only the latter belongs u columns to the right. Each l\ and //, is multiplied in succession by all the A"s wliich lie below it in the same cohuun, but the product by A', is to be written u columns to the right, and that by A'_„ u columns to the left. The sum of the products in any one colinnn gives the coefficient of ^?^ {Ig -\-i'l') in the development of r,*!^, ^) wliich are given in the following table. Tlie terms midtiplied by the time are still to be comptited. They are derived from ('20), wliich may be put in the form Thisexp^ession is computed thus: u Pn *<;" «7.. *•;' /'.,/.•'"' -qJ-V — .t4l)-s;j - 1 •.'44.31 +n->.i8 1 4-.!)!)!! '7 + ;5-.Ms 4- .!»!»«)■ •i — 'io.oo + ;}J.7:) -f 2(1 >i) 2 + .()t>:U -t- 4.()G + smi — 1.8G + ().(»!) + 0.04 We have now ^ /'„/.•':;' = - _|_ OOS.Ct?; \ }r:iii..i-":^ = + -.'nosn -:iqj-r = :.+ '.'().!»:} ; iJ/'iv,./.- -+ -mn r^\> = — 'iU) lit -\- '.'!)«() lit cos If + '>{)(»S1 lit sin g 4- ■<<» III COS '2l/ + ()!)7 /// sin 'i'/ + -',7 COS % + '2i it sill ;{i< 3, -f 2 »< + 24 H< + 14/1/ + 139«< 0, —355045 -354347 oc 1, -I-14S75 —1503 —18411 —1496 2.45551 —2912 +14206 2, — 2,S3 + 49 —1530 —21 2.25448 —039 +457 3. —20 — 1 —112 2 1.97839 —40 —8 —2,-1 — 1 + 1 + 3 — 9 1.7090 + 1 +3 -1, —10 + 8 + 34 —151 1.8098 + 12 —99 0, —9343 —1093 —1112 2!t92 2.00035 —2870 + 10 + 1, -|-17(i557 —9058 + 175?35 —21186 2.18780 +530512 + 100 2, —17508 —312600 —10315 —309359 2.52504 + 1823 —23800 3, —22453 +78204 -22980 +04029 3.28542 —13482 + 18310 4, —4151 + 11044 —5227 + 13804 2.39559 —293 + 104 5, —115 +281 —373 + 901 2.1235 —13 -1-2 — 1 + 1 + 1 2 1.0292 + 9 +1 0, —5(5 — 27G +32 — 100 1.0993 —329 —1.551 +1. +710 +3723 + 1075 + 3728 1.78303 +.5874 + 20114 2, +20180 —13 + 20583 + 484 1.88083 —31000 + 920 3, +7797 +0512 + 8824 -1-0796 2.02348 —0019 —6800 4, + 1534 +0202 + 2054 + 0230 2.22401 +243 —2172 5, + 2071 —5954 + 2134 —5010 2.00780 +527 —421 0, —053 +755 —550 + 494 2.98438 —190 —20 1-3 —151 + 04 —137 + 75 1.5772 —1464 + 053 2, — i-)2 + 221 + 149 4 229 1.03880 + 231 —833 3, +4348 + 15 + 4420 +88 1.71074 —13050 4' 4, + 1504 + 1280 + 1774 + 1314 1.79091 —3225 —3126 5, + 133 -{-085 + 214 +756 1.9045 —18 — 1')43 «, —111 + 232 —109 +273 2.0479 + 144 — 103 7. —103 + 87 —108 + 100 2.2034 + 50 —7 3,-4 —28 + 121 +31 + 122 1.5408 + 181 —038 4, + 1223 + n + 1245 S +41 1.5858 —.5578 — f.8 5, + 485 + 389 + 545 1 4 400 1.6488 —1721 —15(0 <■', +38 ! +201 +03 + 221 1.7225 —50 —590 7, —25 +53 —23 + 03 1.8012 +70 —100 S. —15 + 7 —10 + 9 1.9229 ft,—:) +393 + 12 + 400 + 18 1.4889 —2432 —74 fi. + 175 + 138 4 194 + 142 1.5385 —90S ' —700 7, + 14 +77 -+22 ! 4 84 1.5940 —40 — .UO 8, —10 +20 — 9 4 2» 1,05M<) 1 +40 —70 9, — 5 + 3 — 5 4 4 1.7:!45 ^ims^ I ■itr-Bir r*— "*"•"" wnaniriiriinMiM THE ORBIT OF URAXUS. 49 Pert 111 hations of Lomjitude. The perturbations of the longitude are now to be computed by formulte (24). To do this iu the most simple way we remark that the numbers given on page 42, under the heading '^'^ are those represented in formula (42) by v, and v,. If c*v we put nt = i equation (24) may be put into the form di out we have from (42) „ ill = S( V, sin N— V, cos N). If now we represent the numerical values of cos i|-(^p, already found, by 2 (p, sin .V -f p„ cos N) , and if we substitute these expressions in the above value of . , , the latter will become d'v 'y^, = rr- S Kv. - 2l'.) si" ^V+ (V,. - 2p,) cos N\, where we put for brevity V, = Mn;., The nnmoriral expression for r,"" is given on page 40, and by multiplying the quantities witliin brackets by tiiis expression, after tlie manner explained on pages 40 and 41, we form the terms of ' ,'f. Multiplying each of these terms by its al corresponding value of r, changing ros to s!n and sin to ros, we have the coefficients iu the expressions for I'v given on page 50. Al previously mentioned, before commencing the above computation, I had conii.-''ed all the perturbations of Uranus by the method of "])erturbations of the elements," using tlie formuhr developc^d in my Investigation of the Orliit of Ne|)- tune. The two results are here placed side by sid(>, for the purpose of comparison. The discrepancies in the various coefficients, expressed in thousandths of a second, are shown iu the sixth and seunth cohnuns. It will be seen that the largest discrepancies, and indeed the only ones (with a single exception) exceeding one-tenth of a second, occur in the coefficients of the terms 2/' — / aug %' — /. Here tiie errors are almost certainly in tlie computation from perturbations of tlie elements. Owing to the limg pi-riod of the term 'i(/ — I they would not become sensible iu the course of any one century. 7 April. 1873. ^ 60 THE ORBIT OF URANUS. Pebtl'bbatiu.ns or the Lonuitudk of Uranus produckd by TUB ACTION OF SaTUBN, AND 1 UEI'ENUINU UN THE I'lUST I'UWEB OC TIIK DISTL'BBINU I'llUCKK. From conip. prucuiliiig From purt. of uluiiiviits iu Discrepancy. 0.434294 ip 9, I' 8in coa siu cua .{ill cos cos 8ia ti If fi If It II 0, + io.»(;90< + 10,9045/ .004.5/ 1, — 1.2:10(1/ + 12.231 III — 1.22H III + 12.271)1/ .002(1/ .040(1/ + 13(1/ + 128(1/ 2. — 0.01-2 lit + 0.717 H< —0.072 nt + 0.720(1/ .003 ((/ + !((/ + 9(1/ 3, — O.OOiiil + 0043 «< —0.004/1/ + 0.043 ;i/ 0, '"'*'* —1.541 1, + 8.;") 45 — 4.735 + 2.844 + 1.013 .... .... — 80 — 6 2. + 0.4(>l — 0.1 ('.9 + 0.133 + 0.100 .... — 7 3, -1- 0.028 — 0.005 + 0.013 + 0.014 —1,-1 -f o.o;;*; + 0.039 + 0.032 + 0.005 4 34 _ 1 0, + 1.282 + 0.718 + 1.280 + 0.719 2 1 — 5 — 13 1, —20.817 + 8.522 — 20.873 + 8.595 50 73 + 701 — 100 2, -11.890 -1-143.4(13 — 11.003 + 143.405 797 2 — 45 —1.351 3, + 4!t.30 -|-115.8(i + 49.02 + 110.08 320 220 —103 + 280 4, + 2.1 13 + S.CIO + 2.195 + 5.021 02 5 — 24'+ 01 5, + 0.1 2(i f 0.329 -j- 0.109 + 331 17 2 - 2 + 4 0,-2 + 0.017 — 0.017 + 0,025 — 0.033 8 10 1. + 0.042 + 0.814 + 0.034 + 0.818 8 4 + 7 + I"-. 2, + 4.110 — 0.009 + 4,103 — 0.012 7 3 + 89 + 2 3, + 2.079 — l.(!07 + 2.100 — 1 070 27 G9 + 38 + 30 4, + 0.048 — 1.830 + 0.043 — 1.902 5 73 + 9 + 27 5, + 1.1C3 + 2.950 + 1.274 + 2.991 111 35 + 9 — 24 6, + 0.503 + 0,378 4- 0,550 + 0.445 53 07 2 + 2 1,-3 + 0.034 + 0.012 + 0,030 + 0.015 2 3 2, + 0.037 — 0.041 + 0,014 — 0.050 23 9 + 1 + 1 3, + 0.824 — 0,019 -1- 0,812 — 0.017 12 2 + 19 4, + 0.355 — 0,207 + 0,351 — 0.203 4 4 + « + fi 5, + 0.047 — 0.105 -f 0,039 — 0,191 8 20 + 1 + 3 6, — 0.02r. — 0.003 — 0,028 — 0,000 2 3 + 1 7, — 0.053 — 0.032 — 0.053 — 0.018 14 _ 1 3, 4 + o.oor, — 0.022 — 0,005 — 0.023 11 1 + 1 4, + 0,228 — 0,008 + 0,221 + 0.002 7 10 4- 5 5i + 0.103 — 0.077 + 0.084 — 0,075 19 2 + 2 f 2 fi, 4- 0,013 — 0,044 1- 0,013 — 0,057 13 + 1 T, _ 0.005 — 0,013 — 0.001 — 0.015 4 2 8, — 0.003 — 0.002 + 0.002 3 4 5,-5 + 0.074 — 0.003 + 0.071 3 3 + 2 6, + 0.038 — 0.027 + 0.023 — 0.020 15 1 f I'-f « 7. -f 0.005 — 0.010 -f 0,005 — 0,025 ft 8, — 0.002 — 0.004 — 0,003 2 1 "1 » <4Ki* TUE ORBIT OF URANUS. 61 cos N Perturhationa of Latitude. These are computed from the formuhc (27) and (40), no reductions being made from hk and l/i to hp and hq, but the perturbations of the hititude being computed directly from the former by (40). We have only to represent the expressions for hk and hrihy hk = — 2«„ cos N — 2a, sin N hr, = Sa',, cos iV-|- 2a', sin N and substitute u for n in the equations (40) from which f)3 is computed. The principal stops of the computation are shown quite fully in the following table. The values of c7i 1 1 fJi are first formed from those terms of Ii, on pages 37 and 38, which contain d as a coeffici(!nt. Then, having for each original term of Ii OR _ m' eh dy a, dy all the terms which have the same coefficients of?, and ?.' in N are combined into two «lepoi!(ling on «/ and /' as shown in the case of Ii on page 3(5. The coefficients of these terms, in units of the third place of decimals, are given in the columns licadcd — . Oy The value of ' sin N being formed for each term of R, all the terms depending on the same multiples of ?. and ?.' are combined into two, of which the coefficients are given under tlie proper heading. Tiie terms of (t -\-j)ah sin N being formed in like manner, we have, by adding the last two expressions, all the quantities which enter into the formulic (27). To integrate these equations thus forming the numerical values of (V.- and hv; we have only to multiply each term in the second, third, eighth, and ninth columns of the table by the corresranding values of '" '"' , for which we juay use the value of . '' „ already given. <», cos 4/ sni 1 The quantities giv(>n in the four columns under hk and — hr, show the values of — rt„ — «,., — («',, — rt'., corresponding to each argument. From these the terms of 6;i arc formed by equation (40) with the modification mentioned above. 52 THE ORBIT OP URANUS. PSRTIJllBATIONS OF TUB LATITUDE I'KODUCED BY SaTURN. 3 '' dy 2-'*-«iUiV 'It I ('+»'■'' sln^V :;;=.,/,. X ik — i, W i i' cos sin sin 009 sin cos sin 00s sin It ens tt cos sin It Bin 00s 0, —10.82 +0.008 0.00 + .008 +0.19S 1, — 3.45 — 2.92 —1.94 — 0.11 + 0.17 +0.84 - 1.77 + 0.73 +0.203 -0.172 —0.104 —0.043 +0.246 +0.209 2. 4- 1.42 —11.10 + 1.30 +10.73 + 0.07 +0.05 + 1.37 + 10.78 -0.042 —0.326 +0.042 —0.317 —0.049 +0.0U 3, 4- 1.31 - 1.71 +1.31 + 1.71 + 0.01 0.00 + 1.32 + 1.71 —0.026 —0.033 +0.026 —0.033 —0.008 +0.003 -2,-1 + 1.0 + 0.1 —1.0 — 0.1 0.00 0.00 — 1.0 — 0.1 +0.012 —0.001 +0.012—0.001 0.000 0.000 -1, — e.43 -52.« +6.4 -82. 8 + 0.03 +0.02 + 6.4 —52.8 —0 098+0.804 —0.098—0.804 —0.004 +0.018 0, — 0.97 + 7.06 +1.53 + 5.14 - 4.91 0.00 — 3.38 + S.14 —0.144—0.145 +0.070+0.106 —0.661 +0.678 +1, +46.67 0.00 0.00 +59.45 —0.02 +59.45 — 0.02 +1.482 —1.887—0.001 +0.081 —0.026 2, — 1.84 — 2.81! —0.99 — 0.20 + 0.93+1.32 — 0.06 + 1.02 -0.127+0.197 +0.004+0.070+2.218 +1.92C 3, + 0.90 - 6.98 +0.73 + 0.44 + 0.12+0.16 1 + 0.90 + 6.60 —0.358 —2 m 1 +0.358—2 630+0.091 -0.071 4. + 1.09 — 1.38 +1.09 + 1.38 + 0.02+0.01 + 1.11 + 1.39 —0.056 -0.071 +0.057—0.071 -0.048 +0.068 -1,-2 + 7.6 - 1.5 —7.6 + 1.5 O.UO 0.00 - 7.6 + 1.5 +0 067 +"'"» +0.067 +0 013 +0.(103 +0.004 0, + 13 +10.9 —1.3 +10.7 — 0.11+0.53 — 1.4 + 11.2 +0.013'— 0.112 +0.014+0.116+0.008 +0.034 +1, — 0.27 + 9.03 +1.70 + 0.91 + 1.85—7.35 + 3.35 — 6.44 —0.003—0.112 —0.042—0.080 +0.040 -0.04S 2, -14.51 0.00 o.do — 7.01 —0.21 1 -7.01 — 0.21 —0.230 » +0.112—0.003 1 —0.028 3, — 2.85 — 2.29 —0.42 — 0.19 - 0.97+1.11 — 1.39 + 0.92 — 0.0()2+0.050 +0.0,30+0.020 —0.063 -0.064 4. + 0.43 — 4.23 +0.44 + 3.67 + 0.02+0.22 + 0.46 + 3.80 +0.015+0.146 —0.016+0.134 —0.080 —0 014 6, 4- 0.80 — 0.99 +0.80 + 0.9!) f 0.02+0.02 + 0.82 + 1.01 +0.067 +i> 083 -0.068+0.084 +0.008 —0.004 2,-3 — 1.17 + 0.86 +1.22 + 0.97 + 0.09+0.33 + 1.31 + 1.30 —0.01(1 '0.* —0.011 +0.012 +0.050 —0.046 3, — 8.54 0.00 0.(10 — 4.34' -..« 0.00 —0.090 +0.046 +0.004 -0.010 4, — 2.08 — 1.65 —0.13 — 0.15 — O.fS +0.85 -1.1 + 0.70 —0.026+0.022 +0.013 +0.009 —0.012 —0.015 6. + 0.14 — 2..'i0 +0.24 + 2.00 0.00 +0.22 + 0.24 + 2.22 +0.002+0.041 —0.004 +0.037 —0.010 -o.ooa «. + 0.55 — 0.66 +0.55 + 0.6(! + 0.02+0.02 + 0.57 + 0.68 +0.013+0.015 —0.013+0.016 +0.004 —0.001 3,-4 — 0.72 + 0.47 +0.82 + 0.78 + 0.09 +0.,32 + 0.91 + 1.10 _ 0.005 —0.003 1 —0 006 +0.008 +0.022 —0.020 4, — 4.80 0.00 0.00 — 2.4.S —0.02 — 2.4S — 0.02 — 0.0:!8. 1 +0.020 +0.002 —0.006 5. -,.. - 1.09 -0.01 — 0.10 — 0.06 +0.59 — 0.6" + 0.49 —0.013 +0.010 +0.006 +0.004 —0 004 —0.003 6, + 0.12 — 1.3 +0.12 + 1.0 — 0.02+0.19 + o.in + 1.2 +0.001 +0.014 -0.001 +0.013 +0.0(>4 +0.002 5,-6 — 2.64 — 1.35 +0.04 — l.,35 + 0.04 -0 017 + 0.008 IXl L A" 77 T S"^'"'"' ""•"•' {4 = -4". 77 7- CO. 7. i ^■■■I THE ORBIT OF URANUS. 53 C II AFTER III. PERTURBATIONS PRODUCED BY NEPTUNE AND JUPITER. The perturbations of Uranus by Neptune were originally computed with ele- ments of both planets quite different from those finally adopted. But the last computations, on which the concluded values of the perturbations depend, Averc made with the concluded elements of Neptune found in my investigation of the orbit of that planet.' They are as follows : • > 7t, 4;j 17 .30 0, l:j() 7 33 f. 3;J5 5 39 «?>' 1 47 1.6 «, 7864.935 e. ().00849(J2 log a. l.-l 78141 Mass, TtJiTU rm» (J r (J Hence follow the following functions of the elements of Neptune and Uranus: a = 0.G38195 12° 44' 58" 247 45 20 y = 1 30 29.6 c = sinjy= 0.013161 M= 37.522 (in units of 6th plito of decimals). From these values of the elements are obtained llie following values of the various terms in the development of the perturbacive function, and of v. As the developments have been formed on the same principle as in the case of Saturn, it is deemed unnecessary to give the details of the process. It is only necessary to remark that the indices t' and i arc the coefficients of l' and g respectively, the mean longitude of Neptune, or I', being counted from the perihelion of Uranus. ' Suiithsonian Cuntributions to Knowledgi', Vol. XV. 54 THK OR HIT OF UIIANUS. A CTIIIN or "'x Ni;i-n.Ni!. 1 /<- Ml' dlt '!"='"' X «-r'>< V A. "1 A. 0'" "1 (?P "i aj •' i *'» I'O 00:1 sill h d. 0, +n:ir..(n 4- 0.202 4-3 ;8.2(i 4-.3(18.2(> + 1 — I».(i7 — I.2.'. f 1.01 — 1.23 — (14.80 — 5.89 — 100 94 — 8.39 4- 1 + 2 + 0.41 — 0.0.1 — 0.0 — 0.1 4- 2.98 4- 0.12 4- 3.82 4- O.Oli '4- 5 + ■i U u 4- 0.33333 1,-3 + 0.21 — 001 4- 0.7 — 0.2 4- ].(! 4- 0.2 + 2.1 4-0.1 — 0.40158 — U — fi.21 - O.ii.1 — 4.11 > 4- 1.27 — 211 5 -f- ((.(! — 43.5 — 1.1 1 — 0.()7107 — 1 4- i;u.i!» + (i.:ti» 4-i:i3iis — 0.08 4-509. m 4- 0.77 4-10.'-,7.49 -f- 1.99 — 2.04023 (> 2.'). 4.') -1041 — 18.0 1 4- 0.02 — 84 57 —11.47 — 84.57 — 11.47 4- 1 9(11:17 + « + 1.12 4- 1 19 4- 0.5 — 0.1 4- B.-.;i 4- 0.82 4- 7.31 4- 1.07 4 o.««2;i2 + •■! 0.05 + 0.01 4- 0.02 + 0.02 — 0.24 — 0.32 4- 0.02 1 4- 0.39843 2,-4 + O.S.I — 0.03 4- 2.1 4- 2., 5 4- 0.1 4- 4.8 -- 59.7 — 0.335841 — 3 4- 13.5:1 — 1.01 4- 27.7 4- 3.1 4- 18.7 — 2.1 — 5.5 — 0.504971 t) + 37.'..!I2 + :m 4-751.59 — 0.7! 4-9 7.«2 4- 1.25 4-2491.(19 4- 2.72 — 1.02009 — 1 - U0.090 — 3.1*1 — 117H03 4- 3.20:1 —171.79 -i:i.l:l 4- ,'(931.21 -f3oil.82 4-50.7820 + 3.31 4- 4U 4- 5 9 — 0.4 4- 12.24 4- l-'*4 4- 12.24 4- 1.84 4- 0.98009 + 1 0.09 — 0.01 — 0.0!> 4- 0.02 — 0.51 — 0.08 — 0.(10 — 0.09 -j- 0.49512 3,- f. 4- 0.81 — 0.04 4- 27 4- 0.2 4- 3.0 — 1 4- S.5 — 0.2 — 0.28814 — 4 +■ 12i>9 — 0.11 i + ;i(;.9 4- 3.8 4. 34.5 — 3.(1 4- 73.(1 — 0.7 — 0.40478 — 3 + 2lHi.:)7 — 0.04 +(i01.l:t — 0.42 4-717.94 4- 081 -1-1535.81 4- 0.115 — (I8OOO — 2 lij 3:t — 9.8,S -l.'i0.02 4-lll.M — 190 :!3 —27.51 —(1:15. 2:1 —111.09 — 2 12559 — 1 + ■').ii.'i + 1.5ti + l:i.S — :).l 4- 20.12 ; 4- 4.(15 4- 1.0 — 1.2 1 4- 1.88844 0.24 — 0.07 — 0.6S 4- 0.15 — 1 11 — 0.30 — 1.11 — 0.30 + U. 05378 4 - « 4- 0.7-. — 0.07 + 3.2 4- 0.3 -1- 3.4 — 0.2 4- 0.7 - 0.4 1 - 0.25249 — 5 4- 9.-17 — 0.79 -f 38.:» 4- 4.0 4- :is.5 — 3 9 4- 70.4 — 0.0 ' — 0:1:1777 — 4 4- Ill.l.i - 0.12 4-441.7 — O.H 4-5i2.2i 4- o.:i5 4-9(15.7 — (1.1 ; — O5I01I4 — 3 — ;j8.fi8 — 7.4t — 149 10 4-22.:l7 — 17- 15 —27 95 —419.12 — 74.3(1 — 1.(410; — a 4- ."i.2a-. + 1.^9J 4- 19.. 54 — 5.50 4- 25.:tO 4- 7.19 — 603.3 -185.4 4-25.:i910 — 1 0.3lJ — 0.17 — 1.28 4- 1.49 — 1.90 — 0.70 — 1.21 — 0.37 + 0.90210 6,-7 4- O.!!,"! - 0.07 4- 3.5 4- 0.4 4- 3.5 — 0.3 4- 6.5 — 0.5 — 0.22468 — 4- U.97 — 0.59 4- 3.5.1 4- 3.4 4- 3(1.7 — 3.4 4- liO.9 — 5.6 — 0.281184 — 5 4- (13.03 — O.K! 4-315.3 4- 0.1 4-.3.54.S — 0.1 4-012.0 — 0.7 — 40804 — •'. — 27.43 — 5.32 — I:tt4 4-21.4 — 1 5:1.1! 1 -25.3 —304.8 _ 64. ti ; _ 0.08U29 — 3 4- 4.84 4- 1.81 4- 23.0 — 7.0 4- 27.82 4- 8.57 4- 92.2 ■f 32.7 ' — 2 21043 -- 2 — 0.45 — 0.24 — 2.09 4- 0.90 — 2.77 — 1.14 + 0.48 4- 0.57 4- 1.82073 — 1 ■f 0.03 4- 0.02 4- 0.12 — 0.0(1 4- 0.18 4- 0.09 4- 0.15 4- 0.07 4- 0.0455 «,-7 4- 4.9:1 — 0.41 -f 29 ,8 4-2.7 4- 31.5 — 2.9 4- -19.0 — 4.4 — 0.253S — (1 4- 3li lli - 17 4-217.1 4- 0.4 4-2:111. IP — 0.4 4-387.4 - 1.1 — 0.3400 — 5 19.01 — 3 74 — Ul.ti 4-18.8 -125 7 -214 -22:1.8 — 40.0 — 0.5152 — 4 4- 4.11 + 1..51I -f- 2:1.9 — 7.8 4- 27 7(1 4- 9.01! 4- (13 4- 22.0 — 1.0(12- — 3 0.4i»(» - 0.271 - 2 79 4- 1 :io — .3.42 — 1.55 4- 4i!.40 4- 20.01 4-10 9273 2 + 0.03(i 4- 0.02J 4- 0.20 — 0.13 -f- 0.28 4- 0.10 4- 0.14 -i- 0.00 4- 0.944:1 7,— 8 4- 3.41 - 0.28 4- 24.0 4- 2.1 4- 25.5 — 2.5 4- 37.8 — .3.5 — 2267 — 7 4- 2i>.!l(» — M 4-14H.4 4- 0.4 -f-lilii — 0.(1 4-245.3 - 1.3 — 0.21114 — « 12.1III — 2.. 54 — 811 2 4-15.4 — 98 11 —17 2 — 1(13.1 — 29.8 — 0.411:1 — R 4- 3 40 4- 1.31 4- 2:!.o — 7.7 + 2il.O : -f- 8.x 4- 49.8 4- 18.0 — 0.(i9o8 - 4 0.4x7 — 0.27ii - 3.25 4- i-':i — :i.79 — 1.87 — 12.83 — 0.99 — 2.311)8 — 3 4- 0.044 + 0.o:i4 4- 0.28 — 0.19 4- o.:t7 4- 24 — 0.09 — 0.12 4- '-7577 8,- 9 4- 2.3.-. — 0.20 4- 18.8 4. i.« 4- 19 n i — 2.1 4- 28.1 — 1.2 — 2032 — 8 4- 12.20 — 0.1:1 4- 9ti (1 4- o:i 4-105.0 — 0.7 4-1.54.8 — 1.3 — o,2.'prio — 7 8 M - 1.71 — (18. !t 4-12.1 - 75.5 -l:!.l —117.7 — 21.3 — 0.3423 _ (i 4- 2>:9 +■ 1.05 4- 21.0 — 7.1 4- 2:t.2 4- 8.1 4- 40.0 4- 14.7 - 5205 — 5 45 -0.2i — 3.5 4- 1.8 — 3.90 ! — 1 99 — 8.8 _ 4.H _ 1. 0855 1 — 4 4- 0.0411 + 0.039 4- o.;t7 — 0.2«1 4- 0.45 4- 0.31 — 4..50 — 3.09 -f- 12. 0958 9.- 9 4- 7.0 — 0.1 4- «3.4 4- 0.1 4- (18.2 — 0.(1 4- 9(1.8 — 1.0 — 0.2207 — 8 BO — 1.13 — 51 9 4. 9.1 — 5(1 3 1—9.9 — 84.0 — 15.2 — 0.29:11 — 7 4- 2.117 4- o.si 4- 18.1 1(1.4 4-19 8 1 4- 7.0 4- 31.8 4- 11.7 — 4147 — 6 liO — 0.25 - 3.4 4- 1.9 — .1.78 — 2.02 - 7.1 — 4.1 — 0.7085 1 — 5 + 0.080 4- 0.042 4- 0.43 — 0.32 4- 0.51 4- 0.3G 4- 1.72 4- 1.38 — 2.4308 10,-10 4- 4.1 — O.l 4- 43. 4- 44.3 — 0.8 4- fil. — 0.8 — 0.2040 — 9 — 3.9 — 0.8 - 40. 4- 6. — 42.0 — 7.0 — BO. — 11. — 0.2503 — 8 4- 1.8 + O.li 4- 14.7 — 5.(1 4- 1(1.1 4- (1.0 4- 25.1 4- 9.3 — 3446 — 7 — 0.3 — 0.2 — 3.1 4- 1.9 — 3.5 — 1.9 — 5.8 — 3.4 — 5259 — « 4- 0.0ft + 0.04 4- 0.4(1 — 0.37 4- 0.53 4- 0.40 4- 1.1(1 4- 0.9« — 1.1092 THE on BIT OF URANUS fiS The term of hmg I\ri(Ml, From the cx[)rossioiis for tlio perturbations of Uranus, sijbsoqucntly Riven, it will be seen that several of the terms luive very large coeffieieuls, that of sin (-/'— J/) being nearly an entire degree. Tiie magnitude of most of the terms in which I' is even arises from the near approach to commensurability in the mean motions of the two planets. Twice the mean annual motion of Neptune exceeds that of Uranus by only 30:J".8. The elements ef the orbits of botli planets will there- fore, in consequence of their mutual action, be affected with a slow oscillation, having a period of about 4'2(>6 years. The employment of these large terms and the great inconveniences to which they will give rise, especially in the corrections of the elements of Uranus, may be avoided by the device employed in the theory of Neptune, The following arc the essential featun^s of this method: YJrat, all the perHnbations arising from that portion of the perturbntive func- tion in which the coefficient of the time is 2m'— n or its multiples are considered and developed as perturbations of the elements. Secondly, tlic arbitrary constants to be added to the integrals of these perturba- tions are so taken that the perturbations shall vanish at the epoch 1850.0. In other words, the perturbations in question wiil be treated as producing secular variations of the elements of the orbit, only, instead of being developed in powers of the time, these variations will be retained in their rigorous form. The formuliE for the conq)utations of the perturbations in question, arc as follows : Let m m h cos («" I' -\-il -|-/(j' + j(j) = cos iV be any term of the pcrturbativc function, 7i being a function of a, c, and a. sin 4, = e g = cos 4. tan J 4' For each such term, compute v = n i'u'-\- ill A = 2 !h L=^ cW = ens 4/ ^ E = — ''('.'/+ J ">t 4) T = 1 = l,'^ + (/+y)a/«. M THE OliniT OP U RAN US. The corresponding perturbations of tlic demonts mny then bo put into the form 6 log a = MvA cos N-\- 5»o» il = Mi'L »ii\ N -\- i(o, ehn = Mve IVsin JV-f efi7i„ he = MvE cos N -\- Scq, iy = Mi'f COS N-{- fyo, tan y 5t = Mv T sin iV -\- tun y^To. Here, 5»)o, 3?o» etc., arc arbitrary constants so taken that 5 log a, 3?, etc., shall Tanish at the fundamental epoch. All the terms depending on the same values of i' and / are to be combined into a single one. And it will save labor to make this combination at as early u stage as possible in the computation; that is, to nuiltiply the various values of /j, S^Jl ^JL, _' , and 27 by the sines and cosines of I'u' -t-iw, and afterward proceed Sv 6e Sa with the sums of the products according to the proper modification of the formulir. Thus arc obtained the following long period perturbations of the elements of Uranus : — 3474.3-2 sin (3/' + im.T* mi {At— '2i/) — — 8.97 sin (()/'— Sij) + 4- 0.()4sin(W— 4J/) — 4-consfant=3320".lS. r/)-j- 180.10 cos (2/'— fj) 54.10 cos (4/'— 2//) 5.03 cos (()/'— iiij) 0.53 cos (8/'— 4fj) eSn = — 484.9() sin (2/'— //) + + 38.00 sin (4/'— 2J/) — — 3.01 sin (r)/'—3 is not at present of great importance, because the error will be com- pensated by the corrections of the element during several centuries. Pertuhbatioss or tii« LoNOITUDE OF Uh.\NUH PIIODUCED BY NeI'TUNB. poiH-y. < "i ai'. «i J Discri f 'J sin cos sin cua sill cos sin COS 0, — 0.4 202 < —1 » on.') lit — 1 .isl m —a — 004 (1/ —0.009 nt 0, _\ + O.fittT — o.oss —•2 + o.o4(; — 0.005 —i + o.ooa 1,— T -f 0.147 0.001 .... + 0.140 —0.002 1 1 2 + 2.r)0i» — 0.01 !) .... 4- 2.509 —0.010 9 — 1 + 3!t.(i5S — o.oso .... .... +39.073 —0.0.^1 15 1 + 4. 2;-) 7 — o.r>ii + 4 249 — 0.47S S 33 —1 + 0.2S0 — o.o;i2 ... * + 0.275 —0.032 6 -2 + 0.017 — «.002 2,-4 + 2.ft7S + 0.0 Jf + 2. so -f 0.03 + o.oos —0.002 10 1 :I + 41I.01.5 + o.4l:i + 47.2:' + 0.4 4 + 1.797 —0.021 2 —2 + H40.!t;{ + 7.:1HS -f HO5.04 + 7.43 + 35. .355 — .1.028 0.5 14 —I —347-). 4 + ISO., 'to -3474. 32 +1S0.IO - - 0.700 + 0.095 3S0 ir>5 — i(;2.o7 + S.OI — 101.90 -f S.OI — 0.007 +0.015 43 15 —1 — 9.447 + 0.408 — 9,50 + 0.47 3,— :> __ 0,070 0.000 ... — 0.077 -0.003 1 .T —4 l.ir.j 0.000 ■ . , , — 1.1,53 — O.dll 9 11 —a 17.2S(! + 0.0-2S .... — 17 2H5 + 0.2-29 1 1 —2 — 22.0S5 + 4.o:i7 .... -2-2.077 + 4.(t-J0 H 17 —1 o,(i7:{ + 0.0S2 • . . . . t • . — O.0S2 +0.079 9 3 — 0.0U7 + 0.000 4.-fi . 0.0:50 + 0.00-2 — 0.01.) -f O.OOv — 0.027 —0.003 .1 —5 o.ftr.s + o.o:i7 — 0.25 + 0.04 — 0.315 — 0.007 7 4 —4 — . 7.!tr,8 + 0.7')0 — 4. OS 4- 0.08 — 3. 90S + 0.059 20 11 —3 ..— 7.'>.oo + 12.s:i2 — 09.55 + 11.07 — 5.7.13 -+Iancj. i' 9 sin // COS Biti cos siu COS sin COB b.~ 7 —0.000 0.000 .... —0.015 —0.002 C 2 — (! —0.103 + 0.001! , . , , —0.113 —0.004 10 10 — 5 — 0.!»«(! —0.043 .... —0.004 —0.042 K — 4 -f3.;jfifi —0.(170 t . • • +3.370 — o.(;(i2 4 8 — 3 +3.210 — l.lfiO • . . • + 3.227 — l.lHfi 17 17 — 2 +0.077 —0 017 .... .... + 0.075 —0.002 2 15 — 1 -fO.OOD —0.001 «.— 7 —0 050 -0.004 0.00 0.00 —0,054 —0.002 4 2 — fi — 0.3«7 —0.020 +0.02 —0.01 —0.423 —(,.013 in 3 — 5 -fl.2fil —0.320 + 0.40 —0.14 +0.S.55 — o.i(;o « 11 — i + 7.7HI — 2.S25 +o.«o —2.54 +0.S57 — 0.3IH 124 33 — :i —0.025 +5.073 — S.07 +5.03 +0.017 —0.022 72 C6 —0.437 40.240 —0.42 + o.2i; T,— « —0.027 —0.002 — 7 — o.isn —0.003 .... —0.180 —0.009 3 fi — fi +0.272 — o.om +0.2(iO —0.047 12 1 _ f> —0.571 -< 0.217 — ().53S +0.202 33 15 — 4 —0.450 + 0.250 —0.410 f 0.255 40 5 — a —0.010 -I 0.005 «,— i» -(t.(ll3 0.000 — H — 0.0!(4 —0.002 — 0.003 — <> 007 5 — 7 .) 125 —0,022 -0.115 - •24 10 2 — fi —0.104 ■\ 0,OM2 — 0.03S 4 0.017 —0. 1 45 + 0.055 11 10 — 5 — O.H30 1 (»,4t'.3 — O.fifi -t-0.30 —0.110 + 0.007 (19 06 — 4 + O.G47 —0,501 -fO.«4 —0.53 9,— It —0,040 —0.001 — 0.04^^ —0.005 8 4 — H + 0.0fi3 — O.Oll • • • ■ + 0,002 —0,011 fil — 7 — 0.055 + 0.020 . . . • • . • • —0,050 -t 0.025 4 5 — + 0.OHO — o.o4;t • . • • +0.0«3 —0,054 3 5 _^ r -f 0.050 —0,050 10—10 —0.020 0.0(»0 .... —0.025 —0.005 5 & — 4 0.035 —0. 005 — 8 —0.027 — OOIO — 7 f0.02ti —0,0 IT — fi + 002 —0,070 Thoporturhatioiisof thol()p:!>''ith"isof the radin.s vrctor nro givon in nform similar to thoso of the loDf^itiulc. lliwlcr ^p, we hiive Mio coinpUat' perturbution. Under ^p.j the I'ffoct of tlic perturbations of long period. But luuler ipa wc have only the difference between ^p, and ^p„ it being deemed nnnece-ssary to present in full the perturbations of the radius vector as computed by the other method. Ap, beinir employed in computing ^», may, in fact, be regarded as completely checked ty its affording a correct value of the latter. In the last two columns f^p,. is reduced to common logarithsns by multiplying the coefKcieuts by the modulus 0. i;}4'.'!)4. 60 THE OUHIT OF URANUS. I'KKTL'KUATIU.NS Ut TIIK LodAlllTIIM of TIIK KAIHI.S VKCTKIl OK UlU.Nl 8 IMUH-I CKU IIY KkI'TINK. f 0, 1,-3 2 — 1 + » 2,-4 — ;i 2 —I 3,— ."S —4 _;{ -J —1 4.-fi —4 _4 _3 — r. —r. —4 _3 7,-7 — f. — 5 —4 8,— s —7 —0 —5 9,-9 _s —7 — ••> »P. -f i:i8 + 4 -f fi8 +.')'2:i — (IS — 8 + «4 +i4i(; +-.»(i02r) -I IC.fiS 4-3!) 1 -J — 3 — 3S —■.•-> 7 — 1>S4 -I ir> _ 1 — 10 —•.'.".4 — nr.tt — 143 — 1(17 — 4 — 40 -1-103 -f 3'J — 17 + ■«•-' -f ISO + i:5 — s 4 11 — 17 - 4 -t- ;i — 7 — 19 2 4 ■■! 2 + a + > — c. — 1 — 1 — \i — 179 -I IK! + 194 — (J —at 4 2 — 1 oo — :ioo — 01 — (■.-' + > + -.MI + 1& 4- 1 4I«» +.ir. + « 4 i — 7 — 3 4- 1 — 3 — II — 1 + a *p« -f 92 + 1374 -1-19190 -f 1720 -f 39J7 1 tf. Hill cort — 1 + -' — 12 4 •«•-' — ISO 4-r.35 4 120 — 03 4 194 — ir. s — lis — KiSi — ir.4 4- 10 4-1C.4 4- H — 20 '.>><2 — 70 — (U 4 01 — 11 — 13(i — 7s 4- II 17 32 I(i 1 + « — 4 Mif — I 2 — IS 4 '•• + a + I 4- 6 4- * -\- 3 1 COS 1 4- 00 4- 2 4 30 I-- 27 30 — 3 + I 4- IS L. •»■!.> 1 27 7 1 17 — ; !29 — 24 + 7 r. _ ri9 34 4- 1 17 •f 45 4- 17 — 1 — 7 + U 4 7 — 3 (■ ■'' — 7 + I — 1 4- I 2 ■ 3 « i I 1 — 3 U 2 U — 3 —23 4 1 — 1 — H 4- 4 4- I 4- « -f (i 4 :i 4 a 4- I (I _ I — 5 4- » I ( u. • -T^'iHifiiiflfrarltiiMf'W 1 i 1 1, T A u ■) THE OKBIT OF UUANUS. 61 reilTI.KIIATIUNH ur TIIK liATlTl'DK PUODL'CEI) BV NciTUNB. 1 dit >»' J '* (liD .V 2f I ('■+» r '/"J vr "i XA "iii.V J=«'.„X ik «T tfl 1' 1 COS liu ■Id OM Hill ooa ilu 001 •in CM CO. ■In // sin r, — 21.G5 0.00 -.012 1 + l.ll^,^-2.7:l'— 4.0(1— l.tiUf .01— (Mil — 3.!>ll— l.r.7i— 0.008 +0.021 +0.030 —0.012 + .080 +.020 2 + li».32— U.3!)+1 -f "J.^.s —ii.dl -I- D.i-M — (i.dl _ (i.-i;4 + ('78 + .008 — (104 —1 + &.M'+2.04— 2.12—1 04 — l..');i+0.04 — 3.115 - 1.(10 t— 2.283) [ + 0.K12] t + 1.431) [-0.31)3] + .32(1 +.075 u + 18.7U — a..V2 +l«.5li +i>.;i2+0.08 + ll).(i4+l).32 1 -0.143 —0.072 — 0.04U +0.071 1 + .002 3,-5 + 7.4 '+3.ii ! - 7.4 +3..'. ; - 7.4 +;i.5 +0.017 — O.OOH 1 — 0(117 —0(108 + 002 —4 — 3.011 _2.0ii+ 3.12 2.0 +0.48+0.05 + 3.(!() -2.0 — 0.010 4-0. oou 4 0(111 + OI8l(j + .oi)_.lli4 —3 — 21.4 j 1 10 +7.87 + 7.87 -0.128 40 041 -.(K^+.(l4l •1 — •* + ii.5h+|.7.l— 1.20— 0..14— 1.08 +o.-J(!-_ 3.18—0.28 +0.108 — 0211 - (153 + 0(104 + 303 +.074 — 1 + l:l.ii(; —7.28 +14.tih +7.(KI +0.1^ — 0.04 —14 8il +l!.!lii —0.1 111) — 0.1(1(1 —0 217 + 0.101 — .{190 — .icu — 1.61 +0.22 — l.Sl —0.22 1 1 i 1 — l.il -0.22 i + 0.008 +0.001 + o(Ki» — (1.001 — .013 u 1 4,-5 - 2I». -1.(13+ 2.53 ^1.7 fO..'. + 3.0 -1.7 — (I.1K>8 +0.(814 + (08 + 0(14 4-. 022 —.004 —4 — 17.4 1 +''-8 ' + 5.8 — 0.070 + 023 —Mill- +.(Ki(i —3 + (1.;U + I.34— O..'!!) —0.22 1.07 +0.:10 — 2. .'XI +0.08 -(0.(151 — O.OII (1 (I'ji —0 001 4-.(l4^ -i-.olO -i + 11.42- .'..37+10 57 f 5.04 +0.25 0.07 f 10.82 +4.117 I 1.8Vj) ( l (i5i;| l~2 1J71 l+o 1(77) — (147 — I — 1.72+0.32— 1.72 -0.32—0.112-1 '(.ua _ 1.74— o.iHi 1 1 1 1 1 ' 1 1 + 0.013 +0.(KC +0.013 1 — 0.C02 — 001 8,-e - 2.4 — 1.4o'+ 1.8 - 1.4 +0.B + 2.3 - 1.4 —0.00.11 +0.0(13 + 0(105 ' +0(103 + O12-.003 —5 — 12.(1 (1 1 U (1 -f-4.0 1+ 4.0 — (i.(i:m (I + 013 j —.012 -4 + 5.5h +1.(11- 0.22 -11.04 - 1.75+11.28- 1.117+0.24 + (i.o3(i - 0.(8(5 - (1 olll —0 (Nil — 1(0 — (128 -3 + (1.25— 3. hU+ 7.4.1 f3 5l+0.30— (i.('ll+ 7.73+3.45 +('.1('7 + 0.0(17 +0 l:ii —0 o51( — (i;*4. i,(il -2 — l.li(i+0.33— 1. (id —0.33— 0.(13 +0.01 — i.iiU— 0.32 III i +0.024 +0.wi +0.(iJ6 — (104 + 018 (1 (J,-6 — 8.4 : n ' n +2.8 + 2.80 - 0.022 +0(Ml7 -did .(-.(812 — S + 4.(13 +0.7(1 — 0.03 4-0.04 - 1.4(1 +0.24— 1.40 +0.28 +0 011) —(1.(8)3 — (Kill - 0(01 — (128 — .(H18 —» + 4.(13 - •,;.71( + 5.15 +2.45 +0.31 —0. lo + 5.4il +2 35 +o.o3t + 0.(122 +0 045 —0 (111) — Old —3 — I.4J +0.31 — 1.45 —0.31 —0.04 -(-0.02 — 1.41) -0.21i l+O.llHi) [ + 0.(141] l + " IW] I— 0.038 J + .015^ 7,-7 — S.(! 1 1 +1.1) '+ i.n -0.012 +0.(^4 _.(Kl6+ dOl — tf 4- .•;.ii:l -f (i.iii —1.17 +0.2(1— 1.17+o.-'(i 40.('12 - (1.002 — O 001 —0 (ml _ (108 — ,(KI4 -8 + 2.411—2.(10+ 3.52 + 1.08 +(1.30 —(i.lo-(- 3.82+1. 5« + 0.013 + (i.on 4'i.i2ii — (Hih + .014 -4 - 1.111 +0.27— 1.1'J -0.27 -0.04 +0.02— 1.23.-0.25 -0.021 -(1.(05 —0.022 + 0.004 + .(8)2, 1 8,-8 — 4.0 ! +1.3 ' !f 13 -0.(818 +0.003 — (Ki3 +.002 —7 + 2.711+0 50 —d.ltl +0.1(1 - O.lil +0.1(1 + 0.007 —0.002 + (1.1K12 _,(Ki.i— .ooa — « + 1.411-1.41+ 2.3k +1.14 +0.27 -o.O!" 1- 2.(15 +1.05 + (l.l«lli + (l.(Nl>i + 0.011 — 0(104 + (814 —5 — 0.1(7 +0.22— (i.a7 — 0.22 +0.02— 0.U7— 0.20 1 , . 1 1 -0.(818 -0.(8)2 —0.(8(8 +0 (Ki2 + (8)2 1 b,fular Utriii, M =4- I "25 7' H = - 1 ■ -i:, r .-.HI T Tlio terms iV.- niul fir;, wlii( h iirc inclosed in linieKets, are of very Ion;; period, niul arc tliiTeCore omitted in I'urminy tiie valius of yj in tiie lii>t two (ohimns «9 THE ORBIT OP URANUS. Ihrlu)/ia!ioit8 praluccd by Jupiter. The scries in which these pertmhations are expressed converge so rapidly that I deem it unnecessary to present tiie details of the computation. Thoy have been computed by both methods, and the separate and independent results are given in the following table, where ht\ rcj.. scnts the perturbations computed by the method developed in Chapter I, and it „ those computed by the method of varia.jn of elements. The apparently large discrepancy between the coefficients multiplied by the time arises from the circumstances that in the form of development the mean motion, and hence the mean anomalj ^j)pears affected by the perturbation fiV'.Ql. Accord- ingly when we enter the table which gives the true longitude in terms of the mean anonudy in the form r = ? 4- 2c sin (/ — «)-{- etc., we may consider this quantity 31".2< as a secular variation of / — n produc'ar in v the term it)=^G2'.4c^cos(/ — 7t). In rr, this term is left in its primitive form, while in h^^ t''f" value of/ is supposed to include this term, and the secular terms are only those which arise from the secular variation of the eccentricity and perihelion. It is also to be remark«'d tliat the terms which are inde])endent of the mean longitude of Jupiter, or thos(> in wliich /' = 0, are not comparable, as they corre- spond to slightly diilurent < lliptic elements ia tltu tuu theories. th THE ORBIT OF URANUS. 63 I'EUTIRUATIONS or UllASLS BY JlI'ITKR. 1 »y, t f. Uiff. C08 ^tf 3l»p 1 9 f »iii COS sin COS sin 1 COS coa sia cuu sin 0, 1 2 a — 0. IC.Od/ — 0.010/1/ +ai.2ii(;< +a7.5S5n/ + 2.207/1/ + 0.135/1/ — 0.H;22/t/ — 0.00!t5/i/ — O.OOOti/i/ +31.1ftH2/ — 1.540(;/i/ — O.OSOO/i/ — 0.0054/1/ a 0, 2 a + 25.«57 + I.a!l7 + 0.073 — i.8.59 — l.aVr, _ o.o«7 — o.oao — 0.005 + i.3(;i + 0.080 ... — lOOS!) —401.5 — 10 —4387. — 213.7 — 14.2 — 0.8 — 0.8 + 0.5 j — 32.8+ 1.1 — 1.8| —1,-1 1 2 a + 0.027 + 1.2(!9 — 5a.oc.' — a.4;t'i — 0.14^ + 0.00!) + 0.017 + 0.002 + 1.232 . O.OOJ — 53.0S4 .- 0.0112 — a.5tl5 — 0.047 — 0.104 1 + 0.011 + 0.001 — 0.002 — O.OHJ — 0.050 10 37 20 70 in 2 1 2 10 a 0.0— 0.5 _ 5«.H_ 0.5 + 25H5.7 + 0.1 + l!)5.a+ 5.0 + n.\+ 2.3 0.0 — 25.0 + 1127.2 + 82. s + 7.4 — 0.2 — 0.2 + 2.2 + 1.0 0.-2 1 SI a 4 — 0.027 + 1.1S2 + 0.277 + 0.074 + 0.015 — 0.011 —0.031 + 0.515 + 1.170 + O.oao + o.2r,a — 0.005 + 0.0X3 — 0.003 + 0.014 — 0.014 + 0.515 + 0.037 — O.OOS — O.ooa 4 I'i a 1 3 + 1.4— 0.0 — 50.5 +24. !• + 8.5 + l.H -j- 4.2 + 0.5 + 0.0 + 0.2 + O.f. -24.0 + 3.7 + 1.8 + 0.4 — 0.3 + 10.7 + l.C, + 0.8 + 0.2 1,— a 2 .1 4 5 — ()a2 — 0.005 + 0.025 + 0.01 1 + o.ooa _ 0.034 — 0.025 _ 0.005 + 0.015 — 0.001 + 0.037 — 0.017 — 0.025 — 0.001 — 0.010 + 0.004 7 10 2ti 20 !) 1 n * I'KIITIRII.^TKINS OP TIIK LaTITI l)K. 1 U I' ih »1 tfl t l' 2 a —2—1 — 1 1 2 — ;v-2 2 Hill '/ 1 0.071 —0 012 —O.ooa —O.OOfi — 0.2!t7 —0.1 CI \2.i\\\ 4 0.070 4 0.055 + 0.002 —0 no — 0.014 COS '/ —0.011 —0.075 — (».0I2 40,012 4 l.!»l!) — 0. 101 + tMI21 —0.088 — It.Oll -U.O.'.O COS // 40.030 — 0.012 — (».003 40.000 4(».2!i7 — O.IS'.t 4 2. "128 -j o.oco —0.055 —0.002 —0.10!) —0.011 Kill '/ 4 o.oac. ) 0.075 40.012 4 0.012 -1 1 '.no — 0.1 !I0 —0.002 _0.0><8 —0.011 1 O.OJfi C(I8 // -1 .oi;o — .010 —.002 -( .005 4.001 — .4'.14 — .021 + .002 4 .010 ■\ .017 — .00!t —.002 sill // -4 .or.5 4.047 —.007 —.004 —.003 4 .420 — .013 4.019 —.003 —.004 ( .002 Sr.uhir l.riiis " *'■ \\\Z ««,) — 1.14 1 cos V 64 TUE OKUIT OF UUANUS. CIIAPTKll IV. TERMS OF THE SECOND OltDKll I'UOUUCKl) IJV THE ACTION OF SATURN. PrcVr.uimiry Inrrstiijalhm nf the Orhil of Sulnrn, Tou tlir" arourntr (letcnnination of the pcvtiuhatioiis of a planet it is essential tlia'c tlie fiinctior.s of the time which are suhstitiitcil for the co-ordinates of each ]>la u't in tlie expression of the clisturhing forces shoultl approximately represent the true places of the planet. The tlilferencc! between the true place anis possible free from doubt, we begin with a j)rc- liminary investigation of the orbit of Saturn, the design of which will be to give the co-ordinates of that body in terms of the time with sufficient certainty and accuracy to serve for compiiting the perturbations both of Jupiter and I'ranus, As usual, the first ste]) in thi* investigation will be the determinations of the per- turbations of the planet. •>» M 9 THE ORBIT OP URANUS. w Oeneral Peiinrhationa of Saturn. TIio perturbations produced by Jupiter will be taken from the exhaustive prize memoir of llunsen.* As the perturbations required are those of tlie co-ordinates, it will be necessary to transform those of Hansen into tlie usual form. Hansen gives the true anomaly v in the form V = tj -\- nh -\- «?i sin ( I'EKTtaOATIU.NS Uf TUK LoNOITUne IN ORBIT AND TUK LuUAUlTIlM UlT THK RAIllLS VECTdll or Satlr.n. Ac ion of Ursi III*. I f »»'3 j 1 -f e. cos (i/ + J ii'z) -f •ic, cos 2 (i/ + J nh) j . ,«r=1.021(> n'z Pfttn Or. M ■uii Nuoii. 17 Jl May :n n-)7 Aug. 7 17.VS Aug. 27 17(51 Oct. G 17();l Nov. 1 17(!.> Nov. 2:{ 177:$ Feb. 2() 1780 May 24 17!)4 Nov. k; 180'> Feb. 2:{ is-.»;j Nov. l:{ I8;ii F.'b. 18 18:}8 May 1!) 184.> Aug. 17 l8.-,i Nov. l.» 1S()() F.b. 14 18G7 May l.j nd Fiu'(or. m — 1792.7 — 20:i*<.2 -215;j.2 — 2(»(54.5 -:no7.l — ;{r)()4.() — .'ir).)7.2 —2(540.7 -;i7:n.!) — :M2').r> — ;J0:J(5.8 -;5(571.7 -2777.9 —2220.9 — :{l."»9.:l — ;j4(58.;i — 2227.G The perturbations by Uranus and Neptiuie were computetl from the values of their terms just given. The principal terms, the sum of which make up tlie helio- centric longitude resulting from the adopted elements, are shown in the first of the following tables. In the next table we have after the date the heliocentric longitude from Bouvard's Tables, as deduced from the longitudes given in Airy's reductions of the tireeu- wich Observations, from the Anhonomi'm/ioi Ju/nlnwh for 18;ll, and from the Nautical Almanac. Then follow the corrections, roiighly deduced from ob.serva- tions made near the opposition. Adding these columns, wc have the longitude 1947.7 —.0990 21;J4.2 — .()Go2 2212.') — .0474 2"»4(5.-> +.0244 2880.;} -f.0729 ;j()9").l -{-.1(>82 ;j;J42.o + .0419 28.»S.2 — .09->G :W21.I + .1017 :51S4.7 +.0529 27l(5.;J + .0944 ;w78.-i + .0(5:59 297(5.7 — .08(5(5 2:J42.7 — .0721 2847.0 + .08(5;$ :}l(il.4 + .0740 2;n:5.i — .081*i ,. Tin: ouiJiT OK uiiAXua. •7 from ubserviitiuu. To the ri^^'lit of thoHu uro tlie equations of condition for llic correction of tiic dcnu'iits. IWlurbttioim lijr I.OIIK. of I'lirilivUou. M.'i>n ■ijiiiualjr. Ei|nali 41.2 2311 13 7.1 —5 7 38. 1 —33 58,2 — 3(l.(i + 1.0 — 1 20.5 —11.8 310 Hi 4.1 H.S 47 :il.4 248 25 53.1 —ft 48 30.4 —35 53.2 1 -35.2 + 17 — 1 35.8 —15.0 330 4(1 30.(1 8S 50 7.« 28(1 211 20.8 — ti Itl 1.3 —44 24.(1 - 3.3 + 2.0 — 43.5 — 14.1 8 15 13.5 8H 51 51 « 311 44 13. ti —5 5S.2 —52 37.1 f 11.0 + 2.4 + 43.3 — 4.9 34 43 22.0 Hs5:t;j.j.l :i;if, jV) aii.i _2 41 lo.ii — i58 24.0 , f 10.2 + 1.0 + 1 3tl.(i + (1.7 tl2 12 1.0 8H 5fl :«'.».« (15 40 2.3 -l-t! 35.0 -50 17.2 - -32.0 —0.8 — 1 37.1 + 4.8 150 38 54.0 8!> 5 4:i.4 154 8 4.8 + 2 37 57.7 —44 0.7 i- -53.8 —1.7 + 1 37.5 —14.5 245 8 13.2 8!» 17 50. 7 331 11 10.4 —3 18 5.2 — tI2 17.0 + 0.0 + 3.5 + 1 30.0 — 14.1 50 5 7.9 89 2:» 55.7 50 5tl 17.0 +5 44 4tl.fi _57 5.5 — l8,tl + 2.5 — 1 37.3 + 1.4 154 2.7 8!) 42 7.0 325 22 2tl.O —3 51 58.0 —50 3(1.8 -27.0 —3.0 1-1 21.(1 + 14.7 50 23 3.0 8!» 4H 11. « 54 10 33.1 +5 25 2.(1 —til 11.7 +13.3 +2.5 — 1 33.2 — 7.4 148 21 11.1 8!»54 ir.i 142 44 37.0 -t-3 40 38 1 —4(1 17.0 — 3.4 + 2.1 + 1 20.7 — 4.3 235 34 37.4 90 20.4 231 18 40.0 —4 47 3ti.tl —37 0.0 - -35.5 +0.0 11.8 +14.8 315 52 52.2 no •> 24. ti 310 52 44.8 _4 21 53.0 —52 30.3 + 4.7 —0.0 + 1 8.8 —18.4 44 45 31.3 00 1 2 2H. 7 48 2ii 48.7 +5 1 4tl.4 —57 48,3 f30.3 —3.4 — 1 25.1 +14.1 142 42 31.4 •JO 18 32.8 137 52.« +4 8 31.2 I —37 7.(1 - 1 — 8.2 + 1.3 + 1 21.(1 — 4.1 230 52 50.7 Dat«. Tal.iilir hiiigitU'la. Otis. cor. I.OI1H. from ubixtrvntiuu. KquATioNs or Condition. 1 1751 Mav 31 250 13 38,0 II — 8.5 250 13 20.5 (!3.7 = +0.fl()«, —44*11 +0.nO««' +1.82»'«u 1757 AiiJt, 27 310 17 8.5 — is (1 310 It! 50.5 4(1.4 0.0 4 —30 —1.53 -fl.l4 1758 Aiitf. 27 3.10 47 37.0 — 11.3 330 47 23. ti 47.0 O.Oti _40 — 1.7t! +0.80 17<11 Ocl. C. 8 15 13. S + 0.2 8 15 14.0 0.5 1.03 —30 —1.00 -0.50* 17t;3 No?. 1 34 4.! 42.3 -f-15.3 34 43 57.(1 35.0 l.tl7 —30 —1.(13 —1.31 17(15 X(»v. 2.1 tI2 12 2t».7 + 211. 1 (12 12 41 1 40.1 1.11 — ;iH —0.88 — 1.88 1773 Ffl>. 211 15 It 40 3.8 + IS.S 150 4(t 22.(1 HS.(( I.(l( —28 +1.03 —0.70 1780 .M:iy 24 245 17.tl + T.8 245 25.4 72.2 OOtt — I.s +0,7(1 +1,74 I704 Nov. Hi 5ll 5 38.8 fl4.7 5tl 5 53.5 45. tl 1.10 — ti —1.08 —1,78 lsM2 Ki'l). 23 154 7 2.7 i Kt.s 154 7 13.5 70.8 l,tl5 ^, 2 f 1,85 —0.08 H-23 Xov. 13 50 23 31.5 + 21.3 5tl 23 55. M .52.2 l.tiO + 2ti —1.27 — l.titi 1S31 K.l). is US 22 3S.5 f- 3.tl MS 22 41.5 Otl.4 l.tl7 t 33 +1.75 — l.ltl 1S38 .MttV lit 235 WW 11.2 + 3.4 235 .1(1 14.t; O; a 0.01 } .35 1 1.08 -1 l,5tl 1845 Aii'tf. 17 315 53 42.2 fl5.7 315 53 57.0 (15.7 0.03 + 42 —1,43 +1,28 1852 Nov. 15 44 411 15.4 + 5.0 44 4(1 21.3 50. t) 1.00 1 57 —1,42 —1,53 IXCO Kcl). 14 142 H 111.2 — 13.7 1 12 44 2.5 01.1 i.tts 1 (;5 t I,tl3 —1,35 18)17 Miiy 15 230 54 17 s -1 r,.l 1 2.10 54 52.0 113.2 +0.02 + (12 ^ 1,2,3 ( 1,45 A normal equotion for iV is ohtiiinod hy tnkiii}' tlic sum of all the rqnntions. Tliiit for hi is formed by 8ul)triirtinj; the sum of the first seviMi from the sum of the last seven, and those for F U KAN US. W fiv = 2^e sin « Hill » " tt II S + 2.'). 2 f 21.4 — 2.7 + 0.0 .S + l.ti4< - 1.4H< — 0.H0< — 0.74/ S — J — 28.8 — 34.4 CS — 2./ + 1J3.0 — !l3.f. — M.\ + f)fi.!» rr- s -f H,« f 0.3 + 3.0 _ 0.1 2ir— s + 0.8 _ s.s + 0.1 + 1.4 ■2U — 2S — I.J. 4 f 0.2 — 8.0 + 0.1 :iU— S — «.4 — 27.0 + O.l — 0.7 3U — 2S -1- 1«.3 — 14.« + 8.8 + 7.0 I'EIITL'RBATIONS OF I'llANllS. 1 Argument. t u 1 sin coa COS (in It It n 11 U — n.iu — 0.2f« 4 0.05< — 0.14< U— 8 — 2(I.S + s..--, +3tl.O — 4.0 217— 8 — ll.!» + 14:i.,5 — 2.0 — fi3.8 •.U7— 8 + 411.3 + 11.^.'.) — 4.7 + 13. a 2(7-28 + 4.1 0.0 + 4.2 + 0,1 a (7 — 28 + 2.1 — l.B + l.S + 1.4 4(1 — 28 f 0.6 — 1.8 -\- 0.4 + 1.3 riU — 28 + 1.2 -f ;t.o f 0.4 — i.a Let us now resume the equation '^ ih H i ra 0^> \,^ 70 Til K OllHIT OF UllA NUS. Brginiiinp witli thr last two trrins of tliin cxprissioii, it miiy be sliown at the outset that thry are- (initc iimciiNilih', The vtivvt of llie constant terms in fip and ip' iias alr.Muiy l)een inehuled by correeting the lofjaritbm of the mean distanee by their amount; they are therefore omitted. Tlie Iar<,'e.st remaiiiinf» term is ti4', the square of which k only ()".()•.'. la the proihict ri',Y tiic hirjjest terms urc + 0.014 — O.OlDsinf/ — 0.01 1 sin (% — 20 — 0.011 voH(4(j — 'H) which may be entirely non by the equations (11), with the modifications shown on pages 24 to 27. The derivatives of Jf„ which enter into these e(iuations are formed as follows: If, in the value of Ji produced by the action of Saturn on Uranus, wc consider any term of the form in'h -- cos iV (I, where the accented (piantities always r( -V "1 -'f (/' + /) sin AT "I m' fh „ cos ^V iiiin^ the terms having the saiiie ar;{umeiit, we fiiul tiio foUowiiiji values, omitting those given in Chapter II, and tliose which arc derived from the others by mere addition. The terms of ' ., are also omit- fVV ir-n ted, because they are sensibly the same with those of \ , ,, changing the algc- ' ' t/Vc/V braic sign. 0, dlt 0| rru a, «9'« 1 m' t*v' wi' c^v^v' «•' (5('(9v' 1 9 I' sin CIIH sin COB Hill COS 0,-1 —2 +0.2^*74 -fo.oonii —0.0310 + 0.0310 + 0.2x72 +0.00(i7 +0.2(101 +0.00(10 —0.0322 1, '2 — ;j —0.0103 — :i.4Mii — O.OiK'.M + 0.03S8 —0.. 11 —O.lit.lO +0.4.i04 +0.0173 + 0.0100 —0.0021 —0.4304 —0.0179 —0.0101 — 3.4S10 —0.1039 +0.03N9 +0.0249 _5.2(1S4 —0.0703 +0.0308 +0.1104 + 0.45sn +0.01(13 —3 —3 —0.001.') —0.0.544 +0.4104 —0.0053 —0.0035 —0.0713 + 0.0120 —0.01 111 + 0.0021 + 0. 142(; —0.0171 -f 0.0300 —O.OOHO —O.OCSO + 0.S233 — 0.022(» + 0.01,55 —0.17(10 — 1,(515 +0(I2HH + 0. 00f.fi +0.2403 +0.0014 +0.0547 3,_I —3 —3 —0.0070 +0.0670 +0.2542 —0.0003 — o.o(;5o +0.0171 4 0.lo,'-.7 —0.00(15 —0.0210 -t 0.1013 + 0.7(124 +0(1107 —0.1701 — 1.0S84 + 0.0304 -I-0.2S40 —0.0071 4,-2 —3 —4 — O.OOH +0.0515 + 0.1448 —0.01.10 — O.Oi'.tO —0.0015 +o.o;tfis + 0.1 0H7 — O.OOKi —0.0103 + 0.14.50 +O.57K0 + 0.0213 —0.2144 —0.7(144 +0.0547 +0.2(147 —0.0005 78 TIIK OUWIT OF UK ANUS. «i oJi n. ffit «. irn 1 0 —0.17(1 40.070 — o.col 40.888 —0,227 40.011 -j-o.aia —0.07 ft -i B.I8» 40.100 2,-1 —a 40.084 40.018 -( 0.248 40.472 — 0.2:10 —0.04; 40.100 40.013 — o,:i22 —0.00(1 40.020 — l.OS.'i •'1,-1 —a 40.013 4 o.oTil 40.oo< 4o.O(>8 40.278 40.01;' —0.122 — o.n;i4 40.n'Jl -1 0.21 ft —0.007 -0.049 —0.344 4 0.010 40.028 —0.244 —1.2(10 4,-2 ' 40 oir) 40.001 —0 001 -l-O.li.ft 40.020 - 111 'J 40.042 —0.071 40.001 4 0.01!) — 0.8f)4 Till! iloiivations witli n'spcct to y iiixl tl)(> node \\\\\v hrcu omitted 'ircniiso tlicy lire (jiiite iK^f'nsihle. Tin- terms of hii (lepeniliii;; on tlies<' deriviitives iire f,'iveii liy equation (;J1>. In tlie ease of rramis diMturlu!! by Saturn the largest values of the coelficient» 4,/Moti^; i/.tau Jy, J '^^^■ are only ahotit S)h, .vliile the larjjest roefficieiits in hh, hk\ and }y are less than 10". l^''ne(> the lai'p>st terms in (31) \viil lie > tlu order of niiigiiitiide {)".h miiltiplied •> ■ the mass of Saturn, and may therefore lie omitted entirely. Omitting them the values of hit, h . »niJ ^ ., become w tip iltr^ 6R ^r -f rn h' 4- . ho -4 PR . , f R , , , o'R i'u , , f It . , . in J . dv 'R (Vt'V trR R r-R {R ,V' = . ^W 4- , ,, hv' -\- . . hi) -\ . . ,ho' » All the separate faetors frmii which the second members of lhes( e(|natiniH are formed have alreiuiy been given, lorming their |uodiu ts in tiie way described in Chapter 11, we have the res ilt f: ven in tlie following tallies. The expressions for hR are a;r;ii ged so that the vi.Iiie of P',hR can be olitained (rom them by direct diflereiitiat'on. This is done by disfingiiisliing the time inti'odiieed into R by the co-ord nates of I'ranus from that iiitroilticeil by the co-orilinates of Saturn. TUE OUBIT OF URANUS. 78 2", (fV/ ...•+^»vj u s COS tf U S J II If '. — I.40< — i.3a< + 1. 3—3 -f 37.5 -f7a.3« 2. -1 0.0(!< — 0.2W + 2. -fa, — '.»!I0 — 73 — .'>.^S — 2<;i 0.— 1 — 0,24< + 0.20< +<. + 23 — 42 1 '— ' -f (ta/ -f 0.04< — l.O.V - - iCrf 0, ii — 1;47 — aia + 0.02/ — 0.30< 'i 4 77a7 -\-:,i;\i; «. -f i.'.a — loa 0,-3 + o.n.M — 0.8S» ». + 24 — 211 1,— J — 12.<;'.« f ll.tiO/ 4, 2,— J — 0.2» -f 0.10/ a, -2 — » tilW — 0.5'J/ -8. -i. 6—2 f 00 — (',2.'i + o 1 7 ail l.-'J — I.(i0/ — 1.14/ ,1 + <<1 — 7f; 2,— :» -f 0, •.':.< — (t 211/ ( + •'«, + 24 — (iO f 2.14S —2, :— i — 1. f- 42 f iiias — ia« — M!t f 2 — m — MS f 12 — SO — n,_ t 1, 2, -Ifl — 2.'.0 _ 4 ' \ I!t7 n -no — 4.'.0 — 2.H.'.9 — iNsa — o.s.'.a 2.' a, 1 -1 as I us —a, a— 1 —i'. -f n f till H7S — « — 1.- 0, 1, -M2 1 1 — H — 1' — HO 1 f in f 134 + .13 — 3.H52 —2 sr.2 — 1 S.'.2 0, 1,- -1 1 - no -f 12 — I.K.'.2 —a, 4-1 •> 4- n -221 ( 2S — h a,' 1 1 « 1 -f 'i ! 1 -t l.'lS — 32 — O.S.'.iJ < 148 1 + 2'.t7 -2H« 1 111 10 MV. IbTt. 74 TIIK ORBIT OF U KAN US. U S ». I, (I ■I (I I, 1, I. -a i. ■'t, 4, .■1,— 4 i, 0, 1, -1,-1 0. I •i. 3. it, ;«. 4, f.,— < .'l.— 3 4, I 2 "; *//,« + 1.40/ — 0.1 2* G — 0.0.K I '.'.10/ — 0.0(1« n 4 12 nw ■f- 4H< + l.SiM 4 l.no( — O.fiO* -- tt'iU ) l.4t( — 57/ — o.r,o< — 45 I DiH ( a j I4f) t « —I an — Kl — 4 - MH t lOH I UO — 5a _ I :i.l/ — 0..'.(1/ f 0.04/ _ i.ao/ — 0.510/ I II no/ -i O.tiO/ — 1 77/ — 1 u/ — 0.5 N/ f 0.011/ — I.:ii'>/ — 0.51/ +.119 f :iM» -f 5 — (11 —.•ml — 70 M t ;w — J.Vl — 7 ^ISI 4517 4 4 —S3 —Ian 8 -,,''« in ^v -I \.m — 0.27/ — 0.04/ — 0.00/ } 2.10/ — O.I'M — 0.91/ i l2.no/ ♦ 21/ 4 I.MSM 4 l.(,U — 0.53/ — 0.10/ 4- 1.43/ — O.'i.'./ 0.52/ 4 9 4ll>» I TO It! — 4 4 •■« 4 I H — 14 t 145 — <;o f HI — IH 490« -♦ M2 — 49 — 42 _ l.l.'i/ — o.4;i/ — 0.0/1/ 4 O.OIM — l.Xil/ — 7.« — o.h:u ) 11.02/ 4 0:17/ — 1 75/ — 1.11/ — n.!M 4 0.07/ — I.:i2/ — 5:1/ 4 0.10/ -♦- 5 -( IllO f41'.» — 71 — •,'47 — :I2 — 12 — I 4 4 — no — 13 ) 7'.m ( a53 t H f Ifl — (-,7 -lao — •,114 .-loa 4 0.14/ 4 3.64/ — o.a4/ 4 1 00/ I 07/ f a. 5 4/ — 0.19/ 4 0.84/ — 13.52/ _ 42/ 4 2.7W — 1 «»4/ — m;u — 0.2.5/ I 1.93/ — O.T4« — 0.7 a/ —193 — 49 t 27S — lit I 103 I I — 10 — I1» - 10 I H4 ( M -202 .- 45 t 17 _ .15 — '.12 4 8.21/ -\ 0.110/ — 1.02/ — 12/ I 3.22/ I I Ix/ - 94/ 12.34/ U. I 1/ 2.52/ ■ I 21/ 7.V - 10/ 1.70/ ( Ml — I) 10/ 110 12 ■ 111 -709 5111 ) l-.'5 ( 347 •( <7 — 215 4 3M — 9 4251 4 7 t Nlft f«17 t 2 -» 31 4 41 4 H 4 .''2 4 Us \- J T II K ORBIT 0¥ U K A N U S. 76 f a "/ hl),lt ij «. a';^" 9«. *'?^ m n m t;» m e/p U S J hill It CUM Hill COM 1* Hill n 0, 1 — 1 -ih:i f 8 — H4 .\- 13 1, 1 •.'ir.'.i — 60 fJ477 — 50 — 2'.I47 — ll'J •.', I •» -f- H — 7 + la — I7ii — 17 •'«. — y h l-^ — 11 ) ill — 17 — 20 —2, a— 1 ♦- H4 4 aj f- ii + J!) + 9 + 49 ~i, f mas + »H -t li;4:l + HS i I2ai — ua II, — 114 __ i t 41 -r II 1 1. \- Hit — H!» f sa — Hit — I'til — 112 -i'i, — 4 -1- « — 1 ( » — 11 — 18 —a, a— 1 1 H ( '.J» 1 -••■' 1 2S — 42 } 41 —•J, 1 i.:j - 4s t 1 4a _. ri4 — I'.M — 104 —1, J-47M ^2H0 f477 + 'J7;> + i;.2;i —2711 0, (1 - .-.7 — « — no f 44 —a 4—1 f ^'I _ H4 •f 1> ~ !l| — 't') — ia4 -J — 4IJ } 10 ~-44'.» 1 7 i 7711 — « —I! f i'.iT t -JHIl f aiu ♦ 270 faiii — 2.".8 1, .1-3 - airi f 7 tail — a!ia + 7a4a f r,a!> -+7722 •1, ( l'.tst» — iiiii -\ 1!W7 — 1271 4aao4 «2:>44 •t, t ■iV\ 7sa f ''1 —1172 ( i.-.l f 1011 4, — Hi -HIS — 101 -102 — laii f 1J8 "•. \~i (» fliiil) — »sr, — r.ol — 9i;rt 1, — 77:n ^'•r.lii -77aa t .'Hiort \ 121 an -f HIl'.IH a. —aim — -.'it'i -aa:. —iun ♦ 27a { !I10 a, — 7i — 7H + /)« — iti — (!0 -flO'J —2, .')— a l-lsi) — iJii ♦ i:i',i f f, — aa7 — 37 —1, _i;j.'i — 7a;i — 11 JO —710 — 1222 — 1217 II, II { Hi — r.ii ) ln7 — no 1 1. — r.l t7l'.t — iia + 72.-. + 72 f 1 1 ii'.i )•-'. t 4" t IJ — la — tl — 17 t !i7 —1, f,_j f no .1 ♦ 1.". 11 I mi; I 1 a2 ;» — 127 .'• ) ail 4 —a, tJx-.'a — 7a i f U4S,H — I4.'> —4 la (1 — 7:. 7 .» (lay — isii ♦ ia7 -,'..'.7 — 11 11 — 14111 ~'\. -K'Hja 1 1,111 — WNJ'.I + 11121 — I2'.>7a — mi-.'ii », II 1 7 HI — »!ia t77l 1 IIMI --4, 7 --J ) •-•"(I ♦ * t 'nn + 4-'. — 2it; 1 4M —a, I S(|7 —441 t 7J7 — (ia2 — 112S —977 -J, 1 IX7I - 1 1 IJ \ I'll a -117 4 — a2art — 2111.'. —1, — I'.as — 1117 lll!l.'> — |ltH7 - 207! +'.101 0, 11 II 1 lait 1 (Ml f H4 — 71 oAa 70 TIIK Oil HIT OV UllA NHS. Fii the trrnis of hit intrtMliiccd liy ilu; pcrliirlmtions of Sahim, niimcly, ^ , it/ -^- , Aj)', tlic tlifri'rcutiatioii r<'|ircH«'nt(>*, Klumltl he |M>i'toriii(>(l liy con- RidcriiiK At/ iiiul fif) as t-oiistant, altlit)U(r1i tlicy arc <-\|ir(>NS('(l art n fiiiu-tioii of tlio iiu'un luii^ittiilc of l!raiiiiM, as wi-ll uh o( Saturn. 'I'lic mean loii;;itiiiii' of I'rantw thus iutrodui'cii is tliinfori' rcprcMcutcd hy /", which ia regarded uh constant iu' taking //,/?, and //^only supiMwd to vary. Again, iu the teriiis " ' jf4- ^"•••''kc I'r and A/i reiireHent perturhations of t^'v ' 5p ' ' Iratnis, their roinplete derivatives, with resjXM t to the time, i:r<> to he taiveu. Ihit their exprexiiiuDs eontuin tlie mean lon^'itiuh- of Saturn us well as rrunus. 'llu? mean hmgitudc! of Saturn thiiK introdiieeii is representiMl hy .S*. and iK to he eon- fiidered varialde in ohtaining iy,hll, while S is eonsidi red eonstaat. The ratio of the roeffieient of t to it in tln^ varioUH terms of thin part, of ftit is given to the riglit of eaeli rorres|N)tiding ti rni. 'I'lie vahie of //,A/i' iM'ing onee ohtained, there is no hiUfjiT any disliiutinii necessary hetween /', V\ or l>etween iS' and .S". 'Ihe himdar terms are liu-refore eomltined hy jtiilfing .S" .V; /" r. Ironi llie ul»<»ve vahies of 'J:!)J)) U and 'i,S ' , we hirni the following value of 'V "' H'- '^" J' A/>'. /•'/'+ "' ^V~ and of the other fjuantities whiili enter th( |»er!urhatit»us «d" the ro-ordinates. We liliall l>e;;in with thos<> lernis wliirli depetid mdy on th<- nnihial action of Saturn and I'ranns, lieeansc tiie) are (i w and small, and tiu' only terms whit h are sl■n^illle are those in whiih the rocHMi nt of tin- mean liinj,'itiiile of Saturn is I. \\e shall therefore (online oursi'hes to these. And, instead of enntlii)inj,' the eoii- iliiised furmnhe, Wf shall make the en;!i|)ntation in fnll liy (l-l). ".^Q II {AV "',-.i' f 4'.l-f (1 KVM 1 :i.i... ■O.dC'/ - S-f(Mll.' ~ L'l ) a.iei/ + (,-.'. -siic** 1- II f .' :;7' I |:is , I K-H S '-'It- -1 s,">/ -If.l i •-' llr •J • - fid 1 i -ni — 2s 1 :t.su \— ■>',[) \*\M t t.il :i (i.W ) -.'.'ill- (1 i.!( 4- K.'.— -J 711/ ■ :<•>:> ( ;il; ■ t ;.'|ti-_.i.v.i< 1 7 1 -i-WM :;i t •.'..■!0< — ;i(i ( •i 1 \t 1 14— 1 ti-J/ * —•iwx 1 a l,>< , IJii -' IU ~'X\\'X ) II l.'i/ --l-'(» ) "J 741 i> '■ r>is 1 Hill •» am 1 foi // n ' tt II " '" 1 0—1 { aoi- oiuHtii — o>i:i i imon-.v 1 a oii'j ) a (Midi)/ 1 \ (!'.> 1 ( iKIrtdf 1, t nil , »(Mi:i:(# — (CK; t """i:i:t> 1 M lilt I ii.aaiii,/ 1 (i.'is t a iiihMi/ o ♦ :i:;;i ( imuiti _ iw ( mi.iii 1 IMV.'a \ (>,(IO!f'l ( so:t_.(i (ian7i •'l, __ 1 ir>_ (itin(i-j< ♦ II71I Oi'ir."* f >.'(',!) (1 (Mil W 1 4 II s;tu— (1 iii.Mu 4, _. an i.iii'i |ti l;t'j 11/ 1, i> uta 1 II iMiiiii/ TlIK OUniT OF URANUS, IT The rompiitalitiM of tliiis ni' llic (■((••Hiciciiis, ilio nil (Ui'ii'iits of the tiiiK" ar<' easily oMaiunl liy n-il).stinititi;^ in tlic inlef;mt«'ti per- tiirltatiiMM tl»o vuriwtioui of tlie et;n iilrieity uml iKTiliclioa of Suturii. Thus I htive fuuiitl » n fv := I 0.i)IO;J / sin ('i,j - t) - ().()(»•>* t «os (-2;/ — /) 4-0,0027 I sin (>ij — I) — O.Oiaw t io» {.iy — /) Tlie jjfrefifi'Ht iliserepitury is AmiikI in i1m> coeffii ient of sin {i)ij — t), mnl it anionnts to (I ((();|^/, or iiliont U" I in a lentiiry, IJutj owinj^ to the ;;re,it |Mr;o(i of this term, nearly (iOO years, tliis dift'erenee, thirin^r imy one centurj, v ill lie nearly t'liniinateil llH-on<^'h ihe mean !(>U|U'iliiil< ami nnan tiiotion. It may aUo hi rei)iarki'< ruse the terms derived ftom the pertiir- bations of tlio elements are imdouhtidly thy correet ones, and will therefore ho employed. 'I'he fi>rins whiel, tin' pveci-diii^ iiite^nition fail-i to give, owin;^ to the constant t< mis iiUTodnc( d into ^ls(^ and yii\(^, an I'uund l>y {'22). We tiiiiH have w^; /),,;.;;' r- .x. (»■■.;«» «:;■/,./.•;'=- -fo. '27 r,».^p 1= I ,)[„!' jtr.HC sin ,j - i)".-'! ;iS((»s;/ f (r.il(l()(K».".)sinp<)i'.di remain the same as f;iven on pi;,'e ■'")(). /V, 'iirhiiiiiiiiM i/ijM'iniini/ oil III' jirinliit I nf f/i, imti- s nf Jnplhr mnl St4iirH. Ihe v;»iueiinl'<\/>^/i', <$ ' , ami A , il>'|N'ndiiiK <>h the proUtttt)* of the nuisM s 78 T UK O II U I T OF U U A X U S. of Jupiter and Sutiirn, arc f»ivon on pago 74 The coniptitatioii from tlicso data being coiulucU- c asc of tli<> terms of tlie tirst order, it is not necessary to give niueli more than the results. These are sii nvn in tlie fol- lowing table. The indiees to the left represent tliie <-oefM(!ients of '.lie mean longi- tudes of Uranus, Saturn, and Jupiter, all counted from the (n'rihelion of Uranus. Column I' gives the ratio of the mean motion of Uranub to the coefficient of tho time in each argument. The perturbations of the comnu)n logarithm of the radius vector arc expresM-d in units of the seventh place of decimals. V lu 0.4343 «i> U S J Hin VUH tl fi)» Hill 0, l,~l 1, 3. 4, — 0.23fi4 — 0.30i)f) —0.4480 —0.8127 + 0.002 —0.020 -f 004 + 0.01(1 — 0.0(1.1 — 0.021 u u —2, 2,-1 o! 1. — 0.2!tr,0 —0.4204 —0.7 2.') 4 — 2.(!420 + I.I10!I0 —0.007 — 0. lo.s — (111 +0.1 r,4 4-0 oo,> —.0 (Mil — .(I (107 —0,(112 —0,2117 -0.00,'. — 2 1 1 1 > u —3, 3,-1 -2. —I. — 0.(!.5.')1 — I.81llt7 +2.1115 -1-0 ()7H« fO.Ol2 1 17'> (-0,07s fo.007 — 0,002 — (i.dl.'l fO.:-.12 ) It. 021 1 1 — 2 t :i —J. 4,-1 —2. — 1, -f0.7.'>» -f-(),«.10 -1-0 301 .—0.0.10 f 0.013 1 o.ool 1 0.(I><1 1 0.00.') — 0.003 — 1 f 1 1, 3,-2 •> 4, —0.2170 —0.2771 — :{s:),i —It t.215 _o,o()2 —O.O.'.I —o.olo + 0.(t32 —0.0.12 1 (i,(i:;.'> ( O.'iO t o.o.".2 -• 1 — 1 — 1 - 1 - 1 0. 4,-2 1, ~0,3(;27 — 0..5(i!i2 — 1 3210 1-4 ll.iO —0 010 f o.(i7."> — :;i'.i — 0..M0 —0 001 — I.M — 1 2'.i7 — l,>;t — :> — ,1 '2 1 II — 2 — i! 0, 1. 2, — 0.f)250 — I.IO.M -)-10..'>ir.2 -f0.!1132 -1-0.4773 --0 2ri3 1.133 _o,.'>ii; f II 201! -f 0.012 —0 (131 — o,(;17 (Mi:(2 — ;i -j.ii —(I.I '.12 — 3 — 43 ♦ 2 — 2 ■i 7 — :i(l — 2 —4, 0,-2 —3, —2, — I. 0, — 0.9»'t7 — lH.!l2rtO -)-l.o.''(ris 4o..'>13(t -j-0.33!»3 + 1.824 -t 4O.C..'i0 -(-t;,2:i7 -(-o.4t.7 +(».0(i7 _. ;i 1 (1 -|O..Ml(l — 7.HI".(i _o ."i:!!! —0.017 + 10 ^32 —(13 II 1 *•• 1 l» — 7'.l — 2 -4. 7,-9 —2! — 1. —0, +OMM -|-0.3r)73 -f 2i;32 -f-0. 2(tH4 +0.1724 — tl O.'iO — ii.oti; — 0I'> +0,000 — o.oo;i ) (t (132 1 (i:i2 1 (107 f 1 ( 1 ♦ 1 t 1 ■1 1 ^ TUK OHBIT OF UKANUS. 79 ClIAPTKIl V. COLLKCTION AND TRANSI'OKMATION OF TIIK I'KECKDIXO PJIRTURIJATIOXS OK U RAN US. The tcrniH of tlu.' iicrtitrlmttonM whirh neither contain the clnmrnts of tlin disturldn^ iiIaiictN, nor r, f .11 -illtK Aflioii of Sutiirii, -|-l«) '.WMtl Ai-liuii i>r N<'(ituiic, — M.4Ui'i'ii T..I11I, H ''■''<" 4-j:>.(;ri7 sin 7 4- 1 .;i!)7 nin 2.7 • f «..'i4.'>Hin ^ -fit -Itll hin i!/ -f OCnxin ij -fO.IMf, din iij -)-:i(.H'.)',)nin O'lS ij — 0. Id'.t I OK 'iij ('2) In the value of cos i^p, units of 7th place of decimals. A<-iii>n or Jiip'ti^r, — liKlso — I'.ii con // — :!:i I'n.f 2;/ — 2Kiii(; + I Kin 27 Ariii>n (>r .Siitnrii, — :t.'i4:i — H-l cos j; — l.'>cid(>r tiio first or coiisfmit term in the perturlmtion of ea«h ro-onlinatc. If we sM|i|tosc a change of hi in the mean nio'ion of a |ilauet, the eorrespiindiiig change in i^p will Ix; ip: If, tlieti, we iiicreusi- the mean motion of Iranus Ity 41".1."»1, the cnrrespcmdiiig rliange in Ap will he iHOt.j, imd in cos >^ip, — IHO'Ji'i. Subtracting these from th«' above jK'turb'itioiis, the serular term in the mean motion will ilisaiipear, and we shall hau- for the c(in>r:tMt tiTiu of cos i^Ap j l.-):U This same change in tlie mean motion will produce a sr^nlar term in the ('(pia- tion of the centre of the samf nature witli that prodnceti by the s<'cular variation of the perihelion. The differetices of the values of the secular terms, fouiul b) !u' two inr^phrids emploviil in ('li;ipters H. :md TIT. (trocerd-. iVom thr fut tliaf in the one case tin- i tb'it of tile above term in the nnan motion is included, and in the other excluded. £S 80 T II K U U li I T OF U U A N U S If wo Hiiliiluct tlu! rffcct in qiu-Rtion when iicrcsBiiry, the ri'iniiiii«lrr will In- tlio cflft'ct ut' the hcculur variation uC tin* longitude uC tlic ixriliclion of I'ranus, tu whii'h we Hliall revert presently. lA>t us next introduce Nueli ii ehnngp in the ereentrieity of rninu)* ns Himll |)ro- duee the terni :)4".HiM) Hin «/, and uM-ertain its eti'vct ou the other turins, l-'ur tliiii purpose we luuitt determine te by tite cunditiun 3 (2- 'c»)^c = 34-.Hl)!) whii ch gives ;c=n".404 = . 0000847. A rlnmge of this nmonnt in h will introduee the foUowin;^ t«'rnjs in h nnd ^p h = :14 ".H!)» sin (/ -f 2".()4M itin 'i,j cos ^■^p = *0 — 844 eon ./ — (;".(>M-.> cos fj — 0".-J(i I cos Q;f. COS 4,^p =i: -(- 4 51 1 -f- IGl cos ij -\~ 10 cos '2i/ — IG sin 'j -^ I »in 2j. 0.43429 ^p = IDG!) -{- IS cosy -f 4 cos 2rt prtNluced in ^p by the combined action of Venus, tlie Kartli, and Mars. Tlie effect of each planet is computed by the approximate formula ^p=: J „,'(//;' +„/)aA7). Sinihir ViiiladoiiD. The following inequalities result from the si-cular variations of the ecceiitririty nnd longitude of perihelion produced by each of the disturbing planets, 7' being the time expressed in centuries. From the variation of the eccentricity ArUon of .lti))lt<-r, Ai'timi of Siiliirii, At'liiiti uf Neptune, ti). ■ —I 'in;'/' Hill 7 — ',1 Hi>7'hiii ;/ — o.r)0'.'7'iiiii iii .lij — U (iu-.!7'hiii -.Ig * * .. TUB ORBIT OF UllANUH. 81 Action nf .lupltiT, Actiuii of Suiiirii, Actiuo uf N('|ttum', -[ li T fOK (/ -I- 1 T eon 8<; -f Tif'cuii ity The soculur vnriiition of thv lon|{itiid(> of the {icrihelion is Artinn of .lii|iiti'r, Artioii of Suluni, Action of Ni'|ituno, Total, ■\\t2.\r 4-llH,»y + r.i.ir ««== +2111.(17' Thp rffort of this scnihir varintinii on tlic loiif^itudo niul radius vertor is It II II Action of JopiliT, «('-— ll.4(!rcoM j; — (I.C.TI T wh 2j — 0.0J7 Tcob 3<; Anion of Siiliirii, —W.WTvmij — ll.(!.'il Tion 2riotlir pcrturhationH produced by any planet, nil thoM* in wiiich tlic differcnco iK'twcm the indices •' and i" is tlie same, lor exaniph', in the |H'rturbntiunK of tlie loiij^itiide produced by Jupiter, let us consiiler tlic terms Ap=-}-l.V>(;i)sin( - ;) — 8.495 Bin C^;— -f l.lHisin( g—'il) -f O.ori sin (3y - '2/) — ().(»or) sin {'hj — ;}/) -f ().onsin(4i/ — ;}/) These terms niny be expressed in the form -f 0.002 cos ( — /) — 0.0!)'.>eos(27— /) -fO..')l.'ieos( ij~'il) — 0.00.') cos (% — '21) — 0.001 cos (4«/ — ;j/) >v=ih\\\ij X -f cos;/ X ■ -1-0.094 sin (v — -|- 0.5',»0 sin 'i{ij — — 'i.'i'.Hi sin (./ - /) -4- l.'.»')(isin%- /) -f 0.(M»(i sin ;{(./ - — 4.7r»4eo» ((/ — /) — l.lOHeos'% — /) -f-0.01(icos%— /) — 0.090 cos (,'/ — /) •} 0,."»lOeos'% — /) II May. 1873. 89 T II K O Kit I T F U II A N U H. In gciicrul, u scrius of tornis of the form Soi lin {iA -f «:/) + 2 6, cos (I'A -f inj) -f ia', sill {iA — ««/) -f S t', cos («\4 — «»/), may be put in tliu form \^ (»», — «',) COS iM — 2i (h, — li',) Hill iA ( ttin «(/ -f- I- ("( + "i) "i" '-1 + i (''( 4" ''i) "'" '■'^ { •'•"•"i/- All the fioriodic terms rontaiiiing only if atul / in the nr^iimnits miiy he put into tluH form by taking A=:t/-/, io that the roeffioicntx of Hiii /»■/ anil vm mj may all ho rxprcusnl ns n function of tilt* HJn^lu variable ar^unHMit A. Tlic ]K-rtnrbation8 of the eh'mentfl may be rnlnrcil to perturbations of the civortlinates ex|»rc»sc<(l as the sum of several protiiirts of slowly varying functions into the sines anil cosines of the multiples of s of Jupiter and Saturn, and expressin;r the terms above and below it in each series as coefficients of sin '/, cos ij, sin 2y, and ■ o THE ORBIT OF URANUS, 18 COS 2j/, wo niuy ridiioc iho iiuni»M«r of urKununtB to liglit, niul ihr number of table* to seventeen. Consider, for inutunce, the terniH of the second series, -(U0Hsin(-i/4-'.».V-/) -0.(Mncos(-j/-t-2.S'-J) — 0.0l4sin( 2S-J) -().orico8( '2S-J) 4-().l()4Min( ff-\-'iS-J) — O.'2(i7co8( g-\-'2S-J). These terms may be ullowe«l for by ndding to (r.o.O), (i'.«.l), (t'.c.I), the termi II " (,, ,..0) = — 0.014 sin (2.V- J) - 0.0 1 2 cos {'2S — J) (i'.«.l ) = + 0.2<»0 sin (2.V — J)-\- O.'iTi cos CiS—J) (t'.f.l) = -f- 0.0:0 sin {2S-J) - 0.'i74 cos ('iS—J). From the perturbations of longitude and radius vector already given, we readily find the following values of (I'.c.O), («'.«. I), etc. Adii.n 0/ JiijilfiT. (»'.c.0)=-f53.0fi4 8in A^ —0.004 cos ^4, — 0.277 sin 'ivl, -f 0.0;MJcos'.»/l, — 0.02r>8in;J^l, (».c.l)=-}- 2.226 sin yl , —0.000 cos Ay ((•.«.1)=— 0.001 sin yi, —4.7(54 cos yl, _ 1.2r)(}sin2y1,-f0.510eos2i1, _0.&20sin2il, — 1.108eos2.4, — 0.006 sin :L1, +0.016 cos 3.4, -11.467' -I". 22 7' ((•.c.2)=-|- 0.121 sin i1,— O.OaScoH y1, (r.«.2)=— O-O-'jOsin yl,— O.l75cos yl, -f- 0.012 sin 2.4, -0.01 4 cos 2.1, +0.00H sin 2.4, +0.042 cos 2.4, 4- 0.029 sin ;L4, —0.034 cos 3/1, -f0.034 sin 3.4, +0.035 cos 3.4, -0".6r/' -0".07r («.c.3)=- 0.04 7' (,..^.3)=^ _().()(),-, 7' (().c.O)= + 1127cos yl, + 4 cos 2.1, (p.f.l)=~ 2 sin .4, +.'.7 COM .4, + lO sin 2yl, —23 cos 2.1, + 137' (p.c.2)=+ 7 cos yl, +17' (().«.l)= + 10Ssin yl, + 2co8 yl, + 26 sin 2.1, +12cos2yl, -120 7' (p.«.2)==+ 7 sin yl, -87* IMAGE EVALUATION TEST TARGET (MT-3) /. 1.0 I.I liillll 112.5 |50 Ir- .4^ IM i.': ii 2.0 1.8 1.25 1.4 1.6 : <« 6" - ► Photographic Sciences Corporation 23 WEST MAIN STREET WEBSTER, N.> 14580 (716) 872-V503 ^ t. S'^.y Qa H A 84 TUE ORBIT OP URANUS. («.o.0)=»j (t).C.l) = Action of Saturn, // It '+20"MsinJ, _ 0.062V +8.580 cos A^ — 4.110 sin 2 A, — 0.009 cos 2.4, — 0.824 sin 3^, —0.019 cos 3^, — 0.228 sin 4A, —0.008 cos 4yl, — 0.074 sin bA^ —0.003 cos 5.4, — 0.025 sin 6 A, // " [— 1.032*/ '^ — 2.121 sin 2^, — 0.392 sin 3.4, — 0.109 sin iA, — 0.038 sin 5^1, — U" e»s;:fj».^.(-..)-(i'«:e!«-"-+(+?:»y- —0.793 cos 2i4, —2.421 sin 2^, +2.037 cos 2^, —0.308 cos 3^, —0.226 sin 3^, +0.318 cos 3i4, —0.099 cos 4^, —0.055 sin 4^, +0.097 cos 44, —0.027 cos 6i4, —0.027 sin bA^ +0.038 cos 5J, .iir — 9".18r <"»)H:"::~)"".+t'';:Sr)™^-<""-''=(l''::Sy-".+(+'::"T)-^ —1.813 sin 2.4, +0.631 cos %A^ —0.177 sin 3^, +0.013 cos 3^, —0.044 sin 4^, +0.013 cos 4^4, —0.016 sin bA^ +0.005 cos 54, — 0".65 2' — 0".542' — 0.665 sin 2^4, —1.847 cos 2.4, — 0.081 sin 34, —0.153 cos 34, — 0.013 sin 44, —0.044 cos 44, — 0.005 sin 54, —0.016 cos 54, (t).c.3) = — 2.265 sin 4, — 1.163 sin 24, + 0.026 sin 34, + 0.005 sin 44, + 0.002 sin 54, — 0".04 2' — 0.126 sin 4, — 0.503 sin 24, + 0.053 sin 34, +5.656 ros 4, +2.956 cos 24, —0.063 cos 34, — 0.013 cos 44, —0.004 cos 54, (U.C.4) = +0.329 cos 4, +0.378 cos 24, —0.032 cos 34. (p.r.O) = + 106 sin 4, — a sin 24, (p.e.l): : + 1364 sin 4, 46 sin 24, sin 34, sin 44, + + + + + (p.c.2) = - (p.c.3) = - + +9sr 279 sin 4, — 27 sin 24, + 3 sin 34, f7 2' 61 sin 4, — 24 sin 24, + 761 cos 4, 89 cos 24, 19 cos 34, 5 cos 44, 43 cos 4, 45 cos 24, 9 cos 34, 2 cos 44, 103 cos 4, 9 cos 24, 24 cos 4, 9 cos 24, (u.s.3) = +5.656 sin 4, +2.205 cos 4, +2.956 sin 24, +1.163 cos 24, —0.063 sin 34, —0.026 cos 34, —0.013 sin 44, —0.005 cos 44, —0.004 sin 54, —0.002 cos 54, — 0".03r (u.8.4) = +0.329 sin 4, +0.126 cos 4, -10.378 sin 24, .;-0.503 cos 24, —0.032 sin 34, —0.053 cos 34, (p.s.l) = _ (p..s.2) = - (p.8.3)=- ^ 33 sin 4, 1338 cos 4, + 31 sin 24, + 14 cos 24, + 7 sin 34, . + 5 cos 34, + 2 sin 44, + 1 cos 44, —117 2* — 103 sin 4, + 281 cos 4, + 9 sin 24, + 27 cos 24, + 3 cos 34, —8 2* ._ 24 sin 4, + 61 cos 4, + 9 sin 24, — 24 cos 24, THE ORBIT OF URANUS. 86 Action of Neptune. A, = g-r —0.08 cos ^3 —0.03 cos 2^3 +0.23 cos S.is 4-0.06 cos 4^3 —0.04 cos 5 At —0.01 cos 6^, ,• (w.c.0)=— 39.66 sin A^ —35.36 sin 2^,, +17.29 sin Sylj + 3.91 sin 4yl3 + 0.99 sin 5^3, + 0.42 sin 6^1^ + 0.19 sin 7^3 + 0.09 sin 8.43 + 0.02 sin 9.I3 +11 {v.cA)^ — 6.77 sin A.^ — 0.53 cos A3 — 1.10 sin 2.I3 +0.07 cos 2-13 +23.25 sin 3.43 +4.04 cos 3.43 + 6.05 sin 4.43 +1.06 cos 4^3 — 3.26 sin 5.43 — 0-66 cos 5^43 O.lj — 0.17 cos 6^ lA, —0.05 cos 7.43 8.43— 0.02 cos 8/I3 9/I3 —0.01 cos 9yl3 — 0.80 sin — 0.24 sin — 0.12 sin — 0.06 sin — 0.04 sin lO/l, +1.99945e5i/ (tj.c.2)=— 0.43 sin A^ —0.03 cos A^ —0.03 sin 2.43 +0.01 cos 2.43 +0.75 sin 3^3 +0.08 cos 3^3 —0.10 sin 4^13— 0.08 cos 4.43 —3.20 sin 5.43— 1.17 cos 5.13 —0.83 sin 6.43—0.32 cos 6.4, +0.57 sin 7.43 +0.22 cos 7^ +0.14 sin 8.13 +0.06 cos 8.43 +0.06 sin 9.43 +0-02 cos 9.43 +0.03 sin 10.4, —0.01 cos 10^3 +0.11 722c V A, 3.43 +0.01 cos 4.43 — 0.05 cos 5,43 —0.02 cos 6.43 —0.02 cos 7.43 +0.25 cos 7 A, +0.11 sin 8.4, +0.07 cos S.lj —0.08 sin 9^13-0.05 cos 9/I3 —0.03 sin 10.43 —0.02 cos IO.43 +0.0071 4c V ((7.c.3)=— 0.02 sin +0.04 sin — 0.1 5 sin —0.08 sin —0.02 sin +0.46 sin 3.43 5.43 6.43 (y.s.l)=— 0.49 sin ^3+ 1.75 cos A3 +0.12 sin 2/I3— 2.48 cos 2^ +4.04 sin 3^3 —20.92 cos 3.43 +1.07 sin 4^3— 5.42 cos 4^ —0.68 sin 5.43 + 3.47 cos 5^3 —0.17 sin 6.43+ 0.91 cos 6.4, —0.04 sin 7^3+ 0.30 cos 7^, —0.02 sin 8^13 + 0.12 cos 8^3 —0.01 sin 9.43 + 0.06 cos 9^ + 0.04 cos 10^3 +1.99835Se («7.8.2)=— 0.03 sin ^3 +0.13 cos A +0.02 sin 2.43— 0.16 cos 2^3 +0.08 sin 3.43— 0.60 cos 3^3 —0.08 sin 4^3 +0.15 cos 4J3 —1.17 sin 5.43 +3.22 cos 5^13 —0.32 sin 6^3 +0.86 cos QA^ +0.22 sin 7^-0.57 cos 7yl3 +0.06 sin 8.43— 0.14 cos 8^3 +0.02 sin 9.43— 0.06 cos 9^, —0.01 sin 10.43 —0.03 cos IO.43 +0.117135e (,;.«. 3)= +0.02 cos A^ +0.01 sin 3^1-0.04 cos 3^1, —0.05 sin 4.43 +0.15 cos 4^1, —0.02 sin 5.43 +0.08 cos 5.43 —0.02 sin 6.43 +0.02 cos 6.43 +0.25 sin 7/13—0.46 cos 7^3 +0.07 sin 8.43— 0.11 cos 8^ —0.05 sin 9^+0.08 cos 9^, —0.02 sin \^A., +0.03 cos 10^, +0.007 146e 86 TUB ORBIT OP UUiNUS. Action of Neptune. — Continued. A, = g — 1! («>.c.4)=— 0.06 sin 9^ —0.05 cos QA^ (/•.s.4)= —0.09 sin 10^3 —0.08 cos 10^, +0.00044e.'.».2) const. = = — 0".14. (1) (2) (3) (4) (5) Sum. // // II n // II 1700 +0.11 +0.81 +0.04 —12.46 —0.72 —12.36 1750 +0.07 +0.54 +0.03 — S.31 —0.70 —8.51 1700 0.06 0.49 0.03 —7.48 —0.69 —7.73 1770 0.00 0.43 0.02 —6.64 —0.08 —6.95 1780 0.05 0.38 0.03 —5.81 —0.66 —6.10 1790 0.04 0.32 0.02 —4.98 —0.65 —5.39 1800 0.04 0.27 0,02 —4.15 —0.04 —4.60 1813 0.03 0.22 0.01 —3.32 —0.03 —3.83 1820 0.02 O.IC 0.01 —2.49 —0.0 1 —3.05 1830 0.01 0.11 +0.01 —1.00 —0.60 —2.27 1840 +0.01 +0.05 0.00 —0.83 —0.58 —1.49 1850 0.00 0.00 0.00 0.00 —0.57 —0.71 i8(;o —0.01 —0.05 0.00 -1-0.82 —0.56 +0.06 1870 —0.01 —0.11 —0.01 1.65 —0.54 0.84 1880 —0.02 —0.16 —0.01 +2.48 —0.53 + 1.62 («.f.2) const. = + 0".13. (1) (2) (3) (4) (■^') Snra. // n It It tf II 1700 + 1.00 +o.ns +0.43 +2.20 —0.11 +4.62 1750 +0.07 +0.05 +0.28 + 1.08 —0.13 + 3.29 17 CO +0.00 0.59 0.25 1.55 —0.12 3.00 1770 +0.54 0.52 0.22 1.41 —0.12 2.70 1780 +0.47 0.40 0.20 1.26 —0.13 2.39 1790 +0.40 0..39 0.17 1.11 —0.13 2.07 1800 +0.34 0.33 0.14 0.94 —0.13 1.75 1810 +0.27 0.26 0.11 0.77 —0.14 1.40 1820 +0.20 0.20 0.08 0.59 —0.14 1.06 1830 +0.13 0.13 0.06 0.39 —0 14 0.70 1840 +0.07 +0.07 40.03 +0.20 —0.15 +0.35 1850 0.00 0.00 0.00 0.00 —0.15 —0.02 1800 —0.07 —0.07 —0.03 —0.21 —0.15 —0.40 1870 —0.13 —0.13 —0.00 —0.43 —0.16 —0.78 1880 —0.20 —0.20 —0.08 —0.08 —0.16 —1.19 w> THE Oil BIT OP URANUS 91 (.•..S.3) (1) (2) (3) (4) (S) Jupiter Saturn Neptune Neptune Jupiter & Saturn { Bum. (8W).) (8«C.) (««o.) (long per.) (long per.) // II // // // rr noo -f-0.01 +0.05 0.00 —0.76 —0.70 1750 0.00 +0.03 —0.51 —0.48 17 GO 0.03 —0.40 —0.43 1770 0.03 —0.40 — 0.3T 1780 0.02 —0.35 —0.83 1790 0.03 —0.30 —0.28 1800 0.02 —0.25 —0.23 1810 0.01 —0.20 —0.19 1820 0.01 —0.15 —0.14 1830 +0.01 —0.10 —0.09 1840 0.00 —0.05 —0.05 1850 0.00 0.00 0.00 18G0 0.00 +0.05 +0.05 1870 —0.01 +0.10 +0.09 1880 —0.01 +0.15 + 0.14 («.p.3) (1) (2) O) (4) Sura. // // II // II 1700 -fO.OC +0.06 +0.02 +0.13 +0.27 1750 4-0.04 +0.04 +0.02 +0.10 +0.20 17C0 0.04 0.04 0.01 + 0.09 +0.18 1770 0.03 0.03 0.01 +0.08 +0.i5 1780 0.03 0.03 0.01 +0.07 +0.14 1790 0.02 0.02 0.01 +0.00 +0.12 1800 0.02 0.02 0.01 +0.00 +0.11 1810 0.02 0.02 +0.01 +0.05 +0.09 1820 0.01 0.01 0.00 +0.04 +0.06 1830 +0.01 + 0.01 6.00 +0.02 +0.04 1840 0.00 0.00 0.00 +0.01 +0.02 1850 0.00 0.00 0.00 0.00 0.00 ISfiO 0.00 0.00 0.00 —0.01 —0.02 1870 —0.01 —0.01 0.00 —0.02 —0.04 1880 — O.Ol —0.01 0.00 —0.04 —0.00 For the uadi t;.S VECTOR. 1 t'^ALUES OF (|).( .0) 0) (2) <3) (4) (5) Sum. II // II // // II 1700 +0.4 +3.5 —0.2 + 104 + 13 + 181 1750 +0.3 +2.3 —0.1 +110 7 +120 1700 0.3 2.1 99 107 1770 0.2 1.9 88 5 95 1780 0.2 1.0 77 4 83 1790 0.2 1.4 07 3 72 1800 0.2 1.2 50 2 60 1810 O.I 0.9 45 2 48 1820 0.1 0.7 34 + 1 36 1830 +0.1 0.5 22 23 1840 .0 +0.2 + 11 + 11 1850 .0 0.0 — 1 — 1 18fi0 .0 —0.2 — 11 — 2 — 13 1870 —0.1 —0.5 22 — 2 — 25 1880 —0.1 —0.7 — 33 — 3 — 37 «ll 92 THE ORBIT OF URANUS. (p.«.l 1 count. - + 63. (1) (2) (3) (4) (.'•) Jupiter Saturn Neptune ' NnptuiiB Jupiter&i^atiirn Snm. (sec) (aeo.) (a«o.) (long per.) (long pur.) // // n « It /; 1700 + 180 + 170 +75 +390 —21 +809 1750 + 120 + 117 +50 +302 —25 +627 1760 108 105 45 280 20 675 1770 96 94 40 255 28 620 1780 84 83 35 228 29 403 1790 72 70 30 200 30 405 1800 CO 59 25 170 31 346 1810 48 47 20 139 33 285 1820 36 35 15 106 33 222 1830 24 23 10 73 34 158 1840 + 12 + 12 + 5 + 37 34 95 1850 35 + 28 1860 — 12 — 12 — 5 — 39 36 — 41 1870 24 23 10 80 36 —110 1880 — 36 — 35 —15 —122 —37 -182 (p.r.l) const. — ■ + 73. (1) (2) (3) (4) (5) Snm. // // // // // II 1700 —20 —147 —9 +2241 +104 +2242 1750 —13 — 98 —6 + 1490 + 94 + 1546 1760 — 12 — 88 —5 1346 91 1405 1770 —10 — 78 —5 1197 88 1265 1780 — 9 — 69 —4 1047 85 1123 1790 — 8 — 59 4 897 82 981 1800 — 6 — 49 —3 747 78 840 1810 — 5 — 39 —2 598 74 699 1820 — 4 — 29 —2 448 11 557 1830 — 3 — 20 —1 299 68 416 1840 — 1 — 10 —1 + 149 64 274 1850 61 + 134 1860 + 1 + 10 + 1 — 149 58 — 6 1870 3 20 1 298 54 — 147 1880 4 29 2 — 447 + 51 — 288 (p.s.2 ) const. = +5. (1) (2) (3) (4) tl (5) Sum. n // II ;/ // 1700 + 12 + 12 +* + 28 +61 1750 + « + 8 +fi +21 +48 1760 7 7 5 19 43 1770 6 6 5 17 39 1780 6 6 4 15 36 1790 5 5 4 13 32 1800 4 4 3 11 27 1810 3 3 2 8 21 1820 2 2 2 6 17 1830 2 2 1 4 14 1840 + 1 + 1 + 1 + 2 10 1850 + 5 1860 _ 1 — 1 —1 — 2 1870 2 — 2 —1 4 — 4 1880 — 2 — 2 —2 -6 — 7 "f * THE ORBIT OF URANUS. 93 (p.c.2) const. -+4 (I) (2) W (4) (r>) JupitHr Saturn Nnptuiia Niiptuiie JupitHr jiBatnrn Som. (seo. ) (8B0.) (»O0.) (long per.) (long per.) ti '/ It '/ // // noo —1 —10 +157 + 150 1750 —1 — 7 + 105 + 101 17(10 — 1 _ G !)5 !I2 1770 —1 _ fl 84 81 17H0 — 1 _ 5 74 72 1790 —1 — 4 03 02 ISOO _ 3 52 63 1810 _ 3 42 43 1820 — 2 31 33 1830 — 1 21 24 1840 _ 1 + 10 13 IS.'iO + i 18(i0 + 1 — 10 — 5 1870 1 21 _10 1880 + 2 — 31 —25 ReiJuced E.ipres8iom for ihe Latitude of Uranus. If we represent by F,, T'i, T^, the distances of Uranus from its descending nodes on the respective orbits of Jupiter, Saturn, and Uranus, wc find tlie following perturbations of the latitude, which arc independent of the mean longitude of the disturbing planets. 5/3= — 0.0 11 4< cos F, — 0.0477< cos Fj — 0.0125/ cos Fs -f 0.245 4- 0.386 sin «/ + 0.266 cos g — 0.043 sin 2g -j- 0.006 cos 2ij. To find how far the last five terms may be represented by simple corrections to the elliptic elements, we first represent the cftect of minute corrections to the inclination and node of Uranus as a function of its mean anomaly. Tutting « for the argument of latitude of Uranus, we have to a sufficient degree of approxima- tion 6/3 = sin m6<^ — sin <0 = cos Dfp — sin Diq. And, neglecting (A ^ X sin .s.l), (/>.c.l), etc., the terms , THE ORBIT OF U 11 AN US. 97 Jk (b.c.0) = — e sin u5<^, (h.K.l) = cosu^(J), (i.f.l) = sinut^^), (i.s.'i) = ecostaS<^, (6.C.2) = csinuf^^; Avlicrc Tutting in the above expressions l".->U)T-{-0''.ViT* ,,,3=95^3' + 3459 "7; cos(j = — .0880 — .0167 r, siu(o = + .99Gl — .00157; we find (!>.c.0)=:-(0".ll— 0".06/)7' {h..U) = — (0 ••-'<> - -11")?'- 0".05r» (/>.c.l) - (-i .30 - 1 .24-/) 7'+ .127'" {I>.k2)^ -0.01 7- (&.c.2)= (0.11-0.06.7)7. We have, finallv, to consider the terms of long period in hy; nnd <'/.; which luvvc been omitted froni the periodic perturbations produced by Neptune, in computing the terms of <\i on page 61, and wliich arc as follows: (^^1= l".43cos(2/'— J/)— 0''.39sin{2/'- g) — 2 .12 cos (4/' — 2. = _0"807' ^/• = + 0.277' which arc to be nudtiplied by the factor 1 + ;,. The corresponding perturbation of the latitude will be }^S = sin v'v; — cos V k. Tutting for v its approximate value V =1 (/ -f- (J + 2t' sin g and dtncloping to quantities of the first order with respect to the eccentricities, we have . , . x sin V = sin {g -f ■ c) + e sni (2,7 + 1.>) — f sm u cos V = cos (j/ + u) -i- c cos (2r/ + (j) — c cos u. 13 M»y. 1873. A 98 THE 11 B 1 T O 1' U U .\ N U S. Substituting for o its value, 1'2"45', and for ly; and tJc their above vahics in the expression for 6ti, we find that the terms of S3 in question will add the following terms to (b.c.O), (ft.s.l), etc, (bj;,0) = — .010 ^>7 + .046 Ik = + 0".027' (1 + fi) (b.8.\) = + .975 A>7 + .221 hk = — .727' (1 + n) (A.c.l) = 4- .221 hy, — .975 }k = — .44 7^(1 + ^) (/m.2) = + .040 r^>7 + .011 Ik = — .047' (1 + //) (/>.f.2) = - (b.c.O) = - .027^ (1 + fi) These vahies will be employed in the construction of the provisional ephemeris, but not in the tables. Collecting all three classes of terms discussed in this section, we have the fallowing constant and secular terms in (/>.f.O), [b.n.l), etc. (ft.c.O) = + 0".26 + (-0".09 + O-OS:/)?" (h.8.1) = (— 0".92 — 0".Glr/)7'— 0".05 7^ (h.c.l) = (+ 1 .86 — 1 .68,/) 7'+ .127"' (6.S.2) = —0.06 — 0.057' (6.C.2J = — .01 + (0 .09 — .08 /) T PosUloiia of Uranus resulting from the preceding tlicory. The next step in order is the preparation of an ephemeris of the planet for comparison with observations. As this provisi(mal theory is, for future use, super- seded by the tables appended to the present work, it seems unnecessary to enter very fully into the details of the computation of the ephemeris. The perturba- tions of the longitude, logarithm of radius vector, and latitude, were tirst com- puted by the formulte already given. hv = (tJ.c.O) -\- {v.c. 1 ) cos J/ -f (!'.c.2) cos 2// + etc. , 4" {v.8-V) sin g -\- (v.8.2) sin 2g -\- etc., Jf5p = (p.c.O) + (p.c.l) cos .7 -f (p.c.2) cos 2.7 -j- etc., -f (p.s.l) sin flf + (p-^.S) sin 2^ + etc., S.3 = {b.c.O) -\- (J).c.l) cos^ + {b.s.l) sin g. Each coefficient (v.c.O), {v.c.l), etc., is composed at most of the following quan- tities: 1. The five classes of secular, long period, or constant terms, the separate values of wliich, with the sum of all, are given on pages 89 to 93. 2. Periodic terms due to the action of Jupiter, Saturn, and Neptune, given on pages 83 to 87. 3. Terms depending on the product of the masses of Jupiter and Saturn, given on page 88, omitting those depending on iV„ and N^, because they arc given in column 5 of the terms of the first class. The sum of the perturbations thus computed is given in the third column of the following ephemeris. An approximate value of the perturbations produced by Neptune alone is inde- pendently computed for every fourth date, and tlie result is given in the fourth THE ORBIT OF URANUS. 99 column. The secular and long period terms are liere taken from columns (3) and (4) of the tables on piigcs 89 to 93. The elliptic co-ordinates were then derived from the following elements, which are a little different from those employed in the computation of the perturbations. Elements III. of Uranus. 7t, 168° 15' 12".0 e, 28 25 29 .5 0, 73 11 58.0 4,, 46 20.0 e. .0469436 e, (in sec.) 9682".81 n, 15426.196 log«, 1.2828989 Eed. to Ecliptic, — 9".37 sin 2 (y — 0) The longitudes thus found arc corrected for lunar, but not for solar nutation, and the results are given in the fifth column. The column "correction" arises in this way: after the comparison of the eplie- mcris with observations was nearly completed, it was found that some errors had crept into the former, the most importmit of which was the employment of a mean anomaly, q, corrected for secular variation of the perihelion in the computation of the perturbations from the preceding formula;. As a large portion of the com- putations on the provisional ephemeris had been made by assistants furnished by tlie Smithsonian Institution and Nautical Almanac, I deemed it prudent to make a careful rccomputation of the perturbations for every si.xtli date during the entire period of the modern observations. The longitudes actually printed m the fifth column are the results of the original incorrect computation, while the numbers in the next column sliow the several corrections to be applied to obtain the results of my final revised computation. During the period of the modern observations the ephemeris is computed for intervals'of 120 days, and the selected dates are all exact multiples of that interval before or after the fundamental epoch, 1850, Jan. 0, Greenwich mean noon. For convenience of reference the dates are numbered from an epoch earlier by 212 intervals, and the number is given in the second column. Between 1796 and 1801 no observations wortli using were made on Uranus, the ephemeris has. therefore, not been extended over this iiit(;rval. .L 100 THE ORBIT OP URAxNUS. IleUOCENTKlC El'UEMEIUS OF UlLXNUS FllO.M THE I'BEOKIH.MJ TKOVIiSIONAl, TlIEORY. [T lu longitailea are oorrwtfil fur lunar but not for solar natatioD.] Dat». flrftfiiwioh No. Sum of lieituiba- ApproxiniAlH ptTtiirbutioiiH producHd by Neptune, Longltado. CorrfHition Latitude. Logarithm K.iclins lUtiUU UUOll. tioiis. Tector. / // // / // // / // lf.90, Dec. 23 4-4 28.5 4-23C.1 59 40 35.2 —10 7.8 1.2870828 Dcf. 24 + 4 28.5 59 41 16.5 —10 7.3 1.2876789 1712, April 2 +8 37.9 155 24 42.0 +45 58.7 1.2026650 April 3 +8 .ST.O 155 25 29.0 +45 58.8 1.2626640 1715, Mar. 4 -ffi 5«.3 4-136.2 109 10 21.4 +40 2.1 1.2623890 Mar. 5 -J-r> .58.2 169 11 7.7 +46 2.1 1.2623893 Mar. 10 -f (! 57.5 169 15 0.0 +46 1.7 1.2623911 April 2!) -)-(i 51.1 4-134.8 109 53 42.2 +45 58.0 1.2624006 April 30 f(! 51.1 109 54 28.8 +45 57.9 1.2024102 1748, Oct. 21 —0 28.9 — 65.0 310 13 19.3 —41 27.0 1.3001855 Oct. 22 —0 28.8 316 13 58.8 —41 27.9 1.3001875 1750, Sept. 13 +0 6.3 — 51.3 323 44 14.9 —43 47.2 1.3013714 Sept. 14 -fO 6.3 323 44 54.0 —43 47.3 1.3013731 Oct. 14 fO G.6 — 50.8 324 4 22.1 —43 52.4 1.3014105 Oct. 15 +0 6.6 324 5 1.0 43 52.5 1.3014181 Dec. 3 -1-0 8.0 — 49.8 324 36 49.5 —44 0.0 1.3014850 Doc. 4 -1-0 8.0 324 37 28.5 —44 0.8 1.3014872 1753, Doc. 3 —0 5.6 — 25.1) 336 25 0.6 —40 1.1 1.3025789 Di'c. 4 — 5,6 330 25 39.3 —40 1.2 1.3025790 175f>, Sept. 25 —0 40.5 — 0.4 347 25 34.9 —40 9.2 1.3029303 Sept. 2(i —0 49.5 347 20 13.6 —46 9.1 1.3029363 IICA, Jan. 15 -1-1 24.0 16 13 35.0 —38 36.9 1.3003508 Jan. Ifi -1-1 24.0 16 14 14.2 —38 36.7 1.30034HC 17C8, Dec. 27 -f-1 33.1 30 3 35.2 —27 42.4 1.2955331 Dec. 2f) -fl 33.1 + 100.9 36 4 41.1 —27 41.5 1.2955274 Dec. 31 -f-1 33.1 30 6 5.0 —27 40.7 1.2955216 17fi9, Jan. 15 -1-1 33.5 36 16 4.6 —27 3,.2 1.2954753 Jan. 18 + 1 33.5 36 18 4,5 —27 33.0 1.2954659 Jan. 21 -fl 33.5 + 106.9 3(1 20 4,5 —27 31.6 1.2954566 Jan. 24 -f 1 33.5 3(1 22 4,4 —27 30.3 1.2954476 1781, Jan. 1 1 +:i 57.46 -1-178.9 8(1 36 13.23 +0.16 + 11 3.03 1.2785972 May 1 2 . 4-3 53.41 88 2 34.70 + 12 11.14 1.2780883 Aug. 2it 3 ' -f3 48.58 89 29 7.41 + 13 18.33 1.2775834 Dec. 27 4 1 4-3 43.09 90 55 51.40 + 14 25.14 1.2770827 1782, April 2(J 1 4-3 37.18 +180.4 92 22 4(1.89 -1-15 31.53 1.2765870 Anf;. 24 '1 I 43 30.97 93 49 53.89 + 10 .37.47 1.2760968 Dec. 22 ^ , -1-3 24.52 95 17 12.51 +0.18 + 17 42.88 1.2756124 1783, April 21 8 -1-3 18.03 96 44 42.73 + 18 47.73 1.2751345 Ang. 1!» !) -i-3 11.87 4 180.1 98 12 24.96 + 19 51.93 1.2746633 Dec. 17 10 4-3 6.01 99 40 19.00 + 20 55.07 1.2742012 1784, April 15 11 4-3 0.78 101 8 25.13 4-21 58.09 1.2737412 Aug. I.i 12 42 56.24 102 36 43.29 +23 0.92 1.2732907 Doc. 11 13 4-2 52.82 +178.6 104 5 13,79 +0.23 -1-24 2.46 1.2728469 1785, April 10 14 42 50.58 105 33 56.61 +25 3.14 1.2724105 Aug. H 15 + 2 49.29 107 2 51,36 +20 3.05 1.2719806 Dec. (i 1(1 -42 49., 35 108 31 58,18 +27 1.98 1.2715564 1780, April 5 17 -42 50.53 + 175.5 110 I 16.88 + 27 59.97 1.2711405 Aug. 3 IH -f2 53,29 111 30 47.68 +28 57.02 1.2707299 Dec. 1 in 4-2 57.31 113 30.11 + 0.31 -f 29 53.03 1.2703265 1787, Mar. 31 20 43 2.38 114 30 23.81 +.30 47,92 1.2699294 July 2!) 21 +3 8.51 + 171.0 116 28.62 +31 41,72 1.2095394 Nov. 2(1 22 4-3 15.48 117 30 44.11 +32 34.33 1.2091,551 \ + T H K 11 B I T OF UK AN US. 101 IlKLlOCENTllIO El'llE.MElllS OF UllANUS. — VunlillUed Datrt. Sam of lAPPf'f im»l« ( Logaritlim Ore«iiwicli i-.o. . I p«rturl)iitioii» Loiigitiiile. Correction. Latitude. Radius vector. tui-aii iiuoii. - tlOUB. Neptune. 1 ' " i // o / // / ft 1788, Mar. 25 23 ' +3 23.02 ! 3 31.30 i 119 1 9.79 -1-33 25.72 1.2087775 July 2:! Nov. 20 24 120 31 45.79 34 15.81 1.2084005 25 3 39.82 j -1-105.1 122 2 31.13 4-0.28 35 4.58 1.2080423 1781), Mar. 20 2ri 3 38.07 j 123 33 25.90 35 51.99 1.2070852 July 18 Nov. 15 27 3 57.40 I 125 4 29.49 30 38.01 1.2073353 28 4 5.97 120 35 41.50 37 22.50 1.2009932 1790, Mar. 15 29 4 14.35 -fl57.8 128 7 1.70 38 5.50 1.2000592 July l;5 Nov. 10 30 4 22.48 129 38 29.88 38 47.00 1.2003330 31 4 30.00 131 10 5.37 -1-0.25 39 20.85 1.2000107 1791, Mar. 10 32 4 30.95 132 41 47.99 40 5.04 1.2057087 July 8 Nov. 5 33 4 42.95 -1-119.1 134 13 37.15 40 41.58 1.2054082 34 4 48.11 135 45 32. (;2 41 10.44 1.2051215 1792, Mar. 4 35 4 52.33 137 17 34.10 41 49.53 1.2(i48434 July 2 30 4 50.03 1,38 49 41.78 42 20.89 1.2045770 Oct. 30 37 4 58.32 -fl39.3 140 21 54.47 -1-0.40 42 50.35 1.2043210 1793, Fob. 27 38 4 59.50 141 54 12.11 43 18.00 1.2040700 Juno 27 39 4 59.85 143 20 34.79 43 43.81 1.20,38448 Oct. 25 40 4 58.84 144 59 1.04 44 7.75 1.2030207 1794, F(!b. 22 41 4 50.80 -fl2S.O 140 31 32.77 41 29.74 1.2034220 June 22 42 4 53.48 148 4 7.47 44 49.85 1.2032327 Oct. 20 43 4 49.13 + 122.9 149 30 45. S4 -1-0.47 45 7.99 1.2030577 1795, I'el). 17 44 4 44.20 151 9 27.73 45 24.24 Juno 17 45 4 3S.00 152 42 12.55 45 3S.44 Oct. 15 1801, Jan. 17 4(i 4 31.04 154 15 0.15 45 50.04 02 +3 22.92 3 20.59 179 3 50.23 -1-44 34.51 1.2027599 Mar. 17 03 + 57.3 180 30 54.114 44 12.H(i 1.2028920 Sept. 14 04 3 31.00 182 9 57.40 43 49.24 1.2030302 1802, Jan. 12 ()5 3 30.20 183 42 57.23 43 23.00 1.2031922 Mar. 12 00 3 41.88 185 15 54.08 42 50.34 1.2033599 Sopt. 9 07 3 47.78 4- 43.7 1.80 48 40.54 -fO.35 42 20.97 1.2035387 1803, Jan. 7 08 3 54.01 188 21 35.11 •M 55.80 1.203'' 281 Mar. 7 01) 4 0.29 189 64 18.55 41 22.99 1.2039280 Sept. 4 1804, Jan. 2 70 4 0.43 191 20 50.80 40 48.28 1.2041380 71 4 12.95 -t- 30.7 192 59 30.15 40 11.79 1.2043591 Mar. 1 72 4 18.74 194 31 50.83 39 33.03 1.2045891 Aug. 21) Dec. 27 73 4 24.38 190 4 17.13 + 0.14 38 53.72 1.2048287 74 4 29.34 197 30 30.24 , 38 12.21 1.2050777 1805, April 20 Aug. 24 Deo. 22 75 4 33.' ' -f 18.0 199 8 35.97 37 29.07 1.2053300 70 4 37.^2 200 40 33.91 30 44.40 1.2050037 77 4 40.11 202 12 23.39 1 35 58.15 1.2058808 1800, April 21 Aug. 19 78 4 41.83 203 44 4.25 35 10.48 1.2001074 79 4 42.. '.5 -1- 5.9 205 15 30.11 ' -1-0.21 34 21.31 1.2004031 Deo. 17 80 4 42.19 200 40 58.07 33 30.88 1.2007089 1807, April 111 Aug. 14 Dec. 12 81 4 40.52 208 IS 11.39 32 39.05 1.2070847 82 4 37.43 209 49 13.98 1 31 45.90 1.2074103 83 4 33.71 — 5.0 211 20 0.90 30 51.03 1.2077459 1808, April 10 Aug. 8 Deo. (! 84 4 28.30 212 50 49.02 29 50.12 1.2080922 85 4 21.03 214 21 20.40 + 0.25 28 59.51 1.2084491 80 4 14.32 215 51 41.72 1 28 1.80 1.2088100 1801), April 5 87 4 5.84 — 15.4 217 21 51.97 27 3.03 1.2091945 Aug. 3 Doc. 1 8S 3 50.03 218 51 51.53 1 20 3.28 1.2095829 89 3 47.07 220 21 40.02 25 2.00 1.2099810 1810, Mar. 31 90 i 3 37.10 221 51 19.10 24 1.10 1.2703905 July 29 Nov. 20 91 ; 3 27.15 — 24.3 223 20 47.07 -|-0.23 22 58.08 1.270X094 92 +3 17.52 224 50 4.83 1 4-21 55.38 1.2712377 102 THE ORBIT OF UUAXUS. UkUOCENTHIC Fl'IIK.MElllS OF UllANL'8. — Cunlinufd. Date. Greenwich No. Sum of pfitiiibii- Ai>pioxiiiiatH pfrMirhalioiis pMMiuieil by Loili^itUcle. Corieutloii. Liitltude. Lu}.'HI'ltlllll HailiiiH uuaii iiuuu. tiuUK. veutiir. Nfptuiie. // / ft // 1 II / // 1811, Mnr. 20 93 +3 7.99 220 19 12.04 +20 51.52 1.2716752 July 24 94 2 59.12 227 48 9.00 19 40.95 1.2721215 Nov. 21 95 2 50.91 —32.1 229 10 55.87 18 41.09 1.2725700 1812, Miir. 20 9(i 2 43.45 2.30 45 32.41 17 35.80 1.2730381 July 18 97 2 37.20 232 13 59.13 +0.11 10 29.34 1.2735072 Nov. 15 98 2 31.98 233 42 15.55 15 22.35 1.2739838 1813, Mar. 15 99 2 27.80 —39.1 235 10 21.88 14 14.92 1.2744(i51 July 13 100 2 25.05 23(i 38 18.20 13 7.03 1.2749507 Nov. 10 101 2 22.98 238 3.95 11 58.79 1.2754409 1811, Mar. 10 102 2 22.35 239 33 39.01 10 50.21 1.2759307 July 8 103 2 22.54 —44.0 241 1 4.03 +0.22 9 41.32 1.2704300 Nov. 5 104 2 24.24 242 28 19.59 8 32.19 1.2709372 1815, Miir. 5 105 2 20.58 243 55 23.07 7 22.80 1.2774403 July 3 100 2 30.00 245 22 17.17 (1 13.39 1.2779459 Oct. 31 107 2 34.17 —49.3 240 48 59.79 5 3.80 1.^784533 181f., Feb. 28 108 2 39.00 248 15 31.47 3 54.09 1.2789(00 Juno 27 109 2 44.51 249 41 52.08 —0.01 2 44.40 1.27940S8 Oct. 25 110 2 50.43 251 8 1.42 1 34.82 1.2799725 1817, Feb. 22 111 2 50.75 —54.4 252 33 59.49 + 25.25 1.2804709 June 22 112 3 3.22 253 59 40.00 — 44.19 1.2809792 Oct. 20 113 3 10.03 255 25 21.18 1 53.44 1.28147«8 181S, Feb. 17 114 3 10.03 250 50 44. 4S 3 2,48 1.2819759 Juno 17 115 3 22.90 —54.7 25 s 15 55.79 0.00 4 11.23 1.2824701 Oct. 15 110 3 28.90 259 40 55.14 5 19.73 1.2S29599 1810, Feb. 12 117 3 34.28 2i:l 5 4 2.25 27.83 1.283440 7 Juno 12 lis 3 38.77 202 30 17.04 7 35.52 1.2839317 Oct. 1(1 119 3 42.48 —50.1 203 54 39.42 8 42.77 1.2844143 1820, Feb. 7 120 3 45.03 205 18 49.19 9 49.52 1.284S927 June 121 3 40.37 200 42 40.29 —0.01 10 55. 7(; 1.2853070 Oet. 4 122 3 40.38 20 H .■5il.07 12 1.40 1.2X5.'*3!I1 1821, Feb. 1 123 3 45.07 —50.4 209 30 2.43 13 fi.44 1.2803087 Juno 1 124 3 42.35 270 53 21.. 55 14 10.88 1.2807757 Sept. 211 125 3 3S.08 272 10 27.90 15 14.57 1.2872402 1822, Jiiii. 27 1 20 3 32.80 273 39 22.29 10 17.58 1.2,s77(i80 May 27 127 3 20.18 —50.2 275 2 4.25 —0.10 17 19.84 1.2^81032 Sept. 24 128 3 18.32 270 24 34.00 18 21.32 1.28,S(:214 182;?, Jan. 22 129 3 9.08 277 40 52.38 19 22.01 1.2S90772 May 22 130 3 0.24 279 8 59.05 20 21.89 1.2895311 Sept. HI 131 2 50.30 —50.1 2so 30 54.82 21 20.91 1.2S99820 1824, Jan. 17 132 2 40.10 ■>K1 52 39.71 22 19.10 1.2904315 May Hi 133 2 29.74 2S3 14 14. k; —0.07 23 10.40 1.2908770 Sept. 13 134 2 19.49 2S4 35 38.00 24 12.78 1.2913208 182'), Jan. 11 1.35 2 9.25 —55.5 2S5 50 52.92 25 8.22 1.2917000 May 11 130 1 59.40 2S7 17 57.80 20 2.71 1.2921907 Sept. 3 137 1 50.21 2.SS 38 53.43 20 50.22 1.2920290 1820, Jan. 138 1 41.44 2S9 59 40.04 27 48.71 1.2930570 May 139 1 33.58 —54.5 291 20 1H.03 —0.09 28 40.20 1.2934804 Sept. 3 140 1 20.44 292 40 47.50 29 30.70 1.2938987 1S2'7, Jan. 1 141 1 20.21 294 1 H,.^3 30 20.10 1.2943110 May 1 142 1 14.89 295 21 22.15 31* 8.00 1.2947180 Aug. 29 143 1 10.52 —54.0 29(1 41 27.72 31 55.92 1.2951190 I»ee. 27 144 1 7.22 298 1 25.72 32 42,14 1.2955130 1828, April 25 145 1 4.79 299 21 10.45 —0,07 ;!3 27.24 1.2959002 Au)r. 23 140 1 3.51 300 41 0.14 34 11.25 1.2902790 Dee. 21 147 + 1 3.11 —52.9 302 30.75 —34 54.11 1.29GG49C TIIK OUIJIT OF UIIANUS. 103 m IIkLIoCENTIIIC I-I'IIKMKUIS 01' Utt ANUS.— Continued. Dftt«. Sii ni of A|i|iroxiiiiat« perlartmtiotirt pniilucfil l,y I/ocaritlim Orefiiwicli mtjnn noun No. |iBrturba- tiuiiti. LoDgl uile. Correction. Latitude. Itailiud vector. Ni-piuiia. It // o / " /; / // 182U, April 20 148 + 1 3.(18 303 20 fi. 73 —35 35.77 1.2970111 Aug. 18 un 5.15 304 39 30.08 3(i 1(1.2(1 1.2973030 Dee. Hi 150 7.37 305 58 4(1. 9fi 30 55.(10 1.2977052 1830, April 15 151 10.14 —52.3 307 17 57.34 +0.08 37 33.08 1.2980308 Aug. 13 1.52 13.74 308 37 1.72 38 10.50 1.2983571 Dee. 11 153 17.(13 309 55 59.75 38 40.13 1.2980001 ^831, April 10 154 21.(19 311 14 51.(!8 39 20.49 1.2989038 Aug. 8 155 2(1. or, —52.2 312 33 37.70 39 53.54 1.2992501 Dee. (i 15(1 3(t.2(l 313 52 17.74 40 25.29 1.2995247 1832, April 4 157 34.13 315 10 51.(13 +0.05 40 55.74 1.2997873 Aug. 2 158 37. 50 31(1 29 19.(17 41 24.90 1.3000379 Nov. 30 159 40.39 —52.2 317 47 41.94 41 52.05 1.3002708 1833, .M.ir. 30 KiO 42.02 319 5 57.95 42 19.10 1.3005047 July 28 101 42.8(1 320 24 8.31 42 44.17 1.3007210 Nov. 25 lfi2 42.45 321 42 12.K9 43 7.88 1.3009280 1834, •Mar. 25 1(13 40.92 —52.5 323 11.92 +0.15 43 30.24 1.3011231 .luly 23 1(14 38.01 324 18 5.47 43 51.18 1.3013390 Nov. 20 1(15 34.01 325 35 53.99 44 10.74 1.3014851 1835, Mar. 20 l(i() 2s.7(i 320 53 37.59 44 28.92 1.3010515 July IS 1(17 22.19 —52.0 328 11 10.27 44 45.09 1.3018084 Nov. 15 1(18 14.89 329 28 50.92 45 1.14 1.3019507 18311, Mar. 14 l(i9 (1.50 330 4(1 21.41 +0.19 45 15.12 1.3020!)00 July 12 170 57.31 332 3 48.20 45 27.71 1., 3022205 Nov. !» 171 47.51 —53.9 333 21 ll.7fi 45 38.98 1.302;!4S2 1837, Mar. !» 172 .37.27 334 38 32.35 45 48.84 1.3024012 July 7 173 2(1. (>9 3i5 55 50,40 45 57.28 1.3025050 Nov. 4 174 15.92 337 13 (1.1(1 ~ 40 4.45 1.. 3020022 183.S, .Mar. 4 175 + « 5.23 —55. 1 338 30 20.17 +0.17 40 10.04 1.3027503 July •2 17(1 5.21 339 47 32.7(1 40 14.3(1 1.302S297 Oet. 30 177 15. 7 (i 341 4 43. si 40 17.29 1.3029008 183'J, Feb. 27 178 25. 5 S 342 21 54.35 40 18. S3 1.3029034 Juno 27 179 34.82 — 50.9 243 39 4.45 4(1 19.00 1.3030170 Get. 25 180 43.(18 344 5(1 14.08 40 17.84 1.3030030 1840, Feb. 22 isl 51.42 34(1 13 24.23 +0.17 40 15.28 1.3030990 June 21 182 .. 58.3(1 347 30 34.70 40 11.29 1.3031209 0!t. lit 1S3 4.34 —58.4 348 47 4(1.09 40 O.Ol 1.3031447 1841, !• el). 1(> 184 9.47 350 4 58.23 45 59.35 1.3031528 .luiie Hi 185 13.3(1 351 22 11.79 45 51.:>3 1.3031509 Oet. 14 18(i 1(1.11 352 39 2(1.97 45 41.91 1.3031392 1842, Feb. 11 187 17.73 —(10.5 353 5(1 43.91 +0.15 45 31.07 1.3031104 Juuo 11 188 IS. i:t 355 14 2.89 45 18.90 1.3030817 Oet. il 189 17.35 25(1 31 24.13 4: 5.38 1. .3030351 1843, Feb. (i 190 15.. 50 .357 48 47.74 44 50.50 1.3029707 Juuo (i 191 12.83 — (U1.7 359 r, 1 3.(15 44 34.22 1. 302900(1 Oet. 4 192 9.27 23 42.19 44 10.04 1..302822S 1844, Feb. 1 193 4.7(1 1 41 13.59 +0.07 43 57.59 1.3027207 May 31 194 59.91 2 58 47.55 43 37.23 1.3020173 Sept. 28 195 54.(11 —(14.9 4 1(1 24.34 43 15.53 1.3024947 1845, Jail. 2li 19(1 49.19 5 34 3.83 42 52.43 1.3023588 May 2(i 197 43.90 (1 51 4(1.0(1 42 27.99 1.. 3022095 Sept. 23 198 38.92 a 9 31.09 42 2.22 1.3020472 1840, Jan. 21 199 34.37 —(17.7 9 27 19.04 +0.15 41 .35.11 1.3018722 May 21 200 " 30.55 10 45 9.82 41 0.70 1.3010851 Sept. 18 201 —0 27.21 1 2 3 3.. 39 —40 37.01 1.3014801 104 THE OllBIT OP URANUS. IIeliocentuiu Ei'Iikmeuis op UiiANUS. — Continued. 1 Date. Hum of Approjciiiinte Logarithm Qre«iiwioli uieau noun. No. p«itiirba- tiouH. Heitiirballons pi'odiicetl by Neptniie. Longitiule. noireclloii. LutltuJu. UiKliUll vector. r tf /; o // // / (/ 184T, .Tan. 10 202 — 25.04 13 21 0.27 —40 0.03 1.3012758 M.iy 10 203 24.73 —70.2 14 39 0.21 39 33.75 1.3010544 Si'pt. 13 204 24.95 15 57 3.01 39 0.23 1.3008219 1848, .Inn. 11 ' 205 20.43 17 15 10.40 + 0.00 38 25.40 1.3005790 May 10 1 200 28.95 18 33 21.30 37 49.42 1.3003203 Sept. 7 ' 207 32.75 —72.8 19 51 30.15 37 12.18 1.3000040 1849, .Ian. 5 208 37.59 21 9 65.34 30 33.73 1.2997923 May 5 209 43.47 22 28 19.02 1 35 54.10 1.2995115 Sept. 2 210 50.27 23 40 4S.13 1 35 13.30 1.2992224 Dfc. 31 211 57.74 Y') I> 25 5 22.50 1 —0.08 34 31.47 1.2989252 1850, April 30 212 1 0.18 20 24 2.38 33 48.40 1.2980203 Aug. 2S 213 1 15.11 27 42 48.39 33 4.37 1.2983078 Dec. 2(i 214 1 24.38 29 1 40.87 32 19.00 1.2979881 1851, April 25 215 1 34.33 —77.9 30 20 39.78 31 32.79 1.2970015 Aug. 23 210 1 44.28 31 39 45.91 30 45.40 1.2973287 Dfc. 21 217 1 54.21 32 58 59.55 —0.09 29 57.09 1.290989i 1852, April Aug. 1!) 21.S 2 4.12 34 18 20.89 29 7.71 1.2900430 n 219 2 13.01 —80.5 35 37 50.43 28 17.33 1.2902910 Dec. 15 220 2 23.00 30 57 28.19 27 25.95 1.2959338 1853, April 14 221 2 31.54 38 17 15.00 20 33.02 1.2955701 Aug. 12 222 2 39.58 39 37 10.77 25 40.35 1.2952015 Dec. 10 223 2 40.02 —82.0 40 57 10.17 —0.07 24 40.17 1.2948270 1854, April 9 224 2 52.04 42 17 31.20 23 51.08 1.2944458 Aug. 7 225 2 57.77 43 37 50.24 22 55.08 1.2940583 Dec. 5 220 3 1.00 44 58 31.59 21 58.25 1.2930045 1855, April 4 227 3 4.03 —84.0 40 19 17.49 21 0.51 1.2932045 Aug. 2 228 3 5.35 47 40 13.93 1 20 1.97 1.2928582 Nov. 30 229 3 6.. 32 49 I 21.20 —0.11 19 2.00 1.2924449 185G, .Mar. 29 230 3 3.95 50 22 39.57 8 2., 55 1.2920239 July Nov. 2" 231 3 1.24 — Sli.l 51 44 9.08 17 1.07 1.2915954 24 232 2 57.02 53 5 49.32 10 0.03 1.2911592 1857, Mar. 24 233 2 52.95 54 27 40.00 14 57.72 1.29071.55 ■July Nov. 22 234 2 47.58 55 49 42.98 13 54.71 1.2902043 19 235 2 41.05 —87.9 57 11 50.10 —0.10 12 51.08 1.2898057 1858, Mar. 19 230 2 35.18 58 34 20.10 11 40.89 1.2893395 July Xov. 17 237 2 28.82 59 50 54.59 10 41.90 1.2888059 14 238 2 22.40 01 19 39.71 9 30.91 1,2883853 1859, Mar. 14 239 2 10.35 —88.0 02 42 35.14 8 30.87 1.2878981 July Nov. 12 240 2 10.33 04 5 41.38 7 24.01 1.2874049 9 ' 241 2 5.17 05 28 57.59 —0.13 17.90 1.2809001 18G0, Mar. 8 242 2 0.07 00 52 24.15 5 10.84 1.2804023 July Nov. 243 1 57.10 — S9.1 08 10 0.90 4 3.42 1.2858941 3 244 1 54.37 09 39 47.78 2 55.74 1.2853810 1801, Mar. 3 245 1 52.01 71 3 44.98 1 47.83 1.2848058 July Oct. 1 240 1 51.92 72 27 52.38 _ 39.09 1.2843408 29 247 1 52.43 —89.3 73 52 10.31 —0.19 I-l- 28.00 1.2838257 1802, Feb. 20 ' 248 1 53.51 75 10 38.24 1 37.00 1.2833029 June 20 1 249 1 55.99 70 41 10.05 2 45.40 1.2827788 Oct. 24 ; 250 1 59.54 78 5.40 1 3 53.95 1.2822542 1803, Feb. 21 ' 251 2 3.98 —88.5 79 31 4.97 5 2.40 1.2817290 June 21 i 252 2 9.43 80 50 15.07 10.80 1.2812047 Oct. 19 , 253 2 15.03 82 21 30.01 —0.04 7 19.09 1.2800819 1804, Fob. 10 1 254 2 22.52 83 47 7.95 8 27.25 1.2801009 .luiie 15 255 2 30.20 —^7.2 85 12 .'•)0.83 9 .35.22 1.2790421 Oct. 13 250 —2 38.39 80 38 44.88 + 8 42.95 1.2791203 TUK OKU IT OF UUANUS. 105 IIi-.iiiucKNTUic Ki'iiKMKiiis OF UiiANi 1). — Voniltidfd. Datfl. Sum of ApproxImRte pertarlmtinna prrdiiceil by Nvptniie. Logarithm Oreeuwit'h muftn iioou. No. perturl)*- tiona. Longitada. Correction. Latitude. Radius vector. / tt // / // (/ / If 18G5, Feb. 10 2.'-) 7 —2 47.06 «8 4 00.26 -f 11 52.20 1.2786142 Juno lU 2.')8 2 55.93 89 31 7.17 12 57.54 1.2781063 Oct. 8 2.') 9 3 5.01 —85.2 90 57 35.61 —0.09 14 4.34 1.2776029 1866, Feb. 5 260 3 13.91 92 24 16.00 15 10.74 1.2771044 Juno 5 261 3 22.61 93 51 8.39 16 16.70 1.2766112 Oct. 3 262 3 30.79 95 18 13.06 17 22.18 1.2761237 18CT, Jnn. 31 263 3 38.4i» —82.5 96 45 29.91 18 27.15 1.2756419 Mny 31 264 3 45.25 98 12 59.44 19 31.54 1.2751656 Sept. 28 2(;5 3 50.96 99 40 41.71 —0.08 20 35.37 1.2746948 1868, Jim. 26 266 3 55.67 101 8 36.54 21 38.51 1.2742299 May 25 267 3 69.06 _79.1 102 36 44.31 23 40.96 1.2787709 Sept. 22 268 4 1.02 104 5 5.00 23 42.75 1.2733175 1869, Jnn. 20 269 4 1.71 105 33 38.41 24 43.69 1.2728693 Mny 20 270 4 0.81 107 2 24.67 25 43.81 1.2724264 Sept. 17 271 3 58.59 —75.1 108 31 23.. 55 —0.11 26 43.07 1.2719888 1810, Jan. 15 272 3 54.96 110 34.97 27 41.44 1.2715563 Mny 15 273 3 50.23 Ml 29 58.50 28 38.77 1.2711290 Sept. 12 274 3 44.34 112 59 33.91 29 35.16 1.2707070 1871, Jan. 10 275 3 37.90 — C9.8 114 29 20.79 30 30.41 1.2702903 May 10 276 3 30.56 115 69 19.11 31 24.57 1.2698791 Sept. 7 277 3 22.61 117 29 28.39 —0.1 8 32 17.57 1.2694735 1872, Jan. 5 278 3 14.50 lis 59 48.05 33 9.31 1.2690738 May 4 279 3 6.19 — C4.2 120 30 17.95 33 69.79 1.2686S04 Sept. 1 280 _2 57.05 122 57.87 +34 48.97 1.2682934 The next operation would bo to interpolate these co-ordinates to intervals of time suitable for the computation of a geocentric ephemcris, to correct the longi- tudes for solar nutation, and then to compute the geocentric right ascension and declination. This operation has not, however, been completely carried out except for most of the observations before ISIJO, and for three of the oppositions observed since, the latter being computed only as a check upon the accuracy of the com- parisons. As a general rule, it may be said that wherever a complete geocentric ephemeris, with the heliocentric cphemeris from which it was computed, were available, these ephemcrides were made use of in a manner which will be more fully described hereafter, while, in all other cases, the geocentric places were computed directly. It may also be stated here that Hansen's Tahka tin Solc.il have been adopted as giving the places of the sun to be used in computing the geocentric places. H May, 1873. 100 TUK OIIHIT OF UIIANUS. CIIAPTER VI. REDUCTION OP TIIK OUSKUVATIOXS OK UUANUS, AND THEIR COMPARISON WITH THE I'RECKDINU THEORY. Tub observations of Uranus naturally tlivide thonisclvos into two distinct classes. (1) The purely accidental ones, made jjrevioiis to the recognition of the planet by Ilerseliel in 1781, and therefore witliout any suspicion on tlu; part of the observers that the oltjeet was not u iixed star, and ('2) the systematic observations made since. The first class arc nearly all so uncertain in comparison with the second that I have hesitated over the (piestion of employing them at all. If nothing but a determination of the elements of Uranus were called for, they would certainly not be worth using, since these elements miy be determined with entire certainty from the observations which have been nuide during the entire revoluticm of the planet since 1781. But the mass of Neptune is also to be determined, and it is at least possible that these observations, uncertain though they are, may add materially to the weight of this determination. I have, therefore, determined to include them all, re-reducing them when there seemed to be good reason so to do. The earliest observations are those of Flamstead, ])Td)lishcd in the Ilistoriic Coelestis. The observations themselves, as printed, together with the principal elements for reduction, are given in the following tables. The first colunni of the table gives the name of the star. The second gives the clock time of transit over the wire of the quadrant as given by I'lamstead, The time, it will be seen, is only given to entire S(>conds. AVe must, therefore, exj)ect to find a probable error, of which the mathematical minimum is ()\2o, and of which the minimum we can reasonably expect is much greater. Next we have the apparent right ascensions of the stars as comjjuted. For these data I am indebted to I'rof. Coffin, Superintendent of the American Epiiemcris. The mean ])laces are mostly derived from the " Star Tables of the American Ephcmeris," and from the two Greenwich Seven Year Catalogues, while the reduction to apparent place is made with the modern constants. The fourth coliunn gives the a])parent clock correcticm for sidereal time, in which is inchuled the eflect of deviation of the instrument from the meridian. The clo( k keeping mean time, the errors are in the next column reduced to those of sidereal time at the moment of the transit of Urainis. The next two columns give the corrections for clock rate, and for deviation of the instrument from the meridian, as inferred from the observations themselves, both being referred to the time and position of the transit of Uranus. '» t • ' W.mjH i .A. « R»U ' -lMfflWaj- TIIK ORBIT OF URANUS. 107 '» t In the last column wo Imvo the seconds of conclnd.'d com-ction for clcx instrument to be upplied to tlie observed time of transit of Uranus. k and IG'JO, Ihwmber 23. Star. a Arietis, 7t Arietis, a Arietis, f Arietis, 5 Arietis, n Tauri, Uranus, A Tuuri, Tiino of Tp. Ii. III. n. 7 48 40 8 30 54 8 ;j:} 17 8 40 21 8 52 44 !) 27 47 9 41 4!) y 45 3 U. A. of Htur. II. III. H. (.'lork h. III. B Iliijhl AscciiHion. t". m. «• 58 29.3 1 49 52.1 —5 58 47.9 2 32 8.4 —5 58 45.6 58 33.9 2 34 31.0 -5 58 4(5.0 58 34.7 2 41 38.0 —5 58 43.0 58 33.0 2 54 3.0 -5 58 41.0 58 33.0 3 29 13.0 -5 58 33.4 58 31.1 U. «. —1.1 —0.7 —0.7 — O.G —0.5 —0.1 58 32.4 5() 49.3 58 39.0 3 4(J 31.1 -5 58 31.9 aViiginis,19 4 15 13 8 58.2 -5 55 1G.8 o Bootis, 19 58 31^ 14 1 34.0 -6 50 57.5 Hourly rate of clock, Deviation of instrument for each degree of Z. D., Transit of Uranus, Correction for clock and instrument (mean), Observed 11. A. of the planet. a Arietis, 7t Arietis, a Arietis, f Arietis, h Arietis, >7 Tauri, Uranus, A Tauri, 1690, Dcccmhcr 23. Z. D. observed. Refraction. 2?)° 29' 10' 35 37 31 32 28 31 30 18 55 41 23 15 55 55 20 55 52 35 15 55 +0' 33" 4-0 41 +0 +0 +0 44 35 37 31 3() 84 16 13 20 18 23 21 12 3 Circle reading for Uranus, corrected for refraction, Equatorial point on circle, Docliiiation of Uranus, from observation, Uranns, 6 Virginia, e Virginis, ^ Virginis, 1712, Ajnil h. m. a. 9 35 19 12 19 12 14 51 12 32 11 lil(jht Ascension. li. m. B 12 47 52.4 13 2 31.5 13 20 5.7 Dev, C B. -1.2 + 1.7 +2.9 —0.2 +0.5 —1.3 3i.6 32.9 32.5 33.8 33.0 32.5 0.0 —0.8 33.2 — 0-.6 —0.5 h. 9 -5 3 m. 41 49.0 58 32.8 43 16.2 DecUnafion. Dcfliiuition. 21° 58' 53" 9 4 46 58 4 37 31 40 6 53 Eq. point. 51° 28' 36' 28 40 28 28 28 28 51 51 51 51 51 42 27 12 19 51 28 32 31° 53' 11" 51 28 30 + 19 35 19 +47 33.4 -1-47 40.5 +47 54.7 The discordance of clock errors, and the time whic^h intervened between the transit of the planet and tliat of the first star, seem to render an accurate reduction impossible. 10b TIIK Oil BIT OF URANUS. nir), MarrJi 4. IHyht Aurrnnion T. II. III. K. d Lconis, 1 1 50 lU Urmius, 12 27 1 6 Virgiiiis, 1'2 49 41 11 45 23.G II. A. )l. III. K. 10 45 52.5 V. )i. III. *. 1 4 2ti.5 -1 4 17.4 li. m. 0". 20.5 21.1 Clock time of transit of Uranus, 12 27 1 Correction for clock and instrument, — 1 4 20.8 Ilight ascension of Uranus from observation, 11 22 40.2 1715, March 4. Vcctinalion. Z. I). II. Dec. d Lconis, 40° 19' 40" +1' 0" +5° 8' C Uranus, 46 33 10 +11 iVirginis, 40 13 20 +1 5 14 17 Circle reading for Uranus, Equatorial point. Observed declination of Uranus, Eq. point. 51° 28' 46* 51 28 37 46° 34' 11" 51 28 42 +4 54 31 1715, Mirch 5. Right Ascension T. h. in. fl. rfl-eonis, 11 46 24 Uranus, 12 22 59 b Virginis, 12 45 49 II. A. h. m. 8. 10 45 52.5- 11 45 23.6 0. h. in. R. — 1 31.5 C 25.5 29.1 -1 25.4 h, m. 8. 12 22 59 —1 27.7 11 22 31.5 Transit of Uranus, Correction for clock and instrument, Observed right ascension of Uranus, Tlie large apparent clock rate, and the colons after the time of transit, both throw doubt on this observation. Declination. The circle readings for the stars are the same as on the day preceding, while that for Uranus is 50" less. The declination is therefore 50" greater, or +4° 55' 21". 1715, March 10. lti(jhl Ascension. C. h. m. —0 40 -0 40 d Leonis, p* Leonis, Uranus, b Virginis, T. h. m. a. 11 25 58 11 32 28 12 1 42 12 25 18 R. A. h. m. n. 10 45 52.5 10 52 25.1 11 45 23.6 2.9 59.5 58.1 58.2 Clock time of transit of Uranus, Correction for clock and instrument. Observed right ascension of Uranus, -0 39 54.4 h. 12 —0 39 58.6 11 21 43.4 m. B. 1 42 TIIK OIiniT OF URANUS. 109 UvcUiialio)!. v] tl Loon is, ^i' liconis, Uranus, b Virginis, Z. n. 4(5° ID' JJ/J" 47 M M 40 27 4G 13 2Q 11. + 1' 1" 4-1 ;) +1 1 -i-i Dci'. 5° H' ()" 6 14 17 E.|. rt. 51° 'i< -i'r 51 21' 0!i Circle reading for Uranus, Eciuatorial point on circle. Observed declinutiou of Uranus, 51 46 51 +5 a Lconis, Uranus, V Virginis, 17 Virginis, X Virginis, 1715. Ajn-il 29 T. ii. III. *. 8 42 11 8 50 44 9 G 55 9 43 38: 11 32 48 RitfJit Ascension. n. A. h. 11 ni. •• G 28.3 11 31 14.1 12 8 G.5 13 57 47.G C. Ii. ni. •. +2 24 17.3 4-2 24 19.1 ^-2 24 28.5 +2 24 59.G 28 42 2ir 1 28 49 U 48 C 8.7 [6.4 19.8 33.1 Ti.e discordance of the clock corrections mak.-s a satisfactory determination ot the right ascension very difficult. 1 deem it best to reject the doubt.l o bserva- tiou of 17 Virginis, and the discordant one of . Virguus. 'ihc resuU wdl then be Observed transit of Uranus, '^Jorrcction for clock and instrument, )bscrved right ascension of Uranus, h. in. «. 8 50 44 2 24 17,6 11 15 1.6 Z. D. a Lconis, 43° 52' 40" Uranus, 45 45 30 v Virginis, 43 20 20 17 Virginis, 44 34 10 X Virginis, GO 23 5 Circle reading for Uranus, Mean equatorial point. Observed declination of Uranus, Declination B. -f 0' 55" 4-0 59 -fO 54 -fO 5G 4-1 41 Dpc. 7° 34' 51" 8 6 —8 7 11 53 33 55 42 Ef . Pt. 51° 28' 26" 51 28 25 51 28 39 51 28 G4 45= 46' 29" b\ 28 38 4-3 42 9 The next observations in the order of time are two by Bradley, discovered by Mr. Hugh Breen, but still unpublished. The following are tbe results as given by Mr. Breer in the AstronomiscJie NachricJikn, No. 1463. Mean Time. ^- ^■ 1748, October 21, 7 6 18.4 21 4 37.93 1750, September 13, 10 8 57.8 21 40 0.23 N. P. D. 107 29 104 42 33.9 I no TIIK ORRIT OF URANUS. Mr. Brccn remarks: "The right ascensions are very acciirute. It has been assumed tiiat the N. P. D., on l"i5(), September 13, is identical with // ('apricorni, with which it was compared. Tlie first observation was by the transit instrument, and the second by tlic quadrant." No ground is given for the above assumption respecting tile N. P. D. for tlie second observation ; it may, therefore, be omitted as valueless. In the year 1750 we have also two observatidns by Le Monnier at Paris. For these, and all the other observations by the same observer, 1 shall adopt the results given by Bouvard in the C'oimaissince dia Temps, for 1821, p. 341, with tlie cor- rections indicated by Le Vcrrier, in Connahsance iJcs Temps, for 1849, pp. 125 and 126. The necessary uncertainty of the observations is sucli that, consid: ring that Bouvard reduced them with the star positions of the " Fundumeula," scarcely anything will be gained by a new reduction. 1753, December 3, we have another observation of right ascension by Bradley. I adopt tlie result kindly communicated by my distinguished friend. Dr. Auwers. 1753, December 3, Ii. m. 5 33. R. A. = 22 23 21.59 1756, September 25. Observation by Mayer, at Gottiugen. I adopt the result given by Bessel, in Fundamenta Astronomia:, p. 284. li. m. 1756, September 25, 10 12. R. A. = 348 54.5 Dec. = —6 1 49.4 The following is a tabular summary of the preceding results, with their com- parison with the provisional theory. In the computation of the geocentric place the places of the sun were derived from Hansen's Tables. I am indebted to Pro- fessor Coffin for a duplicate computation of the geocentric places from the pro- visional ephemeris, which was executed by Mr. Joseph A. Rogers. - ate. Right Asceuslon. Deulinatiou. Correction to theory. 1 Obserratlon. i- Observation. £• R. A. Deo. liOng. dl 'ei J s r)-A. ■ ep H II h m s s 1 II II It II IfiflO, Dec. 23 3 43 lfi.2 14.7 + 19 35 19 7 +22 + 12 +24 1.04 +.027 171.'), Mar. 4 11 22 40.2 38.7 4 54 31 48 +22 —17 +28 1.00 Mar. 5 11 22 31.5 29.1 4 55 21 49 +30 —28 + 4* 1.00 Mar. 10 11 21 43.4 41.0 5 4i; 5(! +30 _ 8 +30 1.00 Apr. 29 11 15 l.fi 1.5 -f 5 42 9 U + 1 — 2 + 2 1.04 +.0.30 1748, Oct. 21 21 4 37.03 35.42 +37.0 +39.3 1.015 + .0.50 1750, Sept. 13 21 40 0.23 o / " 57.90 II +35.0 +35.7 1.05 + .022 Oct. 14 324 15 24. f) :47.r, —15 1 40.4 47.0 +37.0 + 0.0 +35.9 1.03 + .045 THE OUBIT OF URANUS. Ill 1 Date. Right Asoenalon. Declination. Correction to theory. • 1 ObserTatlon. o , Observation. b R.A. Dec. Long. 01 m 69 ^ ! / 1 " ; o ' '1 1 // O 1 It tf ti ItoO, Doc. 3 324 34 53.5 15.4 _14 53 20.2 32.4 438.1 412.2 438.8 0.98 +.047 h 111 8 1 s 1753, DfC. 3 22 23 21.(10 19.34 1 o ' " " n5f., Sept.25 348 .54.5,25.0 17()4 .Jiiii 15 12 37 39.0 50.4 433. S 435.0 1.00 + .048 _ 1 49.4 + 4 43 47.2 40.0 50.2 -i-9.5 (-17.4) _3.4 (-V.0) 44,4 1 -0.5 \ -(-25.7 1.05 + .013 17(!8, Dec. 27 31 2ti 52.0 32.0 Dfc. 30 31 20 45. S 38.0 12 15 35.0 12 14 65.4 30.0 55.9 419.4 4 7.8 413.0 1.02 + .045 1709, .Inn. 15 31 22 7.7:55.8 Jiin 10 31 22 23.4 11.1 li 14 2(;.0: 12 14 30.3 29.7 37.0 411.9: 412.3 — 3.7: —0.7 .Inn 20 31 24 0.0:43.7 12 15 19.0 18.« 422.9: 40.2 \ 412.5 1.01 + .049 .Inn. 21 31 24 33.8 14.1 Jan. 22 31 25 4.7 :47.7 Jan. 23 31 25 28.5 24.2 . ^ 12 15 31.8 30.8 419.7 41.0 12 15 45.7 44.7 417. 0: 41.0 12 10 7.5 :59.( , + ^■•^■- 47.9 J AVlu-rc no declination has been observe! Ll.e observed corrections m right ascen- sion have been changed to conretions in h.ngitnde on the hypothesis that the theoretical latitude is correct. The approximate formula is , . (Vx cos h , U-= . ,,, where sin jL cos /; = sin f cos a, f being the obliquity. DISCUSSION OF TIIK MODKUN OnSKIlVATION'S. TtciUctxon of the PiiblUhed Fesnlts of Olmrmilons lo a Uniforvi Sijstem. We have now to discuss the great mass of observations made at the pvincipal observatories of the world since the discovery of the planet by Herschel, m 1 .81 To make all the data of reduction rigorously homogeneous and uniform, it would be necessarv to completely re-reduce the greater part of the observations maue befop. 1850 using the modern values of the constants of reduction and to com- pare each observation separately with the geocentric place deduced from the pro- visional theorv. Such a reduction and comparison would be extremely desirable. Their execution would, however, involve an amount of labor for greater than it is now possible for the author to bestow upon the problem. A^ c must, therefore adopt the reductions which have been already made, applying such systematic corrections for reduction to a uniform system of star places as we have the means readily to determine. No reduced places are employed unless we can find data for some more or less accurate determination of these -^7'=^7';;' '^ "f^; "^^^J necessitates the rejection of a great mass of observations made at the mil o o W - vatories of the European contin.Mit, and published in the A.rononu.cJ.n ^a?• riehten We still have the following rich collection of materials at our disposal . 112 THE OUBIT OF URANUS. 1. Observations at Greenwich, 1781 to 1872. 2 Paris. 1802 to 1827, and 1837 to 18G9. 3 Kcinigsberg, 1813 to 1835. 4 Vienna, 1822, and 1827 to 1839. 6 Speier, 1827-29. G Cambridge, 1828 to 1842. 7 Edinburgh, 1836 to 1844. 8 Berlin, 1838 to 1842. 9 Vulkowa, 1841 and 1842. 10 Washington, 18G1 to 1872. 11 Leiden, 1863 to 1871. 12 Santiago, 1854 and 1855. As to the general distribution of tliese observations in time, we may remark that during the first three or four years the pUuiet was zealously observed at Greenwich. Observations then began gradually to fall off until 1798, in which year we find but one. From this time until 1814 only one or two observations were made at each opposition. They become a little more numerous, until 1829, when there is a sudden increase. Few interruptions have occurred since. With regard to the other observatories it may be said that from 1802 until 1830 there is a gradual increase in the number of observations, and that since the latter year the number of observations is entirely satisfactory. A great number of the observations were reduced with the star places of the Tabula: liegioniontana', and the entire Paris series arc reduced with the star positions of Le Verrier, given in his '■'■Anmilcti de V Obaervaloire Imperial de Paris" Tome II. As a preliminary to the discussion of the systematic corrections to the principal published reductions, I have prepared the following table, showing the corrections which must be applied to the places of the equatorial fundamental stars in the above catalogues to reduce them to the adopted standard, namely. Dr. Gould's coast survey list in right ascension, and Auwcrs' standard in declination. In the table of right ascensions tlie first column after name of the star gives the annual variation of that co-ordinate for the epoch 1860.0, as derived from Le Terrier's tables of right ascensions just cited. Next we have the corrcct'on to this annual variation, expressed in units of the fourth place of decimals, ti reduce it to that given in the " Star Tables of (he American EjJiemeris" the positions in which are founded on Dr. Gould's Catalogue. The fourth column gives the correction to the right ascensions of Lo Verrier for 18G0, in hundredths of a second of time. Subtracting from this column sixth-tenths of the preceding, we have the corre- sponding corrections for 1800. The last four columns give the corresponding num- bers for the right ascensions of the Tabulai RegiomontaiivR. The table of declinations shows, for different epofihs, the corrections necessary to reduce the tabular positions to those given by Auwers in his paper on the declina- tions of the fundamental stars THE ORBIT OF URANUS. 118 • I. RiaiiT Ascensions. Corrections to Ann. var. of _ Corrections to Date. ann. var. IStiU. A nn. var. R. A. B. A. Tah. Keg. 18B0. Ann. var. B. A. 18(iO. R. A. 1800. 18(iO. 1800. _ a Andromeda), 8 +3.0844 + 14 + 2 _ 6 3.0840 + 18 + 8 — 3 y Pcgasi, 3.0801 + 15 + 3 _ G 3.0824 _- 8 — 9 — 4 a Arietis, 3.3C44 + 12 _ 1 3.3036 + 20 + 6 — 6 o Coti, 3.1266 + 10 + 5 _ 1 3.1261 + 9 + 7 + 2 a Tauri, 3.4346 _ 1 _ 1 3.4335 + 10 + T + 1 /I Orionis, 2.8191 + G + 2 _ 2 2.8800 + 3 + 5 + ^ 3 Tauri, 3.1871 _ 4 + 1 + 3 3.7888 _21 — 4 + 9 a Orionis, 3.2460 + 5 + ^ + 1 3.2464 + 1 + 3 + 2 a Gcminovani, 3.8409 — ', + 1 3.8386 + 21 + 18 + 5 a Cauis Min. 3.1 i02 + '^ + 9 + 6 3.1455 + 12 + T fl Gcminorum, 3.6828 + 5 + 3 3.6807 +26 + 13 — 3 a Hvdroj, 2.9485 + 12 + fi — 1 2.9469 +28 + 16 — 1 a Leoui.s, 3.2030 + 15 + 6 _ 3 3.2014 + 31 + 13 — 6 p Lcoiiis, 3.0054 + 11 + ^ — 3 3.0640 +25 + 12 — 3 a Virginis, 3.1495 + 20 + 5 K 3.1497 + 18 + 2 — 9 a Bootis, 2.7325 + 16 + 3 — 1 2.7327 + 14 + 5 — 3 o' Libra;, 3.3044 + 6 + 3 _ I 3.3074 _24 — 6 + 8 — 5 + 3 — 3 — 5 — 5 — 4 + 2 — 5 ^- 3 — 2 _ 8 —12 — 6 —55 —.01 a CoroniB, 2.5318 + 12 + 1 — 6 2.5373 + 17 + 5 a Scrpcntis, 2.9488 + 8 + * — I 2.9513 —17 — 7 — 4 a Seorpii, 3.6654 + 16 + -1 _ 6 3.6672 • Hcrculis, 2.1322 + T + 2 _ 2 2.1319 + 9 a Ophiuchi, a, Lyra;, y Aquilffi, 2.1808 2.0312 2.8520 + 10 + 2 + 9 — 1 + 2 _ fl _ 2 _ 3 2.1183 2.0305 2.8546 +34 + 9 —17 + 15 + 1 — 8 a A(inila!, 3 Aquilaj, 2.9281 2.9466 + * + 5 + 1 + 3 _ 1 2.9281 2.9496 + 4 —25 — 3 —12 o' Capriconii, 3.3338 + * + 2 3.3349 ■ 1 — 6 a Aquarii, a I'iiscis Aust 3.0829 3.3311 + 13 + 8 + ^ _ 4 _ 1 3.0822 3.3326 +-n - 7 —16 a Pogasi, Sum, Mean, 2.9828 + 11 + 251 +8.5 + 81 + .02 — 1 —72 7 — 024 2.9830 + 9 +210 +7.0 1 — 1 +71 + .024 1 15 Uar. 1S73. 114 THE ORBIT OP URANUS. ■ II. DEC1>1NAT10((8. Corrections to Taliu » Regiomontani Correellons to Le Verrier. 1780. 1800. 1820. 1840. 1820. 1840. // n ir If // // a Andromcdfc, -fO.3 +0.2 +0.1 0.0 +0.3 0.0 y Pegasi, —0.2 +0.1 +0.4 +C.3 +0.1 +0.3 a Arictis, —0.3 0.0 + 0.3 +0.5 +0.3 +0.3 a Ccti, —1.0 —0.1 +0.7 + 1.G +0.1 +0.7 a Taiiri, 0.0 +0.1 +0.1 +0.2 +0.1 0.0 |3 Orionis, —0.5 + 0.1 +0.C. + 1.3 +0.3 +0.7 3 Tauri, —0.5 0.0 +0.f. + 1.1 +0.2 +0.5 a Orionis, —1.3 — O.f. 0.0 +0.G —0.2 +0.3 a aemiuornin, —0.8 —0.5 —0 1 +0.3 +0.5 + 1.0 a C<>!!:3 Min. —0.3 0.0 +0.3 +0.G +0.2 +0.5 3 Qcminorum, —0.3 0.0 +0.3 +0.G +0.2 +0.5 a HydroB, —0.2 +0.3 +0.7 + 1.3 +0.3 +0.7 a Lconis, -{-0.3 +0.4 +0.4 +0.5 +0.4 +0.5 Lconis, +0.2 +0.2 +0.2 +0.3 +0.2 +0.3 a A'irginis, +0.3 +0.G + 1.1 + 1.5 +0.5 +0.8 a Bootis, +0.4 +0.4 +0.3 +0.3 +0.4 +0.4 a? LibriB, +0.5 +0.5 +0.G +0.G +0.5 +0.fi a CoroniB, +0.7 +0.(5 +0.4 +0.3 +0.4 +0.4 a Serpentis, +0.2 +0.G + 1.1 + 1.5 +0.5 +0.9 a Scorpii, +0.1 +o.r, + 1.0 + 1.4 +0.4 +0.7 a llerculis, +0.6 +0.8 + 1.0 + 1.3 +0.5 +0.7 a Opbiuchi, +0.4 +0.5 +0.5 + 0.G +0.4 +0.G a Lyns, +0.7 +0.8 + 0.0 + 1.0 +0.4 + 0.4 y AquiliB, 0.0 +0.3 +0.G + 1.0 +0.2 +0.4 a Aquiloe, +0.1 +0.4 +0.7 +1.0 +0.1 +0.4 |3 AquiliB, +0.1 +0.fi + 1.3 + 1.7 + 0.5 +0.8 »* Caprieonii, +0.2 +0.0 + 1.G +2.3 +0.6 + 1.1 a Aquarii, —0.5 +0.1 +0.8 + 1.5 +0,5 + 1.0 a Piscia Aust. +1.0 + 3.4 +3.9 +5.3 +1.7 +2.3 a Pegasi, +0.4 +0.3 +0.2 +0.1 +0.3 +0.2 Mean +0.03 +0.3G +0.C9 + 1.03 +0.3G +0.60 1 THE ORBIT OP URANUS. 115 The coxTection to the reductions to apparent place given in the Tabulae Rcgio- montansc on account of the correction to the constant of Nutation is;— In right ascension : — 0".46 sin ft — 0".l8 sin Q sin a tan S — 0".24 sin Q cos a tan 8. In declination: — 0"18 sin a cos a + 0.24 cos Q sin a. The terms which contain tan S as a factor may be entirely neglected, as they are small, periodic, and contain tan h as a factor which is sometimes positive and sometimes negative. I shall also neglect tlic corrections in declination, as their sum is sensibly 0".21 sin (a - 8) the effect of which will generally be confounded with the accidental errors of observation. The only correction we shall apply on account of nutation is, therefore, ^a = — 0\()30 sin Si. The values of this expression at the dates when it is zero, a maximum, or a minimum, are as follows : — 1778.5 1783.1 1787.7 1792.4 1797.0 1801.7 1806.3 1811.0 1815.6 1820.3 —.03 .00 +.03 .00 —.03 .00 +.03 .00 —.03 .00 1820.3 1825.0 1829.6 1834.3 183S.9 1843.6 1848.2 1852.9 1857.5 .00 +.03 .00 -.03 .00 +.03 .00 —.03 .00 Having adopted this system of standard positions, we may adopt two ways of reducing the observations to it. One is to compare the positions of the stars adopted in the published reductions with the standard, and apply the mean differ- ence to the reduced place of the planet. Another is to make a similar com- narison of the standard cata1.-ue with the positions of the fundamental stars which have been deduced fr- m observations by a system of reduction uniform with that employed in reducing Uie observations of the planet, and to regard the mean difference as a correction appU :.ble to all the positions of the planet. If the standard catalogue and the observations are both free fram systematic error, the results obtained in these two ways should be substantially identical. 1 hese are however, conditions which we cannot expect to find fulfilled. In the follow- in-^ discussions I have sometimes used one, sometimes the other, and sometimes 116 THE ORBIT OP URANUS. Mr combined both, the choice being determined by circumstances. We shall con- sider the different series of observations in succession. Greeincicli Ohaeivatii/ns from 1781 to 1830, These observations are completely reduced by Airy and comp;.red with Bou- vard's Tables, in the work licduction of the Observations of Planets made at the Royal Observatory., Greenwich, from 1750 to 1830, London, 1845, The con- cluded positions given in this work depend mainly on the star places of the TabuliE Regiomontanic, both in right ascension and declination. If we consider the first four oppositions — 1781-1785 — as forming a single group of which the mean epoch is 1783, wo find that the general correction to the TabultE Ilegio- montanse for this epoch is In right ascension, — 0'.030 ; In declination, +0".08. If, on the other hand, we consider only the particular stars compared with Uranus, the result will be a little different. The number of times each of the fundamental stars has been compared with Uranus, and the correction in right ascension cor- responding to each star, are nearly as follows : — a Arictis, JV = = 2 Cor. = -0.09 iVX c = — .18 a Tauri, 2 —.01 — .02 y Pcgasi, 2 -.03 — .06 (i Tauri, 19 +.13 +2.47 a Orionis, 33 +.02 +0.66 a Canis Minoris, 33 —.02 —0.66 /3 Geminorum, 34 —.07 -2.38 a Leonis, 7 —.11 — .77 /i Leonis, 2 —.07 — .14 The mean correction from these data comes out — 0".008, differing by 0\022 from the general mean correction. Our choice between the two (corrections depends on whether we are to consider the relative positions of the Tabultc Regiomontamr, or those of the standard catalogue, as nearest the truth at the epoch 1783, and particularly upon whether we arc to consider the large correction to the proper motion of (3 Tauri as real. In the absence of exact data for settling this ques- tion, the mean of the two results, or — 0'.020, has been adopted. A similar anomaly is exhibited by the declinations. It is probable that the declinations of Uranus during this period mainly depend on stars in the first twelve liours in right ascension, for which the mean correction is about — 0".,'50 instead of +0".08. I have adopted — ()".16. Changing these corrections to lon- gitude and latitude, we have, during the period 1781-1786: — Correction to observed longitude, = — 0".30 ; Correction to observed latitude, — .19. TllK ORBIT OP URANUS. 117 Dtuing the years 1788-1798 the above systematic iliffeience in right ascension does not appear. The most probable correction seems to be A8 = 0".00. Mat. = —()". 10. Aa=— 0'.02"j; Wliencc A long. = — 0". 34 ; Between the years 1800 and 1823 the stars used for comparison are so widely scattered tliat I consider it sate to apply only the general mean correction for the epoch 1813, which is Aa = — '.005 ; Ac^ = +0".()6. Whence A long. = 0".00 ; A hit. = -f .GG. From 1825 to 1830 more than half the weight of the right ascf^nsion comes upon the stars a, (3, and y AquiUv, the mean correction to wliich, during tliis interval, is — 0'.035. The general mean correction at this epocli is -|-0'.002. I think the right ascensions of these three stars in the Tabultc Kegiomontana; are really too great at this epoch by the ehtire diff'ereiue of these results. We may, in fact, hereafter regard the positions of the standard catalogue as sufficiently accurate, 'llie mean corrections to be applied will llien be A'( = — 0\017; A5 = +0".83. Whence A long, =• — O'.O.J ; A hit. = -f<'"-^^- From the year 1831 until the present time the Greeiuvich observations are regu- larly reduced in the several annual volumes of observations. But a reduction of the observations from 1831 to 1835, executed by Mr. Hugh Breen, is given in an appendix to the volume for the year 18G4. The results here given differ from those published by Pond in the several annual volumes for tlie same interval. The right ascensions are altered only by a])plying the constant correction — 0".030, which is found necessary to reduce Pond's right ascensions to those of the Tabula; Regiomontana". This correction I have verified. The mean correction to reduce the right ascensions of the Tabuhr Regiomontana; to oiir standard is at this time -|-0\005. On the other liand, when we compare the concluded right ascensions of stars within six hours of Uranus, as given by Pond in the (ireenwicli observa- tions for 1834, with our standard, we find a mean correction of — ".034 to reduce his positions to the standard, wliieh implies a correction — ".004 to Breen's reduc- tion. The two residts being -|-'.005 and — ".004, I have applied no correction whatever. In the paper in question the declinations are comi>letely re-reduced, using im- proved data of reduction, but, so far as I see, making no changes in Pond's method. The results diff"er strikingly from those of Pond, and suggest the desirableness of a complete re-examination of all Pond's determinations of decli- nation. Having no catalogue of observed declinations of standard stars reduced in this same way, we cannot directly determine the systematic correction to the declinations. I therefore proceed as follows: A comparison of Pond's observed declinations of standard stars with Aiiwers' normal catalogue show that the former require the following corrections near the parallel of Uranus : 118 THE ORBIT OF URANUS. In 1831 — 1".42; 1834 — 2.10. Then comparing Airy's reduced declinations of Uranus with Pond's, we find tlie following mean differences: In 1831, Airy — I'ond = — 3". 18 1834, — 3 .50. To reduce Airy to Auwers we must there apply to the declinations In 1831 + 1".76 1834 + 1 .40. I have regarded the correction -j- 1".60 as applicable throughout the period in question. 1836-72. During this interval the corrections in right ascension have been derived by the following two sets of comparisons: (1) A comparison of the several collected six and seven ycai catalogues with Gould's standard, from which it appears that they require the following general corrections in right ascension : Six year catalogue of 1840 + 0'.047 Six year catalogue of 1845 -f .002 Seven year catalogue of 1860 -f .003 Seven year catalogue of 1864 -\- .022 (2) A comparison of the corrections applied to the right ascensions of the indi- vidual years to reduce them to the standard of the catalogue, as given in the introduction to each catalogue. The sum of these two numbers gives the correc- tions for each year. A sliglitly different method is to regard the above correction for each catalogue as applicable to all right ascensions which depend fundamentally upon that cata- logue. I have sometimes combined both methods so as to derive what seemed to be the most probable result, and sometimes used but one. The corrections to the declinations during tlie interval in qnestion have been derived from Auwers' " Tafeln zur Reduction der DccUnationen verxcJiicdener Stentverzeichnime aiif ein Fundamentdxyfttem" Astronomische Nachncli/en, No. 1536. These tables include the Greenwich seven year catalogue for 1860, when the correction corresponding to the declination of Uranus is about -|- 0".45. The corrections for the previous catalogues vary between 0".35 and 0".68. The cor- rection corresponding to the interval 1861-67 has been derived by a direct com- parison with Auwers' declinations, and the result is -|- 0".44, agreeing with the two preceding catalogues. But, on making a similar comparison with the annual catalogue for 1S69, a considerable change was found, the correction being — 0".17, a change of more than half a second. I shall use this correction for and after the beginning of 1868, as the change is probably due to the introduction of a new constant of refraction in the reduction of the observations for 1868 and subse- quent years. i THE ORBIT OP URANUS. 119 Cambridge, An extended scries of planetary observations was eommcnccd here by Professor Airy, in 1827. Tlie series was continued by him and Professor C'hallis, his suc- cessor, until 1842. During the first three or four years the combined right ascen- sions depend on a few special stars, and mainly on a^ Capricorni. Taking the mean correction to the adopted right ascensions of the stars actually compared as they are given in the introduction to each annual volume, giving to each star a weight proportional to the number of comparisons, the following corrections are deduced : 1828 — OMO 1829-31 —0.10 1832-37 — O.in. In the introduction to the volume for 1838 it is stated that the adopted right ascensions are diminished by the average amount of ()'.()83, wliich would still leave a correction of — ()M07. Actual comparisons in two subsequent years give 1840, A« = — 0'.087 1842, - .069. Although the positions deduced from each year's work were adopted for clock correction" the year following, without any change of equinox, it seems that there was, eifcctivcly, a progressive change of about 0\0l annually in the equmox as " No*" declinations were observed until 1830. On comparing the declinations deduced from several years' work with Auwers, it was evident that the correction increased with the polar distance of the star. The law of increase could be well enough represented by supposing the correction proportional to N. 1 . D. i bus the following corrections were deduced in three different years. 1".78 X N. P. D. jn^egrees 100 1834, h dec. = — 1840, 1842, 1".00 X N. P. D. in degrees 100 1".03 X N. P. D. in degrees 100 From which the correction for other years was deduced by interpolation. But, on applying these corrections, the results were found systematically difterent from those of other observatories, and on referring to Auwers' corrections to Airy s Cambridge Catalogue, it appeared that the mural circle required a large correction near the declination of Uranus during this period. The above results were there- fore altered so as to conform as nearly as practicable to Auwers' law. EiUnhtirgJi. In reducing the observations of 1836 Henderson uses the right ««^">«;°"« °f theTabulffi Ilegiomontana^, to which the general correction is at this epoch + .UU1. 199 THE OIIBIT OF URANUS. But, if wc take only the stars near Uranus, with which the latter was necessarily most frequently compared, the corrections will be negative. C'omparing the con- cluded positions of the otars from a Serpentis through 0'' to fj Orionis, we Hud tho following mean corrections : In right ascension, — 0'.012; in declination, — 0".09. In subsequent years it is stated that the adopted positions of clock stars used each year ore derived from the right ascensions observed at Greenwich, Cam- bridge, and Edinburgli, during the year or the two years preceding, without any statement whether corrections were applied for ditfercnce of equinoxes. In some subsequent years the following corrc-ctions are deduced, sometimes from the adopted and sometimes from tho concluded positions : 1837, Aa = O'.OOO ; 1840, Aa = +0.015; A Dee. = O'.OO; 1844, Aa = +0 .070 ; A Dec. = +0 .45). Far 18. All the positions of planets given by Lc Verricr, in his '■'■Aniiahs tie VOhscrvn- toire Impcmd de Paris: C>i/<6/-f«/t«M«"depcnd both in right ascension and N. P. D. on his adopted positions of fundamental stars, the corrections to which iiuve already been given. As the corrections to the in • • < —.03 .... —0.8 .00 1838 —.04 +0.8 .... .... —.02 1 . . . . —.11 —0.5 +.01 1839 .... +0.T .... .... .... 1+1.0 —.01 .... —.10 —0.3 + .01 1840 .... +0.(! .... .... 1 .00 —.09 —0.2 + .02 +0.1 1841 +0.5 .... 1 .... j .... +.01 —.08 —0.2 —.04 +0.2 ' TUE Oil BIT OF UUANVB. 191 Table or Adoi>tkd Systematic Cobbections.— Co«/tnu»(/. Vuttr. Oraonwluli. I'irU. Kilnlgslwrg. U«rliil. Caiulirlilgs. Kurgh. 1842 1843 1844 1845 184(; 1847 1848 1849-53 1854-55 l85(J-(;o 1801 i802-f.5 l8(!li 18(17 1808 18(i0 1870 1871 1872 As 8 —.02 + .07 + .00 + .08 + .04 + .05 + .05 .00 .00 —.01 —.01 .00 .00 .00 .00 .00 + .01 + .01 + .02 AS II +0.4 + 0.4 +0.4 + 0.4 + 0.4 + 0.4 +0.4 + 0.4 + 0.5 +0.5 +0.4 +0.4 +0.4 +0.4 —0.2 —0.2 —0.2 —0.2 U.2 A. + "01 + .01 + .01 + .01 + .02 + .02 + .02 + .02 + .02 + .02 -1 .03 + .03 + .03 + .03 + .03 +.03 • . . . AS II +0.0 + 0.('. + 0.0 +0.0 + 0.0 + +0.0 +0.0 +0.2 + 0.2 + 0.2 + 0.2 + 0.2 +o.a +0.2 + 0.2 Ao H AS Aa 8 +.03 +.04 + .04 + .04 +.03 +.02 + .01 .00 —.01 At n +1.1 + 1.1 +1.1 + 1.1 + 11 +1.1 + 1.2 + 1.2 + 1.2 Aa 8 —.07 Sillll .00 . . • • .... AS II — o.:i in(fo -j 0.0 A» At 8 — 02 +0.8 + .07 +0.4 + .07 1 ^ 0.6 Wusliington .00 —0.5 .00 +1.1 .00 j +1.1 .00 +1.2 .00 +0.0 .00 +0.4 .00 -4-0.4 .00 -i-0.4 Applyiiij? the prrroding corroctions to tho positions of tho planet as orijjinally Tediux'd and publishi'd, we liavc a sciii-s of observed positions as nearly homo- geneous as it is possible to make them with the means now at our command. Tlio next step in order will be tiio computation of the geocentric place of the planet from the provisional theory for the moment of every observation, to be compared with the results of the latter. The complete execution of this labor, ab initio, is, however, at i)rcsent impracticable, and it is proposed to diminish it by making use of the published comparisons with the older tables. Tliis can be done without danger of serious error, and with all the more case that owing to the great distance of Uranus the errors of the solar tables are, for the most part, without appreciable effect upon tho computed geocentric place of tho planet. Tlie method of making the comparison is different with different series of observations, and each scries must therefore be described and discussed separately. The general plan, how- over, has been to replace observed and computed absolute positions by observed and computed corrections to the geocentric positions deduced from Bouvard's Tables. To carry out this plan it is necessary to have at our disposal an ephemeris botli of the heliocentric and geocentric positions derived from these tables. Tlie corrections to the latter given by the observations are then given by direct com- ])arison. To obtain the corrections given by the provisional theory, the heliocentric longitudes, latitudes, and radii vectores given by tliat theory arc interpolated to the dates of the heliocentric ephemeris from Botivard's Tables, and compared with that ephemeris. The differences are tli(>n chang(Ml to differences of geocentric place by the usual differential formuhr, and thus tlio cornn'tions given by theory are derived. The difference between the two sets of corrections is the difference 10 May, 1873. 122 T II K 11 H IT OF U II A N U S. between the provisional theory niul observation, A roti«b'ns(>(l ■nmmary of the results for euch of the priiieipal tieries of observations is here presented. Oirciivlch, nsl-lSHO. In Airy's rediietions, ahcady refcrvi-d to, we liave given for the moment of each individual obNervation u helioet^iitiie place computed from IJouvard's Tables, nnd the geocentric longitudes and latitudes thenee deduced. The observed right ftscensions and declinations are then changed to longitudes nnd latitudes, nnd thi; apparent error of the tables thence deduced. The means of these errors nre taken for groups of observations, and expressed in terms of tho errors of heliocentric longitude, radius vector, and latitude. The mode in which these means have been treated is fully shown in the following table. The first colunm gives tho mean date of each individual group of observations. The ne.\t three give the mean excesses of the co-ordinates interpolated from the heliocentric ephemeris, p. 100, and corrected for solar nutation, over those printed in the "Compututions of tabular place, etc.," in the Greenwich reductions. In tlie fifth column these corrections nre changed to corrections of geocentric longitude. In the next two columns we have tlie mean corrections to Bouvard's geocentric places given by observation. It is the negative of the mean error of tabular place printed in the " Reductions," corrected by the numbers ulready given to reduce the star places to ii nniforiu system. Then we have the difference between these two sets of corrections, or, the mean correction to the geocentric place of the provisional theory as given by observation. Lastly, we have the differential coefficients for expressing tlie errors of geocentric in terms of the errors of heliocentric co-ordinates taken without change from the Greenwich volume. From ProTinional Theory. From ObiiiTTationa. ~^ rorri-i'tlnn to Mean Date. Corroction to latmlar poaitlou in UreHUwich Kuductiona. Correction to I'lOV. 'i'l»«ory. dl 69 Long. W. Ut. Geoo. Oeoo. H«l. Noof (1(«>0. R. V. long. long. lut. Olw. 1 long. 1. // If ff If II 1791, Oct. 10 — 7.8 • • • • +1.1 — 7.5 — (5.4 + 2.0 4 + 1.1 + 1.5 1.01 — 9 Nov. 13 — 7.5 +1.1 — 7.5 — 4.9 — 1.0 4 +2.0 — 2.7 1.04 — 6 Dec. 27 — 7.3 — i59 + 1.0 — 7.7 — 5.(i — 2.3 5 +2.1 — 3.3 1.05 + 1 1782, J Ml. 31 — 7.4 +0.9 — 8.0 — 4.9 — 1.1 4 +3.1 — 2.0 1.04 + fi Mar. 4 — G.G +0.9 - 7.1 — 4.(i — 0.0 4 +2.5 — 1.5 1.01 + 9 Oct. 10 — 5.0 +0.4 — 5.3 -3.C — 4.0 4 + 1.7 - 4,4 1.01 — 9 Nov 2(i — 5.3 + 0.3 — 5.3 — 0.0 — 5.0 3 —0.7 ~ 5.3 1.04 — 5 1783, .Ian. 11 — 5.0 +0.3 — 5.4 — 3.9 — 3.7 3 + 1.4 — 4 1.05 4-3 Feb. Oct. 24 10 - 4.4 — 3.5 +0.2 —0.3 — 48 — 3.2 — 2.2 — 0.3 — 2.4 — 1.0 8 2 +2,0 — '..6 — 0.7 1.02 1. 00 + 9 —10 +2.9 Nov. 1 — 3.2 —0.3 — 3.0 — 1.1 — 2.3 2 + 1.9 — 2.0 1.02 — 9 Dec. 15 — 3.3 — i2i —0.4 — 3.4 — 2.2 — 3.8 3 + 1.2 — 3.4 1.05 — 3 1784, .Ian. 29 — 3.1 . • • • —0.4 — 3.4 — 2 4 — 2.4 3 + 1.0 — 2.0 1.05 + 5 Mar. Oct. 12 30 — 2.6 — 1.8 .... —0.5 —0.8 — 2.9 — 1.5 — 0.3 — 0.7 — 4.3 + 1.4 3 2 +2.0 — 3.8 1.01 1.02 + 9 — 9 +0.8 + 2.2 Dec. 14 — 1.5 — i28 —0.9 — 1.5 — 2.4 + 2.7 2 —0.9 + 3.0 1.05 — 4 Til 13 OUUIT OF UUANUa. 198 Kiuiii rruvltluual Tliuury. Kruiu OUarTtttiuua. M««u Datii. CurrHCtliin tn tnlmliir pnaltlou In (li'ouiiwiuli iilmirvittioiiii. Carrsatlon to Correilloii to Piov. riieoiy. vl 6-/. dl Lo(i. Lat. n Ohoo. Oeoo. llol. No.of n«oo. H«l. Long. R. V. long. long. lut. UU. long. lat. // It II II II 1785, Jhii. KI — 1.2 — 1.0 — 1.3 — 1.4 — 1.4 2 —0.1 — 0.4 1.05 + 8 KrI.. 1.1 — 1.0 — 1.1 — 1.3 — 1.2 i — 1.1 2 + 0.1 0.0 1.04 + ' Mm. 20 Nov. H — O.fJ + 0.2 —1.1 —1.5 — 0.9 ; — 2.1 + 0.5 ' + 1.2 — 5.2 » — 1.7 2 2 —1.2 — 4.1 1.01 1.02 +10 — 9 +0.7 — 0.2 1788, Mur. l.'l Oct. 2« + a.l + 3.4 — 'i87 —3.2 —3.0 + 1.8 + 4.4 1 — 4.9 — 2.2 3 2 +2^ — 1.7 1.03 1.00 + 9 —10 + 3.9 + 3.0 —0.9 + 1.4 178'J,Jan. 18 + 4.2 -'224 —3.7 + 4.5 + 1U.4 —14.5 1 (+6.9) (—10.8)' 1.00 — 1 Apr. 8 Oct. 31 + ^■1 + 4.8 —3.8 —4.2 + 4.2 + 5.4 + 9.4 + 4.7 — 3.7 — 1.1 4 2 + 5.2 + 0.1 1.01 + 10 —10 —0.7 + 3.1 1.00 1790, Juu. 24 + 4.9 4.4 + 5.2 + 3.1* — 2.6 2 —1.4 + 1.8 1.00 — 1 Nov. 5 + 6.1 - 254 —4.8 + 5.8 + 5.0 — 2.3 3 —0.8 + 2.5 1.00 — 10 1791, Jan. 29 + 5.1 — 258 —6.0 + 5.4 + 4.5 — 3.8 3 —0.9 + 1.2 1.06 Apr. 14 + 5.0 .... —5.2 + 4.5 + 2.7 — 4.8 1 —1.8 + 0.4 ! 1.01 + 10 Nov. 10 + 5.0 - 234 —5.4 + 5.0 + 5.7 — 4.0 2 +0.1 ~+ l74 1 1. 00 —10 1792, Feb. 5 Nov. 15 -f 5.2 + 5.0 —5.0 —0.0 + 5.5 + fi.2 + 4.2 + 3.7 — 4.0 — 2.5 1 3 —1.3 + 1.6 1.00 1.00 -10 —2.5 + 3.5 1793, Feb. 8 t 5.4 —0.1 + 5.7 + 8.6 — 5.0 2 + 2.8 + 1.1 i 1.06 Nov. 14 + 5.0 — 135 — G.4 + 5.9 + 10.4 — 6.3 1 +4.5 + 0.1 0.99 —10 1794, Fob. 15 + 5.7 — 100 —0.6 + 0.0 + 7.3 — 0.3 2 + L3 + 0.2 1.00 Nov. 19 + 5.4 — 07 —0.7 + 5.0 + 4.1 — 5.1 4 —1.5 + 1.6 f 1.00 —10 1795, Feb. 20 Nov. 29 + 5.9 + 5.4 — 40 — 13 —0.9 —7.0 + 0.3 + 5.5 + 0.0 + 3.9 — 8.0 — 3.4 3 2 —0.3 — 1.1 1.06 1.00 —10 —1.6 + 3.6 179fi, Feb. 24 + 5.3 - 8 —7.2 + 6.6 + 4.5 — 4.5 2 —1.1 + 2.7 1.06 1797, Feb. 27 1800, Mur. 14 1814, Mfty 22 + 5.1 + 4.0 — 1.2 + C2 + 224 + 217 —7.5 —8.4 —2.2 + 5.4 + 4.9 — 1.3 + 4.7 + 4.9 + 1.1 — 4.2 — 8.7 + 0.2 3 2 2 ^0/7 + 3.3 — 0.3 1.00 1.00 1.00 0.0 +2.4 ~+ 2.4 1815, May 25 — 1.1 + 214 —1.5 — 1.2 + 1.4 — 1.0 4 +2.6 + 0.5 1.00 1818, June 10 — 0.5 + 413 —0.8 — 0.5 — 1.1 + 4.8 2 —0.6 + 5.6 1.00 1819, Juno 14 — 0.4 + 483 + 1.5 — 0.4 — 1.4 + 3.7 4 —1.0 + 2.2 1.00 1820, June Ifi — 1 + 498 +2.2 — 0.1 — 2.0 + 4.0 2 —2.5 + 1.8 1.06 1823, July 1 0.0 + 538 +4.4 0.0 + 0.2 + 5.0 4 + 0.2 + 0.6 1.05 1825, July 11 — 2.4 + 507 +5.8 — 2.5 — 4.8 + 7.8 2 —2.3 + 2.0 1.05 1820, July l(i — 4.5 + 582 +0.0 — 4.7 — 3.0 + 8.3 4 + 1.1 + 1.7 1.05 1827, July 20 — 0.0 + 02( +7.2 — 0.9 — 7.2 + 9.6 6 —0.3 + 2.4 1.05 1828, July 23 — 9.8 + 710 +7.9 — 10.3 — 0.9 + 0.9 3 +3.4 — 1.0 1.05 1829, Aug. 7 —13.4 + 83.'i +8.5 —13.7 —16.3 + 10.0 14 —2.6 + 1.5 1.08 + ? Oct. 4 —14.2 + SO,") +8.0 —12.5 —13.0 + 9.9 8 —0.5 + 1.3 1.02 + ^ 1830, July 30 —17.8 + 97C +9.0 —18.8 —21.4 +11.1 3 —2.6 + 2.1 Aug. 29 —18.2 + 98!> +9.1 —17.it —20.0 +10.2 5 —2.1 -4- 1.1 Sept. 20 —18.5 + 99i +9.1 —17.2 —19.1 + 9.1 12 —1.9 0.0 Oct. 14 — 18.9 +100f +9.1 —10.8 —17.3 + 9.9 11 —0.5 + 0.8 Nov. 13 —19.3 + 100J +9.2 —16.8 —17.8 -\ a. 3 8 —1.0 + 0.1 124 THE OUBIT OF URANUS. Parh, 1801-1827. A complete reduction of this series is found in Le V'errier's Anindvs lie I'Obser- valoifc Imj}erud (Ic Purin, OhserontioHK, tome 1. No comparison with any ephemeris is given here, nor i« there any complete ephemeris to compare them with. A complete geocentric ephemeris was therefore computed from the provisional theory for the principal groups of the Paris observations. Tlic individual observations being compared with it, the resulting mean corrections arc given in the following table : Menn date. Aa AS N. Mcnn datfi. Aa AS N. 1801, ^larch 24, —".02 + 1".2 2 1813, May 20, +M9 +r.8 6 1802, April 1, + .08 +0.6 13 1814, May 27, + .21 +0.8 4 1805, April 22, + .10 +2 .2 13 1815, May 24, — .02 +2 .2 5 1806, April 17, — .01 -1.6 5 1816, June 1, — .01 +0.8 7 1807, April 28, + .17 +0.4 16 1817, June 5, — .08 + 1.6 5 1808, April 28, + .02 + 1.4 6 1818, June 7, + .12 +2 .2 9 1809, May 5, + .20 +0.1 9 1819, June 18, — .07 + 1.8 7 1810, April 30, + .22 +2 .6 16 1820, June 20, -.20 —2.4 8.5 1811, Fcbr'y 18, + .21 +2 .2 3 1821, June 22, + .05 + 1.0 5 1811, May 17, + .14 +2.6 8 1823, July 18, + .02 + 1.8 5 1812, Febr'y 1(5, + .28 2 1824, July 13, + .04 0.0 7 1812, May 10, + .16 +3 .0 6 1827, July 25, — .05 +0 .6 5 1813, Fcbr'y 25, + .44 3 Total number of observations in right ascension, 175. The observations in this series exhibit numbers of discordances of that class which leave the astronomer in doubt whether the observation should be retained or rejected. This remark applios more especially to the declinations. If we de- termine the probable error of an observation in declination by the condition that it is that amount which the error fulls short of as often as it exceeds, it is found to be about 2". Then, if the errors followed the commonly assumed law of proba- bility, only about one in six of the errors should exceed 4", and one in twenty- three 6". But errors of these magnitudes are much more numerous, the deviations often amounting to six or eight seconds. I nave rejected only a few in which the discordances approached 10". BessiTs IConiifgherg Obxcrrntions, 1814-1835. T have made a complete re-reduction of the right ascensions of this important series, and of most of the declinations. In order to avoid the necessity of ap|)ly- ing systematic corrections. Dr. Gould's right ascensions and Dr. Anwers' declina- tions were used througho)it in these reductions. In this work a selection of the fundamental stars observed by Bessel was made for each observation of the planet, to be used for clock error. These were chosen so tliat the mean of their right ascensions and declinations should be as near as practicable to those of Uranus, a condition, however, which coidd not generally be fulfilled for the decli- nations, owing to the southern position of the planet. Bessel's instrumental cor- THE OUBIT OP URANUS. 126 roctions were applied to his observed times of transit over the mean wire, and the resulting ti.ne was employed as that of transit. Each time, compared with the computed right ascension of the star gave a value of the clock correction, which was reduced to the time of transit of the planet by the known daily rate. It the instrumental errors were always accurately determined, the mean of these clock corrections would be used to obtain the right ascension of Lranus. But it was frequently found that the clock error varied systematically with the declination of the star, so that it was deemed advisable to add to the clock correction a term varying as the simple declination, which was deduced from all the stars, and used to reduce the correction to the parallel of Uranus. It was intended to give the results of this reduction for each observation, but on comparing the results with those of Fleming in the AstronomiscJie Xachnchten, Band 30, it appeared that the results were not materially better than his. It does not, therefore, seem necessary to give more than the mean results for each opposition. . From Bessel's declinations, with the old Cary circle, I was unable to obtain any satisfactory results, owing, apparently, to a wai.t of knowh-dge of some pernl.ar.ty „f tlie instnmunt. Fleming's reductions were therefore adopted, ihey are designated by the letter F in the following list. Mean Corrections to the Prormonal Eplicmeris ijiven ly Bo^scVs Oh^^crvalhns at Koii l/z>ums lu SjH;,rr vou F M. Mwerd, Speyer, 1S29-30, an.l of whi.h the reduced results are found iu tlie Astronomiscbe Naehricliten, Ba-.id 8, S. 2(14. ^ , . , 2. The series by Airy, at Cambridge, commenced iu 1828, and found in the Cambridge Observations. 3. Littrow's Vienna Observations, found in the fiist series of Annahn drr A. A. Stennvartc In Wlcn. The mean corrections to the ].rovI. ional epbenieris given by these series are shown iu the following table. The observations have been divided in the usual Menu ilal?. 1814, May 2\ 1815, May 25, 181(), May 27, IS17, June 0, 1818, June 8, 1820, June 21, 1821, June 23, Aa +M1 + .13 + .()') + .13 + .02 + .02 + .12 Aa +M0 — .05 + .01 + .01 — .01 — .15 —.10 +r'.8 -1 .()/<' — 1 .{)F -2 .4/'' —3 .OF -3 J)F -1 .OF N. J 126 TUE ORBIT OP URAXUS. way into groups of about a month each, and the mean date and mean correction found for each group. The Paris ond Konigsberg results are repeated for tlie sake of clearness. The small figuies show, as usual, the number of observations employed in forming the mean. Aa Ad Date. Observatory. Original. Corrected. 1827, July 22, Speier, — 0M6, — 0M4 July 25, Paris, _() .0:3. —0 .05 +0".53 September 15, Vienna, — 0.11,,— 0.10 .0,. October U, Vienna, —0.18, —0.17 -2.2, 1828, July 25, Kiinigsberg, —0.15, —0.15 -3.5, July 29, Vienna, —0.24, —0.20 -1.4, August 14, Vienna, —0 .13,,, —0 .09 + 1 .1,0 August 27, Speier, —0.10, —0.09 September 18, Vienna, —0.03, +0.01 + 1.0, September 25, Cambridge, _0.05, —O.IG October 17, Vienna, —0 .13,, —0 .09 0.0„ October 17, Cambridge, —0 .02,,, —0.12 1829, August 1, Konigsbcrg, _0.1()„ —0.10 — 1.0 August 6, Cambridge, +0.11, -0.08 -1.1.7 August 28, Speier, _0.04, —0.04 Soptembor 23, Cambridge, +0.21,0+0.05 November 6, Cambridge, +0.25, +0.09 Obnervations from 1830 io 1872. Since the year 1830 heliocentric and geocentric ephcmerides of Uranus com- puted from Bouvard's Tables are at our disposal. We make use of those in the Berlin Astronomisches Jahrbuch for the years 1830 to 1833, and of those in the Nautical Almanac from 1834 forward. The system of comparison is the same as that already explained. That is to say, we deduce separately : (1) Mean corrections to the geocentric longitude and latitude of Uranus hi the ephereris as derived from observation. (2) Mean corrections to the same, given by the provisional theory, as derived from a comparison of the heliocentric positions of that tiieory with the heliocentric positions in the ephemeris. Then (1) — (2) is the correction to the provisional theory given by observation. The process of forming (1) and (2) is shown quite fully in the following jiages. Each individual printed observation was first compared with the printed ephemeris, and a correction to the latter was thence deduced. AVhen this correction was given with the observations themselves, it was of course not recomi)uted, uidess in some doubtful cases. The observations were then divided into groups, usually of about a month each, and coinciding in time with the grouping of the Greenwich results. The mean of the dates and the mean of the corrections were then taken separately fir each group and each observatory. The separate results arc shown THE ORBIT OP URANUS. 127 I J in the proper columns of the following table, under the head "Mean dates," Mean cor. in R. A., and Mean cor. in Dec. These means are those given by the observations as printed, without the application of the systematic corrections on pages 120 and 121. In the columns " Corrected mean" these corrections are applied ; this column would therefore exhibit no systematic differences between the results of the different observatories, unless the observations of Uranus were affected by errors different from those which affect the positions of the fundamental stars. A careful comparison of the differences in various parts of the table shows that this is unfortunately the case. A weight is next assigned to each individual result depending on the number of observations, the general sufficiency of the data of reduction, the mean discordance of the individual observations, and the quality of the instruments. The critical reader will notice a lack of homogeneity among the weights assigned, of which I shall speak presently. The mean of the separate group-results is then taken with regard to these weights, and also the mean of tlie mean dates, using for the latter the relative weights adopted for the several right ascensions. Thus, we have a mean result derived from all the observations for each month, or other group-period, which is written under the horizontal lines. These corrections to right ascension and declination are next changed to correc- tions of longitude and latitude, using for this purpose the following table, which is computed from the formula; of Gauss : cos E = sin E cos a sec b = sin e cos I sec 5 , cos /; . . sin jP cos 5 ^ Al=: , - Aa coso cos b Ab= — cos E cos 5 Aa + sin E Ab. The differential coefficients in this table are expressed as a function of the right ascension of Uranus only, which may be done because, owing to the small iuclina- tiou and great distance of the planet, its geocentric position on the celestial sphere is never more than about 2' from some point of the projection of its heliocentric orbit. The coefficients of A« are multiplied by 15, that the right ascension may be expressed in time. To Convert Eiuiors of Riqht Asoe.nsio.v and Declination op Ukanus into errors op LoNdlTUDE AND LATrfUIlE. When the Ilight Asieiision exceeds 12', enter with R. A — 12^ aii.l oh-iiige the signs of Uie qnanlities (' b , 61 6a od ■J. A. 0" 0"" 10 20 30 40 60 1 10 20 30 40 60 Logniithms of 61 6a +13.8+ +13.9+ 6b 6a -5 97+ 6.96 6 95 5.92 6.87 6.81 -5.74+ 5.66 6.56 5.45 5.33 5.20 61 6S 6b 6b 61 6a 1.1386 1.1387 1.1388 1.1389 1.1390 1.1391 1.1392 1.1393 1.1394 1.1394 1.1395 1.1396 6h 61 \ 6b 6a 66 ' 68 —0.7761 , +9.6000 . 9.9626 —0.7757 "" ,1 +9.5996 ~ „ 9.9626 —0.7743 "~ '... +9.5983 ~ ',, 9.9628 —0.7720 ~ ';; +9.5963 "~ f,' 9 9632 —0.7687 — ii +9.5934 ~ T^l 9.9637 —0.7643 ~ ll +9.5896 _ ;," 9 9645 -0.7588 ,,„ +9.5849 ., 9,9653 -0.7525 — ''■] +9 57114 ~ '{[ 9.9662 -0.7451 ~ I* +9.573(1 ~ 74 9 9673 -0.7 65 ~ ,'; +9.56.-.6 "" r*. 9.96^5 -0.7J70 -,; ;; +9.5573 - :;: 9.9698 -0.71B2 ~},g +9.5479 ^^^^ 9.9711 +0.40— 0.40 0.40 0.40 0.39 0.38 +0.38— 0.3K 0.37 ■ 0.37 0.3(i 0.36 +0.92+ -4 0.92+ 128 THE ORBIT OP URANUS. To CoNVEiiT EiiRons or Uiqiit Asci .\sioN AND Declinatki.n. — C'oiiliniicd. B. A. LogarithniB of 61 ,5ft 61 O^ft ()l eft 5Z 6b da 6a ^5 dS 6a 6a '66 + 0.93+ 21' 0" 1.1.395 -0.7ss therefore does no harm. lesss TUE 0U13IT OF URANUS. 131 T Mean CoiuiiiCTioNS to tiik Kpiiemkhis ok Uranvs in tiik Bkiii-inku Jaiiuducu and the Aaltical Almanac. Obaervatory. [It. A. of Crauus. ] Mean dates. Obaerred oarrections in R.A. Konip;sl)org, Cambridge, [20" 40'°] KOnipKbcrg, Cambridge, [20' 3G'°] Cnmbridgo, [20" 37'"] Cambridge, [20" 3(1'"] Cambridge, [20» 3'i"'] Grcenwieh, Cambridge, [20" M'"} Greenwicli, Cambridge, [20" l>-2'"] Grcenwieh, Cambridge, [20" 50"'] Greenwich, Konig.«berg, Camliridge, Vienna, [21" U""] Cambridge, [21" 12"'] Cambridge, Vienna, [21" 0'"] Cambridge, A'ienna, [21" lO"] 1830 July 29 July 29 July 29 Aug. 12 Aug. 25 Aug. 19 Sept. 19 Oct. 17 Xoir. 14 1831 Aug. 3 A ug. H Mean. Aug. G Sept. 7 Sept. 15 Sept. 11 Xov. Oct. 4 20 ,s — 1.5(i —1.51 -l.f.5 -1.30 -1.4f. — 1.34 —1.30 Oct. 31 1832 Aug. 9 Aug. 10 Aug. 15 Aug. 3 Aug. 12 Sept. 12 Get. Oct. 12 Oct. 1 Nov. Nov. Ifi 9 -1.72 -1.70 -l.f.7 -1.00 -1.4S -1.54 -2.02 -2.24 -1.99 -2.33 -1.97 -1.91 -i.;io -1.S5 -1.89 Nov. 15 No. of obd. 11 9 10 10 13 15 Corrected Observed correotiona in Dec. Corr. to Qeocentrlo Mean. 8 —1.50, — 1.C7, —1.00 -1.05, — 1.52, —1.58 -1.62, —1.50, -1.52, —1.12. —1.86, —1.79" —LOT, -1.70, —1.72 -1.4S. —1.70. — 1.C3 -2.02, -2.24; -2.18, -2.33, -2.19 -2.1c,. -2.09,^ -1.90^ -2.01 -2.04, -1.89, -2.02 +4.0 -I-3.G -f2.5 -f3.5 -f3.1 -fl.7 40.2 -fl.9 + 1.1 No. of Corrected Lougitude. obd. ueau. n -1-4.0 -f3.C -fl.S +5.3 + 4.9 +3.4, + 1.1 4 2.5 15 +2.8 + 2.0 Latitnde. [-19.2] [+ 9.5] —20.8 +10.3 [-18.6] [+ 9..5] —20.8 + 9.1 [-17.2] —21.2 [-16.7] —19.5 [-10.0] —19.8 [—23.8] [+10.0] —23.4 -j-11.1 [—22.0] [+10.0] —22.2 I +11.8 [— 20.7][+ 9.8] —21.0 +10.8 [—28.4] [+10.3] 29.2 [—20.91 — 28.8 + 11.9 [—25.8] [+10.2] — 2G.G +11.1 [—25.2] [+10.1] —27.4 . +10.5 132 THE 11 BIT OF U II A N U S. Mean Coukeutions to the Kimiemeuis or Ukani-s.— C'o»<(Hwtrf Obmirvatory. [K. A. ur Mean dntei. Drauui.] ObBerr>t. 1!» Sept. 11 Sept. 1.S Oet. 11 Oct. 12 Oet. 14 Oet. 12 \ov. \ov. 1« 14 Nov. in 1834 Aug. 1 Aug. 1; Aug. Sept. Sept. 14 10 Hi Sept. 13 Oct. Oct. Oct. 14 l(i 20 Oct. 17 Xov. Nov. Hi 12 Xov. 14 Dec. Dec. Dec. 1835 A ug. Aug. Aug. Aug. 17 20 14 21 Aug. 17 R -2.57 -2.57 -2.40 -2.3.3 -2.33 -2.25 -2.20 -2.37 -2.10 -2.24 -2.58 -3.07 -3.00 -2.00 -2.74 -2.(18 -2.75 -2.04 -2.53 -2.7! -2.55 -3.25 -3.31 -3. 28 -3.30 M<'an. 7 5 11 5 10 4 12 10 11 4 5 18 I'l 4 11 20 1 H -2.57, -2.57. — 2.5'.i., —2.58 -2.3.!, -2.52. -2.50, —2.47 2. 25 — 2.48J —2.43 -2.35, o_28 —2.33 —3.11, —2.83 —3.00, -2^ —2.88 —2.74, —2.87, 4 _ 2.80, —2.83 -2.83, -2.58, —2.77 1 _ 2.00, • 50' —1.4 —1.5 —0.8 -f2.8 -f2.0 + 0.1 -f2.2 + 1.3 1.3 4.4 4.4 1.2 3.1 0.8 1.3 0.7 2.8 1.2 2.1 —2.88 —3.25, ' — fi.O —3.31, —3.47,:— 4.7 — 3.35„ _ 8.8 —3.39 No. of oIm. 4 H 4 13 11 4 5 18 10 10 4 11 8 20 1 Cormotdd uieuii. +"•2, — I .5. —0.0 H-0 8. + 1.2, + 1.2 41-T. -1-1. —0.1 Corr. to Geocentriu Lougltude. Latitude. [—33.8] — 35.0 [ + 10.8] -f 10.9 [—32.2] [+10.8] -33.4 -)-12.3 [-31.2] [+10.0] —33.0 +11.0 [—30.3] [ + 10.3] —0.1 I —32.0 , +10.4 —2.0 —3.5 —2.7 —3.0. 2 5 —2.0 -l-7,„ -2.1,„ —1.9 —2.0,, i_[_40.0] [ + 11.1] —39.7 +11.3 [-30,0] [ + 11.1] —40.5 +11.2 [-37.5] [+10.9] —39.5 i +11.0 ■■' [_3f,.7] [+10.7] 2.0 —38.7 , +11.3 -2., 5. — 1.2i —2.4 -5.4, -5.9., [—30.2] [ + 10.5] —40.3 -f 11.5 :^Aj[_47.1] [+11.3] -5.9 —48.0 +11.8 THE OKU IT OF URANUS. 133 Mean Corhkctions to tub Ki'iiKMKitm ok Uranus. — Continued. Olifler»iitory. [K. A. of Uiauua.] Canibridgo, Vii'iiiia, [22" 1'"] Oi'ot'nwicli, Cambridj^o, [21'' 57'" I Orcciiwicli, C'umlji'iilgc, [21" 57'"] Orecnwicli, Ciimbridge, [22" 21'"] Qi'ccnwirli. Ciunbridifo, I'Jiliidiui'^h, V^ieniia, [22" 20"] Greoinvic'i, Uaiiil)ridj?(', Kdiiiliiii'^li, Vii'iina, [22" l(i">] Qrcpnw it'll, CamhridfTc, Kdiribiii'>;li, Vienna, [22" l;)'"] Green wii'li, Camljrid^c, E)diid)iii'^li, V^ienna, [22" 12"'l Camhrid^o, Edinljurgii, [22" i:{"'] QreiMitt'ii'h, [22" 40'°] Ub8Hrr«(l uorraotions in R.A. , Observed correotiona in Deo. Corr. to Oeocebtrio Mean dates. 1835 Sept. 15 Sept. Sept. Od. Oct. Oct. II 15 10 17 Xnv. 27 Xov. 2() Nov. 2(1 1836 July 22 .Inly 25 July 2;j A lift. Auf;. A..-. AiiK. AngT Sept. Sept. S.pt. St'pt. 24 !J0 Hi 1!» 20 Mean. — :M'? — y.;50 —.3.27 —a. 11 -:!.2i -i.oo -.1.H0 -;{.bii«rv«(l correct! — Ciiiithiticil. Obs«rv«(l corrvvtiouH in It. A. ma ill Uec. Cnrr. to (l«oceiitrio ObHerviitnry. K. A. of Mvau dates. .... 1 Urauua. ) Moan. No. of Ciirrecldd No. of CoiTfcl«d Loiigltudv. Latitude. ulw. Ult'ttU. Muan. " obH. ui«nu. 1887 H H H '/ It Oreonwicli, Ang. IS — 4.;!0 10 -4.33, — 13.4 10 -12..5,,, Cainhritljrt!, Aug. IH — 4.0!l 14 -4.2S, — 12.(i 11 -13.4,, Eiliiihurgli, Aug. 22 —4.40 () —4.40, — 12.7 (i -1-^.T. Paris, Aiij?. 14 —4.34 9 -4.3,3, — 12.0 10 -11.4,„ Vii'iiiia, Aug. 2'> — 4.2'J 4 —4.33,, — 12.(! 4 — 11.(1, 1 [_(i2.9] L + II.7JI [22" Sfi-"] Aug 18 —4.33^ — 12.5 —(14.1 + 12.2 On'cnwifli, Si'pt. 17 4.23 14 — 4.2(!,.i — 13.4 14 -12.5,, KoiiijfttlR'rj;, Sept. 11 4.10 H _4.13„ CuiMl)ri, Sept. 17 —4. or, 14 -4,25„ —11.9 15 — 1-' 7 Kdiiiljurgli, Sept. 10 —4.311 4 —4.39, — 11.5 3 — 11.5, Piiris, Sept. IS —4.20 12 -4.19„ — 12.3 13 — 11 7;., Vii'niin, Sept. l;{ —4.12 5 _4.ir.. — 13.9 6 -12.9, [-(11,7] [ 1 ll.fi] [22" 32"'] Sejit. k; .... 4.21 .... —12.3 —(12.3 + 11.5 Orccnwk'li, Oet. k; —4.13 11 _4.1(V, — 12.9 11 -12.0,, Ciuiibridgc, Oel. 17 — 4.00 10 -4.19. — ll.(i 11 — 12.4,, Paris, Oet. 17 — I.O;-) 4 —4.04, — 11.2 4 -10.(1. Vii'iino, Oet. IS —4.41 2 -4.47, — 15.5 a — 14.5i [-59. S] [H1.4] [22" 2S'"1 Oet. 17 ... —4^ 1(1 " —12.0 —(11.6 -1 11.3 Orcenwicli, Nov. S —4.10 5 —4.13, — 12.4 5 — 11.5. Caiiiliridgp, Xov. 4 — 3.1M1 4 — 4.ir,; — 12.3 a -1.3.1:; Paris, Xcjv. —4.07 3 — 4.0(1' —11.3 3 -10.73 Vii^iiim, Nov. 2 —4.0!) 2 -'■'i —12.9 2 -11. 9j [—59.0] [-111.2] [22" 27"'] Nov. (J — 1.11 —11.7 —(10.7 -f 11.3 Grponwicli, Dee. 2 — 4.().'i 2 — 4. OS, — 14.0 2 -13.1, Cniiil)ri)i};e, Nov. 30 — 3.S7 (1 — 4.0(1.. — 11.(1 7 — 12.4. I'orls, Dee. H —4.03 7 —4.02; — 10.9 7 -10.3; Vifiiiia, Dec. S —4.28 2 — 4.3l; —13.0 2 — 12.0( [_.<,7.7] L+"OJ| [22" 28""] Dec. 5 1838 —4.05 ' ' ' ■ — ll.(! —59.9 + 11.1 Oropnwicl), A ug. 20 —4.72 11 —4.70, —15.9 n -1.5.1,, K(Mii(?sl)crjr, Aug. 2.') —4. (-.5 .5 _4.(1.\ — IS.C, 5 -17.(V, Ciuubridjtc, A ug. 1 '.) — 4.07 11 —4.7 s., — 15.S 12 — 1(1.3,,, Kdinburgli, Aug. 2.T — 4.!)(1 3 _4.1I.V, — 15.0 4 -15.0. Paris, Aug. 20 4. SO 9 —4.79, — 1(1.(1 9 — 1(1.0„ It [-70. 9] r4-ll.7l 1 [22" 51"] Aug. 21 ... —4.7(1 — 15.8 ^-71.4-' l_ 1 -J + 12.3 Qrwnwifh, Sept. 12 4.(12 S -4.(1(1., — 1(1.2 8 -15.4, Koiii(»sl)iTg, Sept. 1)> — 4.tl.'-. u — 4.(17,,, —IS. 5 14 —17.5, raml)rid)?e, Sept. k; 4.(11 11 —4.72, — 14.9 12 -15.4,, Edinburgh, Sept. 15 —4.74 9 -4.7;!, — 15.S 1 — I5.S. Paris, Sept. 12 —4.71 (i —4.70., — IC.O (! -15.4, Vii'iiiia, Sept. 11 —4.71 12 -4.7.3, — IS.l 12 -17.1, Berlin, Sept. t; —4.72 7 —4.74, —17.5 1 -1(1.5, [_70.4]'[+11.7] [22" 47°'] Sept. 1;') .... — 1.70 .... —1(1.0 —70.7 -\-\'i.i) III .,. THE ORBIT OP URANUS. 185 ., Mean CoftRKoxioNs to the Ei'iikmerib or U lANUS. — Continued. 01)imrvnlnry, (U. A. »r Mean (lat«a. Obaervi'd oorreotlona in R.A OI>serTed correotlona In Deo Corr. to Oeooentrio 1 1 UraiiuH. J Mean. No. of CiirroeteJ No. of Corrpoled Longitude batltnde. R uIm. liieau. Mean. obs. lueau. ff ! 1888 8 ff If ,, Orodiwicli, Oct. 15 — 4.fi7 1 -4.71, — 15.5 7 —14.7, CaiiiliridfTc, Oct. 17 —4.49 7 — 4.fiO, — 15.3 G — 15.M, Kdhilnii'^li, Oct. k; — 4.5fi 12 —4.55:, — 15.0 8 —15.0, Piiri.s, Oct. k; —4.(11! 7 — 4.fi5, — 15.9 7 —15.3, Viciitia, Oct. 14 — 4.00 9 — 4.fis', —17.3 9 — 1(>.3, [22" 44'"] Ocl.^K! — 4.fi3 — 10. 2 [— CH.f,] [f 11.5] — C9.7 1 +10.8 Qreciiwicli, \(iv. n —4.52 fi — 4.5fi, — Ifi.O fi — 15 2, Ciuul)ri(l(fo, Nov. 1.') — 4.;i7 10 -4.48, — 15.1 9 — 15.fi, K [ 1 [22" 42-"] Xov. 14 -4.^58 —15.8 — «8.8 + 10.9 Oropiiwicli, Dec. 4.fi0 2 — 4.fi4, — 15.9 2 —15.1, Ciiinliridp', l>cc. 15 — 4.2fi 5 —4.37., — 15.1 7 -I5.fi Eilinhtirgli, iKc 17 —4. IS 3 —4.17; —14.7 1 -14.7; Paris, Due. 5 —4.40 7 4.39, —15.1 7 — 14.5, [— fi5.8] — «5.7 [+11.1] + 10.4 [22" 43"] l>cu. 10 —4. 38 —15.0 1838 Qri'diwieh, Aug. 22 — 5,2S 5 —5.32, —21.3 5 — Sfi.fij Cutnbridgi', Aug. 24 —5. 1 1 S -5.22, —20.7 7 —21.0, Paris, A\ig. 23 ■■. 2 » 3 -5.1 9: —20.7 3 — 20.I3 fidinbiirgli, Aug. 25 —5 . ) 2 —5. 20, [-79.7] —79.7 [+11.7] + 11.0 [23" fi"'] Aug. 23 —5.22 .... . . . —20.7 Grooiiwicli, Sept. 10 —5.14 12 -5.18, —21.0 12 —20.3,, Kolliff.-iblTg, Sept. 12 —5. 1 1 12 —5.13. —21.7 12 — 20.7„ BlTJitl, Sept. 10 —5.10 14 —5.11. —21.1 14 —20.1, Cainliridgc, Sept. 17 '■ —5.12 11 -5.23: —20.0 10 — 20.3,„ Paris, Sept. 14 —5.23 10 —5.22. —20.7 10 — 20.1,„ Ediiiliiirgli, Sept. If, i —5.10 17 -5.15; —20.5 5 —20.55 ViiMiiia, Sept. 17 1 —5.32 9 -5.33, —20.8 9 -19.8, [-V9.0] —78.8 [+ ii.fi] + 11.0 [23" ;i"'] SeptrU —5.17 —20.3 Gn'ciiwicli, Oct. 12 — 5.1H 5 —5.22, —20.2 5 —19.. 5. Caiiiliridjfo, Oct. 1.". —5.07 S —5.18, —19.2 5 —19.5, Eiliidxirgli, Oct. 15 —5,14 10 -5.13, — 19.9 8 —19.9, ViiMiiia, Oct. 10 —5.17 13 —5.18, —20.5 13 —19.5, [_77.4]Y+11.5l| [22" 50'"] Oct. 14 —5.18 —19.6 —78.7 + 11.8 Grccnwii'li, Nov. 12 —4. 98 fi —5.02, —20.2 fi —19.5, Cnrnhridgo, Nov. Ill — 4..S5 fi — 4.9fi., — 19.1 5 —19.4, Paris, Nov. !» —4.99 2 —4.98, —20.4 2 —19.8, Ediiiburgii, Nov. 1 1 —4.90 3 —4.89, —18.3 1 —18.3, 1 r-75.3] r+11.3]| [22" 57'"] Nov. 14 .... —4.97 —19 i j --75.7 1 + 10.4 Grcenwieli, Deo. fi —4.84 2 —4.88, —19.0 2 —18.3, Cambridge, Dec. 15 —4. S3 5 —4.94., —19.1 2 —19.4, Paris, Dec. 3 -4.90 3 —4.89, -17.8 3 -17.2: Ediiibursli, Dec. 2H — 4.9(! 4 —4.95; — 18.5 3 — 18.5, 1 ' [—73.9] [ + 11.0] —18.2 j -74.(1 +11.2 [22" 57'"] Due. 12 .... —4.92 l'.]Cy TIIR OIiniT OF URANUS. MkaN C'llUllKl'I'lilNS Til TIIK I'll'IIK.MKIllH 111' UllANIIH. — O'lilli II liril. Ubattrfntory [It. A of Urnniin. ] Orccnwic'li, Ciiniltriil);)', Eiliiiliiir);li, [l>3'' 20'"] OriMMiwich, Kdiii^slii'r)r, Cuiiiliriil)!0, Kiliiiliiir(;li, I'liris, Berlin, Viuiinii, [2;!'' IH"] (Ireonwicli, Ciiinliridtfc, Kiliiiliiii'gli, I'lirU, IK'rliii, Vii'iiiiiv, [23'' 14""] Orcciiwicli, ("niiiliridgo, Kdiiibui-jjcli, I'liris, ViiMino, [23" 12"] Ort'oii\vi4'li. Ciiinln'idjji', Kdiiiliurj;li, Vii'ima, [23" 12""] Groonwicli, I'liris, [23'' ST"-] Oi'oonwicli, KoiHgsbi'i'g, Berlin, Edinbiirgli, Paris, ViiMinn, [23" 33""] OWwrvoil currHullniin In R. A. OIwitvhI ciiiTi'ilidiiH In I>i'i'. Ciirr tii (Iciii'i'iitrlo No. of I Corri'uti'd LiingltuilH liStituile. oIjm. iiioiiii. -23,!),, -2r).2, -23.7, [_S7.!.] [ + 11.4] 23. !l — S«.8 -j-l''" -23.4. -24.1, -2..,\ -23.(1. -•>:, 3, -21.2, —23.8 -23.0, -^3.7, -23.2,,, -22.7, — 2I..S -21.5, —23.3 00 n --" « -24.4, -22. H, -2.''). 5. [_87.0] [+il.5] —87.5 -f-11.4 —23.3 -22.7, -22.5., —22.8 2.S.1, 2il.O, 28 3 2S.5,„ 2-.U., 2«.r,, 2S.(), 28.4,, 29.3, [_85.8] [ + 11.4] —8(1.3 +11.2 [_S3.7] [ + 11.3] —84.4 -4-10.3 [—82.2] [ + 11.0] —83.7 +10.4 [— !)fi.!l] -2S.. [-nfi..5] — i)(!.3 [ + 11.3] + 10.G [ + 11.4] + l(».4 M * 1 TllK OUUIT Ol' UUANUS. 131 (i 1 MkaN ('.S7 2 —5,, ST, — 2S.2 2 7 —27.2, Kdiiihurgli, Oft. la — (i.OO 4 -(1.04, —27.3 4 1 —27.0. Piiri.s, Oct. 21 —11.10 1 — (l.O'.t, —27.2 1 -2(1.(1, V it'll mi, Oct. l(i —(1.14 "J —(1.14, — 2S.S !) : —27.8. [-94.0] —91.1 [+11.2] + 10.5 [2a'' 2S'"] Oct. IS —(1.0(1 ... —27.4 Grconwicli, Nov. 17 — .'■..n.-> 5 — 5.09j — 2S.0 5 —27.5 BlTlill, Nov. 111 — (1.1;-) 2 -(1.15, — 2s. ^ 2 —27.4,' Etiiiiliiirgli, X..V. 2 ' — f).8!» a — 5.!ia, —2(1.5 4 — 2r,.:t, Yiuiiim, Nov. U — f>.!KS 4 — 5.!)S, —30.0 4 —29.0 [2a'' 2(1'"] Nov. la — (1.01 —27.2 [-92.0] [-1 11.0] 1 —93.3 -f 10.4 1 Grooiiwicli, Dec. 14 —5.8(1 H — 5.!»0, —27.4 8 —2(1.9. Bi'i'liii, ]).c. l.i — ."i.dO a — 5.(10| —25.5 ■ 3 —24.5, K(liiiiiiii';.'li, Dec. 20 — .^.(iH 4 —5.7:1, ■ Puri.s, Dec. 1,^ —5.7s 4 — 5.77j —2(1.7 4 —2(1.1. [2a" 2(;"'] Dee. k; — 5.7s ^ 4 — 2U.¥~ [—90.5] —89.9 [ + 10.7] + 9.8 1842 GrocnwU'li, Aiijr. lit — (1.(12 5 — fi.r,4, 32. 1 (i —31.7, Ciiinliriilcc, AUR. 24 — (i.4s 2 — fi.55, -32.7 2 -33.0, [2a" ol'"] Ang. 21 —(1.(11 't ~32To' [—105.(1] — 103.0 [+11.1] -t 10.0 Orponwicli, Sept, 12 — fi.,W 4 —(1.57, —32.2 4 -31.8, KoniR.sbiT}?, Sept. is —(1.(1!) 10 —(1.110, 1 -:!:;. 4 10 —32.4. Uorliii, Sept. 11 —(1.70 3 -(1.(1!.; — ;!1.7 3 —30.7, I'«ri.>!, iScpt. 14 — fi.(15 5 —(1.(14., —32. 1 5 —31.5, Caiiiliridpo, Sept. 11 — (1.(10 1:5 — (l.(17„ —31.9 14 —32.2,, K(iiiil)iir};li, Sept. 2.! — fi.57 4 —(1.5!), —30.9 4 — 30.(lj Tulkowii, Sept. 17 -(1.(1 a c. — fi.(14, Vienna, Sept. 1',) — c.sl 2 — (l.SI), —33.5 2 — 32.5t [2a" 47'"] Sept. iiV — (l.dG r._l 05.41 T-l-ll oil —32.0 —104.3 L 1 J + 10.3 Orc'oiiwieli, Oct. 20 — r,.(ir, 12 — fi.OS, —31.8 12 —31.4,, Borliii, Oct. 2a — (1.54 4 -(1.5.3-; —30.7 4 —29.7, J'liris, Oct. 17 — (1.(14 12 -(i.(i:i, —31.5 9 — 30.!!,, Ciiinliridirp, Oct. 17 -(1.(10 U — (l.fi7. —31.4 10 — 31.-,„ Kiliiilniri,'li, Oct. 1.") —(1.57 12 — fi..'-)!!,' —31.1 8 —30. .s. Vit'iiiiii, Oct. lU —(1.74 4 — (I.7:!; — 34.G 5 — ;):i.(i, 1 I_r_l 03.01 r-4 11.011 [2a'' 44'"] Oct. 17 .... —(1.(14 ... —31.2 — 103.7 H 10.9 (irrcnwicli, Nov. 2.'! — (i.a(i 7 — fi.as,, —31.0 — 30.r,, Berlin, Xov. H -(1.42 2 — (1.41) — :!0.2 2 — 2!t.2, Cniii1)ri(l!;e, Nov. 17 —(1.40 7 ' —(1.47, —30.7 7 —31.0, iMliiilitirgli, Nov. 111 — (>.a!) =» ; —(1.32, Vienna, Nov. 20 —(1.39 a ! — r,..ss. —30.0 2 — 29.0j ' [_100.S][-f lO.S] — 100.2 -flO.O i [23'' 4.2'"] Nov. I'.l —(1.41 .... 1 — 30.0 18 May, 1873. wmmmm 138 THE ORBIT OP URANUS. MbAN CoUHrCTIONS to TUK Kl'IIEMKRlS »!»• I'llAMS. <\llllinl(n!. [R. A. i.r Urauii.'t.J (llwfrvwl I'orrt'Oliviis in R. A. H«an ilalini. ! Mi'iin. No. of CorrHi'tml (lbs. uiuaii. Ort'fiiwifli, Uorliii, Piiris, Onml)ri(!!?o, K(linbui'(;li, Pulkown, 1842 Dfc. K Dec. Doc Dff. Doc. Dec. [iJJ" 4r] Dec. K) EtiinbiirKh, Greenwich, I'rtriH, [0" C"] Orcenwicli, I'liri.s, Kdinhiirfrli, Piilkiiwtk, ! 1843 I Jiiri. <) AiiR. 20 Au-. 20" Sept. 17 i Sc|.t. l,-) 1 Sept, l.S ! I Sept. 22 » -C,.,-!! -G.Ki -(;.2;j -(',.22 -(;.2.') -t;.2s — ii.:!2 —7. II — 7.20 -7.2.'J 1 ';) -7.17 : II -7.2(i 1 7 -7.1s 10 O.-t. IS Oct. 17 Oct. 17 Oct. 17 \.IV. 20 Nov. 11 Xov. 1.^ Nov. 1.-. Nov. 15 [o" ;i'"] I Sept. ID Green'.vich, Paris, L(ii!|i)lirir||, [23" 5 It'"] Ori'emvicli, Koiii;.;slici'jf, I'ari.s, I'Mliiltiir};!), Oreciiwicii, I'Miiihurj!;,!, I'liris, r2:;'' ,-.(.'"] I Jrcenwicl,, I'arir!, |;)ii 22"'l Orcciiwicii, Kiliiil)iir);li, KOiii)?slicrj?, I'liri.s, [0" IH"-] 1841 •litii. .Ian. I .Ian. Jan. Ansj. I!» A 11,1?. 17 Aug. 18 Sept. 2.") Sept. li» Sept. 17 Se;>t. 10 Sept. 17 -7.10 -7.07 M.OJ -7.01 -(1.S4 -7.0;t 10 12 -fi.7o a -(!.C.7 (1 -(i.72 2 -7.70 -7.7t -7.(i7 -7.(!5 -7.U3 10 II II 10 !0 8 -. 3 -37.1 ■3,-i.l ■34.3 ■34,4 ■3(1.3 ■3:.. 7 -34.1 -.3.-..,-| -3!t,n -40.0 -3<,).!» -40.3 -40.7 -31). 4 14 5 10 10 3 II 10 10 15 Corrnittiil iiii'iin. -2!).S, -30.4] -3(1.3, -30.4. Corr. lo (ivovcnlrio Iiongituile. Lulituil«, —30.2 —31,0 —34.7- -37.1, —35.7 -35.0, -.3(1.2,. —3(1.4, — 35.S !4.i), id. 5, (4.7, 55.2 !3.')„ i3.4, '•'■ ',1 !5.;{,. 31.5 -34. -34. -3!). ), -31), 4, -30,2 -30. -31). -30. -38.; -31). . [— n8.0][-|-l0..5] I — 08.2 I -f D.G l~ !>'7.5]'[ f 1()..3J :— 1)8.2 , -[- 8.(i i [-II4.,3][-FI0.7] -111.1) : -f- !).G [— II4.H[ f lO.d] -1 12. ll' \ 1(1.0 [-1 12.(1] [f 10.. 5] -110.(1 -f i).() [_n().ij[ 1 10.4] — 108.(1 -f 0.5 r— 1011,1 ir f !).!)] — 105.0 )- 8.2 I [-122.3] [h 10,2] — 121,2 -|- 0.(1 ■■• [—122.0] [-flO..(] —120.(1 + D.a THE ORBIT OF URAMJS. 13i) Mean Couiikctions to tiik Kimikmkuis t)F Uh. nvs. — Co'ilinuvd. OlxiorTatory. [K. A. of Uraiiua] Mean datt'S I flrccnwii'li, K(IIiiImii')j:Ii, [0" If)'"] Ori'i'invicli, l'Miiiliur);li, [0" 1-.""] Kdiiiliiirgli, I'uris, [0'' 10"'] (Jrociiwirli, Kiliiiliiir^li, [O*- 11'"] (Jivcir.vicli, (Jri'iMiwirli, Kinii>;sl)>i'}f, I'aris, [o* :j;r] (}i nwii'li, I'aris, [0" iV] (Jri'i'iivvii'li, [0" :;T'"] (In-ciiwicli, [0^ l-.V"] Ol'I'Cllwilll, I'uris, [tf .Ml"] (Iri'i'iiwii'li, Konigsberj^, I'aris, [0" 41'.'"] (Jri'CMwirli, I'uris, 1044 Oct. n Oct. l:! Oct. l.i N(iv. 2(; Nov. 1!» Nov. 2.! Doc. 1 S Iter, i-i Obii«rv«iloorr«ctioii8 in R.A. , Observed corrections in Deo. Muau. -■.7t) , No. of TorrtH'tcd Obl<. UIKUU. l)c( 20 1845 .laii. I.'i .liiii. 14 ,laii. 14 .Vug. 2r. Sept. IS Sc|it. ;!o Sept. 1 4 Sept. Oct. Oct. j Oct. ! Nov. I- 20 Dec. 1 1 1846 Sept. S Si'pt. 12 Sept. II Oct. Oct. Oct. Oct. Nov. Nov. [0'' 41'"] j Nov. s 4 14 10 III 1 I .M -7.44 -7.41 -7.24 -7.10 ^7. IS -7.il! -S.2.'> -S.21 -S.lli -S.24 -S.22 -S.Ol -S.OC, 10 10 6 i lU c —S.SO — S.7:! -S.7I -S.Cl _H.(1!» --s.r,2 — S.4S H 1!) 7 II 8 —7. 1! 4, ^7.r.i, — 7.r.!) -7.:i.s, -7.:(5, — 7.;i7 —7. IS, — 7.0'.>, —7.14 .10, .OS. Iv. No of ('orrec;? I Mean. ' ub». I uioaii. —SI). 2 !t — :!s.,s^ —40.,') 1 ; —40.0, -a',).o —7. — s. s, — s. — s, (•S IT. 1\ 2:1, .17 !•. 00, .OS .OS —7.77 — s. — s. .-;(.. 7 —.'!!). n — ;!s.4 -42.7 -42.(1 -4;!. 4 -4;i.',» -42.2 -4:i.7 —4;!.:! -40.2 _s.7.V, _4C..2 _s.72,„! — 4(i.O — S.73 ; .... _s.7o, — n;.s _s (;i„ -_4( ■ — N.tis, _4(;.', — S.ti;") I ... — s.,^s.^ — S.47, — H.M) - (i.S -4(1.4 9 Corr. to Ouuoentrio Longitude. Latitude. L-I21.f.] [ + 10.2] — 120.0 i -f 11.4 r-ns.2] [-1- '.).!)] — llii.d + H.!< '• [-I1.V7] [f 0.(1] _;i,s 7 -11,1.7 f 7.1 10 a 10 (j — as.o — as.o — 12. a —42.1' —42.4, _4a.a; — 42.r. — ".s,„ — «a. 1,, —42. a —42.0 -114. -112. -i;ii. -120. n- [-lai d] [ -lai -12 ■lao, .I2s ■ 12S ■ 1211 o.r,] 7. a 0.(1] 0.0] 0.1 + [+ « i s II ..•lO.S [-124. S] [ I ~4."..S — 111.:! — I22.S ft..S] S.O 0.(1] 7.0 -,).a] 0.(1 "'[-lao.oj [ j 0.2] 411.1 I —las. 2 ( s.a 7 II -4(1.4, _4a.s. la — 4(1.1 „ [-l;!0.7] [ I 0.2] 4,-,.s I — i;!7.l I s.a (1 ~M 4, I 1 — 4r> s ' I — LJL[-ia(i.M] [f 0.0] _4(i.o ] — ia.">.i I 7.(1 140 TlIK ORBIT OF UKANUS. Mkan Correct.ons to tub Ei.,iKMLtt,8 o. Viij,svs.-Continued. Ob8erT«torjr. [U. A. of Urauus.] Grccnwifli, Paris, [()'■ SO""] Green wicli, I'liris, [0'' 40'"] Oreenwieli, [■Jh ,;,„-| Grcenwieli, Pur-', rjh 2'"] Green Willi, Puris, [0'' ;.:'"] Greeti\vi(rli, Puris, [0" a^-"! Greenwich, Pu-is, [()" 54"'] Greenwieli, Purls, [1" !!»■"] Greenwieli, Paris, Groenwieli, Puris, [!'' ir] Greeiuvieli, Pnris, [I'' O'"! Greenwieli, Puris, [1- aii. Sept. J 2 3 o<-t. Oct. I 12 Oct. 10 \■«,„ —45.5, ,. * [— 1.34..5] — i;J1.7 [— 1.S0.9] lo — 45.0 -42.7. _-44-«. —44.0 p— 129!g —49.0 [—148.4] —144.9 -48,fi„ -'•':''iC-i48.o] —48.fi — 48 S -48.7,, -145.6 [— 14fi.21 -48.8 : —142.(1 —4S.fi, —47.9, —48.2 -4r,.7., —47.0. — 4(1.9 -50,7, —50,',),. [—143.0] —140.fi LatitDdf' [+8.9] + 0.5 [+S.fi] + 7.4 [+8.7] + 0.8 [+8.7] +V.8 [ + «..'■.] + 7.2 [+8.3] -f (1.8 [—139.4] — 13fi.2 —50.8 —51.8, -5I.«, —51.7 —51.9, —51.0. [—150,4] [ + 8.1] + 0.0 [ + 8.0] — 154.1 +7.5 [-15fi.4] — 154.2 "•» , _151.8 — 50.8j —50.7.1 -50.7 [-'•'•>'>■ 2] '[ + 7,7] [-fS.O] + 0.9 [+8.0] +0.3 -148.0' — 50..S, —49.4, ' ^ [-147.7] [ + 7. .5] —50.1 -HO.a', +0,(. +0.2 THE ORBIT OP URANUS. 141 .1 Mean Coukections to the Ei>iiemeri8 ok Uranus. — CoiUinued. Observatory. LU. A. of Uraiiua. ] Oreciiwicli, [!'' ai'"] GiTomviili, I'ari.s, II'' yo'^j Ort'L'invifli, Paris, [1" 2(rl Qrcriuvicli, I'uri.-i, [1" ar] Qrcrinvioli, J'lliis, [!'■ 24'"] Qrccinvich, [I" r.2'"] Orcciuvioli, Paris, [I" 47'"] Grcpriwifh, Pari.s, [!'■ 4:r] Orccnwicli, Koiiifr--l»'r(r, [1" 41"- Orccnwicli, [I" 41'"] Orccinvicli, Koiiijtsltcrg, Puris, [2^ r-] KoiiiKshorg, Piiris, roll 2"] Mean dates. Observed corrections in R. A. Observed correotions ill Deo. 1849 Scjit. 1(> Oct. Oct. 2.S 21 Oct. Nov. .Nov. Mean. s —10.27 — 10.24 — 10.2(i K; ' —10.12 17 —10.22 Nov. 17 Dec. Dec. l:) lii Dec. 14 1850 Jan. 7 •Ian. .') Jan. G Sept. C Oct. 11 Oct. 17 Oct. 1(! \(.v. r. Nov. 10 Xov. 8 Dec. 7 Dec. l!i Deo. 9 1851 Jan. Ill Sept. II Sept. 1!) Sept. 14 Sept. 14 Oct. 22 Oct. 2:; Oct 23 — o.n,') - 0.71 — '.».77 — io.,s;i — 10.02 — I9.S7 —10.70 — 10.«0 — 10..")0 — 10..'')0 —10.21 — 11.42 — 11.41 —11.5!) — 11.7.^ — 11.4(i No. of Correcteil obi. mean. Mean. No of Corrected 1 obs. uiuttii. 8 -10.27 — 10.24, — 10.24^ —10.24 —10.12 — 10.20,, — 10.17 — O.Oi"), -- O.MC. — 0.03 — 0.71, — 0.75^ — 0.74 —10.81! — 10.02. — lO.xri. — 10.87 — 10.70. — 10..s7'j — 10.82 — 10..'-.fi, -10.48, —10.54 —10.21 —11.42, — 11.4li — 11.40 —11.75, — 11.44, —11.50 — 5;i..'5 — 5;{.9 —54.8 —54.4 —54.1 -5.3.1 -54.4 -53.0 -53.0 -53.0 —55.3 —50.3 —.-.5.7 - -.(i.l — .5.-.0 — 'vd.o —54.8 -55.7 -58.4 3 —52.9 Corr. to Oeoceutric Longitude. Latitude. [-f 1(15.1] [-f 7.3] -|-I(i0.8 -fO.2 [_ir,4.4] [+7.4] — i.a.o 4-5.8 [-102.5] —100.1 [— iro.4] — 150.0 r+T.3] -l-c.o [+T.1] -f5.7 "• i [_i:,o.8] [+0.0] 52.5 —153.8 ; +C.2 i i 9 -53.5 [-171.7] '[ + r,..5] . ■ i —108.1 +5.9 —'•'•' [_173..3] [+0.4] . . —55.3 —100.0 +5.3 fi i —55.3, 4 ' —.'•.5.5, '[-172.4] [+0.4] ■55.4 -100.1 I +5.0 54.r., ■ '_■__.'[— 100.0] [ ( 0.3] —54.0 , —105.0 . 5.2 ' 2 —54.4 '[_103..'>] [+0.0] -10O.4 4-3.6 ' [-1S0.0] [ + 5.7] —170.9 I -f4 4 [_ismj [)r..8j —57.4 . —170.1 1 4 4.3 142 TlIK Oil BIT OF URANUS. MkaN ('0ltKEUTItlN8 TO TlIK Kl'IIKMKItIB or UUANUS. — Cimtiliuod. Observatory. IR. A. of Uraiiu:' ) Orpcnwifli, Piiiin, Qreonwidi, Piiri.s, [1" .is™] Orpciiwicli, Paris, [1" 54'"] Orocnwicli, I'liris, [I- o-J"] Orci'Mwii'li, Qri'iMiwicli, I'liris, [•2" 18""] Orpcinvicli, I'liris, Ort'c'iiwicli, Koni)!sl)fr)^, I'urix, [2" 10"] Qrccnwii'li, I'uri.s, Mean ilatei. Observvdcorrtiutiniia ill li.A. <)lid«rvedaorractioD9 in Uecv Mean. 1851 Nov. 3 Nov. a s Nuv. \i)v. 22 .Nov. U .Nov. 21 I>cc-. 22 •J."-. Dfc. 24 1852 .lull. 11 .lull. 14 .lull. 12 .Si'pt. 1 2 Oct. (M. 17 2.! Oct. Ill [2'' 9"] Jreenwii'li, 'liris, [2'' .lO""] Jrconwic'li, furis, [2" J4'"] Nov. Nov. i:i 1.^ Nov. 14 Dec. Dec. Dec. lit Hi p.) Dec lit 1853 .lull. 12 .lull Juii. Sept. Sept. 1.-. 17 in Sept. 17 Oct. Oct. 27 Oct. 22 — ll.-.o — II.4S — II. 4.1 — li.;i;» —II. 1.5 —11.00 — lo.sr. — 1(».S!» — 12. Of, -12.2.! -12.0(1 -11.72 -11.07 -11.77 -11 42 -11.40 No. of obn. -12.00 !, -12. OS 5 -12,00 4 -12. .'is 1 2 -12.f.!l 7 -12.00 r, CorrKuteil luxau. -IL.'iO -ll.40! -11.411 -11.41! -ll.:!7, -11.42 : -ll.lo.l --.57.2 -lO.'.IS I —57.2 Ntan. -.'■)7..'> -iiS.I -iis.;} — 11.(10 -lo.s.v . |(),S7,! —50.5 -10. so —12.00. — 12,2;!. — 12.04 — 12. IS — 12.00 : — 12.00'; — l2.o;i ! -11.72, — 11.75, — 11.7:1 — 11.42, — 11.44 — 11.4! — I2.0tt. — 12.50. 3l2,fi5 -12,00. — 12,04' —12,00 -5(1.5 -5-1.0 —50.0 -50.7 -00.2 — 5S,7 — 50,:l —.50.0 —,'.0.7 —57.0 — 5S.S —50.0 No. of Correi'tail ub«. luuaii. —57.1, -57. .5, —57.2 — 57.;k -57.0, — 50.Sj — .50.0, — 5(iT" — 50.S, —55.0, —50.4 —50.1 — 5S.2, —50.0, — .5S.4 — 50.nj —50.0, —50.4 -5s.:i, -5s.:t, — 5S.4., — 5s.;! —57.1 — 57.1 —50.3, —50.4, — 50.;j -.58.4, -5S.4,_ — 5S.4 C'i>rr. tu (ieooeiitrio Luiigituile. Lulituil«. [-IS1.4] -17S.1 [+•'■••7] 4-4.0 [_I70.0] [ + 5.0] -177.0 -f;!.8 [-175.7] -172.a [+5.:!] -i-3.8 [_l;:i.l] [,5 2] — 100.4 ! -|-;!.I [-1S7.4] -1H4.5 [f4.0] [_ISO.O] [ I 4.7] -1S7.2 f;i.i [-1SS.0] [4-4.7] — 1S5.0 -|2.4 [_IM4.S] [f4.fi] lsi.:i -1-2.8 [_1S1.0] [f4.4] — 170.8 f2,0 [_104.fi] [fl.O] — 101,0 -f '.!- [-100,7] [-1-3.8] -102,7 +1.7 THE ORBIT OP URANUS. 148 MKAN CullllKCTIONti TO THE Kl-lltlMKHIS OF UUANUS. Coutillllfd. Olisfrvatory. |K. A. of Uiaiius. ] Qrcenwicli, 1'uri.s, [2" 31'"] Paris, r^ii St"'! Grociiwicli, I'liris, [2" iir,-] Paris, [2'' 2r."'] Orccnwicli, [2" r,:r] (irccMiwich, Piiris, [2'' SO"] flrccnwicli, Paris, Haiiliuj^i), [2'' 4S'"] (iri'cnwicli I'aris, Saiitiii;^(), [2" -in (jrccnwiili, [2" 4-"J (ircctiwicli, Paris, Orcciiwirli, Paris, Santiajro, (Jrccnwicli, Paris, Santiago, Mean dates. 1853 Nov. 12 Nov. l.") Observed currectiona ill R. A. Observed corructloiiH iu Deo. : Curr. to (ieoceiitriu Mean. | No. of oIm. Nov. Dec. 14 12 1854 Jan. 1.1 .Ian. 20 Jan. l''cl). 15 -12. CI -12.G7 — 12.28 -12.02 -II. s7 -ll.H!) .K cpl. 21 _l.!.l-2 Oct. Oct. 2(! ' _l:!,24 2y — 1;!,12 Oct. 27 Nov. Nov. Nov. 12 l(i 17 — 1:5.22 — I.!. 21 — l;l.20 Nov. li) j ! Dec. n l>cc. It Dec. 1.-. -12.!).'! -l:i,H -I2.s.-i |)c< 14 1855 .Ian. is O.'t. Oct. Oct. Nov. Nov. Nov. l:! 2."i IT II 17 It; Nov. ir> Dec. Dec. Dec. Hi IS -l.i.Tl -I. •!.!»:! -I.'i.s.', -|:!.I12 -l:!.s2 -i:i.4s -l:!..'i2 -Hi .^.1 Dee. 1,S Corrected ; Mean, lui'aii. 14 10 7 I 10 10 3 -12.(l.'),„ -12.(1.5 -12.2C -12.02, -ll.sr., -ll.'Jti -11.87 r> —13.12 -i:i.24, -l.t.lO, -i:i.22 -i;?.22, -i:!.l!t, -l:!.203 -l:!.20 -12.03, -l:f.l2, -!2.S5, — 12.!iO -I2.5.V -i:!.7I„ — l:i.".tl, — l:!.7.s i '— l.'t.s.^, — l.f.c.o, _I.!.S2. — I.!. 7(1 — .'■iS.fi — ."ifl.it — r)it.7 ~M.'.) ■;'i.').S -.'■i7.C. -.'.i;.(; -ri7.2 -.'■.0.4 -50.2 50.0 -50.7 -5S.1 — 57.S 50.0 -5S.0 Noor obn 14 8 — l:t.4S„, —57. .'I '_l:!,50, — 5i;.7 i— i;!.5;i, I — 5S.1 — l:i,50 i .... Correctvd Longituile. Latitude, lueau. -50.0,, -58.(1. -58. -5S.4 -50.2, -57.0, -58.0 -57.G 5S.2 — 5S.7 ' I -5S.7 I a -57.. 5, -57.0, -5 7.. '5 -57.5, -5S.5, -.5H.I. ; —57.8 — 5S.I, -5(;.7i -50.1, i — 5S.0 —5 (',.4 4 — 55 "t 55 "i 55 :i 57 1, 5(; ^. 5(; .li. —5(1.7 10 — 5(i.8, :! —5(1.5, (> -57.5, —57.0 [_inr,.o] [-H!.t] —I ',12.0 1 -I^I.U [_i.,.T4] [+:t.(i] 187.0 -1-1.5 [-188.1] [ + 3..5] — 183.0 . 4 0.1 [-184.5]' [4 3..5] —182.1 I -[0.7 [-201.3] [ + 3.0] —10(1.0 -i-1.7 [-203.7] [-f2.8] lOJI.O -i 1.2 [—203.0] [-I2.T] — 100.1 -i 1.2 [—200.1] [4 2.(1] — 105.3 ; 40.7 [-104. n] [42-''] —100.1 41.1 [—200.2] [4I.X] —204.0 j 4 0.1 r-200.4] [41.7] —205.3 —0.2 [-20(1.2] [41.'1] , —202.1 i —0.1 144 TUE ORBIT OP URANUS. Mean Coruections to the KpiiE.MGiiiii or Ubanus. — Con Hinted. OlpucrTRtory. [II. A. of Uramid.] Grcciiwitll, Paris, [2" OS""] Grcciiwieli, Paris, [3" 27'"] Qrci'iiwicli, Paris, [3" 22"°] Orecnwifli, Paris, [3" K"-] (Irconwicli, Paris, [3" la"] Grecnwii'l), [3" 40'"] Qrconwifh, Paris, KoiiigsliiTg, [3" 41'"] Orci'iiwicli, Paris, KOiiijj.sln'rg, [3" 3Cr] Orccinvich, I'uris, [3" 32™. 2] Orccnwirh, Paris, [3" 3r.O] Qreeiiwich, [4" 3"'.3] flrooiiwldi, Piiris, Ub.-iervt'd uorreotiona v. 17 Nov. IT Dec. 1(1 l)■. '2 — 2S.3 —27.4 —27.9 -29.2 -2.S.0 -28.5 -29.0 -28.0 -40.3 Iiongitnile. Latitude. . ^ —40.3 —30.9 8 ' —30.5 —33.4 i 4 ' —33.0, —32.8 13 ! —32.0,, -34.3 7 -34.0 i 6 —24.5 —32.7 -34. 7„ —33.9, —34.4 -33.9, — 34.4, [—217.9] [—3.1] —215.0 I —0.0 [—220.2] [—4.0] — ?23.4 —0.1 —34.1 —24.1 -20.8, -20. .3, 25.2 —20.3 -27.9,,, -27.2, -27.9 [-227.3] [-4.0] —224.4 —0.4 [—223.2] [—4.0] —220.2 —0.3 [-217.5] [-4.0] —215.1 —0.0 [-220.2] [—4.7] —223.0 —7.1 [—227.0] [-4.7] —225.4 —7.2 I'LL [—224.3] —27.9 —28.4, — 2.S.,5, — 2S.0 ' [—219.3] — 21G.0 -28.0, "^':*''_ [-215.9] -28.2 —213.3 [-4.8] -0.8 [-4.8] —7.2 [-4.8] —7.0 THE ORBIT OF URANUS. 141 Mean Cuiiuections tu tiik Kimiemkuih ur Uhanub. — Continued. ObRerT«tory. [K. A. Ill Urauui..] Paris, Lcydcii, [5" 34""] Greenwich, Leyden, [5" 29~J Washington, Leyileii, [5" as"] \Vii' 41-] Oreciiwlch, Wusliingtoii, [fl" 17"] Orcenwicli, I/cydcii, Wttsliington, [0" ll"] Mean 2 Jan. 17 Jan. 17 Jan. 14 Feb. 14 Feb. 14 Feb. 14 Fub.~14 Mar. 12 .Mar. 9 Mar. 3 Mar. Oct. 9 1.3 Nov. Nov. 9 15 Nov. 14 Dec. Dec. Dec. 11 12 10 Dec. 11 1867 Jan. 21 Jan. 21 Jan. 9 Jan. 19 Feb. l.J Feb. 15 Feb. 18 [C 20™] I Feb. 15 Greenwich, ' Wa.shington, , Mar. Mar. [C 19""] I Mar. 6 Greenwicii, Washington, [fl" 51"] Dec. Dec. 11 18 Dec. 13 Ubavrvvd vorreutloni in R.A Mean R -1(1.20 -lfi.04 -IC.OO -lfl.02 -15.81 -15.81 -15.75 -15.47 -15.49 -15.59 — 15.3fi -15.fi5 -15.72 -10.01 -Ifi.OO -15.87 -15.89 -15.93 -15.82 -15.72 -15. (i5 -15.54 -15.45 -15.32 -15.IJ1 -15.70 No. of obi. 10 6 8 2 10 10 7 2 5 12 4 Correoted u«au. Ubucirved uorreotions in Dec. Mean. — ir..20, — Ifl.OI, — Iti.Ot), — 15.99, —16.09 -1.5.81,. —15.81. —15.72. ^5.79^ —15.47, —15.49, — 15.5(;, —15.50 — 15.3« 7.2 n.3 7.« 7.1 9.1 8.4 8.fl 8.1 9.7 9.0 + 4.4 —15.65, -j- 3.5 —15.72,, 4- 3.8 —15.71 — Ifi.Ol,, -t- 2.S —Ifi.OO, -t- 2.0 — 15.84, -f l.« —15.99 —15.89, — 15.9;{., -15.79, —15.90 — 15.fi9, — 15.fi,5, — 15.51, -15.04 —15.4.5, —15.32, —15.40 — 15.fil, —15.70. — 15.C4 1.4 1.2 0.3 2.4 2.0 0.5 2.9 2.5 8.5 6.9 No. of Corrected uba. mean. 11 8 8 10 9 — fi.S, 1 j — 6.1. 9 1 — 6. .5. 6 — 7.1, — 0.6 Corr. to Oeocentrlo Utitude. Longitude. 8.7, 7.3, 8.6, — 8.2 — T.7, — 8.0, — 9.0, — 8.5 + 5.5 + 3.9, + 4.9, -I- 4.0 10 -f 3.2. 7 -f 3.1, i + l.fi, -\- 2.9 5 12 4 10 3 1.0. 0.1. 0.3, — 0.4 — 2.0, — 0.9, — 0..5, 1.1 2.5, 1.4. — 2.1 + 8.9. + 8.0,^ + 8.5 [—221.6] [. —221.0 7..1] 9.0 [— 217.3]'[. —210.9 - 7-3] — 8.2 [— 212.9]'[- — 212.9 [-209.0] -211.2 7.1] 8.5 [- 7..5] 7.9 [-21.5.1] [ —215.9 -218.2] [- —219.7 7.8] 8.3 8.0] 9.7 [-217.8] [_ 8.0] 218.3 — 9.3 .214..5] [— 7.9] —214.7 — 8.0 [-211.1] —211.3 [—213.1] —215.3 [- 7.8] — 9.0 C- «■«] — 10.4 THE ORBIT OP URANUS. 140 Mean Cobiikctions to the Epiiemkuis or Ukanub. — Continued. ObiierTatory. LK. A. of Unuua.] Qrccrnvicli, [(;" 45'°] Qrpcnwk'li, Lvydcii, VVusliiiigtoii, Piii'is, [•;'' 41'"] LcydcM, Wttsliirigton, [(!'■ 3!)'"] Wiisliiiipton, [7" l(i"'] Wiisliiiijrton, [7" Hi"'] flrccnwicli, Wusliiiigtiin, [7'' lO""] Orpcnwirli, Wusliiiigton, [7'' T)"'] flrconwifli, Wiisliiiigtuii, Paris, [7" l"-] flri'ciiwich, Wttsliiiijjfton, [(i"" 59"°] Oreenwicti, [7'' Si™] Qrconwicli, [7" 25'"] Oroeiiwic'h, Washington, [T" 21-] Mean dstea. Obiertttd oorreotloua in R.A. Mean. 1868 Jan. 13 Fcl>. 15 I'VI). 5 Fub. 13 Ftb. 14 Mur. 1 1 Mill-. 19 Mar. Oft. 15 18 Nov. 7 Doc. Dec. 25 12 Dec. 18 1869 Jan. 19 Jan. 18 Jan. 18 Ffl). Fi'l). Fcl). 14 14 10 Feb. 13 Mar. Mar. 3 7 Mar. Deo. 5 8 1870 Jan. 19 Feb. Feb. Ifi 1(! Feb. IG — 15.C7 -15.43 -15.43 -15.50 -15.41 -15.02 -14.91 -14.57 -14.83 -15.24 -15.1(i -15.28 -15.31 -14.81 -14.87 -14.50 —14.73 -14.58 -14.49 No. of obit. -15.04 • 10 -15.13 13 -15.09 10 10 Corrected luuau. 8 -I5.r.74 —15.4.3. — 15.40, —15.50. —15.38. —15^44 — 14.99, —14.91, —14.95 —14.67 —14.83 —15.24. — la.Ki, —15.20 -15.28, -15.31. Ubaerred oorrootiona in Deo. Mean. -15.31 -f 0.9 + 4.8 + 5.9 -I- 4.9 -t- 4.0 + 3.1 -fl2.3 -j-lfi.2 -fl5.1 + 14.0 -1-12.0 -1-12.9 -15.04,, -1-11.1 -15.13„i -1-11.3 -15.0(1, -1-11.7 -Is. 08 -14.81, -14.87, -1-10.8 -fll.4 -14.84 -14.50, -1-21.1 —14.72. -14.57. -14.49, -14.55 -1-19.0 -fl7.2 -flO.7 No.of oba. Corrected uieau. 10 13 10 Corr. to Quocentria Loiigltade. ' Latitude. -f 6.7. [-213.4] [— 8.7] -215.5 —10.1 + 4.0. + CO, [—210.3] [— 8.0] -f 5.3 -f 4.0, + 4.5, —212.3 -f- 4.0 -1-13.3 -f 17.3 + 14.9. + 15.0. + 15.2 + 12.4. + 13.4. + 13.3 + 10.9,, + 11.7,. + 11. t + 10.6, + 11.9^ + 11.3 +20.9 10 +18.8 + 17.0 + 17.1 — 9.7 -205.0] [— 8.6] -205 5 1 — 9.2 198.3] [_ 8.7] —201.5 —12.9 I- 0.1] —205.0 — 9.3 [—202.0] C-: 200.5] [_ 9.2] 210.3 —10.0 [— 207.1] '[— 9.3] —211.4 -10.3 [—204.7] [— 9.2] -208.1 -10.2 [—201.5] [_ 9.2] —204.8 j _ 9.3 [— 197.1] i[— 9.8] —202.1 —10.3 [—199.3] [_ 9.9] —204.5 —10.8 [—197.2] [_ 9.9] + 17.0 —201.9 —10.9 150 THK ORBIT OF U II A N U S. Mkan CoBBECTioNH TO TUB Ki>iiKMERi8 Of XJRASva.— Continued. Oh»vtJtlotf. [K A. vf VtHUMi.] Orm^nwieli, Wa.sliiii)(ton, [7" 18"] Qropiiwicli, [7" ■17"'] flreonwicli, [7" 41'"] Orpcnwit'h, [7" 3S"'] Orponwicli, [S" 11'"] Greenwich, Wusliiiijjtoii, [8" (;■". 1] flrcenwioli, Washington, [8" l-.O] (JrocMwich, VVusliington, [7'' 57".9] Washington, Uruunwioh, [7" 57'". 3] M«SD >U.ea. Obierruti oorractiona hi R.A. M«au. 1870 Mar. I a Alar. 1 1 Mar. 12 1871 Jan. 9 Feb. 15 Mar. 14 Dec. 21 1872 Jan. A Jan. 21 Jan. 18 Fib. 15 Feb. 21 Feb. 17 Mar. 15 Mar. 15 Mar~15 April 8 April 8 April 8 8 — 14.-->!( — 14. HO — IH.fl!) —13.92 —13.82 — 13.1G -13.32 -13.31 -13.23 -13.24 -13.01 -13.07 -12.79 -12.77 No. of Correi'teil uba. I luoan. 0I> irred correotioD* tn D«a M«an. 14 3 14 10 — I4.2S, — 14.3(», ^14.29 —13.98 —13.91 —13.81 —13.15 —13.31 —13.31 —13.31 —13.22 —13.24 — 13.23 —13.03. —13.07, — 13.05^ — 12.79, — I2.7li, —12.78 -flfi.9 -1-15.9 -f24.8 -I-2I.8 -f20.7 -f 29.2 -f2.S.3 -f.7.8 -f2P..2 -l-2(;.4 -f-25.4 -1-24.5 -f24.9 -f25.0 No of Corrauted ub<. lueitu. 14 + lf..O -|-l(i.3 Corr. to Oeocvutiio Longitude. Utitod*. 14 8 -f 1(1.2 -f24.fl -f21.6 -1-20.5 -1-29.0 -f2S.l. + 28.2, -f28.1 -l-2r..o, + 2«.H, [-103.f.] [- 9.7] —198.1 —10.1 [—190.0] [-10.2] — 19(!.l j —10.7 [—188.8] [—10.1] -194.4 I —11.5 [-184.9] [—10.0] —192.8 I —11.5 [-177.9] [-10.4] -18(5.5 -11.4 -f2(i.3 [—179.8] [—IO.fi] 188.3 —11.3 [_179.3] [—10.5] -180.0 —11.3 -f2o.2. _-\--t^i_ [—170.4] [-10.4] -f25.1 —183.7 —11.1 -f25.3, -f-24.8, -1-25.2 [—172.4] [—10.2] 1 —180.2 I —10.8 1 TUB Oil HIT OF URANUS. Iftl 1 COHIIECTIONM TO HE Al'I'MKI) TO TIIK PolilTKlNll (IF UllANIH IN TriE [iKIILIN jAIIHUlCri AND THE NAuriuAk Almanac to hkuuce them to the I'uhitionh fhom the I'huvihiunal Theouy. 1 Dxte. Htilloo«iitria, Oenoentrlo. 1 n Mf'f 'I'. ib It 1830, July 24 — 18.0 ■rinoi + 0.0 —10.3 + 0.5 Aug. 13 18.3 1007 0.0 18.8 0,5 Sept. 2 18.4 1014 0.1 17.0 0.5 Bupt. 22 18.5 1020 O.l J7.1 9.4 Oct. 12 18.7 1027 9.1 16.7 0.3 Nov. 1 10.0 1031 0.2 16.5 0.2 Nov. 21 —10.3 + 1038 + 0.2 —16.7 + 0.1 1831, July 19 _22.5 + 1112 + 0.5 —24.5 + 10.0 Aug. H 22.0 II 1 0.5 23.7 10.0 Au(f. 2S 22.7 1123 0.6 22.7 10.1 8f'pt. n 22.8 1128 9.6 21.7 10.0 Oct. 7 22.0 1133 9.6 20.0 0,9 Oct. 27 23.2 1138 0.7 20.6 9.8 Nov. 16 23.5 1142 0.7 20.0 9.6 Dec. 6 —23.8 4-1140 + 0.7 —21.0 9.5 1832, Aug. 2 —26.0 + 1108 + 0.8 —28.7 + 10.3 Aug. 22 27.2 1201 0.0 27.9 10.4 Sept. 11 27.4 1202 9.0 26.9 10.3 Oct. 1 27.6 1 205 0.0 26.1 10.2 Oct. 21 27.8 1208 10.0 25.4 10.2 Nov. 10 2H.0 1200 10.1 25.1 10.1 Nov. 30 —28.3 + 1200 + 10.2 —25.2 + 10.0 1833, July 28 —31.0 + 1233 + 10.2 —34.5 + 10.7 Aug. 17 32.2 1236 10.3 33.7 10.8 Sept. 6 32.4 1240 10.3 32.7 10.8 Sept. 26 32.6 1242 10.3 31.8 10.7 Oct. 16 32.0 1245 10.3 31.0 10.5 Nov. 6 33.2 1247 10.4 30.4 10.4 Nov. 25 —33.5 + 1250 + 10.4 —30.2 + 10.3 1834, July 23 — 3S.0 + 1264 + 10.5 —41.2 + 11.0 Aug. 12 38.4 1266 10.5 40.7 HI Sept. 1 38.4 1268 10.6 30.6 11.1 Sept. 21 38.7 1270 10.6 38.6 11.1 Oct. 11 30. 1 1275 10.6 37.7 10.0 Oc.,. 31 30.4 1277 10.6 36.9 10.8 Nov. 20 30.7 1282 10.7 36.6 10.7 Dec. 10 —40.1 + 1282 + 10.7 —36.1 + 10.5 1835, July 18 — 13.0 +l.^oo + 10.8 —47.7 + 11.3 Aug. 7 44.5 1311 10.7 47.6 11,3 Aug. 27 44.5 1313 10.7 46.6 11.3 Sept. 16 45.1 1316 10.7 45.8 11,2 Oct. 6 45.4 1318 10.8 44.8 11,2 Oct. 26 45.0 1321 10.7 44.0 10.9 Nov. 15 46.5 1324 lO.S 43.6 10.8 Dec. 5 —46.6 + 1327 + 10.7 — 42.5 + 10.6 1836, July 12 —51.0 + 1361 + 11.0 —.55.3 + 11.4 Aug. 1 51.3 l:iC.4 n.o 55,2 11.5 Aug. 21 51.5 1365 11.1 54.6 11.7 Sept. 10 —51,0 + 1368 + 11.0 —53.7 + 11.6 162 TUE ORBIT OF URANUS. CoBRECTiONH TO BE APPLIED TO THE POSITIONS OF UiiANUs — Continued. Dr.t«. Heliocentric. Geouentrio. 1 hJ^ Mfip ¥ u hh It II II II i83r>, Sept. 30 —52.2 + 1370 + 11.0 —52.4 + 11.5 Oli. 20 52.9 1372 11.0 51.0 11.3 Nov. 9 53. 1 1375 no .50.5 11.1 Nov. 29 53.9 1377 11.0 .50.4 11.0 I>fC. 19 —03.8 + 1380 +11.1 —49.8 + 10.9 1831, July 7 —58.3 + 1387 + 11.0 —02.9 + 11.4 July 27 MA i:!87 11.0 02.9 11.5 AuK- Hi 59.2 13.88 11.1 03. 1 11.7 Sept. 5 59.2 1389 11.1 02.0 11.7 S.-pt. 25 00.0 1391 11.0 01.3 11.5 Oct. 15 00.0 1.393 11.0 59.0 11.4 Nov. 4 fil.O 1394 11.0 59.1 11.2 Nov. 24 01.3 J 394 11.0 58.2 11.0 Deo. 14 —02.1 -} 1393 + 11.0 —57.1 + 10.9 1838, Auff. 11 — fifi.4 + 1395 +11.1 —71.1 + 11.0 Aug. 31 07.2 1390 11. 1 71.0 11.7 Sept. 20 07.7 1395 11.1 70.2 11.7 Oc-t. !0 08.1 1392 11.1 08.9 no Oct. 30 08.5 1390 11.1 07.0 11.4 Nov. 19 08.9 1388 11.1 00.4 11.2 Dec. 9 09.4 1388 11.1 05.8 11.1 Dec. 20 —09.7 + 1389 + 11.2 —05.4 + 11.0 1839, Aug. —74.7 + 1382 +11.1 —80.0 + 11.0 Aug. 20 75.1 1381 11. 1 79.7 11.7 Sept. 15 75.5 1380 no 79.0 11.0 Oct. 5 70.0 1379 11.1 77.9 no Oct. 25 70.4 1379 11.1 70.5 11.5 Nov. 14 70.8 1379 111 75.2 11.3 Dec. 4 77.2 1378 11.1 74.0 111 Dec. 24 77.0 1377 11.1 73.5 10.9 1810, Jan. 13 —78.0 + 1370 + 11.0 —73.1 + 10.7 June 24 —8'. 9 + 1370 + 11.0 — Sfi.O + 11.1 Aug. 20 ^•z.i 1374 no M7.;; 11.5 Sept. 9 83.4 1370 11.0 H7.S 11.0 Sept. 29 83.5 1377 n.o 80.4 11.5 Oct. 19 84.2 1370 no 85.5 11.4 Nov. 8 84.4 1370 no 83.5 11.3 Nov. 13 85.3 1377 no 82.7 11.1 Dec. 13 H5.5 1378 n.o HI, 5 10.9 1841, Jan. 7 —80. 1 + lu-7 + 10.9 —SI. 3 + 10.7 June 10 -89.8 + 1378 + 10.8 —93.2 + 10.8 Aug. 15 91.1 13S4 10.8 97.0 11.3 Sept. 4 91.5 13H5 10.8 90.9 11.4 Sept. 24 91.9 1388 10.7 90.1 11.3 Oct. 14 92.3 1 390 10 7 94.8 11.2 Nov. 3 92.9 1392 10.7 93 4 11.1 Nov. 23 93.4 1393 10.8 91. H 10.9 Dee. 13 93.8 I.;95 10.7 90.5 10,7 18-i'2, J III). 2 94.3 1 .•19X 10 s S9.7 IIJ.O I'eli. 11 —95.0 -f U04 4 10.7 — S9.0 + 10.3 \\ « THE ORBIT OF URANtJS. 163 CouREtmoNs TO BE AppuKD TO TiiK POSITIONS OF Uranvs Coiilinued Date. Hellootmtrlo. OKOceiitrlc. 1 h?. V^p ^i 11 ih /' ' ft ft II 1842, Tunc 11 — Of). 7 + 1409 + l'.'.(i — 9S.5 + lii.5 Aiiir. 10 !llt.l 14(tS 10. r, 105.3 10.0 jVii)f. ;i() y.t.c 1409 10.7 105.8 10.2 Sept. lit 91t.9 1409 ,M).(J 105.2 10.2 Oct. a l(tO..T 1410 10.5 104.2 10.0 Got. 2!) lltU.'J 1412 10.5 102.6 10.9 Nov. 18 .''11. a 14ia 10.5 100.8 10.8 Doc. 8 l(tl.8 1414 10.5 99.:' 10. (i D.'c. 28 1(12.2 1415 10.5 9S.0 10.4 iH4a, Jul). 17 -1(12.5 ■tl417 -f 10.4 — 97.2 + 10.2 All jr. ft —107.3 + 1422 + 10.3 — M3.7 + 10.7 .Auji. 2.'> i!!-.- 1420 10.2 114.4 10.7 Sept. 14 Kts.O 1421) 'O.l 114.3 10. (i Oct. 4 lOX.f) 1 420 10.1 113.4 10. fi Oct. 24 lOil.O 1419 10.1 112.1 10.5 Nov. l:} Kill. 4 1418 10.1 110.2 10.4 Dec. a IIO.O 1118 10.1 lOsf, 10.3 Dec. 2;j HO.4 1417 10. 1 lOfi.9 10.1 1844, Jiiii. 12 — llit.S -i 1418 + 10.0 —105.9 + 9.8 July ;iO -11.-..1 -t 1 40C. + '•'( — 121.2 ~f 10.0 Aiijjr. lit II.V4 IKia '.1.8 122.3 10.2 i^i\>{. s II. "..8 1 too !l,S 122.7 10 3 Sept. 2s in;, a ia;i7 9.8 ■ 2.4 10. a Oil. ix licit l.'illC. :i.7 i. ..3 10.2 Nin " 117.4 iaii2 o.c, 119.7 1 iO.O Nov. 27 1 17.8 1 .iS9 o.c, 117. (! ! !I.S Dec. 17 lis. a ias4 9.(! 115.7 I 9.7 1845, i1 II II . li IIS.I! lasi '.1.5 111. 3 9.4 Juii. 2(< — llll.l + ia78 + 9.(1 — 1 13.3 + 9.4 A UK. 14 --12a.-) + 1340 + 9.1 — 130.4 + 9.4 J^cpt. a i2a.!» laaa 9.4 l:!l.4 9.8 Sept. 2;t 124.4 la.ao 9.4 131.7 9.9 Oct. l;i 124.9 ia27 9.3 130.(i 9.8 Nov. 2 I2r),a ia22 9.3 129.0 9.7 Nov. 22 12.-). 7 iai8 9 2 1 127.0 9 5 Dec. 12 12(1.2 laia 9.2 125.0 9.3 184P., .lull. 1 l2i;.7 iao9 9.2 123.2 9.2 Jan. 21 — 127.! + iaoa + 9.2 — 121.9 — 9.0 A up. 2'.i — ia2.o -1 125a + 8.8 —1 311.5 + 9.2 .Sept. 18 ia2,a 1249 8.8 l;!9.9 9.2 ■. Oct. 8 ia2.8 124(i 8,8 lao.s 9.3 Oct. 2H iaa,2 1242 8.7 1 ias.4 9.1 Nov. 17 \X).h I2a8 8.7 13(1.4 9.0 Dec. 7 lai.i I2a4 8.7 131.4 8.9 Dec. 27 lai.d i2aa H.C. 132.3 S.7 184T, Jiin. ID — lai.'j 4 i2ao + S.O 1 —130.5 + 8.5 A up. 24 — lait.r) + 1197 + 8.3 — ltd 7 -1 :«.fi Sept. l:! 11(1.0 11114 S.3 14S.0 8.7 Oct. .! '4(i.r> mil 8.3 1 HS.3 8.7 Oct. 2;! — ilO.9 + 11^8 + 8.2 -147.4 + 8.6 ao May, 1 B73. 154 THE ORBIT OP URANUS. Ill Corrections to bb Applied to tub Positions op Uban ua — Continued. D«te. Ileliocentrio. Oeooentrlc. 67i tt Mh^ u II hb It 184T, Not. 12 —141.4 -fll85 +8.2 —145.8 +8.5 Dec. ■^i 141.0 11 83 8.1 143.7 8.3 Due. i2 142.4 1182 8.1 141.4 8.2 1843, Jan. 11 _l42.8 -fll82 +8.1 —139.3 +8.1 Sept. 7 —147.0 + 1105 + 7.6 —150.0 +7.9 Sept. 27 14S.:( 1100 7.0 150.7 8.0 Oct. 17 14M.7 1104 7.0 150.4 8.0 Nov. 6 140.a 1104 7.0 155.3 8.0 Nov. 2(! 140.8 1103 7.0 1.53.2 7.9 Doe. U\ 150.2 1102 7.0 150.7 7.8 1849, .Full. 5 \■^{^^^ 1101 7.5 148.3 7,5 Jan. 25 — 151.3 + 1101 +7.5 —140.0 +7.4 Sept. 2 —155.8 + 1149 +7.0 —104.0 +7.3 Sept. 22 150.2 1150 7.0 105.1 7.3 Oct. 12 15fi.5 1149 7.0 105.1 7.4 Nov. 1 150.8 1148 7.0 104.0 7.4 Nov. 21 157.4 1149 7.0 102,4 7.3 Deo. 1 1 157.9 1148 0.9 100.0 7,1 Dee. .tl I5S.3 1U7 0.8 157.5 0.9 1850, Jan. 20 — 158.0 + 1147 +0.8 —155.0 + 0.8 AiiR. 28 — lo;!.3 + 1135 +0.2 —170.0 +0.4 Sept. 17 103.7 11. {3 0,2 172.0 0.5 On. 7 104.2 1131 0.2 173,3 0.5 Oet. 27 104.7 1129 0.1 173.1 0.4 Nov. 1(( 105.2 1127 0,1 171.8 0,4 Dee. G 105.6 1120 0.1 109.4 0.3 Dec. 2fi 100.0 1127 0.0 100,7 0.2 1851, Jan. 15 100.3 1127 0.1 103,8 0.1 l\\>. 4 —100.7 + 1124 +0.0 —101.8 + 5.9 Sept. 12 — 171.5 + 1109 +5.5 —180.0 +5.7 Oet. 2 171.9 11 Of 5.5 181.4 5.8 Oet. 22 172.4 1103 5.5 181,0 6.8 Nov. 11 172.9 1103 6.4 180.8 5.7 Dee. I 173.3 1103 6.3 178,7 6 5 Dee. 21 173.8 1102 5.2 170.1 5.3 185-2, Jan. 10 174.3 1101 6.2 173.4 6.3 Jan. 30 —174.7 + 10U8 +5.1 — 170.2 +5.1 Sept. — 179.0 + 1074 + 4.7 —180.8 +4.9 Sept. 2(; 179.4 1073 4.0 188.9 4.8 Oet. l(i 179.8 1071 4.5 180,7 4.7 Nov. 5 1H(».3 1008 4.5 180,0 4.7 >■• ■'. 25 180.7 1007 4.4 187,8 4.6 Ito. 15 181.2 1005 4.4 185,3 4.6 185.1, Jan. 4 181.5 1004 4.4 182,3 4.5 Jiin. 24 181,8 1002 4.4 179.0 4.4 Fob. l;i — 182.1 + 1000 +4.3 —170.9 + 4.2 Sept. 1 —185.8 + 1055 +3.9 —192.4 ■14.0 Sept. 21 180,2 1055 3.8 195.2 3,9 Oet. 1 1 — ISC. 5 + 1055 +3,7 —190,7 + 3,9 THE ORBIT OF URANUS. 165 CoRiiECTioNS TO BK Api'UED TO THE POSITIONS OP Ukanus — Continued. Dat«. Heliooentrio. Oaooentrio. 1 ft J/Sp 'i hi It 1853, Oct. 31 —isfi.s + 1054 + 3.fi — 19fi.8 +3.8 Nov. 20 1M7.1 1054 3.5 195.fi 3.7 Dec. 10 1H7.5 10f>» 3.5 193.6 3.B Dec. 30 1«7.9 1053 3.5 190.7 3.6 1854, Jnn. 19 l«S.l 1053 3.5 187.4 3.5 Feb. 8 —188.5 -t-1052 + 3.5 —184.5 + 3.5 Sopt. Ifi —192.2 + 1034 +2.9 —200.5 + 3.U Oct. f, i92.(; 1032 2.8 202.9 2.9 Oct. 2(5 193.0 1032 2.7 203.7 2.8 Nov'. IS 193.3 1032 2.r, 203.0 2.7 Dec. 5 193.7 1020 2.fi 201.3 2.7 Doc. 25 194.0 1025 2.5 198.fi 2.« 1855, Jan. 14 194.2 1021 2.5 195.2 2.5 Fclj, 3 —194.5 + 1018 +2.4 —192.0 + 2.4 Oct. 1 — 198.3 + 1010 + 1.8 —207.9 41.9 Oct. 21 19S.7 1009 1.7 209.fi 1.8 Nov. 10 199.0 1008 in 209.7 1.7 Nov. 30 199.3 1007 1.5 208.4 l.fi Doc. 20 199.fi loofi 1.5 20fi.O l.fi 185i!, Jnn. !) 199.8 1005 1.5 202.fi 1.5 Jan. 2!) 2(10.1 1003 1.4 199.2 14 Feb. 18 — 200.3 + 1001 + 1.3 — 19«.l + 1,3 Oct. 1.^ —203.3 + 9(12 +0.7 -213.9 + 0.7 Nov. 4 203.7 9.-. 7 0.7 214.9 0.7 Nov. 24 203.9 953 0.7 214.5 0,7 Dec. 14 204.3 949 O.fi 212.4 0,fi 185T, Jan. 3 204.4 945 O.fi 209.3 O.fi Jnn. 23 2(14.5 941 0.5 205. (; 0.5 Feb. 12 —204.7 + 'J34 +0.4 —202.2 4 0.4 Sept. 20 — 20fi.9 + 879 —0,2 —214 — 0.2 Oct. 10 207.2 87 4 0.3 217.1 0.3 Oc't. 30 207.5 808 0.3 2 is, 8 0.3 Nov. in 207.8 Kfil 0.4 219,0 4 Dec. fl 208.0 853 0.4 217,fi 0.4 Dec. 20 20S.1 840 0.5 215,0 0.5 1858, Jim. IS 20S.2 S39 O.fi 211,5 O.fi Fell. 7 20S.3 830 O.fi 207,7 fi Feb. 27 —208.4 + 823 —O.fi —204,2 —O.fi Oct. f) —210.4 4 741 — 13 —219,1 —1.3 Oct. 25 210.fi 732 1.4 221,4 1.5 Nov. 14 210.8 723 14 222.2 1.5 Dec. 4 211.0 713 1.5 221.9 l.fi Dec. 24 211.2 704 1.5 219,7 l.fi 1859, Jan. 13 211.3 095 1.5 21fi,fi 1.5 Feb. 2 211.4 «8fi l.fi 213,0 l.fi Feb. 22 — 2I1.« + fi77 — 15 — 209,4 —1.5 Oct. 20 —213.1 4 57 fi —2.1 222, s —2.2 Nov. 9 213.2 509 2.1 224.5 2 2 Nov. 29 —213.4 1 + 5fil —2. 1 —224.8 —2.2 156 THE ORBIT OF URANUS. CuaBECTKlNH TO UK A "i'LiED TO TUB TuMiTioNH OK Uranvd — Conlinuetl. Date. Uelioconlrie. OttovcDlrle. 1 hX Sifif ¥ hi ,M 11 It it If 185», nrc. 19 — 2I3..'> -f554 —2.3 —223.5 —2.3 1»UU, .lull. 8 2i3.r> 54fi 2.3 220. d 2.3 .Inn. 28 213.(5 539 2,3 217.4 2.3 Fob. 17 —213.7 +533 —2.3 —213.5 —2.3 Sept. 24 —214.7 -f 45fi —2.8 —210.7 —2.9 Oft. 14 214.8 451 2.8 223. 1 2.9 Nov. 3 : 444 2.n 2-.?5.0 3.0 Nov. 2H 21;'.. 1 437 2.9 220.8 3.1 Dec. 1H 215.3 431 3.!) 220.3 3.3 1801, .lull. 2 2ir..3 424 3.1 223.8 3.3 Jim. 22 2ir,.3 417 3.1 221.0 3.3 Fib. 12 -215.3 + 411 —3.1 —217.3 —3.1 Oct. 29 —215.fi + ;!3!) —3.7 —225.0 -3.9 1 Nov. 13 2l5.(i 333 3.8 220.9 4.0 Dec. 8 215.7 32!) 3.8 227.4 4.0 Dw. 2s 215.8 321 .1.8 220.3 4,0 1 80 2. .Ian. 17 215,8 320 3.9 223,5 4.0 l'\-l). (i 215.8 315 4.0 220,0 4 1 Ffl). 2(! —215.!) 4-310 —4.0 —210.0 —4.0 Oct. 24 — 21fl.O 424(5 -44 —224.2 — 4.fi Nov. i:{ 21t;.0 241 4.5 220.0 4 7 Dc, 3 21f..O 237 4 5 227.7 4.7 Dec. 23 215.0 2;!2 4.0 227.3 4.8 lues, .litri. 12 215.8 22ti 4.7 225.0 4.9 F«-l>. 1 J 15. 8 21!) 4 7 221.8 4.8 Fob, 21 415.7 213 4.7 218,1 4.8 Mur. 13 -215.7 4-208 —4.8 —214,1 —4.8 Nov. 8 —215.3 4 13!» —5,3 —224.fi —5 5 Nt)V. 28 215.2 133 5,4 220.5 5.7 l>.M'. 18 215.1 n;fi .'.,4 220.8 5.7 18G4, .litii. 7 215.(1 120 ."1.5 225,4 5.8 .lull. 27 214 !t 114 .'■.,5 222,7 5,7 Feb. ii; 214.8 108 5,fi 210,3 5,7 Mtticb 7 — 2U.8 4-103 —5,0 —215.3 — 5,fi Oct. 13 —213.8 4 21 -fi,0 —217.8 —0,1 Nov. 2 213.7 Ifi 0,0 221.3 0,3 Nov. 22 21.3.7 !) 0,1 2 2 to 0,4 Dec. 12 213.fi 4- 1 0.2 225,1 0.5 1865, .lull. 1 213.4 — (1 0,2 224 7 0.5 •Ian. 21 213.2 14 0,3 222,7 0,5 Feb. 10 213 1 20 0,2 2?0.fi 0,4 March 2 —213.0 — 27 —0,3 —215.7 —0.4 Oct. 8 —2ll.fi —103 —0.8 -213,5 —0,9 Oct. 28 211.4 no fi.S 217,2 7,0 Nov. 17 211.3 117 0,9 220,0 7,2 Pec. 7 211.1 124 0,9 222,0 7.3 Dee. 27 210.!) 132 r,!» 222 4 7.3 laort, .lull. l*t 21(1 7 130 7,0 r,-M 5 7.4 Feb. 5 •'10.fi i;5 (I.O 218,8 7.3 Feb. 2.''> 210.4 151 7.0 215.4 7 3 Mur. 17 —210.1 —158 —7 1 -211.2 —7 1 f ^ THE OUUIT OF URANUS. 167 CiiKRECTidNS TO iiE Ai'Pi.iEU TO THE I'oHiTiONS OF I'uAMs — Continued. Date. iI«lior«ntrio. Oeocentrlo. 1 hx II J/^p ft hh II l8Cf., Oft. 3 —208.2 225 — 7.4 —207.3 — 7.4 Oct. 2.1 20S.1 2.30 7.4 211.5 7.0 Nov. 12 207.9 2.35 7.5 214.9 7.8 nee. 2 207.8 241 7.fi 21 7. « 8.0 Dee. 22 207.fi 24« 7. ft 218.8 8.0 ISf.T, .lull. 11 207.4 251 7.fi 21S.5 8.0 Jan. :tl 207.2 25fi 7.7 21(1.7 8.0 F.I,. 20 207.0 2(i2 7.7 213.8 7.9 Mar. 12 —20(1.7 — 208 — 7.7 —200.8 — 7.8 Nov. 27 —203.2 — 351 — 8.2 —211.4 — 8.0 Dec. 17 203.0 358 8.2 213.5 8.0 1808, •la II. (> 202.8 3(15 8.3 213.7 8.7 •Ian. 2(i 2(t2.4 371 8.3 212.7 8.7 Fell. la 202.2 37(! 8.3 210.3 8 Miir. t> 201.8 :',s-2 8.4 207.0 8.0 iMar. 2I-. —201.4 — 387 — 8.4 —202 — 8.4 O.t. 12 — lOH.O — 45S — S.C, — 107.2 — 8.0 XnV. 1 107.8 4(15 8.7 201.1 8.9 N..V, 21 107.5 473 8.8 203. 8 0.1 Kit. 11 107.2 4HI 8.8 20(1.2 0.2 I».r. ::i ioi;.8 400 8.8 207.5 0.3 18(1 it, .Ian. 20 1!h;.4 4!I8 8.8 207.1 0.3 Fell. ii loco 507 8.8 205.4 0.2 Mar. 1 lO.'i.C. 515 8.0 202.4 0.2 Mar. 21 -10,'). 3 — 524 — 8.;) — l!is.4 — 0.0 Deo. (i —1 110.0 — 107.0 Dee. 2 IH'.I.O — (1(15 — 0.4 100.5 — 9.9 Fell. 14 ISH.IJ <17(> 0.4 10H.7 0.0 Feb. 24 ISM fiSCi 0.4 10(1.2 0.8 Mar. ir, 1S7.5 ri07 0.4 103.0 9.0 April ;'» — 1H7.2 — 708 — 0.4 -1x0. 4 — 0.4 De.v 1 — isi.r. — S4(! — O.fi — 1.S(!.5 — 10.0 Dee, 21 isl 1 m5'.» O.fi IHO.O 10.1 1871, •Ian. It l.SO.fl K72 0.7 100,3 10.2 .Ian. 30 170.S 8.M3 0.8 100.1 10.3 Fc'l.. 10 no 1 .X04 0.7 1X8.3 10.1 Mar 11 17X I OOli 0.7 1M5.(! 10.0 Mar. 31 — 177.0 — '.110 — 0.8 — 181.2 — 0.9 Dee. H! — 171.C. — los.l — 10 —177.4 — 10.4 IH'i-i, .Ian. r. 171.0 1005 10. 1 170.3 10.0 Jan. 25 170. 5 1107 lO.U HO. 2 10.0 Fell. 14 100.8 1 1 20 10.0 170.(1 10.5 Mar. r. 100.3 1133 10.1 177.8 10.5 Mar. 2.*» l(l.s.7 1115 10. 1 174.0 10.3 Apri It IC.M.I 11511 10.1 171.3 10 1 May 4 — 1(17.(1 — 11(18 — lo.l — 1(1(1.3 — 10.0 ■fli V0 158 THE lUU T O r URANUS. CIIAPTEll VII. HI FORMATION AND SOLUTION OF THE KQUATIOXS OF CONDITION RESULTING FROM THK riUICKDlNU COMTARISONS. In the preceding cliaptcr wo have ohtaiiicd from observations a series of cor- rections to the {geocentric positions of Vrunns resulting from the provisional theory. The furUier operations are as follows: — 1. To reduce all the corrections in right ascension and declination to correc- tions in geocentric longitude and latitude. Most of the corrections arc already so expressed, so that this reduction is necessary in only a few cases. 2. To find the mean value of the correction in geocentric longitude during each opposition, and to express this mean value in terms of tiie correction to the helio- centric cf)-ordinates. ;{. To express these corrections to the heliocentric co-ordinates in terms of cor- rections to the elements of Uranus and the mass of Neptune. 4. To solve? the equations of condition thus formed. The first of these jjroeesses is too sinii>le to mak(> it necessary to present any details of it. With regard to the second 1 have sought, not the simple correction to the geoeentri*: longitude, hut this correcticm multiplied 'jy smh a factor as it was supposed would make the i)rol)ahle error of tin- correction 0".;"). The equations for expressing the error of geoc(>ntric longitude in terms of trrors of heliocentric longitude and radius vector have been given on page I'i!). The first observation of I'lamstead, p. 107, gives the ('(piation 4- 2v>'=1.04.U + .n'27,^p h?. being the correction to the h error may be estimated at 20", while that of April 2!) is intermediate in certainty. 1'he separate obscTvations give the e = 0.352 2 24 1S02.3 1 =^ + O.S5=r^l.05 13 2) l.S(t5.:5 1 = 4- 0.(i9 = 1.05 13 2G 1N()(;.3 i = + O.20 = 0..>2 5 27 1801.3 1 = -\- 2.34 = 1.045 16 28 1808.3 i t= — 0.0 1 = 0.52 6 29 1809.3 1 -= + 1.80=0.70 9 30 1810.3 1 - + 2.39 = 1.05 16 31 1811.3 I - + 1.49 = 1.01 .^_ ).0l 11 32 1812.4 1 > - + O.S =0.53 8 33 IS 13. J I = + 1.2 =().5:{ 9 34 1814.4 1 -=-h 1.1 = 1.05 15 160 1 r II K OIUMT OF UllANUS. No. Pnle. E<|unlioii» . Xninlier of oltsiTvationa ill It. A. 35 1815.4 3 V=+ 2.1 -=1.58,U 20 36 1818.4* •J = + 0.3 = 1.58 24 37 1819.4 i = — 0.8 = 1.05 11 38 18-20.5 1 = — 1.3 = 1.05 14 3!) 18-21.5 i it = + 0.9 = 0.70 10 40 18:il>.5 i = + 0.9 = 0.70 7 41 18-23.5 .1 = 0.0 = 0.70 11 4'i 18li4.5 1 = 4- 0.5 = 1.05 12 43 18-25.5 ■» = — 0.4 = 0.70 7 44 18-2(5.5 1 = + 0.3 = 1.05 11 45 18-27.7 2 = - '2.7 = *2.07 + 0.04 ,^p 37 4G 18-28.7 -'i = — *2.7 = *2.57 + 0.07 67 47 18*29.7 n = - 2.4 = '2.59 -1- 0.04 61 48 1830.7 n = — 4.9 = 2.56 4- 0.07 73 49 1831.7 2 = 0.0 = *2.oa -j- 0.05 54 60 183*2.7 2 ^^ 2.2 = *2.07 -\- 0.05 65 51 1833.8 n = — 4.1 = *2.58 4- 0.08 88 62 1834.8 n = — 3.9 = *2.57 -j- 0.08 91 63 1835.8 n = — 5.1 = *2.59 -f 0.05 82 64 183(5.8 3 = — 7.0 = 3.11 + 0.08 157 65 1837.8 3 = — 3.r. = 3.11 + 0.01 162 6(i 1838.8 3 = — 1.6 = 3.11 H- 0.06 193 67 1839.8 3 = - 1.2 = 3.11 4- 0.06 170 68 1840.8 3 = — 1.5 = 3.11 -f- 0.06 1*24 69 1841.8 3 = + 0.8 = 3.10 -f- 0.06 108 60 184*2.8 3 = + 1.6 = 3.10 -f 0.06 169 61 1843.8 3 = + 4.7 = 3.12 + 0.04 111 62 1844.9 3 = + 5.1 = 3.11 + 0.03 106 63 1845.9 2 = + 4.2 = *2.08 + 0.02 55 64 184G.9 3 = + (5.3 = 3.10 4- 0.04 98 65 1847.9 *2 = + 6.8 = *2.07 4- 0.03 74 66 1848.9 2 = + 4.4 = *2.()8 4- 0.02 59 67 1849.9 2 = + (5.6 = *2.08 4- <».04 33 68 1850.9 2 = + 7.2 = *2.08 4- 0.01 46 69 1851.9 2 = + «.3 = 2.07 4- 0.04 42 70 185*2.9 2 = + 6.6 = '2.07 + 0.03 54 71 1853.9 •2 = -f 7.9 = *2.09 4-0.01 49 72 1854.9 2 = + 8.9 ::= 2.09 4- 0.02 49 74 1855.9 2 = -l- 8.5 = *2.08 + 0.04 48 76 1 85(5.9 2 = -f 7.9 . = '2.08 + 0,04 45 76 1858.0 n = -f 10.3 : ^ *2.(;i -|- 0.09 66 * The results for ISir, uikI IhU wore ipiiiillcil in this list tlirmigli oversight. J THE ORBIT OF URANUS. 161 Numlirr of No. Date. EqilMinriR. obHiTviitionii ir ill It A. 77 1859.0 2i'' /= + 10.6 = ) ,U + 0.03 -V 58 78 18G0.0 •4 = + 8. -2 — V'.CiO + 0.05 G4 79 IHGI.O •_> = + (i.:j = 2.0!) + o.o;j 41 «() lS()'i.() n = + 7.0 = 2.(iO 4- 0.05 60 81 1H(5:J.0 n = + (>.<» = 2.5!) -f- 0.08 88 82 ls(i4.() ') = + 4.:J = 2.0!) + 0.0:{ 35 8;j i.s,;5.o H = + y.i) = 1.57 4- 0.04 37 84 18()().() n = + 1.1 = 2.(>0 4- 0.06 76 85 1S(H.0 n =; 1.8 = 2.G0 + 0.02 83 8(5 isr.s.o 2 = :j.8 = 2.0!) 4- 0.04 40 87 i,s(;!).() n =: — !).l = 2.01 -f 0.02 r.s 8H imo.o •i :^= y.o = 2.084 + 0.054 31 8!) 1H71.0 u =: — 10.(> = 1.500 -f 0.040 21 !)() 187 2. 1 n == — ii).i = 2.000 -f 0.070 50 inihor of ohsiTX atioi IS ill 11 A., . . . . . . 37()3. 1 -f 0.i»!):J!> CDS c?p 9 6e + 1.099 Oe «/.T de de eiln —2.124 —1.00 u 1.0!)!) +0.0,'!!» 2,124 + 0.001 1.00 —0.02 2 I.01t!» 0,070 2. 1 2:t 0.002 1,00 0.0:1 ■A l.Ollll Oils 2.121 0,002 1.00 O.O.'i 4 l.t»!l!) 0,1. 'iS 2. IIS o,oo:{ 1.00 0.07 T) + 1.011') +0. |;m; —2.114 + 0,004 — 1.00 — 0.00 I.OD!) o,2:i:> 2.110 0,00:) 0.0!) u, 10 7 1.00!) 0,271 2.ior) oooc 0.0!) 0,12 « l.O'.tS 0,.ll.l 2.00!) 0,001; 0.0!) 0,14 '.» roos 0,:!.'i2 2.002 0,007 0.!l!) oin 10 ^ l.OilS -|-o,;ioi — 2.0xr> + o,oos — O.OS —0,17 11 1,007 4.iO 2.077 0,00!) 0. !IS 0.10 12 1.00' ics 2.oi;o 0,010 0.!I8 0,21 l.'l 1.007 o,riO.". 2.o:i0 011 0.!I7 0.22 14 1.0!ll'> .'.44 2,01!) UOll 0.07 0.24 i:. + 1.0!)(! 40,, -.si — 2,0,iS 40.012 —0.07 — 0.2»1 li; I.OO.'i 0,1; IS 2,027 0,01:1 0.!)C 0.2s n 1.0!»r> o,cr,5 2,014 OOll O.OC 0.20 IH 1.004 o,r,o:t 2,001 o,oi:) O.!).") o.;ti 1!) 1.004 0,720 1,0 S7 o.oir> 0.!)r> 0.:!:i iiU + 1.00.J +o.7cr> — 1.07:1 40,01c —0.04 _o.:i4 21 1.002 0,S01 1.057 0.017 0.0:1 o.;ic 22 l.ooi o,n;ic 1,041 O.OIS 0.!):< o.:i7 2:1 1.000 0,><72 i,!i2r. o,ois 0.02 o,:!0 2. 1 000 0,!I07 1 007 0,010 0,01 0,41 2". 4 1 oso 40,012 — is'io 4 0,020 — 0,!ll —0,4 2 2ii LOSS o.07r, l,s72 0,0:>1 0,00 0,4 1 27 1.0H7 1, 010 i,.srp2 0,021 o.si) 0,4.-. 2S l.oHi; i,04;i l.s:!2 0,022 0,HS 0.47 2'.> i(i>4 o.sc o.;'>l 32 l.o.s;i 1 172 1 717 o,o2r> o.sri o.r.:i :;;t 1.0S2 1,203 1 721 0,02c 0.H4 o.:.4 HI l.oso l,2;!;j 1,700 0,02c o..s:i o.r.c ;!■. -t 1,070 -1 1,2c 4 — l,(i7C 4 0,027 — 0.H2 — 0..-.7 lie. l.o-H 1,204 l,r,.M 0.02H 0.x| O.fiO ;i7 1.077 l,:i2:i 1.C2H 0,02s O.XO o.co ;iH 1.07C. 1 :i:.l l,COl 0,020 0.70 o.<:2 :!!i 1.071 I,:i70 i,r.74 o.o:;o 0,7s o, .Ml -f l,oco 4 l,r,.-.i — i.2(;o 40,0:1c —0,04 — 0.7t; r>i l.o:.s 1,072 1 , 22!) o,o;;7 o,c;i 0.77 fi2 l,or.7 1,002 1,107 00:17 0,C2 0.7 s M l,0"i.') 1.712 iicr. ! 0,o;i7 0,C0 0.70 r>4 1.0,-.4 1,7:11 l,i:!:i 1 o.o:;s o,r>!) O.SO .••5 + l,0.-)2 -f 1 7r.o ' —1,100 4o,o;is — o,r.7 — O.Sl f>(l l.o.M 1,7CS I,(m;7 0,0:10 o,ric 0.S2 f.7 lOtO 1,7S5 l,o:i4 o,o:!0 o,r.4 o,x:i f)H : 1.017 1.H02 1.002 0,040 o,.-i:i 0,s4 n;> M"i*i H I,H17 —O.ocs 4 0,010 — 0,.M — 0.s5 THK Oil HIT OF UUA XUS. 1«» d>. 6'/. ('». rV y r9p 9 + 1.044 6e edn —o.nb +0.041 lie edn 00° +l.«a:» —0.50 —0.80 til l.04:i 1.H48 o.ooi 0.041 0.40 0.87 (>i 1.041 l.s(12 0.S07 0.041 0.47 0.87 f.:i 1.040 1.H7.') o.H;i2 0.042 0.4.'> 0.88 r,4 l.o:is l.SSS 0.70,S 0.042 0.44 0.80 «5 + l.o:iii + 1.001 _o.7(i;j + 0.04:1 —0.4 J —0.00 «u-, 1.0 :» 1.012 0.72H 0.04:1 0.41 0.01 fiT 1 ().;:t 1.022 0.(103 0.04:1 0.30 0.01 «■< l.O.il l.o:i2 0.(150 0.041 0.37 0.02 (1» l.o;to 1.012 0.(124 0.044 o.:to 0.0:1 70 + I.02S + l.0".0 — 0.5SS + 0.041 —0.34 — (t.04 • 71 1.(127 l.OiV.t 0.55:1 0.041 o.:t3 0. 05 72 1 o2r> i.0(;7 0.518 0.045 31 0.05 711 1.0 2; J 1.074 0.48:1 0.015 0.20 0.0(1 74 1 1121 I.OSO 0.447 0.0 »5 0.28 0.00 7'> + 1.010 + i.osr> —0.412 +0 1 15 — 0.20 —0.07 70 l.ois 1.000 o.:i7(i U.(>4(i 0.24 0.07 77 l.oir, 1.001 o.:ui 0.04(1 0.22 0.07 7S l.Olt 1.007 o.:j(t5 0.04(1 0.21 0.08 7'.t l.oi:{ 2.000 0.2(10 0.04(> 0. 1 0.08 MO + 1.011 +2.oo;t _0.2:!:t +0.()4C) —0.17 —0.08 Hi I.OIO 2.005 0.1' , 040 0.10 0.00 H-i l.oos 2.007 o.l(i:i 0.04(1 0.14 0.00 8;i l.ooc, 2.007 0.128 0.((47 0.12 0.00 84 1.00.") 2 (»0(1 o.oo.t 0.047 0.11 0.00 «:> + l.oo:{ + 2.005 — 0.057 +0.047 —0.00 — 1.00 Sli 1.002 2.(101 —((.022 0.047 ((.07 1.00 H7 1 IMM) 2.(M(-j +ii.oi:i 0.017 0.05 1.00 8S O.It'.tS 2.000 0.048 0.047 o.o:> 1.00 m 0.007 1.007 082 0.047 — 0.02 1.00 Jto + 0.00.") + 1.00:1 +0.117 +0.((47 0.00 — 1.00 01 O.'.IO.! l.oso 0.152 0.017 -f (1.02 1.00 !t-2 0.002 l.osi 0.1 8(! 0.017 0.03 1.00 ',t:t 0.000 1.07S 0.220 0.047 0.05 1.00 04 o.oss 1.0-2 0.254 (( 01 , 0.07 1.00 05 fO.OHT + 1.0(15 + 0.287 +0.((47 +0.00 — 1.00 Oil 0. OS,-) 1.!I5S o.:i2l 0.047 0.11 0.00 07 (t.OS» 1.050 0.;i54 0.047 0.12 0.00 OS O.OSO I.oi:! o.:i87 0.04(; 0.14 0.00 00 o.oso 1 . o.i;{ 0.421 0.04(1 0. HI 0.00 lIMI +0.070 + 1.021 + 0.45:1 + 0.040 + 0.17 ■ — 0.08 101 0.077 1.014 0.487 0.040 O.IO 08 1112 (t.07« l.OOli 0.510 0.040 0.21 O.08 lo:! 0.074 l.sit:{ 0.551 0.040 0.22 0.07 lot 0.072 l.ss-2 0.582 0.040 0.24 0.07 10.") +0.071 + l.S(iO + 0.(114 +0.045 -fd 20 —0.07 10l> o.oc.o I.S57 0.(145 0.045 0.28 0.00 107 o.oi;s is (4 0.(177 0.015 0.20 0.00 los o.ot;7 >.S20 0.707 0.045 0.31 O.05 loo O.Oli,") 1.S15 0.7:!7 0.014 0.33 0.05 no +o.oi;4 + 1.S(I0 + 0.708 +0.014 + 0.34 —0.04 111 0.0112 i.7sr, 0.708 0.044 0.30 0.03 112 0.0(11 1.770 0.827 0.044 0.37 0.02 U.i 0.0.")0 1.754 0.855 0.04:1 0.30 0.01 114 o.o:>s l.7;is 0.SS4 0.043 0.41 0.01 lir. 0. o.")(; + 1.721 + o.oi:{ + 0.043 +0.42 — 0.00 11 c 0.0;")") 1.704 0.042 0.042 0.44 0,80 117 0.0.")4 l.lisii 0.070 0.042 0.45 0.S8 UH o.o.w l.ddH 0.007 (1.041 0.47 0>7 no +0.0;") 1 + 1.(150 + 1.025 -\ 11.041 + 0.40 _-0.h7 ^:a? m li T Itil T II K O U B I T OF UK A N US. d>. d-K &/. '■'(' ''(' «5|' a 0. de edn d( ve e-/n {■Mt" -fO.lC.O + l.r.:il + l.o:.i + oil +0.f)0 — o.sn 121 0.1»4!» l.i'.ll 1 1.077 0.040 0.:-,l o.sri \.i-2 0.!»H 1..V.IJ l.lol 0.040 :);l 0.M4 IJ3 !t47 l.ii71 1.1211 O.O.'I'.I 0..',4 O..H.J 124 O.it.) i.:..-.i 1.1. -.4 o.o:i<.) u.r)i; 0.^*2 I3.i -f0.'.t44 t l.'i.lO + 1 . 1 ><0 +o.o;iH +».:)7 —O.Ml I2*i O.DI.'t l..')0<.l 1.20.-) o.o:is \).M {).m 127 0.1I42 I.4SS I.22'.t 0.0.17 u.r.o 0.7'J I2S O.'.ltl l.4i:ii 1 I.2...I 0.0.17 U Wi 0.78 1 1't) O.'.MO l.44:i \ 1.277 o.o:i7 u.f-.a 0.77 l.'IU -fl».'.t:il» + 1.421 + 1..IU0 + 0.0.111 + 0.04 — 0.7li III 0.'.t,l7 I.:i!i7 1 .122 o.o;in O.lili 0.7.^ l:iJ O.'.KKi l.:i74 1 :IU o:ir) 0.07 0.74 l;t;j O.'.KIII 1 .l.-iO l.:u;7 o.o;t4 O.II.S 0.72 III 0. !i;i.") l.;i27 l..|M>( O.O.'II 0.70 0.71 i.i;. -|-o.!i:i4 -|-I.;i02 + l.lll!» + o.o;!;i -|0.71 —0.71 I;ir. O.K.I.I 1 277 l.4:;it o.o;i,'i 71 0.70 l:i7 0.'J.12 1 2.-.;i I.4.-.I U.ii:i2 72 o.os l.is (>.!t.ll 1.22S 1 470 O.'ll 0.74 0.07 111) O.'.I.'IO 1.202 I.ISO O.O.'il 0.7') oil) 110 +lt.!l2!l f 1.177 + i.:)Os + 0.0.I0 + 0.7ii — 0.04 141 o.'.tis l.l'il l..'.27 u.(i:io 0.77 0.0:1 142 0.'.I27 1.124 l..^4.') 0.(120 7S o.r>2 lt:i 0.!I27 l.lltM) l..'>r>2 0.(I2S 0.7!» 0.110 1 111 O.llL'li 1 1172 l.'iXO 0.02S O.NO 0..'.!) 11.) )-il.'.l2.') + 1 1114 + I..-.1H; + 11.1127 -t O.Ml — o.;,7 Ill-, O.'.t-.'l 1 Olli 1 i;i:i 0.ll2l-| (I.X2 o.m; 117 0.!ij:| O.'.N!) 1 i;2!» o.ii2r> o.s:i 0..'.4 US o.:ij2 o.!ii;2 i.iiii 0.02.'') O.Ml o.r>;i ll'.l o.;iL'2 (l.'.l.ll l.r..-.!i 0.II2I 0..S5 u..->i ir>o f(l.li2l fo.ooc, + 1.1174 +0.02:1 +o.,s(; —o.'ll) i:.l 0.'.t21 0.S7S l.i-iss l»2:l 0.M7 0.4!l ir>i o.;i2o 0..s4'.t 1 702 0.022 0.M7 0.47 Ia3 0.!M!» 0.S20 1.714 0.021 0.,H.S 0.4:1 l.'.4 oil!'.) 0.7!»2 1.727 0.021 0.M9 0.44 l.i5 -|-0.'.ll<» f0.7ii2 + 1 710 -H» 020 f O.'IO —0.42 I'llt 0.!IH o.7;i;j 1.7;-.l O.OI'.I O.'.H 0.41 1.^7 O.'JIS 0.704 i.7ii;i o.ols 0.111 0.:!!) l.->s o.'.tn o.i;7l 1 77.1 O.OIS 0. O.lll.') l.7s.t 0.017 O.li.l o.:n; I no +ii.:ilil -f-ll.lil.') + l.7'.i:i + 0.011; + 0.'.i4 — o.:ri 1(11 II '.(!.■) o..">s-i 1. HI I.I dill.') O.'.l;') 0.:i.. lr,2 (I.'ll.j ."..").•) I.Hll O.lll.') O.il.'i o.::i n;:{ ((. ;»).■) .'12.') l>20 0.III4 O.'.lll 0.2!l iiii (t.'.ii.-i II. C.ll I.S20 0.01:1 O.tlll 0.2s Id,') f-O.'.Ill -+11.41;.-) + l.s:;i; \ 0.012 + 0.'.I7 — o.2ii Dii; (l.!)ll (1. (:!.•( l.Ni;i O.OII !I7 0.2» Iti7 II. '.Ij 1 II lll.'l I.Mll O.llll 0.07 0.22 ic.s O.'.lj.l o.:!7:i I.S-,;-) OHIO O.'.IS 0.21 KiO It. 11 I.I :!»:t I.HI'mI O.OO'I '.H 0.1 '.I 170 +o'.i|:i +o.:ill + l.sr,-, + O.I)O.S + (I.!IS —0.17 171 '112 0.2><0 1 H70 0.007 O.'I'.I O.ill 172 o.;ii2 24!) l.-<7.^ o.iior. 0.!l!l 14 I7.J 0.!II2 (l.2l!l I.S70 o.ooi; 0.!l!) 12 174 0.!I12 0.1H7 I.HM2 0.00') 0.!I!1 on 175 + 0.1III +0.1. "ill + I.HKt + 0.004 + 1.00 —0.0ft 170 O.'.H 1 0. 1211 l.sstl 0.00:1 1.00 0.07 177 o.'.tn 0.0!I4 I.HS.H 0.002 1.00 O.ll'i 17S '.III o.oi;:i I .SS!» 0.002 1.00 o.o:t I7!» + 0.'.l|| }o.o:il + I.HOO + 0.001 f 1.00 —0.02 Till-: OK HIT OF UK AN US. 160 In the oqiiutions of coiulitioii tea yenrs lias bivn ndoptrd for the unit of time, ill onlcr to iiiuki- tlic gcMunil valiii.> of tlio cuctKciciitH uh iii'iiily iqinil as possihlr, iiiul llic lime liiis hi'cii (duiitid from tlio cjiorli 1N;J0,(), in order to liave tlie jiosi- tive and negative values of t in tlie efinutioiis mon; nearly i»aluiired. To distin-- giilHli theHu values of lie and iii they aru marked with an accent. 'I'iiis unit of •) time gives (),HJ)1 1 for the value of ' in arc, wliencu ('li 'I'lie e«|iiations of condition are now formed by putting in the preceding equa- tions for heliocentric longitude and ratlins vector 6X 67. f'?. , . {jT. fu, h-K = Th + : fit- + . " ,v + T e\n + : , V vv oe t:a.1 O'i • (Je I'H ve ' «<',T 1^ I'or the coefficients " have beiii taken one-hiiiidr<'dtli the iierturbatioiis of da' ' loiif,'itu(l(! jModiucd by Neptune, as given in the heliocentric ephunuris at the end of C'lia[it 0.10 — 1.11 -t-o.oi —0.10 +0.1-2 = + '2 .2 a o.;ji - '>..>;» -|-0.:{:{ -f ().")! -0.-2-2 = +12.8 4 0.3-2 — 'i.'t'i 4-0. •27 -f0.">8 —0.18 = +11.8 T) (».;«) — 2.:Ji +0.1-2 +o.r)0 — 0.00 = +ll.() (1 ().1!> — 1.40 —0.01 -fO.:{0 0.00 = + 5.0 7 0.1!) - 1.18 — O.'.'i) +o.-,';j + 0.-2-2 = +4.8 8 1.41 — ().7rj — '.'.(iO —0.7") +-2.47 = +28 y l.OO — ;■).() 1 — 1.08 —0.7-2 + 1.8") = + 1.1 i(» 1.07 — 4.01 -1.04 — o.so + 1.8.7 = +1.7 11 l.Oi) — 4.S7 — l.S,") — 1.04 + 1.8.") = — .") li o.-;j — ;{.()() — 1.04 —0.07 + 1.10 = + 1 .0 V.i o.:)4 — '.».•.'() —0.71 —0.78 +0.82 = +1.4 11 ().;m — 2.10 — 0.08 —0.81 + 0.81 = — .0 1.-) 0.7:J — 'IS{\ — O.S-2 -l.Ki + l.o:J = — .9 i(> o.:n - 1 ;{!) — o.;u; — o.(;o +0.40 = - .2 n 0.."»() — '.'.O") —0.47 — 0.0"» -f0.70 = — .4 IS o.:n — l.:5:J — O.'JO —0.0") +0.44 = + .0 1!) 0.7:) — '2.M -0.:J8 -l.:]4 + 0.8-2 = - 1 .0 !"1 166 T il E O K H 1 T O V U R A N U S. 20 0.56 y — 1.89^, -0.22,V -1.04.S« +(V58^m' = — 0".9 21 0.5H - 1.92 -0.13 -1.12 +0.55 ' = — (» .5 22 0.;J8 — 1.15 +0.09 —0.74 + 0.-24 ^ - .2 23 o.;}8 - 1.11 +0.15 0.73 +0.21 = - .3 24 1.15 - 3.18 +0.65 —2.13 +0..")0 = + .6 25 1.13 - 2.81 + 1.!1 — 1.91 +0.19 = + .5 26 0.56 - 1.33 +0.62 —0.89 +0.05 = +0.1 27 1.12 - 2.54 + 1.38 — 1.68 0.00 = + 2.1 2S 0.56 - 1.21 +0.75 —0.79 —0.04 =.- 0.1 29 0.70 - 1..V) + 1.09 —0.97 —0.10 = + 1.6 »0 1.11 — 2.19 + 1.73 -1.32 -0.-23 =-+ 2.2 31 1.09 - 2.04 + 1.81 -1 14 —0.-29 = + 1 .3 32 0.5(5 — 0.!>8 -H»-06 -0.51 —0.18 = + .7 33 0.55 — 0.!>1 -I i.OO —0.43 —0.21 -+ 1.1 34 1.08 - 1 CO + 2.0'< -0.71 —0.46 = + 1.5 33 l.(i:< - 2.37 + 3.11 —0.8-2 -0.74 = + 2 .(» 35' 1.07 - 1.45 + 2.08 —0.38 —0.52 = + .4 35* i.oe — 1.33 +2.10 - 0.-22 —0.55 = + .8 30 1.58 - 1.S3 + 3.18 - O.tiO — 0.S5 = + 0.3 3-J 1.05 - 1.11 + 2.10 +0.08 —0.59 = — .8 38 1.04 — 0.99 + '>.07 +0.-2(} — 0.59 = — 1 .3 39 0.69 — 0.58 + l.3() +0.27 —0.39 =z + .8 40 0.69 - 0.51 + 1.33 +0.38 —0.39 = + .8 41 (».<)S - 0.44 + 1.29 -( 0.47 -0.39 .0 42 l.Ol - ().,")6 + J.88 +(t.si —0.59 .4 0.3 4:) 0.67 - (».30 + 1.20 + 0.64 —0.38 —. — .5 44 1.00 — 0.35 41.72 ) !.09 -0.57 =^4-0 .2 45 1.95 — 0..")0 +3.21 4 2,39 -1.12 — — 2 .5 46 2.42 - (>.:n +3.77 +3.21 -1.3(i =-- — 2 .() 47 2.42 - 12 +3 ,-,3 + 3.52 -1.37 =^— 2.4 48 2.39 + 0.12 + 3 i +3.(i9 — 1.33 = - 4 .9 49 1.91 -I- v».28 ^2.;i7 4 3.14 -1.07 rrr _ 0.1 50 1.91 ;- 0.48 + 2.15 +3.30 — 1.08 = — 2 .4 51 2.37 + 0.S3 +2.31 4 ! 32 -1.34 = — 4 .5 52 2.3(J + J. 06 + 2.02 44 u — 1.36 = - 4 .3 53 2.37 + 1.32 + 1.68 44.62 -l.:57 ■=.— 5.5 54 2.85 + l.S(> + 1.66 45.(11 -I. (IS = — 7 .5 55 2,84 -f 2.17 + 1.22 -] 5.7 1 -1.71 = - 4.1 5(J 2.84 f- 2.44 + 81 45.84 -1.74 = — 2.1 57 2.8? + 2.71 + 0.K» 45 8H -1.78 = - 1 .7 5S 2.83 + 3.0() — 0.()(» 4 5..SS -^ 1.S2 = - 2 .0 59 2.82 + 3.28 -0.41 45.S4 — l.s(; :-- f .3 60 2.83 + 3.57 — (».N1 45„si - l.iU =- + 1 .2 61 2.85 + 3.00 -1.27 4 5.78 — 1.98 - + 4 .4 62 2.85 + 4.21 -1.74 -f 5.(il -•2.02 == + 4 .M 63 1.91 + 3.02 -1.42 43,(i9 — 1 .39 = + 4 .0 64 2.85 + -l.-M -2.50 45.:;2 -•2.11 = + 6.0 65 1.91 4- 3.3!» — l.'tV' 4 3.J(i ^1.17 = + 5.6 66 1.92 -f 3.61 -2.18 43.3-2 -1.52 = + 4 .3 67 1.93 + 3.8-2 -2.40 + 3.17 l..-)6 =3 4 () .() 68 1.94 + 4.05 — 2.65 4 2.0s - 1.(10 .-:- f 7.3 69 1.94 + 4.20 — 2. S3 +2.80 -1.61 == -( (i .5 k^ ■'«--: THE Oil BIT OF URANUS. 1()7 70 1.}).-.,V -t- 4.44^h' -3.()4'e -f2.59c,^n — l.(58,'iH = + (5".8 71 l.»8 -h 4.72 -3.23 +2.37 -1.73 ' = + 8.1 72 i.})y + 4.93 -3.44 + 2.15 -1.7(5 = +9.1 7:J 1.99 + 5.13 —3.58 + 1.91 -1.78 = +8.7 74 •i.oo + 5.35 —3.73 + 1.(54 — 1.80 = + 8.1 7.) '2.M + 7.02 -4.N3 4 1.73 —2.30 = +10.(5 7(J 2.54 + 7.34 —4.9(5 + 1.33 —2.29 = +10.9 77 2.5(} 4- 7.(54 -5.97 +0.97 —2.31 = + 8 .2 7s 2.»»7 + (5.38 -4.14 +0.47 — 1.8(5 = + (5.3 7i» 2..">!) + S.25 —5.20 +0.21 —2.32 = + 7 .0 m 2.()(> + H.51 -5.21 -0.15 —2.29 = + (5 •<> 81 2.11 + 7.U -4.19 -0.51 -1.N3 = + 4 .3 S'> l.fiO + 5.5(i -3.13 — 0.5M -1.3(5 = + 3 .9 Kl 'Hil + 9.59 -5.12 — 1.37 —2.20 --+ 1.1 84 2.G8 + 9.!>2 — 5.00 — 1.80 -2.14 = - 1 .s 85 2.17 4- 8.22 —3.92 -1.75 -1.(57 = — 3 .8 m 2.73 + h).()l -4.97 — 2.59 -2.01 = — !).0 87 2.20 + 8.78 — 3.(i;j —2.33 — 1.52 = — 9 .5 88 l.(>5 + (5.74 — 2 58 - 1.9(5 — 1.08 = -10.5 8!) 2.77 + 11.(51 — 4.04 -3.(51 -1.(59 = — 19 .9 2S3.(54,V + 41l.3(i,^u' — 151.()3^; 414.3(5 + 1(519.14 — (5N9.ll 151.(53 — (SsD.ll + 557, S2 247.23 + 2(!0.2(5 + 3S.»5 ri(5.0:{ - 43(;.(t2 + 122.S8 'I'lic lollowiiifj arc tlu' iipproximiifc normals to which these r()!iatioiis fii\e rise. Inaccuracies lieiii;,' delected ill .^cMTal of the e()liatiolls ot" conditinil alter tiiese normals were formed, they uov,-n ([iniitities deduced frc.n tlicse ufU'iiials were snlisti- tnted ill tiie , whih in c<'!uini'. ?! this residu.d is dividetl hy/'to oltain the roidiial correction cd' the loii:,'itii(ie itself 'the \alues of the factor* / are found with tin original (upuitious on pa),'es 159 and 1(50- Cj?«» 168 THE OKIIIT OF URAXUS. . of Eq. Year. f)l 1 1691.0 -0.5 -10. 2 1T1"».'2 —O.S — 10. ;{ IHS.H I n'lO.H T) 17 •»:{.}) (i 17,6.7 +1.5 -f O.M — o.-i + ().() - 1.0 7 1 7(;!).0 H 1 7S-.>.0 )) ns;t.() 10 ITSI.O 11 nsvo 12 i;j u i:. Ki 17 IS 1!> 20 21 ;^2 ;j.v 1 7SS.0 1 7Si).0 175)0.0 nin.o n!»2.o 17!):l.O 17!M.O 17!).").0 1706.0 17!»7.1 22 1S()0.2 2:{ I MO 1,2 24 I s( )•,'.:{ 2.") lso.V;J 2(1 ls(Ki.:j 27 ls07.;{ 2H 1M0H.:{ 20 lson.:J :J0 isio.:} ;Jl isll,;} 1 SI 2.4 isl:).4 IMI.I IHI.VI l^K! t — 0.2 -fO.l — O.j +0.(i — 1.0 + 1.2 ■fl.7 — (t.5 -0.7 0.0 -0.1 -0.6 — 0.") ~o.(; -1.4 — 1.4 —0.6 -1.4 -1.0 O.l 1.0 -f-0.1 -0.1 —<».!) -0.;) -fO.l —0.4 o.:» - (».H — 1.0 + 0.2 — 0.;> + 0.6 — 1.0 + l.S + U.4 — 1.0 - 1.0 0.0 0.2 -I I •■■» - 0.i» - 1.0 - 1.2 — 4.2 - 4.2 — 0.6 — 1.4 - 2.0 - O.l - 2.0 + 0.2 - 0.1 - Us* — o.rt •f 0.2 - 0.4 — 0.2 — «.H So. of K(|. Ycnr. ;n ;js ;i!) 45 46 47 4S 4!) 50 51 52 1S17.4 ISIS.4 1SI!).4 1.S20.5 1N21.5 40 1S22.5 41 l,sv';j.5 42 1S'.'4.5 4:{ 1,S25.5 44 1S26.5 1S27.7 1S2S.7 is-,';). 7 1s:M»,7 |s:tl.7 ls:t2,7 lN:!:t,H ls:n,M 5;{ ls:i.-,..s 54 l.s;{(i.S 55 ls:n.S 56 ls:{,s.H 57 ls:i!).S 5.S l,s|(),S 5!) ls||,S 60 1S4'.'.,S 61 lH4:J.s (»2 1S44,!» M IM.V!) (it 1sk;,!» 65 lsl7,i) M ls|slf 67 lH4i).i) 6s 1M.V),<> 6!) 1M51.0 70 Is.V.'.O 71 ls.-,;l.i) '3? -0.;} —0.7 -1.4 — 1.4 +0.0 + 1.4 + 0.6 +0.7 +2.0 + 1.0 + 2.5 + 2.S -f 0.4 + 41 + 5 +0.1 — 1.4 -:1.4 —0.6 +0.5 -0.4 -1.7 — 0.:j —0.5 + 1 .6 +0.7 i o.(; -0,4 + 0.6 l.() +0.2 |-o.:j -1.2 - 1.2 -0.2 fit n — ().;) — 0.5 — 1.4 — 1.4 + 1.4 + 2.1 + 0.0 + 1.2 + 1.0 + 2.0 + 0.5 + 1.0 + 1.1 f 0,2 + 2.0 + 0.!) 0.2 0.0 0.6 1.1 — 0.2 + 0.2 — 0.1 — 0.6 — 0.1 — 0.2 + 0.5 + 0.2 + ().:} — o.l + o.:i — o.s + 0.1 + 0.2 — 0.6 — 0.6 — O.l THE O U m T OF U K A N U S. 169 Ni). Ill' Ki]. Yi'iir. NO i.sC);;.o ni H Vi IH.-)!.)) + 0.5 + 0.2 r.\ IH.V).!) 0.0 00 14 isr.G.'J —()..■) — o.-J 1") is.ys.O — O.'i ~ 0.1 It; l.s.-,!).() 1 1).!) -f o.i ^ •» 1 1 Ihdd.O - 1.1 — o.i "S 18(U.(> — 0.(> — (),:{ 1!) iw;-,>.() — o.(; - O.'J 40. v' -\- 0.1 No. of V.i\. Yi-'iir. Pl u HI is(;-i.o +0.7 + 0.4 S-2 isd.j.o + 2.-i + i.r. s:{ 1,S(!().0 + 1.0 -f- 0.4 Nt 1M)7.0 + ()..-> -t- o.'i 8.5 INIJS.O + 0.4 •1- 0.'v» Hi l.S(i!).0 -l.iJ — O.H NT IHIO.O -o.l 0.0 HM 1.S71.0 -l.'i — o.s W) 187 '2. 1 +0.4 + o.'i .\ simple •(limcc at tlic roiirsc of tlio rcsiilimls shows (1) lliat tlnir i>rol»iiliIc visliic i.s fonsidcialiiy !j;i(atnti''t'-\ tlw oliservatioiis show a de.«':» I".!»0.0 ; 0.(i'.» d <»■«<» 170.'i.O — O.."),") t 0.11 ISO •.>.(> — l.'j:. + ,."» 1.00 + (!.:il isio..-. o.:!7 +0.:j'2 IMI .'. 0.117 -\ 0.',':l islii,.-) o.:n XS^AH •Xi M..y 1873. Ynir. 18'.>».8 -I l.:,o +0.1 () ISJO.7 + 0.01 +0.10 |8:r..'.> — 0.'.>7 + 0.00 I8;i!).s — 0.17 + <>.i>7 184 4.8 ■1 0.1 1 +o.o:J isio.o — o.-Jl +0.11 is.-, 1.0 .... 0.1 1 +0.11 Isdo.o — 0.1 :J + 0.0!> ls(i.',.o ■f ().:{() +0.10 ls-,0.0 — O.'Jl ri <».n M i 'IF = k no THE ORBIT OP URANUS. |i^ A Miiiplc glance at the* rcsitluiils il hIiowji that tlicy nro much {greater Ihuii the purely lu'cidi'iital residuals rehulting iVoni tlie theory of least sciuares. We may divide th(> putisible cause!* of these sytiteuiatic enttrs iutu three classes. 1. iSi/nlt iiiallc h'rroiM of Olitnrnitwn. — These may result from deviation of the line of ct>llimatiou of the iustruuuMit from u true great cireh-, or from any i»ecu- liarify of thseivntory diiring the yem^ preceding and tnllowiiig, seem to be afieetid witii some constiint err.n in U. .V. of about 'i". I tind that this disc ppaney can be attributed only to the original observations. 'i. JilrmiM III ihi Tliriiin i-nitiintii il.- ■\'\\i'> attraction of an nnknuwn planet. NN'ith regard to the probability of tin >e ditl'i rent siinrees of ernii' it may bi' rem iiKed that errors of C(im|iutation seem pnssibli' imly in the terms of the seennd order, that the mass (d' Satnru is taken from the exhaustive disenssiun of the Saturnian sjstem by IJessel, in which an error suffieient to intluenee tin' theory of I r.uins si mss higldy improlmble, atid that a trans- Neptunian planet large enuiigh to produce a sensible deviation of the orltit of Iranus fntm an ellipse in the cotuNc (da eentnry vvimld be too large to have es<'a|ie(l detection. The ehoii'e of tlie elliptic (liinenls(d' I raiins and N( jitniie is snrh that the terms of the serond order, due to the ai/ ir/iir/i Tlnori/iinil O/tsrrriitiiiH iiri' idih- jMiri'l.-~l\\ the methoil adopted for comparing tbeorv and olisirs in the adopted positions of llie Inmlanniilal stars, and indeed in nearly ai! the ilata of ritha (ion. In tlie lale v .irs they arise principally from the great magtiitiwle (tf the correction to Ho.ivanl's tables, and the cotiseqnent rapid ( liange of the correii' rides, which make the dcti rminaJio!; of the corrections J^l sumI N from theory and obsrr\ation somewhat niieertaiii. l'".rrors jVoni ftii-< source will uecettsarilv Iw in put of a systematic < haracter. and. in view of tie : possibilitv, I re!.,'ret not bating been able to completely re-r<(lue.- all tin- observations before INIO, and to compare all since dircctlv with ephennridt s eoinpnti'd IVom llw pr.nisional theory. In ordir, however, to test fjie ipiestion whetlnr tliej are siMsilde, I leixi' prepired an e|iloMneris from tlie pro\isiftnal theorv for the three relent oi'.positioiis of I'^ii! 2, \h{,} ;[. aiid In 7-.', and compind it dir.cily with ! Til K 'I U MIT UK I' UA N US. ,171 , till' ulisrnaUniiN. 'I'hc iMcfiu r)M'«-ti(iMs ill <;('occtifric loii^^itiidc for •;i'(tiij>s of (il»^:l .! ") (••i) +'r.« K/'.-r •i.!> •i . K ;f ,0 :i.l, 2 » •.',1 , Mcjm I -.Mi) f'i.()i 0) i;;> -1 '->".»> -r •/'.«. •J .;{ 2 ,:», t.' 1 •,' 0, ;{ .;> •> .*». •J .)i • » . « , -f -J .ti.'i -l-i.5! H7'i (1) (•-') __H.;i — H.l, -7. a -7.5, -11 -7.7, —7 .H -7 .«, — 7 .(ij —7 .82 A systciniitir ilHI'i r iicc of 0" IC) woiilii >i riii to lie indiciiti'd, iiiul on nrcomit of it II con'cctiiid iif II", 10 was iiiipl'i'd, to !ii>' i»;iiij>aii,'^oii.'> of tlic la.-l frw jrais in I'liniiid^ titr »'i|iiati!>iis I'f t'diiihtiiiii. Ill uiu of tilt' po.ssiliility of >yst(!iialir i'rri)i< fmni tlii-. .soun-c it may he coii- Kiili'icd that too f^rcat rclativi' wii-^iii !ia,s In t u ;!s,siirii(.(l to the rtsult.s of ilic later oljscrvatioii.H. If tin- ii'siiliials aiisf froiu i rrois of coiiniarison and of (Ihoij, tliiir jirohahli' Jiia;,'iiituili' is iwarly as j.'i-('at ai iuir ipiu Ii a,>. at aiintlii-r. It may llicrc- ton- 111- niti'i'cstin;; to iiKin'.n' what r»'s', >ti'iii of Diraii rrsidiiaU will riiahlc ns to liisciiss tills ([ncstion i|iiiti' ra'-ily. In llir Hist solution wi' sliall rcji'i t the rrsnlts from I'lamsti'ml's olwcrvations, Mwinf^ to tliiir assurni Mn< 'itaintj, and thosr fmiu 1,1' Monnicr's of \'l\'.). nwjn;;; to iIh' ]iii<..ilil<' nia!;!ilJ!i-!iii I i.l -:!.!» -(>.() "•.'.0 fi.'~' r--^-0.iM 1 ):i -U -f-O.I.) -t>.l fO.o = -1 .'.>;■) 1 M ~'iA\ -Hl.;J - I.M o<» =. — 1 (10 •I 1 1 -i:i -1 1.7 1 ;i .» = - ;(7 I 1,1 1,7 * 'H) o.l tl.) = - 0.:!: l 1,0 11 ri\ (0.1 -tl.O ss - 0,:i7 I 1,1) ,v -) l.H 1 0,0 0.(t ass -•{ 1 ,."iO •» 10 0.0 1 1.4 f 1.4 - 0.:, — iO.OI :) o;j ho..-, 1 O.H 4»M 0.5 = -o.'j; :) (•.:» f (>,!» ^01 ,'<• 0(i r^ -0 17 :i !.(» 1 1 ( — «ii -i I.IJ -0.7 = 1 O.H :» 10 ! 1 '» -ft? f Mi - JI.M =:- -0 •.>! :t 10 4*^.> 1.7 -fl.l (».«> = on ;j 10 fiU - 'i.O f l>. 1 040 ~l.i -OM = —0 .;»l a 17-2 T II K O II It I T O F V K A N V S. (living tlios«> uinofcon cqtififioiis «>i|iiiil wci^'lits, we liavr tliv second of flio fi)lIowiii<,' soliilions, anil tiic second »»f the series (if residuals tiie Hrst eorres- pDiidinj,' to liie |triinitive Nolntion. Solvin-; liieni a},'ain and assijrninf,' tlie wcijjlits attaehed to the respective e.inations, wiiich 1 jndyo to he inose to wliich th<'y are entitled when a liberal allowanec is made for systematic errors of observa- tion an -jo.-i ~ ^2 .'2 I we have the second solution, and the third series of ri'siduals. 6*e, - (r.;!<) i*ni — (» .;w i*e —(>.:};) efi^n I .'i.j Ay _ 1 .()-2 m' Vear. 1 a r il ^J I 1) » 7 II lU.SIIH M.S. (-) -0.21 -0.19 -0.1 -) +0.1!) —0.1!) a; 1 ii;*)i.ii -10. ^11. -I-.'. ni.vi -10. - !). ~ 7. n:)i.i + :U + 0.0 + 'i.O IKiil.O - 1.0 - '.>.() — 1.!) ns:{.:{ — O.IS — o.:m - 0.17 17!K).(» -f o.d? ( <).!):{ + O.H!) n!).V(» d.,').". — 0.0!) — 0.18 |s(i-,'.(» - 1. •.'.'. — 0.7S - O.MT iN)(i.:» ~ l.(M> — ().(\!) - 0.73 l^'KI..'! - o.:J7 — 0.10 — 0.05 Is 11.:. — ().:n 0."J(i — O.'iH isl!)..-, — o.:t7 — o.;U - o.;n INJ4.S + l.:.() + 1.4(1 f 1.41 1.S-MI.7 -f o.!)l -i- 0.!)0 "i- o.si is:{:).-.» — 0.'i7 - o.4;j — ().4fi 1n:$!).s - 0.17 — 0.5(i — 0.48 1S44.M -f O.U — o.:Jl — 0.18 |S|!).«t - O.'il — 0.70 - i)M |s.")l.!> - «.I4 — i)M — ().;}« IsiiO.l) — M.l:{ — o.*i:{ - 0.14 |s(i,-,.(» + *.:*« -f o.io + OM'i lHl((M — »:*1 -f O.su -1 U.t4 ir TIIK ORBIT OF URANUS. 173 It will 1)0 socn th;it the effect of these changes of weights is, that the older observations are a little better, and the later a little worse represented. I concuivc tliat our choice must lie between the first and third solutions, tiie first being the more probable if we conceive the outstanding residuals to b.; due to errors of observation only, and tlie third if we suppose; tlieni e(pially due to errors of com- putation. On the whole, I consider the mean of tin; two to ho. about the most probable, and this will giv(! the mass of Neptune very near the rouiul nu.aber 1 i;>ioo which will be adopter! as the definitive value. T!ie definitive corrections to Elements III (p. i>'J) will then be be' {\X-W) h (1N.>()) - '.V'At - li.4.> - 4.44 - 4.1i - O.'i.j - o.i:n f'firi'trlioim la l/ii Iinl'niutiua and A'w/c. These corrections have been derived entirely from the n\oderu observatiems, the ancient oiu's lieing too micertain to add anytliing to tlie weight of the re-sidt. The mode in which the correclif>n to tlie latitu 13 0.3S o.o:J 1 S3 1.7 (-1.0 23 0.«7 0.50 17M4.0 i7s.').0 —0.1 13 10 0.5d 0. so O.Sj is.ii 7 Is;t;t.7 +0.0 -to.n 20 f)4 —0.90 U.93 —0.44 0.37 I7SM.0 — 1.0 5 +0.71 + 71 1.S34.7 + 0.3 02 0.95 0.31 17H!t.O H-(l.!l 1 ,1 i> « 0.77 o.r,4 1S3.-I.7 +o.a 71 —9.97 —0.24 1 ilMI.O 17',ll.0 -1 •-'.(> -1 l.U 4 7 0.s| O.St; 0..'iS 0..-.I ls.|(;.7 ls:l7.S + 0.1 0.0 13.'i 154 0.98 0.99 0.17 O.IO li'.fi.O -|1.S :i 1 O.'.HI + u Is.lS.S — 0.-2 1S2 — 1.00 —0.03 I7!l,».0 ntu.o -f.'.'J 3 0.03 0. '.I.I o.:(7 30 is.m.s 1S40.S —0 4 —0.3 142 lOli 1.00 0.99 + 0.03 0.10 Kitr.o 17!t(;.0 4.;i 1 1-^ 13 0. !l i 0.31 ls4r..o — 11 02 0.S7 0.50 lH(ir>.;t iHO(i.:t -l-l.l — 1.4 3 r. -t(>.S3 0.7S — 11. "ii; o.i;3 isn.o |sls —1.1 — 0.0 CO .'>(; — 0.H3 0.70 + O.51; 0.112 lso7.;i -fl.C, d; O.Ti 0.1,0 Is 10.0 — 1.0 ■M 0.74 o.(;7 1X(IS.,| + I.S n 1 7 —0.74 Is.Ml.'.l —0.7 4l> — O.C.O 40.72 0.77 l.s(i'.».;t + l.;i •1 O.l'.O (ISO 1 s.-, 1 . — 1 . •-• 3."» O.IU IMIO.,! 4 3.7 K! o.:>3 O.S.) 1S.-.2.0 — 1.3 49 0.59 O.Hl Hll .! -|:i.4 11 t (.-, 0.s!( Is.-,:!.o — l.S 4S — 0.53 + 0.S5 o.ss Mli'.4 4;i.(; •; !>.:;>< (1.03 1 s.i 1 — 1.0 47 0.47 lsl;i.4 4-J.» t; O.JO 0.01) Is, -,.-1.0 — 1.3 40 0.41 0.91 |HU,4 -1 •J.S i.t f o.-.'i' — OS Is.Mi.O — 1.1 41 —0.34 4 (1.04 0.97 1M|.>.4 ■l:.'.!' ii; It 0.00 Is.iS.d — 1.7 (;.') 0.20 isic.i -|l.:j H o.im; 1.00 IS.iO.O — !.7 fiii 0.19 0. 9» 1HI7..'. -f.'.4 13 — III — 1 no isr.d.o — 1.0 M —0.12 4 0.09 1 110 isis.r> + I.J 5J2 0.110 1.00 lsi;i (1 —■'-.i 41 0.05 1 Hi '.!.;, + 1.7 7 0.17 0.00 lsr.:!.o --1.0 .'^.2 40.02 1.00 lH;!l».r> + 1.0 — I'l —0.07 lsc,:(.(i — 1.7 H3 + 0.10 + 0.99 lx-.'l.'> + l.:i 10 ::i O.O.'i Isr, 1.0 ■>-2 39 O.IS 0.99 l>s-2-2.:> + 1.S 7 0.3S O.OJ lsi;.'i.(i — \.l 37 0.25 0.97 \x-2:t.:> +o.r, 7 —0. 1.^) — O.Sft l>-r,c,.i» —0.9 72 + 0.33 0.40 40.05 0.92 isoi.r. — 0.-2 11 o.r.i o.si; 1 si; 7. 11 — 0.(> «3 lH->:,.:, + 1.5 5 0..-.7 O.si ll'.CS.l —0.0 32 0.47 O.SS IH •.'(;. 5 + 1.-J 7 — o.r.:{ —O.TS lsr,!l 1 — o.s (i.'i +0.54 + 0.S4 H:J7.(; -1 1>.0 ■I I'lS (1.73 IsTO.l —0.4 32 o.i;o o.so Is2s.(i — l.h 7 0.73 o.r.s IS71.1 —0.7 21 4 0.(;(i 0.75 ls2".t.7 i +(!.!» 9 — 0.7S — O.f.3 Is7:J 1 —0.4 47 + 0.72 +0.(19 It will be rcmomlicrcd tliut the oliscr^'orl (Iccliiiiitions have, ns for 11s imssililc, liccn reduced to .Viiwers' .standard. \N'e li.ivc no |Misitive proof that this »t correction to tlie position of the orhit, will leave a constant residual. Now, we can best ihtrriiiine lliis residual by includinjj it us uu unknown ipiantity in oiir eipiations. THE ORBIT OF URANUS. 1 i-> Aj^uin, the error of the stnndnril is not nrccssnrily coiistniit, Imt miiy contiiin a term iiriiporliuiial tu tliu tiiiii', arising from erroneous proper motions of tlie stundaril stars. 'I'lierefore, imstead of siipposing tlie residual eoiiMtaiit, we shall suppose it of the form a -j- hi. Kaeli observed correction to the theoretical latitude will then give the cqiuition, bin «.^<^ — cos uiphO -\- a -(- h 7'= h^i. To facilitate' tlie solution of these e(|natioiis they have been di\ided into groups, each group usually comprehending three oppositions, and comliined into a single etpiatiou multiplied hy such a factor as would make its proltalde error half a second. 'I'Ik! factor hy which the correction of the latitude is multiplied in the equation is the same with tin; coeflieieiit of ii. The year 1H4().0 is taken as tin- epoch for A. 'I'hus we have the followin*': KglATIciNS or I.ATITI UK. t)iitf:< of o|i|iniiHlnnii. No. of op|>. KlIIMlllMI. ITMS.O-S.l.d 4 i^f — o.'.ifi'y 1 1 .On — n.fifc : —\.r, I'SS.O-'.II.O 4 OS — o.i; ^ 1.0 — O..W( .. 4 1.1 I7'.>2.0 Itt.O :i o.."> —0.2 -|0.5 —0.2 ^ -lO.!! 17!l.">.0 !I7.I ;i a.:! 0.0 4 »."> —0.2 . p».7 isno 2 o-.> .1 ;i (1.0 ■fO.IJ -fl.O —0.4 0. 4 1.0 lso.''i,;t 07. :l :! o.s -1 0.(1 ^ 1.0 —0:1 = 4 "7 |HOM.;j 10. :i ;i 0.(1 -I 0..S 1 1.0 — o.;i - 4 2.(1 isii.:t-i.: J ;i 0.4 -1 0.9 1 1.0 —0,1 - 4:1.1 isn.»-n;. J .•t 0.2 • tl..'') -11.;^ —0.4 : f.TO isi7.(-i!i.ri :( —0.1 • fl..'-) +1..'-. — o.:i 1 4.S 1S20..'. •Jl'..''> ;: — o..'> 4 1.4 4 !.•'■. — o.:i - 12.0 is-.';i. .'■>-•-'.■)..". :! -~0..'> ■1 O.ft 1 1.0 —0.2 >a 1 0.(1 ls-_'r,,.'i I's.c, >*i -0.7 -1 0.7 i 1.0 — 1 - I 0.(1 1 s I'll, r,;; 1.7 •■t -1.2 .|0.S -1 l.Ti —0.1 - 4 1.1 ls:!J.7 :;i 7 :i -^2.S -|l.l -|:i.o —0.2 -^:-\\.^ Ih:-,;-, 7-::7.s ;{ — 2.;» 1 O..'. -) .'1.0 —0.1 -. -, o.:i |s:js.,s-io.s ;i -;! () —0. 1 +:i.o 0.0 m, !) isil M-|;;.s ••1 — -J.'.i -0.7 4:1.0 4(U — — 2:1 isu.s.ic.s .! —•-'.7 -l.:i -1 ;i.o 4i».2 = -2.7 1S47 S-t!M> •> — 2. « — 1.!) -1:1.0 4 o.;i : — ;i.o \ 4 o.(i ^:— l.l Treating these eiiiiatioiis liy tlir iiictlnxl of least s([iiiiiis, we find the normal e({uations If It n 'I X ():5.C.0,^./) + {\.\ruf„\0— rri.li),,,— l.OSAr-. 4. t>s.;{:l (;. l.>.^<^ -I (i!»..'i!>r/»V) — .V.MiO.e — 1 I. :.■)/< I lll.d'i li 'it 1.O8A0 - 1 l.'v';"».i' tiliii'N III' •( .mil /' lliilli nil III. it lit ihr . |iiirli |M|II \invrls' i'i|l|iiliil llll llrrllllMtliillN nil' tiMi en Ml, III lir< liHltll |iiil. || tlir nliln liiniNiiir<4 III ilri lllliilliill liiiil lirrli i ii||i|MIMl)lr III |ilri IMiHl U llli IIiiim' niiiilr ill till' |Mrsi'lll. lillir, Mini ll' till- |iiisMli|r iii'iiKilii riliil in llir irilmril ll)^llt iiii rlisiiiiis linij lirni Clllrlllll) rlllllllliilril, I sliiiiilit ir;;.ilil llii . ililrl liiMliltliill lis rlilltliil tii riMINJlli'l'illlli' Mi'l;;lll III \lr«\ iit' lltr <;ir,ll lllli rl t.illll \ ill llir ilri ll|l;|l Inlis |i|r\|iM|-i to IN'.'II, it run III' n'l^.inlril ns liiilr mmi' lli.in u iiiii.';li iillrin|il iil ii ilrlriniiniilinii. I'm lliii Irii-tiin till' lii-^l h\ii iimniiil ri|n.itimis Iium' lin ii shImiI, Ii.uiii;; h imil !• milrlri ininiitr, so IIS III hliinv tin \iiIm s nl ,\i^, miil i/iAll in Iriins nl tin sr i|niiiitilirM ||. Mill III- Nirtl lllilt llilil Ki' ni';;li'i till .1 iinil /< rnliii'U, till' tilllli' III' Ai/i umilil IllMr lirrll stiliillrl li\ '.'Ii, Mini llnit nl i/i,\|/ Mihillri li\ IT |S tllMli IIiiim- iiiIiiiIK inn rlllili'il \'. llir nliM'M iitimis » itli tlir N\ ,isliin;;lnn Tl iin'.ll ('llilr, iinil tllnsr N«|||| lln' rnlKiMtii \ rilii ill ( 'mil', Imlli liiilii.itr im linir.i'.i' nl' \in\il't' |iii|.il ill'.liinri s, I --ll ill hiKi' Im llir ill lililllM' I I'l III llnii'i In I lir lIli llll.lt Inll imil liiiili' lllir.i' u lili li inlinn limn tin' iilin\r \,ili|i I' ii iilnl /', m. W\ .Nl/. i/..N() I II ll I I ..-., Tlic rnllnwilli; l.ililc -.li.nv^ tin' ii-.ii|ii,iK nl' llii' i'i|ii;ilinns, miiiI tin' iiii'mii mil- Ntiinilllif^ iniii rlimis to tin- l.ililinli , \ I | win n tin' i i.ni liiili i| \ .\\\\i ; nl .N miiiI ./..Vl >' 'Mill /• .III' iitl liM'il, iinil ('.M Ntlnn n ,inil /> iiir sn|i|in/ iV.\ I 1^ I ••.'i I I '.^ I o.:t o."» I I •-' I I '1 I 0.1 I IT 1 0,7 o.:t TM 10 OK KIT OK II II A N VH. m ^ riir |h:io ls:i:i \K{V, |m:j!» I Ml? |H|.. IHIM IH..I iN/il |M.'»H I HI. I I Ml. I |H)1V I Ml ill iMiJ IIi'kIiIiii.Ih (1) (•!) II It Hi \ •.' |o:i 1 0<) o;i 1 0.1 .. 1 |0.l 00 1 li in.1 |0!l 1 OM 1 o;i 1 o.:t |0l 1 o.:i 1 1 10 1 .. 10 1 1 1 1 1 J 1 ,, 1 "» 00 1 O.V ¥ /I Ci) 1 MM 1 10 1 :{0 0.00 1 10 O.ICt 1 o:i on 0.1 :t 1 ii:i 000 1 ''0 1 •,»;{ 1 :io 1 :io 1 10 1 10 1 o.o:i 1 10 -o.n o:i:( - o.rio o:i:i 1 :n 1 0,.'.7 1 17 1 :i:t 0.00 1 o,;':J 'I'lic hiiiii t.r lln' st|iiiiirM I.I' llii' i( siiltiMl'i In ill (III lii>l ni'.r I7".!M, mill in llir Hiriiiiil '.'.ill, Ki. Ilial till' iiilMHliirlKiii iil i( mill /' llhiki s a lli riiliil illl|ili)Vi'lili'lll. lit ||>i' |i'|iri'si liliillnll III llir l.li.ilMltli.llS. I liiiM' nut iilli iii|ili il II ti;'iiinii'. in\i slii'iiliiiii iif lln |.ri.l.Hlilr iiini iif miy of till -.1' ir.iill-i lor llii 11 M'.iin lli.il llii' talin -1 nl llir |.iiil>.ilili' ritm ill iliii ilili' liy llii; iiii'tlinil id II Mst si|nmrs Hi.iilil, III a iiisr liKr llir |iii .i III, lit iniiri ly nnliiiNlwnrlliy. II is, IliiWiM I Ml\ ill Mialilr Ihal \M' '.lliilllil \»- alilr III riillll s • jl|il);nirlll III' llli- niiii il linU 111 tlir iini'.^. nl' Nr|.lmir. I'mni llir la. I sy.li in nl i i|iniliiiiis ul ininli- tliill llir >alilr nl' »' nillli'H mil, willl llir Wril-lll :i lit, ni liiailv lllil liNsij^'lli il In llir nil III naill nl' i aill IiM' \ial-. i.I' iimililli nl.srivalinlis. Ki7,'aiillli(,' lliisr riMllls iih null |ii lull III, llii II nil all i mil Mnlllil lir al.iilll 0"..'., sn llial. llir |i|iilialilr rum nl ^,' Wiilllil lir 0.., anil llial nl ii »»nnlil lir 00... nr alinlll , ,', „ llir ilililr Iiiiisn nl" Nrlillllir. A lilnlialilr illiil ili.lMil llnni llir nii;;inal i ijlial iiili> Wnlllil liaxr III 1 II ininll s Ill T. I wlirll, ill llir la. I rijiial lull ., \Vr allow lor llir s\.lrlliall(; « liailM'Irr nl" llir rrsiillials, il mil lit Im^n. IT \Vr sli|i|.iisr llir llirnl) 'n lir iiill'iil, I n.lirrivi' \vr iniiy I'ailh i slimali llir |ili.lilliilr i itin III llir inil-iM III' Nr|.liiiii In lir > III' ils I lilllr miliiinil. anil ils |.nssili|r rlliil lun nr llin r linirs (.'iratrr. H Mil ir JM any ninr nr iniiirrlri Imii in lln llimi), lln i iinr may In niinli lai;.;i i. •i\\ M»y,IH7:l m VQ <^ /2 / ^ #./ ^ #5. w fi: .'%- /^ M o 7 IMAGE EVALUATION TEST TARGET (MT-3) 1.0 m m 2.5 2.2 I.I 1^ 12.0 1.8 Photographic Sciences Corporation k A O V «> #> '^^ 33 WEST MAIN STRIi T WEBSTER, N.v •tSSO (716) 873-4503 3 sin iV„ — 3 .87 cos .V^ + 8.06siniVj— 8. 38 cos .V, These terms, together with the arbitrary corrections of the elements whi'ih have been applied to make them very small at the epoch, may be replaced by tin follow- ing corrections to the elements: ,y = — 0".55 sin iV,, — 0".03 cos N, + 40 .05 sin iV, — 10 .50 cos N., 4--27".-27 — ll".7-.>7'. ,Vt =+ 2 .09 sin .V, + 1 .94 cos N^ ' — -i.lSsiniV,-}- 3.71 cos iV^7 + 1".'28. a- = 4- 1 .3'2 sin iV„ -1- 2.32cosiVr„ 4- 3 .08 sin iV, + 2 .21 cos JV, fiv = 27 sin iV, + i04 cos iVj + 76 (in luiits of the 7th decimal). The amount of the perturbations of the elements for every half century, from the year 1000 to 2200, is given in the following table. Cohinm(l) gives the per- turbations by Neptune, Saturn, and Jupiter, computed from the expressions m 1' m- m 184 THE OR BIT OF URANUS. on pago 182; column (2) those just given depending on tlic product of the masses of Jupiter and Saturn. Year. hi Ih hk hv hx h ^'\, C?) (•>, (2) ^'l (2) (!) (2) II 1000 +2050.31 +100.09 + 42.00 +1.94 —389.51 +G.08 + 1955 —140 —2.00 + 1.20 1050 1841.17 149,08 27.79 3.00 375.05 0.08 1882 150 2.10 1.37 1100 l(i38.70 130.39 14.03 4.99 300.31 0.82 1802 169 2.12 1.64 1150 1444.87 122.87 + 1.45 0.01 343.49 6.47 1717 178 2.08 1.60 1200 1200.24 108.85 9.90 0.00 325.27 5.04 1026 183 2.02 1.77 1250 + 1085.7(i + 94.07 — 19.84 + 0.75 -305.09 +4.42 + 1529 + 183 —1.94 + 1.80 1300 921. 7(i 80.07 28.34 0.49 284 90 2.80 1420 180 1.84 1.91 1350 7(!9.32 07.12 35.27 5.90 252.90 + 1.10 1318 173 1.70 1.92 1400 fi29.00 54.35 40.50 5.08 239.78 —0.57 1205 101 1.64 1.91 1450 501.39 42.02 44.11 4.16 215.08 2.17 1080 147 1.37 1.85 1500 + 387.18 ^- 32.11 — 45.83 +3.24 —190.00 —3.52 + 903 + 129 —1.19 +1.74 1550 28G.05 22.78 45.03 2.42 104.84 4.52 835 110 0.99 1.60 1(100 200. fi5 15.38 43.40 1.80 138.31 6.12 704 90 0.80 1.42 1G50 129.43 9.34 39.20 1.41 111.22 5.30 608 08 0.01 1.20 1700 73.40 4.80 32.79 1.23 83.70 5.09 430 48 0.44 0.95 1750 + 33.00 + 1.90 — 24.10 + 1.23 — 55.90 4.50 + 288 + 29 —0.27 +0.60 1800 8.51 0.33 _ 13.20 1.30 — 27.93 3.81 + 146 + 11 —0.12 +0.34 1850 0.00 0.00 0.00 1.52 0.00 2.00 — 5 0.00 0.00 1000 7.04 0.71 + 15.05 1.03 + 27.74 2.10 - 145 18 +0.10 -0.30 1950 31.47 2.23 33.70 1.59 55.09 1.34 290 26 0. 10 0.73 2000 + 71.43 + 4.25 + 54.20 + 1.34 + 81.81 0.73 — 433 — 32 4 0.20 —1.10 2050 127.33 0.52 77.12 85 107.77 0.34 574 30 0.20 1.47 2100 198.80 8.75 102.40 +0.13 132.07 0.14 711 27 0.17 1.83 2150 285.72 10.59 1.30.15 —0.77 150.33 0.11 843 20 0.11 2.17 2200 387.11 + 11.77 + 100.15 —1.70 + 178.48 —0.19 908 — 9 +0.02 —2.48 Mean Elements af Uranns. If, instead of the elements of Uranus affected by the great inequalitj', we wish the absolute mean elements, these are to be obtained by adding to the elements already given the constants applied to the perturbations hi, hh, hk, and hn to make the perturbations vanish at tiie epoch 1850.0, and also the corrections (p. 113) which we have subtracted from the elements and added to the perturbations to reduce the latter to a small quantity during the period for which the tables are likely to be used. We thus find the following mean elements : Elements V of Uranus. Epoch, 1850, Jan. 0, Greenwich mean noon. l/ongitude of the jierihelion Mean longitude at epoch, Longitude of the node, Inclination of the orbit, Eccentricity, Eccentricity in seconds, Mean motion, Log mean distance (uncorrected). The same corrected. True mass of Neptune, 170° 38' 48".7 4-8()i)8". fi 29 12 43.73 + 2811.4« 73 14 37.6 + 29 .6« 46 20 .92+ 0.38/t .0463592 — 5236,a 9562".27— 108".b« 15424.797 — 0".838« 1.2829251 + 179h " 1.2831223 -j- 179;t 19700 *f4ai- THE ORBIT OP URANIJS. 165 Supposing the ninss of Noptunc to be imccrtnin by oiu'-fiftietli of its entire amount, wliieli is quite possible, it will be seen the longitude of the mean j)eii- helion is from this cause uiicevtain by more than two minutes, the mean longitude of Uranus itself by nearly a minute, and the mean motion by nearly two seconds in a century. It will be seen that the logarithm of the mean distance just given docs not accurately correspond to that of elements I V' jjIus the constant term of iSv X 0.-i343, as it should. This ditt'erenee arises from the rejection of the terms of the second order in hr), which can not affect tlie geocentric longitude of the planet by a tenth of a second for a number of centuries. It is to be remarked that these mean elements arc those to be used in the general theory of the secular varintion of the planetary orbits. Cviirhuhd Theory a/ l^raiiKx. The elliptic longitude and radius vector of Uranus, affected by the secular and long period perturbations of the elements, will be given by the following equations. Tut I, = iij + f(„ (j = l — 7r,„ k = c, + tf>\ the zeros mdicating elements IV, and i7i, fil; and &l being the perturbations of these three elements just given. Then Elliptic longitude in orbit = Z + |2-]^e=-f ^e' I |/.sinj/-Acos^J + i 1 i! ~ 04 "' } I ^'' " ^'''^'^ ''" '^^ ~ ^^''' ~ ^'"^ '"' '^'A . Nepcrian logarithm of >• = n -|- I*?- -|-^,\,e' — I 1 — j^«'' I I /'Cosr/ + 7t sin^ J - { 4 - 2I '"' ] 1 ^'' ~ ''"'^ "^"^ ^^ + ^'''' '"' ^^ } 24 May, 1873. 'V T 1 I I I M '0 17 li . 1 I I 1 186 THE ORBIT OP URANUS, ~ 24 { ^'''' ~ "^'''''^ ""^ ^^ "^ ^'^''''' ~ '*'^ *'" ^^ } — Ij J I (/.' — Gi7i^ 4- A') cos 4j + (.4/.-7t — 4M') sin iy I In computliifif these oxprcssions it will bo sufficient for scvorul centuries before or after 1850 to develop h, H; and <*/ to tlieir first dimensions: it will, liowever, l»e more convenient to correct the mean anomaly ij for the perturbation hi before obtain- infj the equation of the centre;. Dcvelopin<; the perturbations of 7i and k to terms of the first order, we have for the effects of the perturbations of those elements: fk (('.c.l) = -(*2-^f',r)<7« {v.8.2) = (Jo - y^o' ) '^* (r.s.3)= ^^c,,-Uc 13 „„ (r.c.;J) = — "4 <'"' «« 103 (i-.c.4) = — .j^V ' ^« (p.f.O) = hn -\- (\,()7c (p...l) = _(l_|,„^),vA (p.c'.l)-=-(l-yv)^/^ (p.c.2) = — 2^0 '7'; 17 8 17 (p.«.3) e.'^h (p.c.3)=-- J^e,^7^ These coefficients for p must, of course, bo multiplied by the modulus 0.434394 to reduce the perturbations to those of the common logarithm of the radius vector. *£i4." THE ORBIT OF URANUS. 187 Among the elliiitir terms may l)c> iiuliulcil the effect of the following minute constants introdjiced by tlie peitnibutions. (ns. 2) =_()". 144 (w;.2) = + 0.l:{() 0.434;} (p.f.O) = -f l\)V2 in units of the 7th place ().4;34;j (().«. 1 ) = -j- 63 of decimals. 0.4343 (p.f.l) = + 73 0.4343 (p.«.2) = 4-5 0.4343 (p.r.2) = 4-4 This term (p.c.O) is that added as a eonection to the logarithm of the mean distance. To the coefficients (I'.sA), (r.cA), etc., arc still to be added the following periodic terms : — 1. The periodic terms due to tin; action of Jupiter, given in Chapter V, omitting the terms multiplied by T, which are inchuled in the perturbations of the elements. 2. The periodic terms produced by Saturn, including those terms multiplied J)oth by 7' and by sin A.^ or cos A.,, b»it omitting those multiplied by 7' only for tlie same reason as in the case of Jui)iter. 3. The periodic terms produced by Neptune, multiplied by the factor 0.8(5294 on account of the correction to tlio mass of that planet, and omitting the terms multiplied by hi, he, and elg. 4. The periodic terms multiplied by the product of the masses of Jupiter and Saturn, given on page 88, omitting the terms nndtiplied by the sine and cosine of N^ and N-, because tiiey are replaced by tiie terms of hi, hh, and hi; given on page 183, and tabulated in the columns headed (2) on page 184. The result will bo the same whetlier we employ the terms of (r.c.O), (r.s.l), etc., given at the bottom of page 88 and tlie top of page 89, omitting the nundwrs in the columns 2 on page 184 from the expressions on page 18G, or whether we include the latter and omit the former. The true anomaly of Uranus will then be : , 0, and t will be found by successive api)roximations from the formuke 188 THE OIUUT OF U II A NUS. (;U), corrcctiiig tlic dutii for tlic new nuiss of Ncptiiiu'. \Vc shall also use the saiuo niotiou of the (.•clii)tic adoptud on j). !).j. \\C liuvo thus: til — r.:>:\ ^= + 5.4:}4-0".;J87'. at lit lU = — 5.17 :=_4(;.7S-f 0.1'27'. Asa first approximation wc liave = T = 7:J^ 14' 8" — 31G!)".'i7' vahies of hr^ and i^x to be used in tliose expressions are those the expressions for whicli are given on page 97, and wliich are tabulated in the last two columns of the table on page 184. The following tables are bused on the elements and theory laid down in this chapter. Ul' 190 TUE Oil 15 IT OF UUANUS. CHAPTER IX. GENERAL TA15LES OP UUAXUS. Emnmration of the Quantifies contained in the several Tahlca. The first six tables are designed to give the values of tlic three arguments of the elliptic motion, g, w, and 0, and of tlie nine arguments of the tables of pertur- bations. The argument u is, however, diminislied by 3', the sums of the constants added to the perturbations of (i-.c.O) to make these quantities positive, and by 10", the constant added to the reduction to ecliptic. The expressions for the argu- ments of perturbations are as follows, the mean longitude of each planet, counted from the perihelion of Uranus, being represented by the initial letter of the planet. All these arguments are expressed in units, of wliich GOO make an entire circum- ference, so that each unit is 3G'. The time t is counted in Julian years from the fundanu iitul epoch, 1850, January 0, Greenwich mean n. on. Ar; 1 = J— u = 219. 1 0( + 43.44028/ 2 = S — u = 577.34! 1+13.22717/ 3== U- X = 88.88-i t+ 3.50035< 4 = J- 2S = 497.(5 + 9.8445/ 5=: 3,^'- U- - ,7= 79.8 + 3.3825/ () = 46' — nr- J= 57.1 + Kl.OlO/ 7 = '2J - 3,S" - -3r7=238.7 + 18.()33/ 8 = 'iJ — \S - 2f/=2()1.3 + 5.4058/ y^ IS - '2 J - 3^=130.9 + 19.992/ Table I gives the corrections which must be applied to the values of the argu- ments at any time during the nineteenth century to reduce them to the corresponding time in any preceding or following century between the Cliristian era and the year 2300 Since o and each contains a term proportional to the square of the time, the correction for these quantities is not constant during each century, but is of the form ij + cj'T* 6) and J being constant during each century, and T being the fraction of the century counted from its beginning. Table II gives the value of 7, u — 3', — 10", and the above nine argiiments for (ircenwich mean noon of Jan. of each leap year from 1752 to 1948, and for J iniiary — 1 of the years 1800 and 1900, corresp'Muling to Decemb(>r 30 of the years 1799 and 1899. Tlu: corrections for the perturbations of long period are not k3 THE ORBIT OP URANUS. 191 applied in this tabic. The numbers at the bottom of this table, in the line AfJi, sliow the variation of the corresponding quantity in I'iO days, for the epoch 1850.0. In tiie line "Factor 7'" is given the change of tliis variation in a century, while Alii is the second difference for intervals of 120 days, liy means of tliese num- bers, when the arguments are computed for any date, their values for other dates at intervals of 120 days may be found by successive addition. Table III gives the motion of tiie several arguments between the epochs of the preceding table and the zero day of each montii in the course of a four-year cycle. The variable motions, w and 0, correspond to the epoch 1850, and rigorously they each require a correction for any other four-year cycle tlian that between 1848 and 1852. But, owing to tlie small inclination of the orbit of Uranus it is not neces- sary that either « or should be exact, if only their sum is exact. Tlie column 0' of this table, therefore, gives the correction wiiicli nuist be applied to the motion of at the end of a century (1050) in order that, being applied to alone, (j -)- may be exact. Tliis correction is, in fact, tiiat for the secular variation of the precession. Tables IV and V give the motion of the arguments for days and hours. The motion for hours is, liowever, not necessary in the case of any argument but (j, as all the others can be readily enougli interpolated to fractions of a day. Table VI gives the corrections to the arguments on account of the terms of long period from 1000 to 22()(T. The terms in question are, in the case of Jupiter, tlic great ineciuality produced by the action of Saturn, in the case of Neptune tlie great inequality produced by Uranus, and, in the case of Uranus, the iiu'qualitics in the mean longitude tabulated in the preceding chapter. The numerical expres- sions are U= 0.5:J5 sin (110° 21' + 40^ 45' 20" T) hu=n The corrections to the several arguments are b arg. 1 = (7 U—H b arg. 2 -rV S ar?' 3 = (V ,").V=1.75,y Xo correction to the mean longitude of Saturn is applied, all its inequalities being taken account of in the terms of the second order. The corrections, expressed in seconds, have been reduced to units of the argu- ment by dividing them by 21(iO". Outside the limits of the table these corrections must be computed from their formuhe. Table VII gives the equation of the centre, and the elliptic part of the logarithm of the radius vector. No constant is ajiplied to the former, but the latter is dimi- nished by .0()0:M00, the sum of the constants added to (ji.r.O) in Tables VIII, IX, X and XVII. 192 THE ORBIT OF URANUS. The formulae for the Tables are Equation of centre = 193iV2".()() sin -j- 5G7 .24 sin 4- 23 .05 sin 3j + 1 .07 sin 4 450 — 445 (p...2) + i 450 — 446 The perturbation constant of (p.c.O), being added to log a in forming tiie elliptic radius vector, is not included in this table. Table XVIII gives the reduction to the ecliptic — 9".37 sin 2>,. The constant 10" is added to make the numbers always positive, wliich constant has been already subtracted from (). Table XIX gives the principal term of the latitude 4(i' 20".54 X sin ». 'fable XX gives the coefficients {h..s.l) and (h.c.]) for the jjerturbations of the latitude produced by Jui)iter. They are given by the formuhc (-/>.«.!)=: 0'.65 cos (,/— r+ 40^) (h.c.l) = .65 sin (./— U -\- 40^) The ronstnnt 0''.70 is added to make all the numbers of tlie tabl(> positive. THE OUBIT OF URANUS. 195 Table XXI gives the conospouding coefficients for tlie action of Saturn, com- puted from the expressions on p. 87 witli the addition of the following constants. Const, of {h.c.O) = 0".lO (i..v.l).-=;}..3() (i.e. 1) = 3.10 (i.«.2) = .20 (6.C-.2) = .20 Tabic XXII gives the coefficients for the action of Neptune from the formuhe on p. 87, all the numbers l)eing multiplied by the factor ().8();3 to reduce them to the adopted mass of Neptune. The following constants are added : To (i.f.O) .... 0".06 (i.s.l) 1 .00 {h.c.l) 1 .20 (/>.s.2) .20 (A.r.2) 0.20 Table XXIII gives the secular and long-period terms for various epochs com- puted from the formula? of p. 189. The sums of the several constants added in the three preceding tables are here subtracted again so that these expressions become h.rJ) — 2c= 0".10-0".12r— .01b^>7 + .046,^x {/t.f.l) ~^c^ — o .00 — .227' — 0'.orj7'- + .975^>7 + .221Sx (A.c.l) _ vc = _ 5 .00 + 2 .47T+ .12'/'^ + .22bV (/>.«.2) — 2c = — .40 — .01 7' 4- .046,V + -Ollf^K (/AC. 2) — 2c = — .41 + .127'+ .OIU*? — M68x .9'iobx Prcccjyfs for the nse of tli( Talks. Express the date for which the position of Uranus is required in years, months, days, and hours of Greenwich mean time, according to the Julian Calendar if the date is earlier than 1500, according to the Gregorian Calendar if it is later than IfiOO, and according to either calendar between these epochs. Enter Tabj I with the beginning of the century, and take out the values of g, «, J, 6, 0', and arguments 1 to 9. Multiply w' and (/ by the fraction of a century corresponding to the date, and write the products with their proper algebraic signs under o and 0. If the calendar is the Julian, the century marked ,7" must be takcMi, and if the Gregorian, that marked G. Between the dates 1752 and 1951 it is not necessary to enter Table I at all. If Table I was not entered, enter Table II with the yenv, (U* the first preceding year found therein. If Table I was entered, enter Table II between the year 1800 and 1896 as if the niimber of the c(>ntury wer<> changcul to 18. Take out th(! values of centuries and fraction of a centiny of the actual date after 1850, and the l)rodiHt must be included witli the corresponding quantities, (r.'i.l), (r.c.l), etc. Before 1850 this product will always be negative; aftmward always positive. All the quantities taken from these tables are positive except (f.«.4) and (r.c.4) in Table IX, which are negative. Add up all the partial values of (r.c.O), (r..'».l), etc., thtis obtained from Tables VIII to XVI, and from their sum take the corresponding quantities obtained from Tabl<> XVll by interpolathig to the date. The rc^quired quantities arc all given in Table XVII A ; Table XVlIa being only an expansion of a i)art of XVIli for the present century. The final values of (r.s.l), (r.c.l), (r.s.2), etc., (p.s.l), (p.*".!), etc., thus obtained are to be multiplied by the sines and cosines of the correspond- ing multiples of g, in doing which four place logaritlmis are sufficient if the coni- jMitation is carefully mad(\ Tlie products are then all added togetlii^r, and to f/, w, 7s', and (r.c.O); in the; case of r, and to log. r, (p.c.O) in the case of p. That is, we are to form the expressions; u==.c.2) cos Qg, and add it to the principal term of the latitude, with regard to the algebraic signs. Tlie sum will be the heliocentric latitude of Uranus above the ecliptic of the date. "SViien an ephemeris of Uranus is to be conq)uted for a series of years, some modifications may be introduced, which will save the computer labor. In the first place an ecpiidistant series of d.ites being selected for computation, it will be suffi- cient to compute g, u, 0, and the arguments for every sixth, eighth, or tentli date, and to fill in the arguments for the intermediate dates by adding the nearly con- stant ditterences corresponding to the adopted intervals. The agreement of the numbers thus obtained for the last date with those found by the original computa- tion will prove the whole process. This interval may be as great as l'>() days without detracting from the accuracy with which the places for the immediate dates can be interpolated, and the diff'erences for this interval may be deduced from the numbers at the bottom of Table II. If these numbers are used without change the values of w and for the last date muy not always come out right. But these errors, if less than a second, will be of no importance if the one quan- tity comes out as much too great as the other is too small, and they may be avoided entirely by making a small cliange in the constant difterence to be added. Tables XI to XVI, inclusive, need be entered only for every third or fourth date, and the sums of the quantities can be then interpolated to every date, and added up with the corresponding quantities from the other tables. Again, it will be found convenient to compute the sum of the small terms (y.s.3) sin % + 0'<^--^) ^^^ '^.Z + ("•^''•*) ^'" "^'^ + {r.cA) cos 4r/, as well as the corre- sponding terms of the radius vector, and all the terms of the latitude, not for the dates adopted, but for every fourth entire degree of g. Having a series of values computed in this way, the sum can be interpolated to the value of g corresponding to the date. To facilitate the formation of the smaller products for entire degrees of //, a table of products of numbers by the sine and cosine of every degre(> is appended to these tables, by which the products in question can be formed at sight ji 198 THE OUBIT OF URANUS. wlioncvor the coefficient to bo nmltiplicd is less than 3'2". Tiie Viihies of these coofficieiits, (f.«.;3), (r.r.'A), etc., corresjioiuling to the entire dej^ree.s of //, may ho eitlier foiinod by interpohition at sight from tiiose corresponding to tiie dates of computation, or the vahies of the arguments 2 and !i corresponding to tin; required degrees of ij nniy be computed, and the values of {r.x.'.i), etc., correspond- ing to these values of the arguments may be taken from Tables IX and X, while Table XVII must bo entered with tlie corresponding dates. If tlic lieliocentric ephomeris is coinput(;d for ten years at a time, tlio last of those modifications in the mode of computation will greatly facilitate the computa- tion of tiie smaUer terms. We first lind tlie date, and the values of arguments 1, 2, and 3, to one jdace of decimals, for some entire degree of y preceding that wiiicli corresponds to the first date, and tlien find the dates and the values of the arguments corresponding to successive values of (/, differing by 2"^ or 4°, until we pass the last date of computation. We then take out the values of (?\«.3), (r.c.3), ((•.«.4), ((•.c.4), (p.«.3), (p.c.3), (/(.c.O), (ft.fl.l), {h.c.l), {l)..i.2), and (/>.c.2), with these values of the dates and iirguments, form their ])roduets by the sines and cosines of th(> corre- sponding multij)les of ij by means of tiie supplementary tables, and add tiie proper products together so as to fori:; three small tables witli tj as the argument. These terms are then iuteipulated to the values of (j corresponding to tlie original dates of computation. As a iinit e.\ampl(< of the use of the Tables we will compute the heliocentric co-ordinates of Uranus for Greenwich mean noon of tlie date 1753, Dec. 3. In computing the argninents we sliall make use of Talde I, tiiongh it is not necessary to do so. The computation of the arguments is as follows: u u 7 Arg. 1 Tabic 1, 1700 PriHliicl hv a.5:!02 Tablu Il,"ls52 111, Y. 1, Dec. ]V, 3 iliivs VI, 175:! 02 o / 2-.ll 31 n 7. 0.73 43. 3S (>.70 J2.30 30.48 :!50 04 / // 7 5.33 +3.24 50 2.03 1 0.73 0.2(1 o t 350 2'J 73 14 // 11.4 4 4 450.002 300.010 83.252 0.357 0.507 22S 44 S 12 2 108 30 35. 1 1 35.50 0.15 1753, Dec. 3 94 7 11.50 72 44 17.87 240.218 Arg. 2 3 4 5 7 8 9 401 177 38 Tabic I, 1700 II, 18,52 HI, y. 1, Dec. 1 IV, 3 (lavs ' VI, 1753.0 477.310 3.785 25.340 0.100 —0.014 500.548 240.075 05.880 G.700 .020 + .024 210 517 10 202 S7 130 00 32 537 270 30 59.4 272.1 10.3 0.0 175.3, Dec. 3 352. CI 7 1.52 355 201 240 341.8 10 THE OUBIT OF TJRAXUS. li)9 (vr.O) (r.i.l) (I'.c.l) (i'.«.2) (v.c.2) (.'.s.S) (r.0.3) (r.xA) (r.r.i) ,,,,/, " '. " " " " 0.37 0.31 CO.'M i-M.H'i 1.42 11.07 0.32 1.04 0.04 5.3a 0.70 1.1'J — O.O;: —1.04 Table VIII 83.58 9.97 6.0.1 IX 21.32 22.24 223.49 X 71. OS 10.02 is. 30 XI .13 .47 XII .13 .3(1 .10 XIII .or, 11 .01 XIV .01 .04 .03 XV 2.38 1.40 1.12 XVI .12 .09 .11 V 178. <;» 53.31! 250.24 Table XVII — 2!)2.r),5 —140.91 —230.19 -j-109.35 (;7.!I3 244.58 2.12 12.2(1 —140.73 —131.03 —9.41 — (i.M5 — 72.80 -1-113.55 —7.29 -|-5.41 Tabic VIII IX X XVII (l>.''0) 251 1080 320 1003 1104 2707 (p.x.l) 188 741 259 118S —1321 — 133 (P.''.l) 32 300 222 (p.s.2) (p.*-.2) 021 21 014 —500 + 114' 042 —412 -f230 507 42 039 —300 +279 —0.03 {,,.... 3) 154 r I —08 +50 +0.00 (,,.c.3) 111 HI —04 l0(r(l'..S.l)— 2.37H7 lipjr sill +9.2004 luhr(p..s.l) -2.124 o / 30 108 91 1 7 2 — 1 1 9 U K (•!.<'.0) (r.,s.l)siii j?(ii.<'.2 + 2. lojr cos 2,7 + 0. lui;0,.r.2) + 2. 0,552 0041 440 XX ".00 X.\I 3.01 XXII _ .84 4.81 7 .80 6^59' XXIII —4.28 — .88 —0.43 + .53 + . 08 + (f. 1 0" (h.r.\) (6.,s.2) (/».c.2) 0".02 5 .88 0.41 1 .30 0.18 0.17 _0.I7 0.34' — 0.51_ ""—0.17 Long, true Kq. 330 25 20.75 Tabic (I'.f (p..S ((>.(• (p..s- (p.. (p..s (p.r log VII <•) 1 ) **'" .'/ . 1 ) COS (I .2) .sin 2;/ .2) cos 2.7 .3) sin 3f/ 3) cos 3;/ I — 1875.90. 9 u Arg. 1 Table 11, 1872 III, 3 Y. Doc. IV, 15ihiy« VI, 1875.90 a 1 It 314 25 6S.70 IG 4(! CC.7C 10 33.. 50 2.20 95 1 9 3 2 If 5.04 4.00 1.30 / 1/ 73 20 40.0 1 12.7 0.7 7 > 574.815 170.072 1.784 + 0.420 147.097 For 1875, IH'o. 15 331 23 5.10 95 11 41.00 73 22 0.15 Arg. 2 3 4 5 C 7 8 9 Tubloll, 1872 III, 3 Y. Doc. IV, 15 (liiv.s VI, 1875.90 208.328 51.785 0.543 — 1 320.055 105.888 13.705 0.144 + 2 114 39 154 13 423 05 1 49 73 1 380.3 21.1 0.2 577 78 1 For 1875, Doc. 15 179.739 l5;j 107 489 123 401.0 50 11. For 1878, April 3 = 1878.20. 9 M Arg. 1 148.570 97.043 .357 .421 Table ir, 1870 III, 2 Y. April IV, 3(1 VI, 1878.26 o 1 n 331 31 18.70 9 37 53.02 2 0.70 2.02 95 / // 11 42.33 1 11.23 0.20 / ;/ 73 22 1.(1 41.7 0.1 5 5 For 1878, April 3 341 14 21.04 95 12 53.82 73 22 42.93 240.997 Arg. 2 3 4 5 G 7 8 9 Table II, 1870 III, 2 Y. April IV, ;!(1 VI, 1878.20 321.237 29.732 .109 — .001 351.077 179.889 7.808 .029 4- .002 153 22 108 7 175 489 37 123 42 401.9 57 12.1 45 i 1 For 1878, April 3 187.788 175 520 105 414.0 102 We now fill in the values of (7, the arc;nments 1 — 9, and the times with which Table XVII is to be entered, for the intermediate dates, by adding the nearly constant differences dcdnced from the numbers at the bottom of Table II. The seconds of > /' II II /' /,, .(..3) — 1.14 — 1.13 — 1.13 — l.ir, — 1.23 — 1.33 — 1.48 (•' .c'.:t) — 2.4fi — a-.-io — 2. CO — 2.7(1 — 2.88 — 3.01 — 3.10 (l .^.4) -f o.r.o -f O.fiO + 0.,'')0 + 0.55 + 0.50 + 0.43 + 0.30 (l .f.4) — 0.10 — 0.20 — 0.30 — 0.40 — 0.47 — 0.57 — 0.04 (l .X.3) Bill 3;; + 1.14 + 1.12 + 1.11 + 1.11 + 1.12 + 1.15 + 1.20 (t .(•.;i) COS It;/ .00 — 0.20 — o.,^5 — 0.85 — 1.17 — 1.50 — 1.80 (l .0.4) sill 4;/ — .,11 _ .M — ..17 — .55 — ..50 — .42 — .34 (l .r.4) ecis 4f7 4- .0,'j + .07 + .07 + .04 — .02 — .10 _ .19 Siiiu + O.fiS + 0.37 + 0.00 — 0.25 — 0.57 — 0.87 — 1.19 It will bo s(>en tliat wo hiivo licro romputod twicu as many numbers as arc nocos.sary to intcrpoliito with all attainable aixuiacy. Tiio rest of tiio computatiim is fully jjivcn on the four followin«,' paj^os. First wo have the values of ij and the nine arf,'umonts for the intermediate dates, filled in by suecessivo additicm of the nearly constant diflerence. The ar<,'uinents thus obtained for the last date may be compared with those just cimiputed on the pre- ceding page. The numerals in tin; first columns of the sceticms of computation following indicate the argum(-nts with which tables arc entered to obtain tlie separate values of the quantities (r.c.O), (r.«.l), (r.c.l), etc. Tlio negative terms in Table XVII being taken from the sum of all the periodic terms from Tables VIII to XVI with argument 1 to 9, we have the final values of {r.c.O), {v.s.l), etc. The final computation of the products («.«.(') sin i;/, etc., and the addition of the separate terms which make up the three co-ordinates, are shown on page 1205. The expressions c.O, .s.l, etc., are employed for brevity, instead of (r.c.O), {v.s.l) sin , l>H«. 16 I87'l, Apr i:i Auk M Di'c. » Xm, Apr. H Aiii.' H !>..•, 4 1>7K. Apr 3 l»75.tf5D. / l»7(i.'>3. lb7ii lili. lb7i).U4(l. 11)77. ilijtf. / In77,r)li7, f lh77. !•-.'>. / / / o / / Hal 23. 081! 332 47.554 331 12.021 335 30.480 337 0,!»57 338 25,425 339 49.803341 14.3(ll| ArK. 1 U-.(t!l7 ini.ac.s 175.(13!) I80.!(l(t 204.182 218,453 232.724 240.0!t5 2 32ll.H,'ii'> 32.'). (Mil aj',i.34(l 333.(102 3as.(i;is 342.383 34(1.720' 351,(174 a iT.i.-ai) 1H0.«.S!( 1^2.038 l83. 1M8 ls.»,y;i8 1.S5.488 18(1.038 187.788 4 \:>:i 150. 150. 1(13. ico. 109. 172. 175. 5 lea. • • • . • • . . . • 175. ts 4S1). 4!H. 500. 505. 511, 5io. 521. 527, 1 i-.'a. 12!(. 135. 141. 117, 153. ICO, 100, 8 4U1.C 403.4 405.2 407.0 408,7 410.5 412.3 414.1 •1 (D.O.O) 1 6(1. (13. (10. 10(1.2!) 75. 103.08 82. 88. 95.25 95. 89,02 101. 83.21 los.ol 107.74 0!l,il8 2 ir,..so 14.71 13.(1(1 12.(15 11,08 1((,70 0,00 0,00 S 72.4a 73.(10 74.77 75.04 77.10 78.20 70.42 80.50 .,■.;> .58 .58 .57 .50 .50 .55 ,54 l> .(12 .02 .02 .03 .03 .03 .03 .03 7 .07 .07 .0(1 .00 .00 ,00 .05 .05 8 l.CS 1.(15 i.(ia 1.00 1.58 1.55 1.53 1.50 (.'.<-.0) .011 I'jH.di) .011 li)8.4(i .00 107.10 .00 lft4.02~ .08 .08 .08 .07 175.05 101.07 180.55 181.18 (r.«.l)l fi.84 7.(18 8.40 8.00 9.44 9.74 9.01 0.08 :! 132.20 12.').48 118 82 112.21 105.07 99.2» 92.89 80.09 SVC. i .IS . 20 •)•) .23 . 25 .27 .29 .31 :i l.(l!l 1.07 1.08 1.12 ].10 1.20 1.41 1.50 4 .13 .12 .11 .10 .10 .00 .09 .OX f) .!'.» .10 .10 .10 .lit .20 ,20 .20 ii .04 .04 .04 .05 .05 .00 .00 .07 7 .14 .14 .13 .13 .12 .12 .11 .10 H 1.28 1.28 1.27 1.2(1 1,20 1.25 1,24 1,23 y z .OS 142.17 .08 13r>.28 .08 130.34 .08 ,07 118.34 .07 112.33 .07 100.27 .07 124.30 100.20 Tab. XVII (r.,..l) (u.e.l)l — ir)4.21 —153.84 _lf,;j,47_ — 23.13 0.38 — 153.10 — 28.74 —152.73 —152.30 —151 99 — 45.72 — 151.(12 — 51.33 — 12.04 — 17.5(1 fi.03 — 34.30 — 40.03 5.fi4 0.07 0.85 0.01 0.S2 0.59 •2 5.01 fi 00 7-31 8.00 10.80 13.00 15.49 18.25 soc. •! .CI) .71 .72 .74 ,75 .77 .79 .so ••i 23.20 24.00 24.05 25.82 20,08 27.53 28.37 20. 1 4 .47 .48 .40 .50 .50 .51 .frtJ .52 f) .37 .37 .37 .:;7 .37 .37 .37 .37 (i .1(1 .17 .17 .18 .18 .18 .18 .18 7 .1.-. .14 .13 .13 .12 .12 .11 .10 8 0. (iC. • 0.(14 0.(13 0.01 0.00 0.50 0,57 0.50 .07 3(!.42 .07 38.70 .0(1 41.21 .00 .((5 .04 .04 .03 50,50 43.08 40.00 50,02 53.20 TuIlXVII — 20.'). hi; — 20(1.08 —20(1.20 —200.51 —200.73 —200.04 —207,15 —207,37 (f.r.l) — 1(',!).44 — 1(17.38 —105.08 —102.53 — 150.83 .20 — 150.92 — 153.89 — 150.78 (W.S.2) 1 .or. .10 .15 .21 .31 .34 .37 2 54.12 40.37 44.78 40.37 30,15 32.11 28.30 24.70 sec. 2 .3!) .42 .44 .47 .40 .51 .53 .50 3 2 2.05 2.19 2.31 2.45 2.00 2.70 2.91 32,08 3.07 28.70 5r..fi2 52.08 47.08 43.50 30.50 35.00 Tuh.XVII —1.34.34 —134.32 —134.31 — 80.03 —134.20 —134.27 — 04.77 — 134.20 — 08.57 — 134.24 — 102.i0 —134.22 — 105.52 |(.-..s.2) — 77.72 — 82,24 _ 00.79 T H E II IJ IT OF U 11 A N U S W3 l)M», iHT.'i, D.'O. ID Ih7li,l»,'j6. Ih7i). Apr. l:(, Aux. 11 I Of,'. » Itilii.M.i. \ lb7il UU. I lf .31 2"> 34 .711 0.23 32.(14 -131). .'>7 -IU3.03 (u.0.3) 2 3 Tab. XVII (o.f.3) (u.hA) 2 3 (v.sA) 0.02 0.H7 7.70 — «.02 —1.13 8.19 1.34 4.. '^3 -7.0(1 O > O / 132 47.. '154 334 12.02133:) 3(1. 4S0 .30 2H.1)3 ."(1 (1.40 3(1.30 — 13(i.fi0 —100.20 o.so 0,01 7. HO — H.02 — 1.12 3.12 1.34 4.4(1 —".(((1 — 2.r)3 1.4S — OSS -|-o.(;(» —2.00 1.4S — O.SH -f 0.(10 (u.c.4)2 3 (■U.C.4) (p.<.'.0) 1 2 3 Tab.XVll (ff^) (p.».l) 1 .J 3 S Tab.XVll (p.S.l) 0.S3 — 1.00 —0.17^ 1230 OS 11 0(18 (p.c.l) 1 2 3 Tab.XVll (p.c.l) 2307 248 2S.J-) 173 32,')(1 — 10S4 + 1272 123 12(1(1 CO 14,'>5 -1077 0.77 —1.02 -0.2') 10(13 00 10 0(18 2140 240 2S22 KIH 3230 -10S7 1243 .20 32.71 ,77 (1,5(1 40,33 — 13(1,(10 — 0(1.27 112 1202 66 13H0 -lOSl f..H4 _ 0.04_ 7.78 —8.01 — 1.13 3.05 l.>14_ 4,30 —7.0(1 —2.(17 .20 36.73 .78 6.70 44.50 —13(1.62 — 02.12 6,78 0,08 '^^ 7.76 —8.01 ~^1.15" 2.08 1.34 4.32 —7,06 —2.74 1.47 — 0.H8 -+-0.50 0.72 —1.03 —0.31 800 lol 9(18 "l077 522 — 601 230 2S07 1(15 3202 -1080 1213 07 1138 67 1.302 -1086 - 684 1.45 -0.80 -f 0.5(1 0.66 —1.04 -0.38 741 104 967 ~i82^r 221 2780 161 lh77, Apr. H I l»77.l!tl!». I AiiK. II lh77 f.l>7. IIhii. 4 Ml MM. o / I o > o / U37 0.057 338 25.425 339 49.893 .20 40.03 .70 6.85^ 48,86 —136.64 — 87.78 6,70 1,01 7.71 — H.Ol — 1.20 .30 45.35 .79 (1.99 63.43 —136.06 30 40 03 .HO 7.10 68.13 —136.67 — 83.28 — 78.64 6.61 1.04 6.51 1.08 2,00 1,33 4,23 -7.0(1 -2.83 1,43 — o.si) 4(1.54 0.61 —1.05 —0.44 504 108 s 967 1677 211 2760 158 3171 -1001 1180 3138 -1 003 1145 83 1074 67 1224 —1000 — 766 69 1000 68 1146 —1094 — 848 7.65 —8.91 2.82 _l,33 4,15 —7.0(1 —2.01 7.50 —8.01 "—1.32 2.75 1.31 4.06 -7.0(1 —3.00 1.40 —0.01 +0.40 0.57 — 1,0(1 —0.40 460 113 8 967 1548 203 2748 155 3106 — 1006 ~^ 1110 55 047 70 1072 -1098 — 926 3.36 —0.03 + 0.43 0.51 — 1.07 —0.56 314 118 8 06(1 I»<78, Apr 3 IH-M 2!i4. 341 14.361 .31 54.60 .NO 7,20 63.00 — 136.69 — 73.69 6.37 1,11 7.48 —8.01 1436 104 2723 151 3068 -1008 1070 -1.43 2.67 1.29 3.06 -7.06 -3.10 1.33 _ .04 +0.30 0.47 — 1.08 —0.61 246 126 9 966 1347 187 2606 148 3031 2000 1031 41 886 72 31 827 74 909 -2002 -1003 932 2006 —1074 ilftl i-'77. Apr. 8 lt>77.2ii». o ' 337 0.!)57 204 24 228 —460 —232 Ann. 6 1877.597. Dfo. 4 1877.925. 1878, Apr. 3 1878.264. 9 (p.8.2) 2 3 S Tab. XVI I (p.8.2) o / 331 23.0S(i 174 2fi o r .338 25.425 213 24 o / 339 49.!- 93 223 23 o / .341 14.361 232 23 200 —459 20C —459 2i4 —459 220 —459 —239 237 —460 ~— 223~ 246 —460 255 —460 —259 —253 —245 576 34 —214 —205 (p.c.2) 2 3 S Tab. XVI I (p.c.2) 559 33 5fiS 33 584 35 6!)2 36 600 37 607 39 614 40 592 -404 + 128 601 —464 610 —464 619 —465 628 —465 n;3 637 —465 646 —465 654 —466 137 146 154 39 —101 172 181 188 (p.s.3) 2 Tab.XVII (p.8.3) (-^.^.3) 2 Tab.XVII (p.c.3) 28 —101 31 —101 34 —101 43 — 101 47 — 101 52 —101 57 —101 — 73 — 70 — 67 — 62 — 58 — 54 — 49 175 —102 73 — 44 155 — lOJ + 53 0.12 0.11 0.07 0.30 159 —102 57 162 —102 166 —102 169 —102 67 172 —102 177 —102 75 60 64 70 (6.C.0) 2 .'t Tab.XXIIl (6.C.0) 0.12 0.11 0.07 0.13 0.11 0.07 0.13 0.11 0.07 0.31 0.14 0.11 0.07 0.32 0.14 0.11 0.07 0.32 0.15 0.11 0.07 0.15 0.11 0.07 0.30 0.31 0.33 0.33 (6.S.1) 1 •2 3 Tab.XXlIl (6.S.1) 0.30 0.12 l.ll 1.53 —5.23 —3.70 1.21 O.'.X! 1. 01 0.22 0.09 1.11 1.42 —5.24 —3.82 0.16 0.06 1.12 0.12 0.05 1.12 0.08 0.04 1.13 1.25 —5.24 —3.99 0.06 0.04 1.13 1.23 —5.25 4.02 0.05 0.05 1.14 1.24 —5.25 0.06 0.06 1.14 1.26 —5.25 — 3.99 1.34 —5.24 3.90' 1.06 1.17 1.00 3.23 —4.43 —1.20 1.2!) —5.24 —3.95 —4.01 (6.^.1)1 2 3 S Tab XXIII (6.P.1) 1.14 1.06 1.00 0.!)8 1.28 0.99 3.25 —4.43 — 1.18 0.04 0.15 —0.48 —0.29 0.34 0.10 —0.38 +0.06 0.!10 1.39 0.99 3.28 —4.42 — 114 0.80 1.51 0.99 3.. 'id 4.41 —1.11 0.70 1.63 0,98 3.31 —4.4 1 —1.10 O.fil 1.75 0.98 3.1:i —4.45 -1.27 3.20 —4.44 —1.24 3.34 —4.40 — 1.06 (6.8.2) 2 3 Tab.XXIIl (6.8.2) 02 O.K! — 0.4S -0.30 0.31 0.10 — 0.3>i 4-0.03 0.02 0.16 —0.48 —0.30 0.03 0.16 -0.48 —0.29 0.04 0.15 —0.48 —0.29 0.05 0.15 —0.48 —0.28 0.06 0.15 —((.48 ~— 0.27 30 0.11 —0.38 +0.09 0.07 0.15 —0.48 —0.26 (6.C.2) 2 3 Tab.XXIIl (6.„.2) 0.32 0.10 —0.38 0.33 0.10 —0.38 f0.05 0.34 0.11 —0.38 0.35 0.11 —0.38 0.36 0.11 —0.38 40.09 +0.04 +0.07 + 0.08 THE ORBIT OF URANUS, 205 Date, 1 187.'), D™. 15 ; Ih". , April 13 1 AnK- 11 Dtto. 9 1877, April 8 A UK- •! Deo. 4 1878, April 3 IbTS.aOO. 1 lbTti.2»3. 1 lH7ti.(il2. lb7li.94U. lS77.2lii). 1877.697. 1877.925. 1878.2S4. r / 1 ° ' / o / / o / O 1 9 331 23.080 333 47.554 334 12.021 335 30.489 337 0.957 338 25.425 339 49.893 341 14.301 Iog(u.8.1) — i.osor. —1.2445 —1.3042 —1.4585 —1.5305 —1.0024 -1.6001 —1.7104 8111 g _9.(;ho3 — 9.(;(;oi —9.0380 —9.0159 —9.5910 —9.50.55 —9.5375 —9.5073 l0g(p.8.1) +3.1045 3.0944 3.0838 3.0719 3.0588 3.0453 3.0294 3.0132 log(u.e.l) —2.2290 —2.2237 —2.2177 .^2.210 —2.20.30 —2.1957 —2.1872 —2.1784 cos 7 + 9.9434 9.9491 9.954 9.9594 9.9041 9.9085 9.9725 9.9703 log(p.f.l) —2.718 —2.779 —2.835 —2.884 —2.928 —2.907 —3.0013 —8.0310 l0)r(«.,S.2) — 1.890.') —1.9151 — 1. 9370 —1.9580 -1.9700 —1.9937 —2.0093 —2.0233 sill iff —9.9247 — 9...102 —9.8941 —9.8703 — 9.85(;0 —9.8350 —9.8111 —9.7840 log(p.s.2) —2.413 —2.403 —2.389 —2.378 —2.305 —2.348 —2.330 —2.312 log(u.,..2) — 2.01(;7 —2.0009 —1.98.35 —1.9044 —1.9434 —1.9202 —1.8951 —1.8074 CDS 2(/ +9.7335 9.7049 y.7932 9.8189 9.8421 9.8030 9.8821 9.8994 K>fr(p.<-.2) +2.107 2.137 2.104 2.187 2.212 2.230 2.258 2.274 o ' n ■ '/ O f / '.• t ft / n O / II 1 II A!7 1 24 28.0(15 1 24 2S.O00 1 24 ':>..000 1 24 28.007 1 24 28.008 1 24 28.008 1 24 28.009 a 331 23 fj.lO 332 47 33.225 334 12 1.291 3,35 30 29.357 337 57.424 338 25 25.492 339 49 53.500 341 14 21.63 u 95 11 41.00 95 11 61.40 95 J2 1.81 95 12 12.21 95 12 22.01 95 12 33.01 95 12 43.42 95 12 53.82 E —2 42 49.15 —2 35 33.18 —2 28 10.i;3 —2 20 41.70 —2 13 0.93 —2 5 20.43 — 1 57 40.55 — 1 49 49.00 c.O 3 18.i;9 3 18.40 3 17.10 3 14.02 3 11.07 3 6.55 3 1.18 2 55.05 8.1 6.77 8.03 lO.Oi 11.87 13.43 14.72 15.70 10.51 e.l —2 28.72 —2 28.80 —2 28.02 —2 28.00 —2 27.13 —2 25.93 —2 24.43 —2 22.80 «.2 1 5.34 1 0.88 1 7.87 1 8.28 1 8.12 1 7.40 1 0.13 1 4.20 r.2 — 50.27 — 68.31 — 59.. '^O —1 0.71 —1 1.03 —1 0.70 — 69.87 — 58.45 (3 + 4) 0.40 0.25 0.03 — 0.19 — 0.41 — 0.03 — 0.83 — 1.00 11 03 .'■)3 2.28 05 24 57.89 00 50 59.12 08 29 5.08 70 1 17.15 71 33 33.48 73 5 54.37 74 38 19.30 e 73 22 0.15 73 22 0.20 73 22 12.37 73 22 IS. 48 73 22 24. CO 73 22 30.71 73 22 30.. s2 73 22 42.93 R 2.5,-i 2.91 138 47 7.00 1.2(!44735 3.25 3.01 3.98 4.37 4.79 6.22 143 1 7.61 Ldiigitudc l.n 15 5.01 140 19 14.74 141 51 27.77 143 23 45.73 144 50 8.50 140 28 35,98 l"f? '•o 1.2i;4-392 1.2042190 1.2(i39778 1.2037480 1.2035319 1.2033280 1.2031370 c.O 2307 2140 1977 1821 1077 1548 1436 1347 s.l —009 —508 —528 —487 —447 —408 —309 — 332 e.l —458 —535 —015 —097 —780 —801 —941 —1017 S.2 + 218 200 192 179 107 152 138 12: C.2 + 09 80 91 101 113 120 138 149 8.3 + 73 09 05 CO 64 49 43 37 C.3 + •* 1.2048990 8 13 19 U 29 80 41 log r 1.2040135 1.2043391 1.2G40774 1.2038293 1.2035954 1.2033701 1.2031720 1 H / II ' // > It ' // 3« + 41 30.04 42 8.4S 42 .'IS. 54 43 0.78 43 33.14 43 57.02 44 20.23 44 41.19 c.O 4().:!0 0.30 0..I1 0.31 0.32 0.32 0.33 0.33 .'.■l + 1.77 1.75 1.70 1.03 1.57 1.48 1.38 1.29 e.l —1.11 —1.1(1 —1.08 —1.07 — 1.05 —1.03 —1.03 — 1.00 8.2 +0.25 0.21 0.23 0.22 0.21 0.19 0.17 0.10 0.2 Latitudo + 0.02 0.03 42 9.70 0.03 42 39.73 0.04 43 7.91 0.05 0.06 0.07 0.07 44 42.04 + 41 37.87 43 34.24 43 68.04 44 21.15 li: 206 THE ORBIT OF URANUS. TABIiE I. CoUEKCTIONS OF AKOUMENTS TOR PAST AND FUTURE t'ENTURl E8. Arg. 1 Century. 9 u u 6 0' O / tt t // // o 1 It tt OJ 207 15 59.32 343 52 17.36 + 108.00 351 C 26.89 —148.32 408.924 100 275 45 34.18 344 40 54.12 102.00 351 34 55.39 —140,08 552.952 200 344 15 9.04 345 41 24.88 96.00 352 3 32.13 —131.84 96.980 300 52 44 43.90 346 35 49.64 90.00 352 32 17.11 —123.00 241.008 400 121 14 18.76 347 30 8.40 84.00 353 1 10.33 —115.36 385.036 500 180 43 53.02 348 24 21.10 -1- 78.00 353 30 11.79 —107.12 529,004 COO 258 13 28.48 349 18 27.92 72.00 £j3 59 21.49 — 98.88 73.092 700 326 43 3.34 350 12 28.08 60.00 354 28 39.43 — 90.64 217.120 800 35 12 38.20 351 6 23.44 00.00 354 58 5.61 — 82.40 301.148 900 103 42 13.06 352 12.20 54.00 355 27 40.03 — 74.16 505.170 1000 172 11 47.92 352 53 54.96 + 48.00 355 57 22.69 — 05.92 49.204 1100 240 41 22.78 353 47 31.72 42.00 356 27 13,59 — 57.08 193.232 1200 309 10 57.64 354 41 2.48 30.00 356 57 12.73 — 49.44 337.200 1300 17 40 32.50 355 34 27.24 30.00 357 27 20.11 — 41.20 481.288 1400 80 10 7.36 356 27 40.00 . 24.00 357 57 35.73 — 32.96 25.310 1500,1 154 39 42,22 357 20 58.70 + 18.00 358 27 59.59 — 24.72 169.344 15000 154 32 39.89 357 20 57.89 18,00 358 27 59.08 — 24.72 168.1.55 1(100 22.1 2 14.75 358 14 4.05 12.00 358 58 31.18 — 16.48 312,1.H3 1700 291 31 7.37 359 7 5.33 0.00 359 29 11.47 — 8.24 450,092 1800 0.00 0.00 0.00 0.00 0.00 0.000 1000 08 28 52.63 52 4S.67 — 6.00 30 56.7V + 8.24 143,908 2000 130 58 27.49 1 45 31.43 — 12,00 1 2 1.83 16.48 287,936 2100 205 27 20.11 2 38 8.11 — 18.00 1 33 15,08 24.72 431.845 2200 273 56 12,74 3 30 38.78 — 21.00 2 4 30.57 + .32.96 575.755 TABLK 11— A HaUMKNTS FOR THE BEQINNINQ OK EACH FoURTH Year 17 52—1948. Year. 9 U Arg. 1 o 1 ti o t ft O t tt 1 , 1752 160 15 8,10 94 6 10,48 7r 43 42.30 102.101 1756 177 23 31.10 94 8 17.46 1 *d 44 56.25 335.8(12 1760 194 31 54.09 94 10 24.43 72 40 10.22 509,023 1704 211 40 17.09 94 12 31,39 72 47 24.21 83,384 1708 228 48 40.08 94 14 38.34 7i "H 38,20 257.115 1772 245 57 3.08 94 10 45.28 72 49 .52.21 430.907 1776 ■ 1780 263 5 20.07 94 18 52.22 72 51 0.23 4.068 280 13 49.00 94 20 59,14 72 52 20,27 178.429 1784 297 22 12.00 94 23 6.05 72 53 34.32 352,190 1788 314 30 35.05 94 25 12.96 72 54 48,38 525,951 1792 331 38 58.05 94 27 19,85 72 56 2,40 99,712 1796 348 47 21,04 94 29 26,74 72 57 16 54 273.473 1800 6 55 1,.>S0 94 31 33,53 72 5H 3(1,59 447,115 ■ TlIK ORBIT OF URANUS. 207 TA15LE I . — Continued. Century. 2 3 4 5 6 7 8 9 OJ 191.528 299.485 280 512 102 61 470.0 15 100 314.245 49.520 04 250 503 124 410.6 214 200 430.902 399.555 449 589 424 188 351.2 413 300 559.079 149.590 233 327 285 251 291.7 13 400 82.396 499.025 18 05 146 314 232.3 212 500 205.113 249.600 402 403 1 378 172.9 411 fiOO 327.830 599.095 187 142 468 441 113.5 10 700 450.547 349.730 571 480 329 604 54.1 209 800 573.204 99.705 356 218 100 507 594.6 409 900 95.981 449.800 140 550 51 31 536.2 8 1000 218.098 199.835 624 295 512 94 475.8 207 1100 341.415 649.870 309 33 373 157 410.4 400 1200 404. 1?2 299.905 93 .•^il 234 221 357.0 5 13110 680.849 49.940 478 109 95 284 297.6 205 1400 109.506 399.975 202 448 660 347 238.1 404 1500 J 232.283 150.010 47 186 417 411 178.6 3 1500CJ 231.921 149.914 47 186 417 410 178.3 2 1000 354.038 499.949 431 524 278 473 118.8 201 1700 477.319 249.975 216 202 139 537 59.4 401 1800 0.000 0.000 1 11100 122.081 350.025 385 338 461 63 540.6 199 2000 245.398 j 100.000 109 76 322 127 , 481.2 398 2100 308.079 460.080 554 415 183 IflO 421.8 697 2200 490.700 200.111 338 153 44 253 ' 362.3 196 TABLE U. — Continued. Year. 2 3 4 5 6 229 7 8 9 1762 481.104 346.855 133 348 213 331.6 678 1766 534.013 369.856 172 362 296 287 353.2 58 1760 586.921 373.858 212 375 362 302 374.8 138 1704 39.830 387.869 251 389 429 436 396.4 218 1768 92.739 401.860 200 402 495 511 418.1 298 1772 145.047 415.862 330 416 562 585 439.7 377 1776 198.666 429.803 369 429 28 60 461.3 467 1780 251.465 443.865 408 443 95 134 482.9 537 1784 304.373 467.866 448 467 161 209 504.6 17 1788 357.282 471.868 487 470 227 283 526.2 97 1792 410.191 485.860 527 484 204 358 547. S 177 1790 403.100 499.870 566 497 360 433 569.4 257 1800 515.972 513. S62 6 611 427 607 691.1 337 208 THE ORBIT OF URANUS. TAHLE II. — Continued. Year. V (J Arg. 1. / /' t i> / tt 1800 5 55 1.8m 94 31 33.53 72 58 30.59 447.115 1801 23 3 24.80 94 33 40.39 72 59 44.71 20.876 1808 40 11 47.79 94 35 47.25 73 58.84 194.638 1812 57 20 10.79 94 37 54.09 73 2 12.98 368.399 18IG 74 28 33.78 94 40 0.93 73 3 27.13 542.100 1820 91 36 56.77 94 42 7.76 73 4 41.30 115.921 1824 108 45 19.77 94 44 14.58 73 5 65.48 289.683 1828 125 53 42.76 94 46 21.38 73 7 9.67 463.444 1833 143 2 5.76 94 48 28.18 73 8 23.88 37.205 1830 160 10 28.75 94 50 34.97 73 9 38.10 210.966 1840 177 18 51.75 94 52 41.75 73 10 52.33 384.727 1844 194 27 14.74 94 54 48.52 73 12 6.58 558.488 1848 211 35 37.74 94 56 55.28 73 13 20.84 132.249 1852 228 il 0.73 94 59 2.03 73 14 35.11 306.010 185G 245 6a 23.7b 95 1 8.77 73 15 49.40 479.771 1800 203 46.72 95 3 15.50 73 17 3.70 53.532 1804 280 9 9.71 95 5 22.22 73 18 18.01 227.293 1868 297 17 32.71 95 7 28.94 73 19 32.34 401.054 1872 314 25 55.70 95 9 35.64 73 20 46.67 674.815 1876 331 34 18.70 95 11 42.33 73 22 1.03 148.576 1880 348 42 41.69 95 13 49.01 73 23 15.39 322.337 1884 5 51 4.69 95 15 55.69 73 24 29.77 496.098 1888 22 59 27.68 95 18 2.35 73 25 44.16 69.860 1892 40 7 50.68 95 20 9.01 73 26 58.57 243.621 1896 57 16 13.67 95 22 15.05 73 28 12.98 417.382 1900 74 23 54.43 95 24 22.20 73 29 27.36 691.024 1904 91 32 17.43 95 26 28.83 73 30 41.81 164.785 1908 108 40 40.42 95 28 35.44 73 31 56.26 338.546 1912 125 49 3.41 95 30 42.05 73 33 10.74 512.307 1916 143 57 26.41 95 32 48.64 73 34 25.22 86.068 1920 160 5 49.40 95 34 55.23 73 35 39.72 259.830 1924 177 14 12.40 95 37 1.81 73 36 54.23 433.591 1928 194 23 35.39 95 39 8.38 73 38 8.75 7.. 35 2 1932 211 30 58.39 95 41 14.94 73 39 23.29 181.113 1930 228 39 21.38 95 43 21.48 73 40 37.84 354.874 1940 245 47 44.38 95 45 28.02 73 41 52.40 628.635 1944 262 56 7.37 95 47 34.55 73 43 6.97 102.396 1948 280 4 30.36 95 49 41.07 73 44 21.56 276.158 A '1' 1 24 28.007 10.411 6.100 14.2715 Factor T 4-0.222 —.020 + .027 —.0012 A.'S + .0007 —.0001 +.0001 THE Oil BIT OF URANUS. 209 TABLE U.— Continued. 2 3 4 5 6 427 7 8 9 337 1800 r)ir).972 513.802 5 511 507 591.1 1804 r)08.881 527.803 45 524 493 582 12.7 417 1808 21.790 641.805 84 538 500 50 34.3 497 1812 74.098 555.806 124 .551 20 131 .55.9 577 181(1 127.007 609.808 103 605 93 205 77.5 67 1820 180.510 583.809 202 578 159 280 99.2 137 1824 233.424 597.870 242 592 225 354 120.8 217 1828 2.S0.333 11.872 281 292 429 142.4 297 1832 339.242 25.874 320 19 358 503 104.0 377 1830 392.150 39.875 300 32 425 678 185.7 457 1840 445.059 53.870 399 40 4 91 52 207.3 537 1844 497.908 07.878 439 59 557 127 228.9 17 1848 550.870 81.879 478 73 24 201 250.5 97 1802 3.785 95.880 517 87 90 270 272.1 177 1850 50.094 109.882 .557 100 157 350 293.8 257 iHfiO 109.002 123.883 591! 114 223 425 315.4 337 18(!4 li;2.51I 137. S85 35 127 290 499 337.0 417 1808 215.420 151.880 75 Ml 350 574 358.0 497 1812 208.328 105.888 114 154 ( 23 49 380.3 577 isto 321.237 179,889 153 108 489 123 401.9 57 1880 374.140 193.890 193 ISl 555 198 423.5 137 1884 r 427.054 207.892 232 195 22 272 445.1 217 1888 479.903 221.893 271 208 88 347 400.7 297 1892 532.872 235.895 311 2:>2 155 421 4X8.4 37C 1800 585.780 249.890 350 235 221 490 510.0 450 11)00 38.053 203.889 390 249 287 570 531.0 530 11)04 91.502 277.890 429 2i;2 354 45 553.2 16 11)08 144.470 291.891 408 270 420 119 574.9 96 1912 197.379 305.892 508 290 487 194 590.5 176 1910 2.50. 2S8 319.894 547 303 553 208 18.1 256 1920 303.197 333.895 5S7 317 20 343 39.7 336 1924 350.105 347.897 20 3:!0 80 417 01.4 410 1928 409. OH 301.898 05 341 152 492 83.0 496 1932 401.9.! -;..900 105 357 219 500 104.0 576 1930 514.831 389.901 144 371 2S5 41 i 120.2 56 1940 507.740 403.1^2 184 384 352 110 147.8 130 1944 20.049 417.994 223 398 41S 1 90 109.5 216 1948 73.557 431.905 1.1500 202 411 485 205 0.1 , 191.1 1.78 296 6.6 4.3457 3.2 1.1 5.4 — .0001 -f .0002 S 7 June. 187 3. 210 THE ORBIT OF UIIANUS. TAI5LK III. — IlEDiiCTioN OF THE EpociiH AND Akqumentb to tue beuinmnu or EACH Month IN A Cycle of fouu ykaus. YearO January Ft'bruary March April May June July Au^cust St'ptemljer October Xovcinber Ueccuiber Year! January Fi'liruary March April May Juno July August Scptoniber October (» Nuvembor Ueceiuber Year 2 January February March April May June July August September October November Ueceiuber Years January February March April Jfay June July Auirust S('|)teniber October November Ducembur 9 0.00 21 411.24 42 14.00 1 4 3.24 . 9 10 10 II 11 11 12 12 14 14 14 15 15 10 Hi IG 2o 4(> 8 2!) 51 12 ;54 55 10.24 60. 4S 0.48 55.71 44.95 51.95 41.19 48.19 4 17 37.42 4 39 2(1.1)6 4 59 9.19 5 20 58.43 42 3 25 40 8 29 51 12 59 20 41 3 25 40 8 29 15 37 58 20 42 3 25 4U 5.43 54.07 1.07 50.90 40.14 47.14 30.38 43.38 8 34 32.01 8 50 21.85 9 10 4.38 9 37 53.02 0.02 49.80 60.80 40.09 35.33 42.33 31.57 38.57 12 51 27.80 13 13 17.04 13 32 59.57 13 54 48.81 55.81 45.05 62.05 41.28 30.52 37.52 20.70 33. 7C 0.00 2.09 5.21 7.89 10.60 13.19 15.79 18.48 21.17 23.77 20.40 29.06 31.75 34.44 30.87 39.56 42.10 44.85 47.46 50.14 62.83 55.44 68.13 1 0.73 1 3.43 I 6.11 1 8.64 1 11.23 1 13.83 1 l(i.52 1 19.12 1 21.81 1 24.50 1 27.10 1 29.79 1 32.40 1 35.08 1 37.77 1 40.20 1 42.89 1 46.50 1 48.18 1 60.79 1 63.48 1 66.17 I 68.77 2 1.46 2 4.06 // 0.00 1.68 3.05 4.63 6.15 7.73 9.25 10.83 12.40 13.93 16.61 17.03 18.01 20.18 21.61 23.18 24.71 20.28 27.81 29.38 30.96 32.48 34.00 35.59 37.16 38.74 40.10 41.74 43.20 44.84 40.30 47.94 49.52 51.04 52.62 54.14 1 1 1 1 55,72 57.29 68.72 1 0.29 1.82 3.40 4.92 6.50 1 8.07 1 9.60 1 11.17 1 12.70 & 0.00 0.00 0.00 O.Ol 0.01 0.01 0.01 0.01 0.01 0.02 0.02 0.02 0.02 0.02 0.03 0.03 0.03 0.03 0.03 0.04 0.04 0.04 0.04 0.04 0.04 0.05 0.05 0.05 0.05 0.(]^ 0.06 0.06 0.00 0.00 0.00 0.07 0.07 0.07 0.07 0.07 0.08 0.08 0.08 0.08 0.08 0.08 0.09 0.09 Arg. 1 0.000 3.087 7.136 10.823 14.391 18.078 21.647 25.333 29.019 32.587 30.274 39.842 43.529 47.210 50.640 64.233 57.801 01.488 65.056 68.742 72.429 76.997 79.0S4 83.252 86.939 90.020 93.956 97.643 101.211 104.898 108.400 112.153 115.840 119.4(17 123.094 120.662 130.349 l:i4.o;i(! 137.366 141.053 144.021 148.308 151.8(6 155.563 169.249 102.S17 100.504 170.072 TlIK ORBIT OP URANUS. 211 TAULK III — Cuntinued. 2 3 4 5 6 7 8 9 YearO January 0.000 0.000 0.0 February 1.123 0.297 1 1 1 0.5 2 March 2.209 0.575 2 1 3 3 0.9 3 April _ 3.25a 0.872 2 1 4 4 1.3 5 May 4.382 1.159 3 1 5 1.8 7 June U 6.505 1.457 4 1 7 7 2.2 8 July C.5!)l 1.745 5 2 8 9 2.7 10 August 7.714 2.042 2 10 10 3.1 12 Scptombcr 8.837 2.339 7 o 11 12 3.0 13 October «J.y23 2.02G 7 3 12 14 4.0 15 November 11.040 2.923 8 3 14 15 4.5 17 December 12.132 3.211 9 3 15 17 4.9 18 Yearl January 13.254 3.508 10 3 17 19 5.4 20 February 14.377 3.805 11 4 18 20 6.9 22 March 15.301 4.073 11 4 19 22 0.3 23 April 1C.514 4.370 12 4 21 23 0.7 25 May 17.000 4.058 13 4 22 25 7.2 27 June 18.723 4.955 14 5 24 27 7.0 28 July 10.809 5.242 15 25 28 8.1 30 August 20.932 6.539 15 5 20 30 8.5 32 September 22.054 5.830 10 28 31 9.0 33 October 23.140 0.124 17 29 33 9.4 35 November 24.203 0.421 18 30 34 9.9 37 Uecomber 25.340 0.709 19 32 30 10.3 38 Year 2 January 21). 472 7.000 i^O 1 33 37 10.8 40 February 27.505 7.303 20 7 34 39 11.3 42 March 28.009 7.571 21 7 30 40 11.7 43 April 29.732 7.808 22 ■y 37 42 12.1 45 May 30.818 8.150 23 8 3S 44 12.0 47 Juuo 31.941 8.453 24 8 40 45 13.0 48 July 33.027 8.741 25 8 41 47 13.5 60 August 34.150 9.038 25 9 43 48 13.9 52 Sei)tembcr 35.272 9.335 20 44 50 14.4 53 October 3(1.358 9.022 27 9 45 51 14.8 56 November 37.481 9.919 28 (•..49 35 22.14 84 37. '.)7 33 54.00 +33 10.95 32 2(;.71 31 43.41 31 0.37 30 17.01 +20 35.15 2S 53.01 2'< 11.20 27 20.71 2(i 48.5U +2ii 7.71 25 27.27 2t 47.1(1 21 7.42 23 28.00 +22 40.0!) 22 10.51 21 32.31 20 54.57 20 17.23 -10 40.31 10 3. S3 is 27. so 17 52.21 + 17 17.0'J + lfi 42. n IC. S.25 15 34.55 r- 1.31 14 2S.(;3 + 13 5r,.44 13 24. 7(! 12 53. (;i 12 22.00 11 52.00 + 11 23,35 10 54.35 10 25.02 5.S.04 9 30.74 4.01 37. SM 12.34 47.30 23.01 + r. 59.20 (; 311.14 n 13.(11 5 51.00 + 5 30.40 -0.014 0.021 0.020 0.03S 0.04S -0.0.50 — 0.(172 O.IISO 0.702 0.71S -0.735 0.752 0.770 0.700 0.800 -0.820 0.848 0.800 0.883 0.899 -0.914 f 0.927 0.039 , 0.950 I 0.000 '—0.907 ' 0.972 ! 0.975 ! 0,977 I 0.977 j '—0.974 ! 0.907 I 0.059 0.948 j 0.030 —0.021 0,003 ! 0.8S3 O.SOl 0.837 — O.SIO 0.780 (.748 0.714 0.078 —0.041 0.(101 0.500 0.518 0.475 —0.431 ().3S0 0.340 0.293 —0.247 -1.024 1.003 0.083 0.902 0.942 ■0.022 0.902 0.882 0.802 1 0.842 : -0.822 0.803 0.783 0.704 ! 0.745 I -0.720 0.707 i 0.089 j 0,070 i 0.052 I -0.034 0.010 0.598 0,5S1 0.5U4 —0.547 0.530 0.513 0.407 0.4lil —0,405 0.140 0,433 0,417 0.402 — 0.3S7 0.372 0.358 0.344 0.330 —0.317 0.303 0.200 0.277 j 0.204 —0.252 0.240 0.228 0.210 0.205 —0.104 j 0.1S3 ! 0.172 I 0.102 — 0. 153 Year. + 1.702 1.750 1.720 1.0S4 1.040 + 1.014 1.570 1.544 1.500 1.471 + 1,430 1 . 405 1.370 1.337 1.304 + 1.270 1.237 1.200 1.172 1.141 + 1.110 1.07 ■> 1.040 1.017 0.980 +0.957 0.028 0. 898 O.S70 0.842 +0.814 0.7 SO 0.758 0.730 0.704 +0.(;77 0.051 0.020 0.002 0.578 +0.555 0.530 0.508 0.485 0.402 +0.441 0.420 0.399 0.378 0.359 1550 1500 1570 1580 1500 1000 1010 1020 1030 1040 1050 1000 1070 lOsO 1090 1700 1710 1720 1730 1710 1750 171.0 1770 1780 1700 1800 181)1 1802 1803 1801 1805 1800 1807 180S 1809 1810 1811 1812 1813 1814 1815 1810 1817 1818 1819 1820 1821 1822 1823 1824 +5 4 4 4 3 II 9.73 49.70 30.:!2 11.57 53.48 +3 30,03 3 19.20 3 3.14 2 47.09 2 32.89 + 2 18.77 1 1 1 -fl 1 52.55 40.40 29.05 18.32 8.28 58.92 50.24 42.2(1 + +0 +8!'84 8.49 8.15 7,82 7,50 +7.18 0.87 0.57 0.27 6. 31.90 28.30 22,44 17.22 12.08 — jj •34 •33 •32 — 31 •30 •30 .2<) • 2y +5.00 _ 5.41 '^'^ 5.14 --7 4.88 -^^ 4.02 -'^ .98 +0,340 1825 0,322 1820 0,303 1827 0,284 1828 +0.208 1829 +4.37 4,13' 3,s9 3,00 3.44 + 3.23_ 3.02 • 2. 82 ■ 2.03 ■ 2.44 • +2.20_ 2.09 1,92 1,70 j + 1.01 .24 .24 •23 .22 .21 —0.200 0,153 0,1 (h; 0.000 —0,013 + 0.034 0.078 0,120 0.102 0.203 +0,242 0,278 0.312 0.345 0,375 fO,403 0,429 0,452 0,472 0,489 +0,503 0,514 I 0,522 0,527 0,520 +0,528 0,528 0,528 0,527 0.527 +0,527 0.520 0.520 0.525 0,525 + 0,524 0,523 0,523 0.522 0.522 + 521 0.520 0.520 0.510 0,518 17 ■»7 .16 •■5 — li + 0.517 0.51(1 0.515 0.514 0.513 +0.512 I 0.511 I 0.510 i 0.509 —0.508 -0.144 0.134 0.125 0.117 0. 108 -0.100 0.092 0.085 0.078 0,071 -0,004 0.058 0.052 0,040 0,041 -0,030 0,031 0,027 0.023 0,019 -0,015 0.012 0.009 0.007 0.005 -0,004 0,004 0.004 0.004 0,004 —0,003 0,003 0.003 0.003 0.003 —0.003 0.003 0.003 0,003 0.002 -0.002 0,002 0,002 0,002 0,002 -0,002 0.002 0.002 0.002 0.002 -0.001 0.001 0.001 0.001 -0.001 + 0.252 0.230 0.210 0.205 0.189 + 0.175 0.101 0.148 0.135 0.123 +0.112 O.lOl 0.091 0.081 0.072 + 0.003 0.054 0.040 0.030 0.032 + 0.020 0.021 0,017 0.013 0,010 +0,007 0,007 0,007 0,007 0,007 + 0,000 0,000 0,000 0.000 O.OOG +0.005 0.005 005 0.005 0.005 + 0.004 0.004 0.004 0.004 0.004 + 0.003 0.003 0.003 0.003 0.003 +0.002 0.002 0.002 0.00 a +0.002 214 TUE ORBIT OF URANUS. TAULIO y\.—(!i>iilii,ti,il. Toar. 1830 1831 1S32 1833 1834 1835 1830 1837 1838 1839 1840 1841 184-2 1843 1844 1845 184(! 1847 184 8 1841) 1850 1851 1852 1><53 1854 1855 185fi 1857 1858 1859 18(10 18(il I8fi2 18()3 I8(i4 1805 1800 1807 1808 1809 1770 1871 1872 1873 1874 1875 1870 1877 1878 1879 1880 1881 1882 18.S3 1884 .oo .00 0.00 0.00 0.00 , + 0.01+-°' 0.03 ■°' •03 +.04 04 + 0.00 0.10 0.14 0.19 -"S 0.24 +0.30 •05 .06 0..'i7+-°7 ((.45 0.53 0.02 +0.71 0.81" 0.92 1.04 I.IG oS .08 .09 .09 .10 .11 .12 .12 •13 + 1.2!) , 1.43 +-'4 1.57 -li 1.72 -5 1.88 -'f .16 + -''^*+ 17 2.21 I ' 2.39 2.57 +2.70 1.32 — "* 1.19 '-^ 1.07 •'•' 0.95 •'" .11 +0.84 0.74 0.04 0.55 0.47 + 0.39 0.32" 0.20 0.20 0.15 +0.11 0.07 0.04 0.02 0.00 —.10 . 10 ,09 .08 .08 -.07 .oO .06 ■05 .04 —.04 •03 .01 .02 .00 .18 .18 .19 .20 +2.90 3. 1 3.37 3.59 +3.82 , •^■' ^ +.24 + .20 .21 .22 +0.507 0.500 0.505 0.503 0.502 +0.501 0.500 0.498 0.497 0.495 +0.494 0.493 0.491 0.490 0.488 +0.487 0.485 0.483 0.482 0.481 +0.479 0.477 0.470 0.474 0.472 +0.471 0.409 0.407 I 0.405 0.403 +0.401 0.459 0.457 0.455 0.453 +0.451 0.449 0.440 0.444 0.442 +0.440 0.438 0.430 0.433 0.431 I +0.429 I 0.420 0.424 i 0.422 I +0.419 ■ +0.417 ! 0.414 I 0.412 ' 0,409 +0.407 I —0.001 0.001 0.001 0.001 O.OUi (I -0.001 O.ool -0.001 0.001 0.001 0.001 0.(»01 -0.001 o.ool 0.001 o.ool -0.001 -0.002 0.002 0.002 0.002 -0.002 I +0.002 0.002 0.002 0.002 0.002 + 0.001 0.001 0.001 o.ool +0.001 0.001 0.001 o.ool o.ool +0.001 0.001 o.ool 0.001 0.002 +0.002 0.002 0.002 0.002 +0.002 +0.003 0.003 0.003 0.003 +0.003 Yuur. 1 885 1880 1887 18S8 1889 1890 18111 1892 I81t3 1894 1895 1890 I8II7 1898 1899 1900 1901 1902 1903 1904 1905 1 900 191(7 19((8 1909 1910 11(20 1930 1940 11(50 1 90(( 11(70 198(( 199(( 2000 2010 202(( 203(( 204(( 2050 2000 2((70 2((s(( 2((1I0 2100 2110 2120 213(( 2140 2150 2100 2170 2180 ! 2190 I 2200 4.79 5.05 + 4.00 , ^4.30+'^* 26 27 +5.33 , 5.59+ ', 5.87 •-•'^ (1.10 0.45 .29 .29 •3° 7.00+ '»' 7.37 7.09 8.02 3' 32 33 3i + 8.. 35 , 8.08+'-^-' 9.0:; -^^ II. 3h -^^ 9.75 :j; + !((.! 2 '"■•''" Is •30 .40 10.88 11.27 11.07 +0 12.07 10.47 21.55 27.30 33.71 +0 40.70 48. r)3 50.113 5,98 15.08 1 1 + 1 1 1 20.03 37. ((3 48. Cm 2 0.94 2 13.85 + 2 27.38 2 41.53 2 50.29 3 11.05 3 27.01 + 3 44.17 4 1.32 4 19.07 4 37.40 4 50.31 +^ 1.5.74 35.72 5 50.24 17.29 +0 38.88 +0.104 ((.402 0.31(9 0.397 0.394 +0.391 0.3.S,S 0.3S0 0.3^3 0,380 +0,377 0.374 0.372 0.309 0.300 +0.303 0.3(;(( 0.357 0.353 0..350 +0.347 0.344 0.34(( 0.3;!7 0.334 —0.331 0.21(8 0.204 0.229 0.193 +0.155 0.115 0.074 0.((32 +0.010 —0.053 o.((i(i; 0.14(( O.IS.'J 0.220 —0.208 0.3((1( ((.349 0,388 0.427 —0.404 0.491( 0.5,32 0.503 0.593 — 0.020 0.045 0.008 O.OKK ■ -0.709 -0.002 +0.003 0.((((2 . 0.001 0.002 ! 0.004 0.002 0.002 —0.003 0.0((3 0,003 ((.((((.3 0.003 -0.0((3 0.0((3 0.0((1 0.0((t 0.0U4 -0.001 0.((((l ((.'(((I 0.0(15 0.((l(5 -0.0((5 0.((05 0.(l((5 o.((i(i: o.((((i; -((.0((0 0.0((S 0.010 0.((12 0.((15 -0.018 ((.((22 0.02(; 0.030 0.((35 -0.040 0.045 ((.050 0.050 0.002 -0.008 0.075 0.08 2 ((.081) 0.090 -0.104 ((. 1 1 2 0.120 0.12^ 0.137 -0.140 0.150 0.105 0.175 -0.1.84 0.((04 0.004 +0.0((4 0.005 0.0)(5 0.0((5 0.((((5 +0.000 0.0((0 0.000 0.000 0.007 + 0.007 0.007 0.((((8 ((.0((8 0.008 +0.009 0.009 0.)(10 0.010 9.010 + 0.010 0.013 0.017 0.021 0.020 + 0.0,32 0,039 0.040 0.053 0,001 +0,009 0,078 0,088 0,098 0,108 +0,119 0,130 0,142 0,155 0,108 + 0,182 0,190 0,210 0,225 0.240 +0,250 0,273 0,289 0,300 +0,322 s?^m TllK OR HIT OF UIIANU8. 2ir» TAHI.K VII , — KvjLA-niiN or Centue and rulNCU'At . Teum c )f Lou. r. Q E I.(.(? r ■ ;/ E Log r 1 1 It II 1.26 / II II 1.26 j 0' 00.00 S9.R. 59.8 1 IS9I6 360' 10' 69 20.71 58.70 58.67 22488 ' 350° 10' 611. si 18910 1 6(1' 10' 1 26.41 22008 I20 ' 50' 2U 1 69. (i2 IS!) 19 3 40 20 1 1 24.08 22729 121 40 3U 2 59.42 59. So ISII24 S 30 30 1 2 22.71 58.63 22862 '23 30 4U 3 69.23 59.81 |S!l.-il 7 20 40 1 3 21.30 58 59 22977 '25 20 6U 4 69.03 59.80 59-79 18940 9 II 10 60 1 4 19.S6 58-55 58-52 23104 127 ■30 10 1° 5 68.82 is 061 360° 11' 1 6 IS. 37 58.48 58.43 23234 340° W (i 68.(11 59-79 1S1I04 '3 60' 10' 1 1(1. S6 23305 '3' 60' 2U 7 68.39 59- 7« 1S1I79 '5 40 20 1 7 16.28 23498 ^n 40 30 H 68.18 59-79 ISillXi 17 30 30 1 8 13.08 58.40 5836 230.33 >35 30 40 !» 67.9li 59-7« 19016 '9 20 40 1 9 12.01 23770 '37 20 50 10 67.74 59-78 59-77 19030 21 23 10 60 1 10 10.30 58-32 58-27 23909 '39 142 10 2^ 11 67.51 59-76 1 9059 358' 12 1 11 8.03 58.2.? 24061 348° 10' 12 67.27 190S.4 25 60' 10' 1 12 o.sc, 24194 '43 60' iiU 13 67.02 59-75 19111 27 40 20 1 13 6.(H 58. 18 58.14 24339 '45 146 148 40 30 14 60. 70 59-74 19140 29 30 :!() 1 H 3.1s 24 4 85 30 40 16 60.49 59-73 19171 3' 20 40 1 16 I.2S 58.10 24033 20 50 lU 50.22 59-73 59-71 19204 34 10 50 1 16 69.33 58.05 58.00 24784 '5' '53 10 8' 17 65.93 19238 357' 13' 1 10 67.33 57.96 24937 347° 10' IS 66. 03 59-70 19276 37 60' 10' 1 17 66.29 26091 f ' 1 '54 60' 20 1!) 66.33 59-7° 19314 39 40 2(( 1 IS 63.20 57. 9> 57.86 57 Si 57.76 26247 '56 '58 161 162 '64 40 30 20 66.01 59.68 59-67 59.66 1 9366 41 30 . .30 1 19 61.0(1 26406 30 40 21 54.08 1 9397 42 20 40 1 20 4S.S7 26600 20 60 22 64.34 19442 45 10 60 1 21 40.03 26728 10 59-64 47 57.72 4' 23 53.98 59-63 59.61 UH89 356° 14" 1 22 44.36 26892 346° 10 24 53.(11 19638 49 60' 1(1' 1 23 42.01 57.66 20057 165 168 60' 20 26 63.22 196S9 5« 40 20 1 2t 39.03 57. 6i 2(i226 40 30 2i; 62.81 59-59 59- 5« I!M;42 \\ 30 30 1 26 37.19 57.56 20395 170 30 40 27 62.39 19097 55 20 40 1 20 34.70 57.5' 20600 '7' 20 60 2'i 61.96 59-5" 19764 57 10 60 1 27 32.10 57-46 20740 '74 176 10 59-54 S« 57-41 5' 29 61.49 19812 60 355' 15' 1 2S 29.67 20910 345° 10 30 61.01 59-52 19S72 60' 10' 1 2!) 20.92 57.35 27093 '77 '78 60' 20 31 60.62 59-5 » 19934 62 40 20 1 30 21.22 57.30 27271 40 30 32 60.01 59-49 19999 65 66 69 30 30 I 31 21! 47 57.25 27461 180 30 40 33 49,48 59-47 2(1006 20 40 1 32 IS. 07 57.20 27034 183 185 20 50 34 48.93 59-45 20134 10 60 1 33 16.80 57. '3 27819 10 59-43 7' 57.08 1S8 6' 35 48.30 20206 354' 16' 1 34 12.88 28007 344° 10' 3(1 47.77 59-41 2(1278 73 60' 10' 1 .36 9.90 57.02 56.97 2S190 1 89 60' 20 37 47.16 59-38 59-36 2():!62 74 76 40 20 1 30 0.M7 28380 190 40 30 3S 4<1..-,1 2(1128 30 30 1 37 3.77 56.90 28678 192 30 40 3!) 46. S4 i)9-3? 20607 79 81 20 40 1 3S 0.02 56.85 56.79 5673 28772 '94 196 20 60 40 46.16 59-31 20688 10 60 1 38 67.41 289(18 10 59-29 82 199 7' 41 44.44 59.26 2(1070 85 353° 17° 1 39 64.14 56.67 29107 343° 10' 42 43.70 2o"66 60' 10' 1 40 60. Si 2930V 200 60' 20 43 42.94 59-24 20S41 86 88 40 20 1 41 47.43 56.62 56.55 56.48 56. »3 56-36 29508 201 40 30 44 42.16 59-21 59.18 20929 30 30 1 42 4:!. 98 29771 203 30 40 46 41.33 2 1(1 -JO 9' 20 40 1 43 40.40 29970 205 20 50 4(i 40.48 59- '5 21112 92 10 60 1 44 80. S9 30183 207 10 59- '3 94 209 8' 47 39.fil 21200 96 98 352° 18' 1 46 33 25 56.30 56. -'3 56.17 30392 342' 10' 4S 38.71 59.10 21302 60' 10', 1 4(1 29.66 30003 211 60' 20 49 37.78 59-07 21400 40 20 1 47 26. 7 S 30810 2 '3 40 30 60 30.82 59-04 21499 99 30 30 1 4X 21.96 31030 214 30 40 51 36.83 59.01 58.98 21001 102 20 40 1 49 IS. 06 56. 10 31240 216 20 50 62 34.81 21706 '04 10 60 1 60 14.09 56.04 31106 219 10 5S-9S ic6 '55-97 221 9° 53 33.70 58.91 2IS11 loS 351° 19' 1 61 10.00 31080 341' 10 54 32.07 21919 50' 10' 1 62 6.90 55-90 31008 222 60' 20 56 31.55 58.88 22029 1 10 40 20 1 63 i.so 55-84 32132 224 40 30 5G 30.39 58.84 58.81 58.77 22140 1 1 1 30 30 1 53 57.67 55-77 32358 226 30 40 57 29.20 22264 «'4 116 20 40 1 64 63.27 55-70 55-64 32685 227 20 60 58 27.97 22370 10 60 1 55 48.91 32814 229 10 5874 1 18 55-56 231 10^ 59 20.71 224SS 1.26 350° f7 20° 1 50 44.47 33046 126 340° II 216 TllK OllUlT OF URANUS. 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(U41J 5''o .-iO :!0 ft JO 10.01 7.16 00771 50.-; :jo 40 r. s j:i.ft7 16 ()6 16. So (!400:! 5S1 JO 40 ft JO 17.01 7.00 O.S., ♦oo:!t;o 5'b ♦OOOCft sofi JO ftO ft S 40., '17 «ftft74 5S1 10 ftO ft •JO J (.77 10 1 6. 6 ( 5'^' 6.67 505 70' 5 1 8 ft7.01 ('.(ilftii 1.27 290 80 ft JO ;ii.44 ♦iMftr.o *1.2P 280° 7 T H K ORBIT OF URANUS 219 TAIU.K \'\\—0 82° 10' 20 :io 40 50 83° 10' 20 :io 40 50 84' 10' .'id 40 50 85° 10' 20 •M 40 50 8G" Id' 20 ;io 40 5() 87° 10' 20 :io 40 50 88° 10' 20 :to 40 50 89^ 10' •io no 40 50 80' 20 20 20 20 20 21 21 21 21 21 21 21 21 21 21 21 21 21 :tl.44 :n.ii4 41.2S 50.45 5(!.4f! 2.:io l.'.ir, i:i.4<> IH.SO 2:t.!>7 2H.1I7 n.f.sl ;is.4s 42.'.lil 47.:t:t 51,50 55.51 511, :!0 :i,04 (;.5r, 11. 01 I :;.()'.> ii;.il is,'.)7 21.r,(', 21, 10 2l'i,.i5 2S,75 :!0,7H ;{2.(;4 :t4.:!4 :t5,s7 .17.21 lis, 1 1 :i!i.4s 40.:i5 41. on 4 I. (SO 22 II. '.I'.) 22 42.21 22 12. 2i; 22 12.15 22 41.SS J.» 41.15 5 -zi 40. S5 5 22 4<».()0 5 22 :i'.(. 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ID 5 22 15.;tl 5 22 12. 2(! 1 5 22 0.05 4-5" 4 ,il 4 '7 .).ol .V«.S .,.68 .V.i.S .,.i,S 3 "- 2..S6 2.6y 2.y, 2.20 2.05 I..S(> 1.70 '-.S.i ' ,57 1 . 20 I o( 0..S7 0.71 0-.S4 o V> O. 22 0.05 O. 1 I 0.-7 "•1.1 o.Oo 0.76 o';.i 1 .01; I. t2 i,5,S 1 74 2.07 2.2.t 2 ,V/ 2.56 2.72 2.SS 1.28 ol5<;o 02155 02750 o:!:i4<> OltlMl OI5:ifl 05i:tl 0572(1 0(i:i22 0(1017 07512 ().sl07 0^702 ((0207 00sii;i KIIMS IHw:t I Kill) 1 227 I 1 2-ti;i» i:it(;5 1 KICO 1 1(155 15250 I5S4(! 1(1111 170.111 17(i:il IM22ii IS.S20 10115 200 1 II 201IOI 21 I OH 21702 22:!SI1 22its0 2:157 I 211(17 217(11 25:!5I 2511 H 2(1511 ■.;7l:!l 27 7211 2s:i!ll 2S1)II 21150 1 :s(ioi)(> iiodss :!12S0 :il.s72 :i24(i4 ;;::o55 :!:!(! »7 ill 2:1s ;!IS2H :!54I.S :ti'(i(m :t(;5ii7 :ni^7 123 SO.S 5V.S 5'/' S'JS 50.S 595 595 5'/> 505 595 595 595 595 59^' 595 595 59^' 595 595 595 595 595 595 59^' 595 595 595 595 59 1 595 595 591 59 1 59 1 591 59 1 594 59.5 59 1 59.1 591 59.1 $')' 59.! 59.: 59.1 59-! 592 592 592 59-' 59" 592 59' 59" 590 5''<9 590 280 50' 40 .•10 20 lU 279' 50' 40 :iO 20 10 278 50' 40 30 20 10 277' 50' III 30 20 10 276 50' 40 :!0 20 10 275 50' 40 30 20 10 274 50' 40 :to 20 10 273 5ii' 4(1 :io 20 10 272 50' 40 30 20 (0 27 V' 50' 40 :;o 20 10 270 K 90" 10' 20 30 40 50 9r 10' 20 30 40 50 02" 10' 20 :to 40 50 93' 10' 20 30 40 50 94' 10' 20 30 40 50 95" 10' 20 ;io to 50 96° 10' 20 30 40 50 07° 10' 20 30 40 5(1 08° 111' 20 30 40 50 99' 10' 20 30 40 50 100 i 22 I 22 > 22 1 21 5 21 5 21 5 21 5 21 5 21 5 21 5 21 5 21 5 21 5 21 5 21 5 21 5 20 5 20 5 20 5 20 5 20 5 20 5 20 5 20 5 20 5 111 5 10 5 111 5 10 5 10 5 10 5 10 5 10 5 IS 5 IS 5 l.S 5 IS 5 IS 5 IS 5 17 5 17 5 17 5 17 5 17 5 17 5 111 5 1(1 5 1(1 5 111 5 15 5 15 5 15 5 15 5 15 5 II 5 I I 5 II 5 II 5 It 0.05 5.(1H 2.15 5S.45 51.50 50. 5H 4(1.40 42.0(1 37.57 32.01 2S.01I 23.11 17.07 12.(17 7.20 I.5S 55.70 4 0.S5 43.74 37.4 s ;!I.(I5 24.17 17.72 10. H2 3.7(1 5(1.51 40.1(1 41.(12 3:!.o:t 211.07 IS. 0(1 O.sil 1.5(1 5:!. (Id 41.42 35.(11 211.115 17.53 S.2I1 5s. S3 40.24 311.. 50 211 (10 10.5! Ii.:i:t 5S.1M1 4s. 44 .■!7.7(1 2(1.02 15.03 4.7s 5:!. IS 42.(12 30.41 IS. (11 (1.72 51.111 4 2.41 :io 11:1 17.111 4. so ,r.i7 ,^•5.1 3-7° 3.X6 4.01 4.1.S 4. 54 4-49 4.66 4.«2 4.9H 5- '4 5 •,■50 5'47 5.62 5-79 5'9l 6.11 6.26 (-.4,5 6.5,S 6.90 7.o(j 7.22 7-.5« 7-54 7.6.) 7..S6 ■S.oi K.I 7 «..5,5 8.50 8.6.) 8.81 8.(/) 12 7 4.5 59 71 90 10.06 io 2 r 10. -,7 10.52 10.68 I0.8.I 10. ()9 11.15 • ' .50 I l..|6 11.61 11.77 I 1.92 12.08 12 2.1 I2..?8 12.5.1 I 2.69 1.28 37187 :!777(l 3S3(!5 38051 311:43 40131 40710 41307 41H04 424S2 4:10(10 43(1.55 44242 44S28 45114 4(1000 4I15S.^. 47170 4 7755 4s:i;{0 4sll2:i 40507 500110 501173 51250 5Is:js 52420 5:tO(l2 5:f5s:i 541(13 54743 55:12:1 55II02 5(14 Si 570(10 571138 5821(1 5M71I4 50371 5004S (10525 (IIIOI (11(177 (12252 H2S27 ii:t4(il (i:i075 (14548 (15120 (1511112 (1112(14 (ir,s:tr, (17 107 (17077 (1S547 110117 (loiisii 711255 7IIS23 71301 71058 i?.t; 270" 5«9 50 5«9 40 5«9 30 5«9 20 588 10 5. 88 268' 58S bU 5«7 40 5,88 30 5«7 20 586 10 5«7 2«H' 5 86 50' 586 40 586 30 S«5 2U 5«5 10 S«5 267^ 5'^4 5(1' S«4 41» 5«4 30 5«.5 20 SH.5 10 S«1 266" S«2 50' 5«2 40 S''^-' 30 5«' 20 580 10 5,80 5 So 579 579 579 57« 57« 573 577 577 577 57'-' S7'-> 575 575 571 574 57.5 57-> 572 572 572 57' 570 570 570 5'>9 $(>') 568 568 5 ''7 , 1 -i 265° 50' 40 30 20 10 264° 50' 40 30 20 10 263" 50' 40 30 20 10 262" 5(1' 40 30 20 10 261" .50' 40 30 20 Id 200 220 THE ORBIT OF URANUS. TABLE \\l. -Continued. E 100° 10' 20 30 40 50 lor 10' 20 ;to 40 50 102' 10' 20 30 40 f)0 103° 10' 20 30 40 SO 104° lu' 20 30 40 50 105° 10' 20 30 40 50 106° 10' 20 30 40 50 107° 10' 20 30 40 50. 108° 10' 20 80 40 50 109° 10' 20 30 40 50 UO^ 14 4.80 13 51.96 13 38.96 13 25.81 13 12.51 12 59.06 12 45.45 12 31.(19 12 17.78 12 3.72 11 49.50 11 35.14 5 11 20.62 5 11 5.95 5 10 51.13 5 10 36.16 5 10 21.04 5 10 5.77 5 9 50.35 9 34.78 9 19.06 9 3.19 8 47.17 8 31.01 14.69 58.22 41.61 24.85 7.94 50.88 6 33.67 6 16.32 5 58.82 5 41.17 5 23.37 5 5.42 5 6 5 4 .'■)9 4 59 4 59 4 .^S 4 5S 4 58 4 57 4 57 4 57 47.33 29.10 10. -:i !)2.19 33.52 14.70 55.74 3(>.63 17.38 57.99 38.45 18.77 58.94 38.97 18.H6 5 s. 60 38.21 17.67 56.98 36.15 15.18 54.07 32.81 11.42 4 56 49.88 2.84 3.00 3»5 ,?-45 3.61 ,V7<5 4.06 4.22 4- 36 4-52 4.67 4.S2 4-97 5->- 5-27 5-42 5-57 5-72 5-87 6.02 6.16 6.32 6.47 0.61 6,76 6.91 7.06 7.21 7-3S 7-5° 7-6.S 7. So 7-95 S.09 S23 8.38 8.53 8.67 8.82 8.96 9.11 9-25 9-39 9-54 9.68 9-83 19.97 20. 1 1 20.26 2°-,l9 20.,S4 20.69 20.83 20.97 21. 1 1 21.26 2' -39 21 54 Li>;f. r 1.23 71958 72525 73091 73656 74221 74785 75348 75911 76474 77036 77598 78159 78720 792SO 79839 80398 80956 81514 82071 82628 83184 83739 84294 84849 85403 !S5'.I56 86508 87059 87610 88160 §8709 89258 89806 90354 90902 91449 91995 9-.'540 9;i(),s5 931:29 94172 94715 95257 9579S 96338 96S78 97417 97955 98493 99030 99566 *00101 *n(l636 *01170 *01703 ♦02236 *(»2768 *(»3299 ♦03829 ♦04359 ♦ (tJssH I •1.29 S^'7 566 565 565 564 563 563 Sf'3 5f'2 562 561 561 560 559 559 55S 558 557 557 55^' 555 555 555 554 553 552 55' 55» 55° 549 549 548 548 54S 547 546 545 545 544 543 543 542 541 540 540 539 5.?8 538 537 5,^6 535 535 534 533 533 532 S\^ 530 530 I 529 1/ 260' 110° 50' 10' 40 20 30 30 20 40 10 60 259° 111° 50' 10' 40 20 30 30 j 20 40 10 50 258' 112° 50' 10' 40 20 30 30 1 20 40 10 60 25T 113° 50 10' > 40 20 30 30 20 40 10 50 l256° 114° 50' 10' 40 20 30 30 20 40 10 50 255° 115° 50' 10' 40 20 30 30 20 40 10 5u 1254^ 116° 50' 10' 40 20 1 30 30 20 40 10 50 253° in° 60' 10' 40 20 30 30 20 40 10 50 252° 118° , 50' 10' i 40 20 1 30 30 20 40 10 50 251° 119' 50' 10' 1 40 20 1 30 30 1 20 40 1 10 50 2.'J0' 120 ! a E 49.88 28.20 fi.39 44.44 22.34 0.10 54 37.72 54 15.2! 53 52.55 53 29.76 6.83 4 53 4 52 43.76 52 20.55 51 57.20 51 33.72 51 10.10 50 46.34 50 22.44 49 58.41 49 34.24 49 9.94 48 45.50 48 20.92 47 66.21 47 31.36 47 6.38 46 41.26 46 16.00 4 5 50.61 45 25.09 44 59.44 44 33.66 44 7.74 43 41.69 43 15.50 42 49.19 42 22.74 41 56.16 41 29,45 41 2.ro 40 35.62 40 8.52 39 41.28 39 13.91 38 46.41 38 1S.7S 37 51.02 37 23 13 55.12 26.98 58.71 35 30.. il 35 1.78 34 33.13 34 4.35 3:! 35. M 33 6.41 32 37.25 32 7.96 31 38.55 21.68 21.81 21.95 22. 10 22.24 22.38 22.51 22.66 22.79 2--.93 23-°7 :J3.2i 23-35 23.48 23.62 23. 7O 23.90 24.03 24.17 26.05 4 31 9.61 L(i(r. r 129 04888 05416 \ 05944 5 06471 5 06997 \ 07522 5 08047 0857 1 ^ 09094 ^ 09616 \ 10137 ^ 10658 ^ 11178 11(;97 5 12215 5 12732 5 13248 5 13764 5 14279 14793 5 15306 5 1581H 5 16329 5 16>i40 5 5 17350 17859 \ 18367 \ I8f<74 \ 193.><0 \ 19886 ^ 20391 20895 \ 21398 ^ 21900 ^ 224(11 \ 229(11 ^ 23401 . 23S<.|<) ^ 24397 Z 24.s-.,4 ^ 25390 ^ 258.S5 ^ 263S0 , 26S74 ^ 27366 ] 27S57 \ 28348 ] 28,><37 'I 29325 29812 ] 30299 '; 307.X5 ] 31270 Z 31754 ] — ■" 4 327 1 9 J 33200 \ 336S1 ] 341(;l J 34640 ;; 3.1II7 1.29 \- i !i THE OllBIT OF UK AN US. 221 TAULK Vn.—Conlinued. 1520° 10' 20 30 40 50 121" 10' 20 30 40 50 122' 10' 20 30 40 50 123' 10' 20 30 40 50 124° 10' 20 30 40 50 125° 10' 20 30 40 50 E 31 !).01 30 39.35 30 9.56 29 3fl.fi5 29 9.(il 28 39.45 28 O.lfi 27 38.75 27 8.22 2(i 37.50 21! (1.78 25 35.88 25 4. 85 24 33.71 24 2.44 23 31.05 22 59.54 22 27.91 4 21 50.10 4 21 24.29 4 20 52.29 4 20 20.18 4 19 47.94 4 19 15.59 126° 12 4.35 10' 11 30.37 20 10 5(1.27 30 4 10 22.00 40 9 47.73 50 9 13.29 127^ 8 38.74 10' 4 8 4.07 20 4 7 29,29 30 fi 54.40 40 ft 19.40 60 5 44.29 128' 10' 20 30 40 60 129' 10' 20 30 40 50 ,') 4 3 3 2 2 1 59 59 Log. r 43.12 10.53 37. S3 5.01 32.07 50.01 4 15 25.84 4 14 52.55 4 14 19.14 4 13 45.02 4 13 11.98 4 12 3S.22 3 58 9.00 33.72 58,27 22,72 47,05 11.27 35.38 59.38 23,27 47.05 10,73 34.29 130' i 3 57 57.75 29.66 29.79 29.91 30.04 30.16 30.29 30.41 .so- 53 30.66 30.78 30.90 3103 3114 3«-27 31-39 3i-5» 31-75 31-87 32.00 32.11 32-24 32-35 32-47 32-59 32-70 32.82 32.94 33-06 33-17 33- 29 33-41 33-52 33-<>4 33-76 33-87 33-98 34- to 34-21 34-33 34-44 34-55 34-67 34- 7« 34-89 35-00 35-" 35-23 35-34 35-45 35-55 35-67 35-78 35-89 36,00 36. 1 1 36.22 36-32 36-44 36-54 1.29 35117 35593 3(;009 30544 37018 37491 37903 38434 3(S904 39372 39840 40300 40771 41230 41700 42103 42025 43080 43540 44005 44403 44919 45375 45829 40282 407.35 47180 47030 4S080 48534 48981 4!I428 49x73 .•.0317 50701 51204 51045 520S5 52524 52902 53399 53834 542l!8 54702 55134 j 555(;5 i 55995 1 50424 50^52 57279 57705 5sl.i0 5S554 58977 59399 59820 00239 00057 01074 01489 01903 1.29 476 476 475 474 473 472 47« 470 468 468 466 465 465 464 463 462 461 460 459 458 456 456 454 453 453 45 « 450 450 44S 447 447 445 444 444 443 441 440 439 43^ 437 435 434 434 432 43' 430 429 42S 427 426 425 424 423 422 421 419 418 4'7 415 414 240' 50' 40 30 20 10 239' 50' 40 30 20 10 238° 50' 40 30 20 10 23T 50' 40 30 20 10 23U' 50 40 30 20 10 235° 50' 40 30 20 10 234' 50' 40 30 20 10 233' 50' 40 30 20 10 232' 50' 40 30 20 10 231^ "'0' 40 .30 20 10 230" n 130' 10' 20 30 40 50 131° 10' 20 30 40 50 132^ 10' 20 30 40 50 133' 10' 20 30 40 50 134' 10' 20 30 40 50 135' 10' 20 30 4J fO B 3 57 57.75 3 57 21.10 3 60 44.35 3 50 7.48 3 55 30.61 3 54 53.43 3 64 10.25 3 53 38.97 3 53 3 52 3 51 3 51 1.58 24.08 40.48 8.78 3 3 3 49 50 39.97 49 53.00 15.04 3 48 30.92 3 47 68.70 3 47 20.38 3 40 41.95 3 40 3,42 3 45 24,79 3 44 40.00 3 44 7.23 3 43 2S.30 3 42 49.27 3 42 10.14 3 41 30.91 3 40 51.58 3 40 12.15 3 39 32.02 3 3S 3 38 3 37 3 30 3 30 53.00 13.28 33.40 53,54 13,52 35 33.41 136' 3 34 53.21 10' 3 34 12.90 20 3 33 32,50 30 3 32 52.01 40 3 32 11.42 50 3 31 30.73 137' 3 30 49.95 10' 3 ;!0 9.08 20 3 29 2S.11 30 3 28 47.05 40 .3 28 5.90 50 3 27 24.00 138' 10' 20 30 40 50 139° 10' 20 30 40 50 140° 3 20 4.3,32 3 20 1.S9 3 25 20,37 3 21 3S,70 3 23 57.05 3 2.! 15.20 3 22 33.38 3 21 51.41 3 21 9, .35 3 20 27.20 3 19 4 4.90 3 19 2.03 3 18 20.22 36-65 36.75 36.87 36-97 37.08 37-«8 37-28 37-39 37-50 37.60 37-70 37-81 37-9" 38-02 38.12 38.22 38.32 38.43 38-53 38-63 38-73 38-83 38.93 39-03 39- "3 39-23 39-33 39-43 39-53 39-62 39.72 39.82 39.92 40,02 40.11 40.20 40.31 40.40 40.49 40.59 4^.69 40.78 40.87 40.97 41.06 41.15 41.24 41.34 41.43 41.52 41,61 41.71 41.79 41.88 Log. r 1.29 61903 62317 62729 63140 63550 63959 64300 04773 05179 65583 65987 06389 60790 07190 67588 07985 08382 68777 69171 69504 69955 70345 70734 71122 71509 71895 72270 72602 73044 73425 73805 74184 74501 74937 75312 75080 70059 70430 7 0800 77109 77537 77903 78208 7S032 7S995 79350 79710 SOU75 80433 80789 81144 81498 81S51 82202 82552 414 ;12 411 410 409 407 407 406 404 404 402 401 400 398 397 397 395 394 393 391 390 389 388 387 3S6 384 383 382 381 380 379 377 376 375 374 373 37» 370 369 368 366 365 364 363 361 360 359 358 356 355 354 353 35 > 350 41.07 8''90I 349 42,06 83249 348 42.15 83595 346 42.24 83910 345 42.33 84284 344 42.41 343 84027 1.29 230' 50' 40 30 20 10 229' 50' 40 30 20 10 228° 50' 40 30 20 10 227" 50' 40 30 20 10 226' 60' 40 30 20 10 225' 50' 40 30 20 10 224° 60' 40 30 20 10 223° 50' 40 30 20 10 222° 50 40 30 20 10 221° 50' 40 30 20 10 220' I i ! \ IIP 222 THK ORBIT OF URANUS. TAUl r K VII . — C'tiiil iiiiK'd. 9 E Loff. 9 E I'|>K- r Off/ ff 1.29 n t tr 1.30 140° 3 IS 20.22 84027 220' 150° 2 33 20.93 02798 262 260 210° 10' 3 17 37.72 42-5° 84908 341 50' 10' 2 32 39.79 47.14 030(;0 50' 20 3 l(i 55.13 42-59 85308 340 40 20 2 31 52.59 47-20 03320 40 30 3 l(i 12.45 42.68 85647 339 30 30 2 31 5.32 47-27 03579 259 258 256 30 40 3 15 2!l.(18 42-77 85985 336 20 40 2 30 17.98 47 34 03837 20 50 3 14 4U.83 42-85 80321 10 60 2 29 30.58 47-40 04093 10 42-94 335 47-47 25s 141° 3 14 3.89 80050 219° 151° 2 28 43. n 04348 209° 10' 3 13 20.87 43.02 80990 334 50' 10- 2 27 55.58 47-53 47.60 47.66 04002 «S4 50' 20 3 12 37.78 43- « I 87322 332 40 20 2 27 7.98 04854 252 40 30 3 11 54. 5G •43-20 87053 33^ 30 30 2 20 20,32 05105 251 30 40 3 11 11.28 43.28 43-36 87983 330 20 40 2 25 32.59 47-73 05355 250 1 20 50 3 10 27.92 88312 329 10 50 2 24 44.80 47-79 47-85 05004 249 10 43-45 327 *47 142° 3 9 44.47 88039 326 218° 152° 2 23 50.95 05851 208° 10' 3 9 0.94 43-53 88905 50' 10' 2 23 9.04 47.91 00090 245 50' 20 3 8 17.32 43.62 89290 325 40 20 2 22 21.00 47-98 00340 244 40 30 3 7 33.f)2 43-70 89014 324 30 30 2 21 33.02 48.04 00583 243 60 40 3 (1 49.83 43-79 43-86 89930 322 20 40 2 20 44.92 48.10 48.16 00824 241 20 50 3 G 5.97 902 ' 321 10 50 2 19 60. 7G 07004 240 238 10 43-95 1 48.22 143° 3 5 22.02 9057'. n° 153° 2 19 8.54 48.28 07302 237 236 234 233 207' 10' 3 4 ;i7.99 44-03 9()S95 -■^o' 10' 2 18 20.20 07539 50' 20 3 3 53. .H8 44- 1 1 91212 3'- 40 20 2 17 31.92 48.34 48.41 48.46 07775 40 30 3 3 9.09 44.19 91528 3'6 30 30 2 10 43,51 08009 30 40 3 2 2.^). 42 44.27 9IS42 3'4 20 40 2 15 55,05 OS242 20 50 3 1 41.08 44-34 44-43 92155 313 311 10 50 2 15 0.53 48. 5 2 48.58 08473 231 230 10 144° 3 5n.r,5 92400 216° 154° 2 14 17.95 48. 6 J 08703 22S 206° 10' 3 12.14 44.51 92770 310 ,Ml' 10' 2 13 29.31 08931 50' 20 2 59 27.55 44-59 9;iOS5 309 40 20 2 12 40,02 48. 69 09158 227 226 40 30 2 TiS 4 2.89 44-66 9;!393 308 30 30 2 11 51, SO 48. 76 09384 30 40 2 .-.7 5S.14 44-75 9:!700 307 20 40 2 11 3,05 48.81 4R.87 09008 224 20 50 2 57 13.32 44.82 94005 305 10 50 2 10 14.18 09831 223 10 44.90 304 48.92 222 145° 2 50 28.42 44-98 94.109 215' 155° 2 9 25,20 10053 220 219 205° 10 2 55 43.44 91012 3,°3 50' 10' 2 8 30.28 48.98 10273 50' 20 2 54 5S.39 45-05 94 913 301 40 20 2 7 47.24 49-04 10192 40 30 2 54 13.2(! 45- '3 95213 300 30 30 2 G 5.-<.15 49.09 10709 217 216 214 30 40 2 53 2S.05 45-21 45.28 9,"i511 298 20 40 2 9,01 49.14 10925 20 50 2 52 42.77 li.">S08 297 10 50 2 5 19.81 49.20 11139 10 4;,- 36 296 49.26 213 146° 2 51 57.41 90104 214° 156° 2 4 30.55 11352 212 204° 10' 2 51 11.97 45-44 90398 294 50' 10' 2 3 41.24 49-3' 11504 50' 20 2 50 2 7 14750 10 149' 2 38 8.85 46. 73 ♦01197 i 211' 159-= 1 49 .35.12 14944 1S6 185 •83 182 180 201° 10' 1 2 37 21.02 ♦014 07 270 ; 50' 10' 1 48 44,91 50-21 15130 50' 20 2 30 34.81 46. Si 46.87 ♦O1730 26() 40 20 1 47 54,05 50.26 15315 40 80 2 35 47.94 ♦02004 26S 30 30 1 47 4,34 50-31 15498 30 40 2 35 I. 01 46.93 ♦02270 266 ! 20 40 1 40 13.98 50.36 15080 20 50 { 2 34 14.00 47.01 ♦02535 265 ! 10 50 1 45 23.58 50.40 15800 10 47.07 263 5... 44 •79 160' 2 33 20.93 ♦0279^^ 210' 160 1 44 33.14 10039 200° 1 „ 1 ♦1.30 '< !/ 1 ISO __ THE Oil 11 IT OF URANUS. 223 TABLK VII .— Cont inited. 9 E Log. r K Log. r 1 Q 1 II >t l.SO 1 n II 1.80 IQO^ 1 44 33.14 10039 200° 170° 52 57.33 24087 89 88 86 84 83 82 190° 10' 1 43 42.05 SO'49 10210 ' 77 76 50' 10' 52 4.81 52-5' 24176 50' 20 1 42 52.11 50-54 S°-5S 50.62 50.O7 50-71 10392 ' 40 20 51 12.28 52-53 52-56 52.58 52.60 52.62 24204 40 40 1 43 1.53 1 41 lO.'Jl 10500 ' 10739 ' 74 73 30 20 30 40 50 19.72 49 27.14 24350 24434 30 20 50 1 40 20.24 1C310 ; 71 70 10 50 48 34.54 24517 10 161° 1 39 29.53 17080 68 67 66 64 63 62 199° mr 47 41.92 52.64 52.66 52.6.;j 52-70 52.72 52-74 24599 80 189° 10' 20 30 1 3,S 38.7S 1 37 47.98 1 3G 57.14 5°-75 50.80 50.84 50.89 17248 ' 17415 17581 50' 40 30 10' 20 30 40 49.28 45 50.02 45 3.93 24079 24758 24835 79 77 75 74 73 50' 40 30 40 1 3G 0.25 17745 20 in 44 11.23 24910 20 50 1 35 15.33 50.92 50.96 17908 J 10 50 43 18.51 24984 10 162° 1 34 24.37 18070 60 198° 172° 42 25.77 52.76 26057 188° 10' 1 33 33.37 51.00 18230 ' 50' 10' 41 33.01 25128 71 50' 20 1 32 42.33 5' 04 l.')388 ' S^ 40 20 40 40.24 52-77 26198 70 69 67 65 64 40 ao 40 I 31 61.24 1 31 0.12 510') 51.12 51.16 51.20 18545 ' 18700 57 55 30 20 30 40 39 47.45 38 64.04 52-79 52.81 _ _ y _ 25207 25334 30 20 50 1 30 8.90 18854 54 53 10 50 38 1.82 52.82 52.84 25399 10 163° 10' 1 29 17.70 1 28 20.52 51-24 19007 19158 51 197° 50' 173° 10' 37 8.98 30 10.12 52.86 52-87 52.89 26403 25525 62 61 187° 50' 20 1 27 35.25 51-27 19308 50 48 40 20 35 23.25 25580 40 30 1 20 43.93 51-32 19450 30 30 34 30.30 25045 59 58 56 30 40 1 25 52.58 5«-35 19003 47 46 20 40 33 37.45 52.91 25703 20 60 1 25 1.19 5'-39 19749 10 50 32 44.53 52.92 25759 10 S»-42 44 52-93 55 164° 1 21 9.77 51-46 19893 193° 174° 31 51.00 25814 53 52 50 48 47 46 186' 10' 1 23 1S.31 20035 42 50' 10' 30 58.00 52-94 52.96 25807 50' 20 ; 22 20.81 5'-5o 20170 41 40 20 30 5.70 25919 40 30 40 1 21 35.28 1 20 43.71 51-53 5>-57 51.60 51-64 20310 20454 40 138 ■•^6 '35 30 20 30 40 29 12.73 28 10.75 52.97 52.98 25909 20017 30 20 50 1 19 52.11 20590 10 50 27 20.70 52-99 53-00 2(!004 10 165° 10' 20 1 19 0.17 1 18 8.80 1 17 17.10 5'-67 5'-7° 20725 20858 20090 32 195' 50' 40 175° 10' 20 20 33.70 25 40.75 24 47.72 53-01 53-03 53-04 53-04 53-o6 53-07 20110 20154 20197 44 43 42 40 39 37 185° 50' 40 30 1 10 25.30 5'-74 21121 31 30 30 23 54.08 20239 30 40 1 15 33.59 5^-77 51. So 5'-83 21250 29 20 40 23 1.04 20279 20 50 1 14 41.79 21377 27 26 10 50 22 8.58 20318 10 166° 1 13 40.90 5 1. 86 21503 194° 176° 21 15.51 S3-08 53.08 20355 35 184° 10' 1 12 58.10 21028 '25 50' 10' 20 22.13 20300 50' 20 1 12 0.20 51.90 21751 '23 40 20 10 29.35 20424 34 40 30 1 11 14.27 51-93 51.96 51.98 52.02 21873 22 30 30 18 30.25 53-'o 53-'o 53-12 53-12 20450 32 30 40 1 10 22.31 21993 20 20 40 17 43.15 20480 30 20 50 1 9 30.33 22112 '9 118 10 50 10 50.03 20515 29 28 10 167° 10' 1 8 38.31 1 7 40.27 52-04 52.08 22230 22;!40 ri6 193° 50' 177° 10' 15 50.01 15 3.78 53-13 53 13 53.14 53- '5 53-15 53-15 20543 20509 26 2 z 183° 50' 20 1 54.19 22400 114 40 20 14 10.05 20504 23 23 40 30 1 2.08 52.11 22573 '■3 30 30 13 17.51 20017 30 40 1 5 9.94 5-'-' 4 52.16 52-19 22084 ( 1 1 20 40 12 24.30 20039 20 20 50 1 4 17.78 22793 109 108 10 50 11 31.21 2G059 19 10 163° 1 3 25.59 22901 192' 178' 10 38.00 53- J 6 53.16 5316 20078 1 7 182° 10' 1 2 33.37 52.22 23008 107 50' 10' 44.00 20005 16 60' 20 1 1 41.13 5 =-24 23113 ■05 40 20 8 51.74 20711 ■5 13 1 J 40 30 1 48.80 52.27 2.3217 104 30 30 7 68.58 20720 30 40 59 50.57 52-29 23320 'O,? _ 20 40 7 5.41 53-17 20739 20 50 59 4.25 52-32 52.35 23421 lol 100 10 50 12.24 53-17 53-17 20750 10 10 169° 58 ii.no 23521 191° 173° 5 10.07 53-18 20700 8 181° 10' 57 19.53 52-37 23019 98 50' 10' 4 25.89 20708 50' 20 50 27.13 52-40 23710 97 40 20 3 32.72 53-17 53-18 53-18 53-18 20775 7 40 30 , 55 34.71 52-42 23811 95 30 30 2 39.54 20780 5 30 40 ' 54 42.27 5244 52-46 23904 93 20 40 1 40.30 20784 4 20 50 53 49.81 23990 92 10 60 53.18 20780 2 10 52.49 91 53-18 1 170' 52 57.32 1.30 190° 180° 0.00 20787 1.30 180° 28 °'5^ 0.29 4.20 0.10 21 2200 213 141 33 72^80 °-5^ 0.31 4.20 0.10 0.21 2202 214 142 34 73.32 III 74.34 +°- 0.33 4.25 0.10 0.21 2258 210 142 35 0.35 4.25 0.10 0.22 2254 217 142 30 0.38 4.24 0.10 0.22 2249 218 142 37 74.85 °- 0.40 4.24 0.10 0.22 2245 219 143 38 75.30 °-5 0.43 4.23 0.10 0.23 2240 221 143 39 0.30 70.87 +°--2° 0.40 4.23 0.09 0.23 2230 222 143 40 0.49 4.22 0.09 0.23 2231 223 143 41 0.52 4.22 0,09 0,23 2220 224 144 42 77.37 °-i:° 0.55 4.22 0,09 0,24 2221 225 144 43 78.80 0.58 4.21 0.09 0,24 2210 227 144 44 0.01 4.21 0.09 0,24 2210 228 144 45 0.04 4.21 0.09 0,25 2205 229 144 4(i 79.35+°- 49 0.08 4.20 0.09 0.25 2200 230 145 47 79.84 °-49 80.32 °-4f 80.80 °-48 0.48 81.28 , 81.70 +°-48 0.71 4.20 0.09 0.25 2194 231 145 48 0.75 4.20 0.09 0.20 2188 232 145 49 0.79 4.20 (.09 0.20 2182 233 145 50 0.82 4.20 0.08 0.20 2170 234 140 51 0,80 4.19 0.08 . 0.20 2170 235 140 52 82.23 °-47 0.90 4.19 0.08 0.20 2104 230 140 53 82,70 °-*7 0.94 4.19 0.08 0.27 2158 237 140 54 83.17 °-+J 0.46 0.98 4.19 0.08 0.27 2151 238 140 55 83.03 1.02 4.19 0.07 0.27 2145 239 140 5li 84.10 +°-*7 1,07 4.19 0.07 0.28 2138 240 140 57 84.55 °-45 85.01 °;f 1.11 4,19 0.07 0.28 2132 241 140 5S l.lfi 4,19 0.07 0.28 2125 242 147 59 85.40 °-*5 0-4.1 85,91 1.20 4.19 0.07 0.28 2118 243 147 r.o 1.25 4.19 0.07 0.29 2111 241 147 TUE OllBIT OF URANUS 225 TA15LE VllI, Ano. \.—C inlinii.d. Arg. no (ii.f.O) Diff. (o.s.l) (v.c.X) (».«.2) It 0.07 (i'.f.2) (p.r.O) (,,.«. 1) 1 (p.O'.l) II II 85.01 . 1.25 4.10 II 0.29 j 2111 244 147 (ii 80.30 +°-'*5 1.30 4.19 0.07 1 0.20 2104 244 147 (12 80.80 °-*^ 1 34 4.20 1 0.00 0.29 2007 245 I 148 03 87.24 °-*-* 1.39 4.20 0.00 0.29 2000 240 148 04 8 .08 °-*-* 1.44 4.20 0.00 0.29 2082 247 148 05 0.43 8S.11 . 1.49 4.20 con 0.30 I 2075 248 148 Oli 88.53+°-*' 1.54 4.21 0.00 0.30 2007 248 MS 07 88.00 °-« 1.00 4.21 0.00 0,30 2059 249 148 OS 80.38 °--*' 1.05 4.21 0.05 0.30 ' 2052 250 148 0!) 80.80 °;]J 90.21 , 1.70 4.22 0.05 0.31 2044 250 1 148 70 1.75 4.22 0.05 0,31 2030 251 149 71 00.02+°- 4' 1.81 4.23 0.05 0.31 i 2028 253 149 7i 91.03 "-4' 1.80 4.23 0.05 0.31 ; 2020 252 149 7:i 91.43 °»° 1.92 4.24 0.05 0.31 I 2012 253 149 74 91.83 °-+^ 1.98 4.24 0.04 0.32 2003 253 149 7.1 0-39 92.22 , 2.03 4.25 0.04 0.32 1995 254 149 7."> 02. 02+°- 4^ 2.09 4.20 0.04 0.32 1980 254 149 77 ot.oo °-^i 2.15 4.20 0.04 0.32 1978 255 149 7'f 9.!.3S °--^f 2.21 4.27 0.04 0.32 1000 255 149 7'.» 80 93.7li °--]^ 0.38 94.14 , 2.27 2.33 4.2s 4.29 0.04 0.04 0.32 0,32 1000 10.--)2 25 G 250 149 149 81 94.51 +°-.>7 2.30 4.30 0.03 0,32 r.M3 257 149 82 04.87 °--5^ 2.45 4.31 0,03 0.33 1934 257 149 83 95.23 °-^^ 2.51 4.32 0.03 0,33 1925 257 149 84 95.50 °-^^ 0.36 05.05 , 2.57 4.33 0.03 0.33 1010 258 149 85 2.04 4.34 0,03 0.33 1000 258 149 80 9!. 30 *-°--^5 2.70 4.35 0.03 0,33 1807 258 149 87 9;.04 °'5-* 2.70 4.30 0.03 0.33 1SS8 259 149 8S 01!. OS °--' + 2. S3 4.37 0.03 0.S3 1878 250 149 8!> 07.32 -35 97.05 , 2.80 4.38 0.02 0.33 1809 259 149 00 2.90 4.30 0.02 0..33 1859 200 148 01 07.07 +°--'^ 3,02 4,41 0.02 0.33 1850 200 148 02 9^.20 °-3-' 3.00 4,42 0.02 0.33 1840 ac'o 148 03 os.oi °-3^ 3.15 4,43 0.02 0.33 1830 200 148 94 OS. 02 °-3' 0.31 90.23 , 3.22 4.45 0,02 0.33 1820 200 148 05 3.29 4.40 0.02 0.33 ISIO 201 148 01") 90 53+°- 3° 3,35 4.4S 0.02 0,34 l.'^OO 201 148 07 00. S3 °-3° 3.42 4.40 0.02 0.34 1790 201 148 OS 100.12 °"} 3,49 4.51 0,01 0.34 1780 201 m 00 100.70 , 3.50 4.52 0.01 0.34 1770 201 147 100 3.03 4.54 0.01 0.34 1700 201 147 101 100.97+°-; 101.25 ° :, 3.00 4.50 0.01 0.34 1750 201 147 102 3,70 4,57 0.01 0.34 1739 201 147 103 •"'.'V2 : 3.83 4,50 0.01 0.34 1729 201 140 104 3.00 4. .01 0.01 0.34 1718 201 140 1 05 102.04 , 3,97 4,03 0,01 0,34 : 1 708 201 140 100 102.29+°- S 4,04 4,05 0.01 0,34 i 1097 201 145 107 102.54 °-',^ 4,11 4.07 0.01 0.34 lOSO 201 145 lOS 102. 7S °- 1 4.18 4.09 0.0 1 0,34 1070 201 145 109 103.02 °;;J 103.25 4.25 4.71 0.01 0,34 1005 201 145 110 4.32 4.73 1 0.01 0.34 1054 201 144 111 103.48+°- '5 4.30 4.75 0,01 0.34 1043 201 144 112 103.70 °-" 4.40 4.77 0.01 0.34 1032 201 144 113 103.02 °-" 4,53 4,70 0,01 0.34 1022 200 143 114 10.13 -; 101.34 , 4.00 1 4.81 0,01 0.34 1011 200 143 115 4.07 4,83 0.01 0.33 1000 200 142 lift 104.54+°-^° 4.74 4.S5 0,01 0,33 15SS 200 142 117 104.74 °-"° 4.81 4.87 0,01 0.33 1577 200 142 118 104.03 °-'^ l«^ll 0.8 4.88 4.90 O.Ol 0.33 1500 259 141 119 4.05 4.92 0.01 0.33 1555 250 141 120 105.29 5.01 4.94 0,02 0.33 1544 250 140 1 19 July, 1878. „! \ if! !«i I ■ I 226 THE ORBIT OF UIIANUS. TAIILF VIII, A no. 1. — Continued. Ai'n. (iJ.c.O) Diir. (i>.».l) {v.cA) (V.H.i) (0.C.2) (p.f.O) (p.».l) (P.f.l) t, n n tt tt tt 120 •"■'■'•2« ,„ ., 5.01 4.94 0.02 0.33 1544 259 140 121 105.40+°- 5.08 4.97 0.02 0.33 1533 259 140 122 105.03 °-'/ 105.79 °'^ 105.95 °"^ o> 1 ^ 6.15 4.99 0.02 0.33 1521 258 139 12;j 5.22 5.01 0.02 0.33 1510 258 1 39 124 6.29 6.04 0.02 0.33 1499 258 138 135 100.10 , 100.25+°- ;S 5.30 6.00 0.02 0.33 1487 257 138 12(! 6.43 6.09 0.02 0.33 1470 257 137 127 100.39 °-^^ 6.50 6.11 0.02 0.33 1404 257 137 12S 100.53 °- j 6.57 5.14 0.02 0.33 1453 250 130 129 100.05 °;;; 100.78 6.04 6.16 0.02 0.33 1441 250 135 IM 5.71 6.19 0.02 0.32 1430 250 135 l.Jl 100.«9+°-" 6.78 5.21 0.03 0.32 1418 255 134 l;!2 107.01 °-'* 6.85 6.24 0.03 0.32 1407 255 134 l:!.l 107.11 °-'° 5.91 6.20 0.03 0.32 1395 254 133 131 107.22 °-" 0.09 6.98 5.29 0.03 0.32 1384 254 132 135 107.31 , fi.05 6.32 0.03 0.32 1373 254 132 13(! 107.40+°-°'> fi.lt 6.34 0.04 0.32 I.'IOO 253 131 137 107.49 °°'l 107.57 °-°'^ fi.18 6.37 0.04 0.32 1349 253 131 13S 0.25 6.40 0.04 0.32 1337 252 130 139 107.04 °-°/ 0.06 C.31 6.42 0.04 0.32 1325 252 129 140 107.70 107.70+°°^ 107. S2 °-°'^ 0.38 6.45 0.04 0.32 1314 251 129 141 fi.44 6.48 0.05 0.31 1 302 251 128 142 G.51 5.51 0.05 0.31 1290 250 127 143 107..S7 °-°5 0.57 6.53 0.05 0.31 1278 250 120 144 107.91 °-°-t 0.04 107.95 fi.G4 6.50 0.05 0.31 1207 249 120 1!5 0.70 5.59 ( 5 0.31 1 255 249 125 lie 107.98 +°-°3 0.70 6.01 0.06 0.31 1243 248 124 147 los.oi °-°-< 0.83 5.04 0.00 0.31 1231 248 123 US 10X.03 °-°^ fi.H9 5 07 0,00 0.31 1220 247 123 149 108.05 °-°^ 0.00 0.95 5.09 0.00 0.31 1208 247 122 150 108.05 , 7.01 5.72 0.07 0.31 1190 240 121 ir.i 108.00+°°' 7.08 6.75 0.(»7 0.31 11S4 240 120 152 108.05-°°' 7.14 5.78 0.07 0.31 1172 245 119 153 108.04 °-°' 7.20 5. SO 0.07 0.30 1101 244 118 154 108.03 °-°' 0.02 7.25 6.83 O.OS 0.30 1149 244 118 155 i-^.Ol 7.31 6.80 0.08 0.30 1137 243 117 15r, , 'T.98-°-°-' 7.37 6.S8 0.08 0.:i0 1 1 20 243 110 157 107.95 °-°-] 7.43 5.91 0.09 0.30 III4 242 115 15S '"'•"» TA 7.49 6.94 0.09 0.30 1102 241 114 159 107.87 °-°! 0.05 107.77 °-°S 7.54 6.97 0.09 0.30 1090 241 113 IfiO 7.00 6.99 0.10 0.30 1079 240 112 1(11 7.00 6.02 0.10 0,30 1007 9 to 112 1(;2 107.70 °-°/ 107.04 °-°^ 107.56 °-°^^ 0.07 107.49 7.71 0.04 0.10 0.30 1055 239 111 103 7.77 0.07 0.11 0,30 1044 23S 110 1G4 7.82 6.10 O.ll 0.29 1032 238 109 ir,5 7.87 0.12 0.11 0.29 1020 237 108 ii'.a 7.93 6.15 0.12 0.29 1009 230 107 107 107.31 °-°^ 107.22 °-°? 7.98 6.17 0.12 0.29 997 230 100 lOS 8.03 6.20 0.12 0.29 !)S0 235 105 109 8.08 6.22 0.13 0.29 974 234 104 170 107.01 8.13 6.25 0.13 0.29 903 234 103 171 100.89-°-'^ 8.18 6.27 0.14 0. 29 951 2.-13 102 172 100.77 °-'-' 8.23 6.29 0.14 0.29 940 232 101 173 100.05 °-" 8.28 6.32 0.14 0.29 928 232 100 174 100.52 ° '•' 0.14 100.38 8.33 6.34 0.15 0.29 917 231 99 175 8.37 6.30 0.15 0,29 900 230 98 170 100.24-°-'-* 8.42 6.39 0.15 0.29 894 230 97 177 100.10 ° ';} 105.94 °-'^ 105.78 °-'^ 0. 16 8.40 6.41 0.10 0,29 8s:i 229 90 17S 8.51 6.43 0.10 0. 29 872 228 95 179 8.55 6.45 0.17 0.29 801 228 94 180 105.02 8,00 6.47 0.17 0.29 850 227 93 THE OIllJIT OP URANUS. 227 TAlUiK VIII, Ann. \.—C onlinued. Arg. (u.c.O) Diir (1..8.I) {>y.v.\) (.'.«.2) (y.f.2) (f.c.O) (p.O.l) (p.C.l) 180 105.f.2 8.00 n 6.47 0.17 0.29 850 227 93 181 105.45-°- 7 105.27 °' ^ 105.09 °- '^ 8.04 6.50 0.17 0.29 839 220 92 182 8.08 0.52 0.18 0.29 828 220 91 183 8.72 0.54 0.18 0.29 810 225 90 184 104.90 °;;j 8.7(1 C.50 0.19 0.29 805 224 89 185 8.80 fi.57 0.19 0.29 794 224 88 18(> 104.51-°'^° 8.84 C.59 0.19 0,29 784 223 87 ls7 104.31 °-\° 8.88 6.01 0.20 0.29 773 222 80 188 104.10 °- 8.92 0.03 0.20 0.29 702 222 85 18« 103.89 °:^; 8.9(i 0,05 0.21 0.29 751 221 84 190 103.07 „ ,, 8.99 6.07 0.21 0.29 740 221 83 191 103.44-°-^ 9.03 0.08 ..21 0.29 730 220 82 192 103.21 °-\\ 9.07 0.70 0.22 0.29 719 219 81 193 102.98 °':'; 9.10 0.71 0.22 0.29 708 219 80 194 102.49 „ ,. 9.13 0.73 0.22 0.29 698 218 79 195 9.1 7 0.74 0.23 0.29 688 217 78 I9f> 102.24 -°-^S 101.72 °,, 9.20 0.70 0.23 0.29 677 210 77 197 9.23 0.77 0.24 0.29 607 210 70 198 9.20 0.78 0.24 0.29 057 215 75 199 101.18 , 9.29 6,79 0.24 0.29 640 215 74 200 9.32 6.81 0.25 0.29 030 214 73 201 100.90-°-^„ 9.35 0.82 0.25 0.29 620 213 72 202 100.(12 °-^'* 9.38 6.83 0.25 0.29 610 213 71 ms 100.33 °'9 9.41 0.84 0.20 0.29 600 212 70 204 100.04 °-"9 O.JO 9.43 6.85 0.20 0.29 590 211 69 205 99.74 9.40 0.80 0.26 0.29 680 211 08 20li 99.44 -°-3° 9.48 0.80 0.27 0.29 577 210 07 207 99.13 °-5' 9.51 0.87 0.27 0.29 507 210 00 208 98.82 °--5' 9.53 6.88 0.27 0.29 557 209 05 209 98.5(- °-^- 0.32 98.18 9.55 0.88 0.28 0.29 548 208 64 210 9.58 6.89 0.28 0.29 538 208 63 211 97. 85-°' 3^ 9.00 0.89 0.28 0.29 629 207 02 212 97.52 °-3- 9.02 0.90 0.29 0.29 520 200 01 2l;5 97.18 °\] 9.04 0.90 0.29 0.29 510 200 00 214 9(1.84 "-^1 911.49 9.00 0.90 0.29 0.29 601 205 59 215 9.0s 0.91 0.30 0.29 492 205 58 21(1 9(1.1 4 -°-3S 9.70 0.91 0.30 0.29 483 204 57 217 95.79 °-35 95.43 °-t 9.71 0.91 0.30 0.29 474 204 50 218 9.73 0.91 0.31 0.29 404 203 55 219 95.0G -3; 9.75 6.91 0.31 0.30 450 202 54 220 9.70 6.91 0.31 0.30 448 202 53 221 94.32-°-" 93.94 °-3-^ 93.5(! °-3« 9.78 6.90 0,31 0.30 439 201 52 222 9.79 6.90 0.32 0.30 430 201 51 22;] 9.81 0.90 0.32 0.30 422 200 50 224 «^-^^ c:5 9.82 6.90 0.32 0.30 413 200 49 225 92.78 9.83 6.89 0.33 0.30 405 199 48 22« 92.39-°- 3^ 9.85 6.88 0.33 0.30 398 198 47 227 91.99 °-^° 9.80 0.88 0.33 0.30 390 198 40 228 »!-'"•» o\i 9.87 0.87 0.33 0.30 382 197 40 229 90.77 9.88 6.86 0.34 0.30 374 197 45 230 9.89 0.85 0.34 0.30 305 196 44 231 90.30-°-^' 9.90 0.84 0.34 0.30 357 190 43 232 89.94 °-*^ 9.91 0.83 0.34 0.30 849 195 42 233 89.51 ° •♦3 9.91 0.82 0.34 0.30 342 194 41 234 89.09 °--*^ 0.44 88.r,5 9.92 0.81 0.35 0.30 334 194 41 235 9.93 0.80 0.35 0.30 327 194 40 23() 88.22 -°-*3 9.94 0.79 0.35 0.30 320 1 93 89 237 87.78 °-4;^ 9.94 0.78 0.35 0.30 312 192 88 238 87.34 °-44 9.95 0.70 0.35 0.30 305 192 38 239 80.89 °-'*5 0-45 8(1.44 9.95 0.74 0.30 0.30 298 191 37 240 9.90 0.73 0.30 0.31 291 191 36 ! 1; j'l ^.1 11 228 THE ORBIT OP URANUS. TAHLE VIII, Alio. l.—CoiUinm- l(1.83-°-3'> 8.92 1.21 0.37 0.11 410 97 72 377 10.44 °-." Ifi.Ofi °-^'J 15 -i °-^'^ 0.38 15.30 8.91 1.18 0.37 0.11 418 9(1 73 871 8..S9 1.10 0.37 0.11 420 95 71 371) 3S0 8.88 8.87 1.13 1.11 0,37 o.;i7 0.11 0.10 435 414 91 9:1 75 76 8S1 14.93 -°" 14..',7 °-'^ 13.85 °»^ 13.50 8.80 1.08 0.37 0.10 4,^2 92 77 8S2 8.85 1.00 0.37 0.10 401 91 7S 3S3 8.83 1.04 0.37 0.10 470 90 78 3S4 3M5 f:.82 8. SI 1.02 1.00 0.37 0.37 0.10 0.10 479 488 89 88 79 80 3S(I 13.15-°'" 8.79 0.98 0.37 0.10 497 87 81 3s; 12.80 °--^3 8.78 0.90 0.37 0.10 50(1 8(1 8:' 3HS 12.47 °-]l 8.77 0.94 0.37 0.10 514 85 8' 88 y 12.13 III 11. Si 8.75 0.92 0.37 0.10 525 84 3!)() 8.74 0.90 0.37 0.10 534 82 391 11.48-^-" 8.73 0.88 0.37 0.10 544 81 8(. 392 11.10 °- 8.71 0.87 0..37 0.10 554 80 86 393 10.85 °- 8.70 0.85 0.3(1 0.10 5(14 79 87 394 10.54 °;]; ^0 -^ 70 8.(18 0.84 0.3(i 0.10 5f3 78 88 395 8.07 0.83 0.3(i 0.10 5s;i 77 89 39(i 9.93-°-^° 8.05 0.82 0.3(i 10 593 76 90 397 9.(14 III 8.(!.<; SO 3(1 0.10 (103 75 91 39 S 9.. 35 °- ^ 8.03 0.79 0.3(5 0.10 013 74 92 399 '^.00 0.7.8 0.30 0.10 C23 73 93 400 «-T!^ 0,, 8.5S 0.7T 0.30 0.10 033 72 9.3 401 8..52-°- 8.. 57 0.70 0.3(1 0.10 043 71 94 403 «-2'^ olJ 8.55 0.7('. 0.30 0.10 653 70 95 4113 v.fls °- ! 8.53 0.75 0.30 0.10 604 69 96 4U4 ^■7=' L'^ 8.51 0.74 0.35 O.IO 674 C8 97 405 7.47 8.49 0.74 0.35 0.10 684 67 98 40(! 7 .)2-°='S 8.47 0.74 0.35 0.10 695 66 98 407 (iMs °-^-* 8.40 0.73 0.35 0.10 700 65 99 40 S fl.75 °13 8.44 0.73 0.35 0.10 71(5 64 100 409 G.51 °-^-* 0. 22 8.42 0.73 0.34 0.10 727 63 101 410 0.29 8.:!9 0.73 0.34 0.10 737 62 101 411 G.()7-°" 8.37 0.73 0.34 0.10 . 748 61 102 412 5.85 °" 8.35 0.73 0.34 0.10 759 60 10.^ 413 5.04 °" 8.33 0.73 0.34 0.10 770 59 104 414 6.14 °-'^ 0. 20 8.31 73 0.34 0.10 781 58 104 415 5.24 8.28 0.73 0.34 0.10 792 57 105 410 5.04-°-^° 4.80 °''^ 8.2(1 0.74 0.33 0.10 803 56 106 417 8.24 0.74 0.,33 0.10 814 55 107 41S 4.C.7 °"' 8.21 0.74 0.33 0.10 825 54 107 419 4.33 8.19 0.75 0.33 0.10 836 53 108 420 8.17 0.76 0..33 0.10 847 52 109 TIIK OniJIT OF URANUS. 231 T.VHLK VIII, Ami. 1 —f iiiUniird. Arg. (h.K>:2) (,.r.0) (P.-.I) (p.C.l) n tf ft n tf 420 <-''3 „,, 8.17 0.70 0.33 0.10 847 53 109 421 4.1(1-° 'J 4.00 °'^ 8. It 0.70 0.33 0.10 858 51 110 4'2!) 8.11 0.77 o.;;3 0.11 870 50 110 4-i:t 8.85 °-\l 8.00 0.78 0.33 0.11 881 50 111 424 8.0(1 0.79 0.33 11 893 49 111 4'2.'> 8.04 0.80 o.;;3 0.11 1)04 48 112 421! ^■*'^ oA 8.01 0.81 0.31 0.11 010 47 113 427 3.29 °^ 7.98 0.83 0.31 0.11 037 40 113 428 3.1(1 °\l 7.95 0.83 0.31 0.11 938 45 114 429 3»^ o.W 7.!I2 0.85 0.31 0.11 950 44 114 430 2.n3 7.89 0.80 0.31 0.11 901 44 116 431 2.S2-°'" 7.80 0.87 o.;io 0.11 073 43 110 4;t2 2.72 °''° 7.83 0.89 0.30 0.11 984 43 110 4:t;{ 2.(12 °'° 7.80 0.!(0 0.30 0.11 990 41 117 434 2.53 °°2 o.o8 2.45 7.77 0.93 0.29 0.11 1008 40 118 435 7.74 0.94 0.29 0.11 1019 40 118 4;t(J 2.37 -°°'] 7.71 0. !(5 0.29 0.12 1031 39 119 4;{7 2 i)() °o8 7.07 0.97 0.29 0.12 1043 38 119 4:i.s 2:2.3 °°'^ 7.04 0. 119 0.28 0.12 1054 38 120 43!) 2.1(1 °°7 0.0s 7.(10 1.01 0.28 0.12 1000 37 120 440 2.11 : 7.57 1.03 0.28 0.12 1078 30 121 441 2.00 -°°S 7.54 1.05 0.28 0.13 1089 35 121 442 2.03 °°^ 7.50 1.07 0.27 0.12 1100 35 123 443 1.98 °°-* 7.47 1.09 0.27 0.12 1113 34 123 444 l.!)5 °°^ 0.03 1.92 7.43 1.11 0.27 0.12 1124 33 123 445 7.39 1.14 0.20 0.12 1137 33 123 4411 , ,,,,-0.02 7.3(1 1.10 0.20 0.12 1149 32 123 447 1.88 °°' 7.32 1.18 0.20 0.12 1100 32 124 44S 1.87 °°' 7.28 1. 21 0.20 0.12 1172 31 124 44!) 1.87 °'°° 0.00 7.24 1.23 0.25 0.12 1184 30 125 4:.o 1.87 , 7.20 1.20 0.25 0.12 1190 no 125 451 1.88+°'°' 7.10 1.28 0.25 0.12 1208 30 125 453 l.!l() °°- 7.12 1.31 0.24 0.12 1220 29 120 453 1.92 "■°-' 7.08 1.33 0.24 0.12 1 232 28 120 454 0.03 1.98 , 7.01 1.30 0.24 0.13 1243 28 127 455 7.00 1.39 0.23 0.12 1255 28 127 45(; 2.01 +°°.^ 0.90 1.42 0.23 0.12 1207 27 127 457 2 0(1 °°-^ 0.91 1.45 0.23 0.12 1279 27 128 45H 2.11 °°3 0.87 1.47 0.22 0.12 1291 20 128 45!) 2.1(1 °°^ 0.07 2.23 0.83 1.50 0.23 0.12 1302 20 128 4(->0 0.78 1.53 0.22 0.12 1314 20 128 4(i! 0.74 1..50 0.21 0.12 1320 25 129 402 0.(19 1.59 0.21 0.12 1338 25 129 4(13 2.44 °'°7 0.0. > 1.02 0.21 0.12 1350 25 129 4(14 2.53 °°'^ 0.09 2-''2 ,„„„ 0.00 1.05 0.20 0.12 1301 25 130 4f,5 0.50 l.fiS 0.20 0.12 1373 24 130 4r>a 2^1+0.00 0.51 1.72 0.20 0.12 1385 24 130 4fi7 2«2 °- ! 0.40 1.75 0.19 0.13 1390 24 130 4fiS 2'92 : 0.41 1.78 0.19 0.13 1408 24 130 4(19 304 ::;-; 0.37 1.81 0.19 0.13 1420 23 131 ■170 3.15 0.32 1.84 0.18 0.13 1431 23 131 471 •J .)s+°'.'5 0.27 1.88 0.18 0.13 1 1 13 23 131 472 3.41 °'-^ 0.22 1.91 0.18 0.12 1 455 23 131 473 3.. 55 °'4 0.17 1.94 0.17 0.12 1400 23 131 474 3.(19 °'-* 0.14 3.83 , 0.12 1.98 0.17 0.12 1478 23 132 475 0.07 2.01 0.17 0,12 1489 23 132 47G 3. 98+°- '5 4.14 °'^ 0.02 2.04 0.1(1 0.12 1501 23 132 477 5.90 2.07 0.10 0.12 1512 23 132 478 4.31 °;7 5.91 2.11 0.10 0.12 1524 23 132 479 4.48 °'7 0.17 4.05 5.80 2.14 0.15 0.12 1535 23 132 480 5.81 2.18 0.15 0.12 1540 23 132 iMii: .i * )■ H i IP f i i! ii ;! 233 THE ORBIT OF URANUS. TABLK VIII, A KO. 1. — ContinvriK A.V'. (f.c.O) Diff. (v.x.l) (•'•'•. 1) (U.S.2) (o.r.2) Cp.c.o; (f)."!) (P.e.l) If II tt $t ft ft 480 -;Uo..8 5.81 2.18 0.15 0.12 1540 23 133 4Sl 5.75 2.-1 0. 15 12 1558 23 132 4,S2 5«=2 nl^ 5.70 2.24 0.15 0.12 1509 23 133 483 5.21 ""^ 5.05 2.28 0.14 0.12 1 5S0 23 132 484 ^•^" .r^ 5.59 2.31 0.14 0.12 1591 23 133 485 5. CI , 5.54 2.35 0.14 0.12 ! 002 24 133 4Si; 5.81+°" 5.48 2 38 0.13 0.12 1013 24 133 487 fi.03 °- 6.43 2.42 0.13 0.12 1 024 24 133 48S f'-'-^l o. 5.37 2.45 -.1.13 0.11 10,35 24 134 481) 6.70 , 5.32 2.48 0.12 O.ll 1040 25 134 490 5.20 2.52 0.12 0.11 1057 25 133 491 f;.!l3+°-'-5 5.20 *2.55 0.12 0.11 1008 25 133 492 7.17 °-^-t 5.1;-) 2.59 0.12 0.11 1079 25 133 493 7.41 "•'■> 5.119 2.02 0.11 0.11 1090 25 133 494 7.(!(! °%5 O. 20 6.U3 2.00 0.11 0.11 1700 20 133 495 7.92 , ' . 8.1,S+°-^^ 8.-I4 °-^^ 4.98 2.09 O.ll 0.11 1711 27 133 49G 4.92 2.72 0.11 O.ll 1721 27 133 49T 4.80 2.70 0.10 0.11 1732 28 133 498 8.71 °H 4.80 2.79 O.IO 0.11 1742 28 l.i3 499 8.99 ""-'^ 0.27 9 20 ^ , 4.75 2.83 0.10 0.11 1753 28 133 500 4.09 2.80 0.10 on 1703 29 133 501 9.55+°-^'^ 4.03 2.89 0.10 o.ll 17-4 30 133 502 9.84 °-'' 4.57 2.93 0.09 0.10 '784 30 133 503 10.13 °'^'^ 4.51 2. 91'. 0.09 0.1 1» 1794 31 133 604 10.43 °'^ 10.74 , 4.44 2.i'9 0.09 0.10 1804 31 133 505 4.39 3.02 0.09 0.10 1S14 32 133 &0I> ll.0;-.+°'^' 4.33 3.0;! 0.09 0.10 1824 32 133 507 11. 3(, °'' 4.28 3 09 0.08 0.10 1834 33 133 508 11. (!8 °y- 4 22 3.12 0.08 (1. 10 1844 34 133 509 12.00 '"■•'^ °J3 4.10 3.15 0.08 0,10 1S53 35 133 510 12.33 , 12.r,.!+°-|3 4.10 3.18 0.08 0.09 1 803 !>;"» 133 511 4.04 3.21 0.08 0.09 1873 30 133 5i: 13.00 °34 8.93 3.25 0.07 0.09 lss2 37 133 ;-.l:i 13.34 l\\ 3.92 3.28 0.07 0. 19 1892 38 133 614 13.G8 °;j^^ '4.03 , 14.39+°'^" 14.75 °'^^' 15.11 °-|^ 8.80 3.31 0.07 0.09 1901 39 133 515 3.S0 3.34 0.07 0.09 1910 40 133 510 3.74 3.37 0.07 0.09 1920 40 133 517 3.08 3.10 0.07 0.09 1 929 41 133 518 3.02 3.13 0.97 0.09 1938 42 133 519 3.50 3.. 0.05 , 0.07 2040 55 132 531 20.21 +°-'' 2. MO 3.77 1«.05 "0.07 2048 50 132 532 20.(12 °-" 2 80 3. so 0.05 0.07 2050 67 132 533 21.05 °« 2.75 3.82 0.(i5 0.07 2(103 58 132 534 21.47 °-»' 0.43 21.90 , 2.09 3.84 0.05 0.07 2071 69 131 535 2.03 3.87 0.05 O.OT 2079 01 131 531) .,J ;,4+o-14 2.57 3.8i) 0.05 0.07 2080 62 131 537 "VT °-!^ 2.52 3.91 0.05 0.07 2094 03 131 538 23.21 °^* 2.40 3.93 0.05 0.07 2101 04 131 539 23.. IC, "■'•S 0.44 24.10 3.40 3.90 0.05 0.00 2108 66 ; 131 1 540 2.35 3 OS 0.05 0.00 2115 67 131 THE ORBIT OF URANUS. 233 TAHLK VIII, A 10. 1. — Concluded. Aig. (o.e.O) Diff. (u.s.l) (...c.l) (0.8.2) ] 0>.6'.2) (p.c.O) {p.»l) (p.o.l) ft ff fr ft // 640 2*10. ,A 2.35 3.98 0.05 0.00 2115 07 131 f)41 2.2'J 4.00 C.')5 0.00 2122 08 131 542 2.24 4.02 0.05 0.00 2129 70 131 643 2.18 4.04 0.05 0.00 2135 71 130 644 2.13 4.C.' 0.05 O.OG 2142 73 130 545 2.08 4.07 0,05 0.06 2149 74 130 51() 2.02 4.0!) 0.05 0.00 2155 75 130 547 27.;i2 °-^^ l.i)7 4.11 0.05 0.00 2102 77 130 54S 2«-2T o:£ 1,'J2 4.13 0.05 0.00 2108 78 130 l>i'J 1.87 4 14 0.05 0.06 2174 80 130 550 2S.75 o 1.H2 4.',; 0.05 0.00 2180 81 130 551 2!t.2.i+°-*° 1.77 4 1: 0.05 0.00 2180 83 130 652 21)71 0-48 1.72 4,19 0.00 0.00 2191 84 130 653 30.20 °-»9 1.07 4.20 0.00 0.00 2197 8". 130 564 30.0'J °-'9 0.49 1.C2 4.i« 0.00 0.00 2203 1 130 555 31.18 , 1.57 4.23 0.00 0.00 2208 89 1.30 550 31.07 +°-»9 1.52 4.25 0.00 0.00 2214 90 130 557 32.10 °-*'} 1.48 4.20 0.00 0.00 2219 92 1:^0 558 32.00 °-5° 1.43 4.27 0.00 0.00 2224 94 130 551) 3U.j; "50 .-!.•!. 07 . , 1.38 4.28 O.oG 0.00 2229 95 130 SCO 1.34 4,29 0.00 0.00 2234 97 i;io 5(il 34. 17+°- 5° 1.2!) 4,31 coo 0.00 2238 98 l:w 5(>-i 34.08 " 5' 1.25 4.32 0.07 0.00 2213 100 130 5(i.'i 35.1'.. "^^ 1.20 4.32 0.07 0.00 2248 101 130 5(;4 35,70 °-^; 1.10 4.;i3 0.07 0.00 22.'! .i 103 130 5i;5 1.12 4.34 0.07 0.0() 2250 105 130 5or> 30.73+°-^ 1.08 4.35 0.07 (..'<; 2201 100 130 507 37.25 °-5^ 1.03 4.:!0 0.07 0.00 2205 108 130 50S 37 70 °-^ 0.!)'.) 4 37 0.07 0.00 2209 110 130 50'.) 3s.2',t °-:- o.;if; 4.38 0.07 0.00 2272 111 130 570 38. SI , ^ 0.02 4,38 0.07 0.00 2270 113 130 571 3!..3»-*-°-''-5 0.88 4.39 0.(17 0.00 2280 115 130 572 3U.S0 °-^^ 0.84 4.40 0.07 0.07 2283 110 130 57:5 40.3!) °-^^ 0.80 4.10 0.08 0.(17 22x7 118 130 574 40.:.2 :^ 0.77 4,41 0.08 0.07 2290 120 131 575 0,73 4,41 0.08 0.(17 2293 121 131 5 10 41.!)8+°-^:-^ 0,70 4,(2 0.08 0.07 22:10 123 131 577 42.,^2 °- 1 0.i;7 4,(2 0.08 0.07 22;i9 125 131 57 S 43.05 °^ 0.03 4.42 0.08 0.07 2302 120 131 579 «-^'» :ii 0.00 4.43 0.08 0.07 2304 128 131 5S0 44.12 , 0.57 4,43 0.08 0.07 2.307 130 131 5S1 44.00 +°-"!t 0,5 4 4.13 0.08 0.08 23(19 132 131 5.S2 45.20 "-y^ 0,51 4,(4 0,(19 0.08 2312 l:!3 131 5s:i 45,71 °5» 0,4 8 4,(1 0,(19 o.os 23 1 I 135 131 5S4 40.28 °?» 0.40 4,44 0.09 0.08 2310 i;{7 132 5S5 40.83 0.43 4,44 0,09 0.08 2318 138 132 5S0 47.37 +°-^> 0,40 4.4) 0.09 0.08 2319 140 132 5.S7 47,1)1 °5^ 0.38 4.!. 0.09 0.09 2321 142 l:!2 5SS 48.10 °-5.> 0.30 4,4i 0.09 0.09 2322 144 1:12 589 49.00 °5^ 40.55 , 0.33 f '■■( 0,09 0.09 2324 145 132 500 0.31 ■' , i i 0,09 0,09 2325 147 132 5 '.11 50, 10+° 5' 0.20 4.41 009 0.09 2320 I '9 132 5!)2 50,0 4 °-5> 0.27 4.44 0,10 0.09 2327 150 133 5li;J 51,10 °5.S 0.25 4.44 0,10 0.10 2328 152 133 5i»4 51,74 °ll 52.20 , 0,j3 4.43 0,10 0.10 2329 154 133 5!>5 C .!2 4.43 0,10 0.10 2330 155 133 5!)0 52.s4+°- O.jl 4.43 0.10 O.li) 2330 1.^7 133 5117 53,38 °.';4 O.H 4.43 0,)0 0.11 2330 159 l:i3 5'.18 53, '.)3 °-^5 0,17 4.43 0,10 0.11 23:! 1 101 134 5il!» 51.4S °?5 0-5S 55.03 lO 4.42 O.IO 0.11 2:131 102 l:!t 000 0.14 4.42 0.10 on 23:tl 104 i;u it 30 July. 1873 23 i THE ORBIT OF URANUS- t ;li (., I!!i TAHLR IX, Ami. 2— VCI'ION oi- Saturn. 1 Arg. (c.r.O) Dill. 1 (i.'.s.l) Dili. Sco.var. {r.c.D Dili, g L'c.var. It (r.*i.2) Diff. S t'c.var. (v.u.2) Dill". Sec.var. II II // // /' // // /' // /' " 38.54 , 139.94 , 2.53 293.78 , 0.50 182.02 , „ 1.77 244.07 0.12 1 S8.63+°-°9 141.38 + ;-'| 2.52 293.81 +°°3 0.55 183.80 + '-'" 1.70 244.10-°-57 243.52 °-f^ 0.12 2 38.72 °-°5 142.82 -^l 2.50 293.83+°-°^ 0.54 184.90 '-'^ ) 74 0.12 a 38.81 °°9 144.26 '-^^ 2.49 293.83 °°° 0.53 180.12 '■'" 1.73 ^^ I o.u 4 38.90 °°l 145.70 '•'»■* 2.48 293.82-°°' 0.52 187.27 '-'5 1.71 0.11 0.08 1.44 0.03 '■'5 5 38.98 , Kf.14 ... 2.47 293.79 „„. 0.51 188.42 , 1.70 0.11 (i 39.07+°°'^ 148.58+ -y 2.40 293.75-°-°:^ 0.50 189.50 +''-^ 1.09 O.ll T 39. 1(! °-°l 150.01 1 •♦•? 2.45 293.70 °°^ 0.49 190.09 '-'^ 1.67 0.11 8 39.24 °°l 1-'1« 11 2.44 29,3.03 °-° 0.48 191.82 ''3 l.Ofi 0.10 » 39.33 '■'■°'> 152.89 -^j 2.43 293.54 °°^ 0.47 192.94 '■'- 1.04 0.10 o.oS >-4J 0.10 I. II 10 39.58 °°^l 154.32 2.43 293.44 0.40 194.05 1.03 23S.48 .„ 237.sO-°-f 2.-J7.11 °-^y 0.10 11 155.7(! + '->-> 2.41 293.33-°" 0.15 195.10 + '" 1.01 0.10 1:2 157.19 '•^■5 2.40 293.20 °'-' 0.44 190.20 ■•'° 1.00 0.10 13 39.C7 °-°J, 39.75 °°^ 158.(i2 ■•« 2.39 21)3.00 °'j 292.90 °''' 0.43 197.35 '"^ 1.58 2.-50.41 °-'° 0.10 14 100.05 '•■♦•> 2.38 0.42 198,43 '•°^ 1.57 235.09 °-7- 0.10 o.og '•43 0.17 1.08 0-73 15 39.84 ,„o 39.92+°°^ 1(51.48 , 2.30 292.73 . 0.42 199.51 , 1..55 234f»P „ ,, 0.10 11; 102.90 + '••*^ 2.35 0.41 200.58 + '-°J 201.04 '"^ 1.53 234.22-°- t 0.09 IT 4»-0» Irl 104.32 '■•♦^ 2.34 292!3"t °--' 0.40 1.52 2;!3.48 °-7-* 0.09 IS ^o-ic III 105.74 '•'^ 2.33 292.13 °-' 0.39 202.09 '■°5 1.50 232.73 °-75 0.09 19 107. IG '••*^ 2.32 291.90 °-'i 0.38 203.74 '"S 1.49 231.90 °-'^ 0.09 0.09 1.42 0-^5 1.04 0.78 20 --+:■- 108.58 2.31 291.05 , 0.37 204.78 , 1.47 231.18 0.09 21 109.99 + '•■♦' 2.30 21)1.39-°-^'^ 0.30 205.sl+'-°3 1.40 23().;i9-°-?'^ 0.09 22 -"^ H 171.40 '■->' 2.29 2:)1.12 °-'7 o..;5 200. S3 '•°- 1.44 229. CO °-l'^ 0.09 2;{ 172.81 ' •*• 2.27 ■'9,1 S3 °-"> 0.35 207. S4 '■°' 1 . 13 22s. SO °-f 0.l(» 24 *^-^^ °nn 174.22 '•^' 2.2>> 290:53 °-.'° 0.34 208.85 '■°' I 11 227.98 °-'^' 0.10 0.09 1.40 0.31 1.00 0.83 2J -:-+o:^ 175.02 ,. . 2 25 290.22 0..33 209.85 , 1.40 227.15 ^„ 0.10 2(5 177.02+'-'° 2.24 289.89- °"-'' 0.32 2I0.84+°-W 1.39 220.31 -°-''» O.IO 27 4tJ-«9 0°^ 178.42 '-t" 2.23 289.,55 °-5;f 0.31 211.S2 °-'>^ 1.37 225.40 °if 0.10 2s ^"•'•"^ 000 ! 79.81 '••''> 2.21 289.19 °--''' 0.30 212.79 yi l.3(; 224.00 °-^ S23.74 °it 0.11 29 4107 12 ;sl.20 ■•■''> 2.20 2^8.82 °-" 0.30 213.70 yi 1.34 0.11 0.09 > 39 0.38 °'J5 0.87 :iO ^'^S+o:^ 1S2.59 , 2.19 2SS.44 0.29 214.71 , 1.33 .).)0 ,;)7 0.11 31 183.117 +'--^^ 2. IS 2SS.04-°-»° 0.2S 215.00 +°-9S 1.32 221:09-°-^ 0.11 ;i2 *»:i5 °'° 1S.V35 '••'' 2.1; 2S7.(;3 °-*' 0.2s 210.00 °'^-^ 1.30 221.10 °-'''^ 0.11 ;!:} ^'•4* o,o 180.73 '-3^ 2.15 -S7,20 °-^^ 0.27 217.53 °'^^ 1.29 220.20 °-*^° 0.12 34 ^l-Sl on 1S8.10 " 2.14 2S0.70 °-^j 0.20 21S.44 y\ 1.27 219.28 °-'^' 0.12 0. 10 '•37 0.4O o.yl 0.92 35 3(> "°'>*r: H9.47 , , r 190. S3 + '•■''• 2.13 2.11 si;-°-^'^ 0.20 0.25 ?i:!;S+°r 1.20 1.25 21s. 30 „ , 2 17. 43 -°-'-'-' 0.12 0.12 37 ""> oZ 192.19 '-'^ 2.10 2S5.35 °-''^ 0.24 .,.., ,, °-^') 1.23 2IC.49 °'>^ 0.12 3S :i:s B 193.55 •••'? 2.09 2S4 80 °j 0.23 ^'^■■'^ 0.86 1.22 215.55 yj 214.59 °-''^ 0.97 0.13 39 194.90 1: '33 2.07 2S,.35 m 0.23 1.20 0.13 40 ^B 190.25 , 2.00 2.S3.S4 22 223.70 , „ •>■>, ci 4-0.85 .1.19 213.02 0.13 41 197.59 + '■•'■* 2.1(5 283.31 -°-53 0.22 1.18 212.05 -°-''7 0.13 42 198.93 ' ■'"* 2.03 2S2.70 °-55 0.21 225.45 °-^^ 1.10 211 07 °''"'* 0.14 43 *'^*i o.\, 200.20 '■■'■' 2.(12 2S2.20 °'5 0.21 ■).),; .>^ 0-3 1.15 2I0.0S °-')') 0.14 44 4^.55 rt , . 201.59 '•■'•' 2.00 281.03 °-57 0.20 227;ll °-^J 1.13 209.08 '■°° 0.14 0. 1 1 '•32 °S') o.bi 1. 01 45 42.98 „.,, 43.09 0.,, 202.91 . 1.99 2>*1.04 0.20 227 02 ^•',^72+°-''° 1.12 20s. 07 0.!5 4(1 204.23 + ''^* l.'.i7 280.41-°-^'° 0.19 1.10 207.00-'°' 15 47 205.54 '5' 1.90 279.S2 °';' 0.19 229.51 °-''> 2:!(i.29 °-7';^ 231.07 °■7^^ 0.76 l.((9 200.04 '■°' 0.15 48 200. S5 '■■^' 1 14 2T9.20 °-^^ 27S.5G °^-> O.H l.((7 205.01 ""-i 0.15 49 208.15 ' '° 1 93 0.18 1.00 204.57 '•°-' ••°5 0.10 50 43.20 , 209.44 1.92 277.91 , 0.17 231. H3 ,^ ,. 1.04 203.52 0.10 51 43.32 +°' = 210.73 +'■*:> 1 91 277.24 -'^f 7 0.17 232.58+°- 75 1.(13 2((2.17-'°P 0.17 52 "•« 0',: 212.01 '•-'„ 1.89 270.50 °'!^ ((.10 233.32 °-7» l.((l 201,41 '•°'^' 0.17 53 43.54 °- ' 213.29 ' ^;^ 1.88 2V-..87 °-^'> 0.10 23l.((5 °-73 1.(10 200.34 '■°l 0.18 54 43.05 °" 214.57 ' 1.87 275.17 °-7° O.lfi 234.77 °l' 0.99 1911.20 '"'^ 0.18 0.12 1.27 0.72 0.71 1.08 55 43.77 , 215.^4 1.H5 271.45 0. 15 235.48 , , 0.97 19S IS 0.19 5(! «•"«+!■ 1 217.10 + '-^'^ 1.84 273.72-°-" 0.15 2;i0.l8+°J° 230.H7 °p 237.51 °f7 23<.20 ° '" 0.(15 0.90 197.09-'°'^ 0.20 57 44.00 °- 2H..I5 ''^S 1.82 27' 98 °'''* 0.15 0.95 19C.((0 '•°'' 0.2(( 5S 44 12 °" 21 9. CO ' '^ l.Hl o-T)'.,.! 0.75 0.15 0.94 191.90 '■'° 0.21 59 44.21 °" 220. '<4 '"^ ■ -3 1.79 27i:4o °;:; O.lt ((.92 1:13.79 '" 1. 1.> 0.21 CO 44.35 2:;2.il7 1.7S 27((.C9 0.11 .»;;■.; S.'» ((.91 192 >;7 (i..'2 THE ORBIT OF URANUS. 235 TARLE IX, Aro. 2 . — Con billed. Arg. (u.«.3) (i).c.3) (U.8.4) (U.C.4 (p.c.O) (p.s.l) (p.C.l) (p.s.2) (p.c.2 (p.K.3) (p.c.Sj 11.40 ft 14.53 tr 1.58 // 1078 182 1513 711 306 137 85 1 11.52 14.48 1.59 1.66 1079 182 1526 711 303 137 85 9 U.fil 14.43 1.00 1.05 1080 183 1539 710 293 137 85 3 11.75 14.38 1.01 1.04 1080 184 1551 709 295 137 85 4 11.87 14.33 1.62 1.03 1681 185 1504 70 J 291 137 85 5 11.98 It. 28 1.03 1.62 1682 187 1.577 707 288 137 85 6 12.10 It. 22 1.04 1.01 1083 188 1590 700 284 137 85 7 12.21 14.10 1.04 1.00 1083 190 1003 704 281 138 84 8 12.32 14.10 1.05 1.50 1083 191 1015 703 278 138 84 9 12.43 14.04 1.66 1.58 1682 193 1028 701 274 138 84 10 12.54 13.97 1.07 1.50 1082 195 1641 700 271 138 84 11 12.05 13.91 1.68 1.55 1081 197 1054 099 267 138 84 12 12. 7G 13.84 1.09 1.54 1080 199 1006 09 ( 264 138 84 l:i 12.87 13.77 1.09 1.53 1079 201 1079 090 260 138 84 U 12.97 13.70 1.70 1.52 1078 203 1091 095 257 137 83 15 13.07 13.03 1.71 1.50 1077 200 1704 691 253 137 83 |{> 13.18 13.50 1.71 1.49 1075 20 < 1710 692 250 137 83 IT 13.28 13.49 1.72 1.48 1074 211 1729 091 240 137 83 IS 13.38 13.41 1.73 1.47 1072 21 1 1741 6-19 243 137 82 19 13.48 13.33 1.74 1.45 1070 217 1754 087 239 137 82 20 13.58 13.25 1.74 1.44 1008 220 1700 085 230 137 82 :>1 13.(18 13.17 1.75 1.43 1000 223 1778 0^'3 233 137 82 •2i 13.78 13.09 1.75 1.42 1004 227 17V0 681 230 137 81 2:j 13.S7 13.01 1.70 1.40 1002 230 n02 07.1 220 137 81 24 13.97 12.92 1.7G 1.39 1000 23 4 1S14 C77 22J 137 81 2r) 14. on 12.83 1.77 1.37 1058 2:!7 1 820 075 220 137 81 2fi 14.15 12.74 1.78 1.30 1050 241 ls:!s 073 217 137 81 27 U,2t 12.05 1.78 1.35 1053 245 Is5(t 071 214 137 80 2S 14.33 12.50 1.79 1.34 1051 219 ls(;2 00 i 211 137 80 29 14.41 12.47 1.79 1.32 1048 253 1874 GOO 208 137 80 no 11.50 12.37 1.79 1.31 1045 258 issi; 004 205 137 79 ;ti 14.5S 12.28 1.S0 1.30 1042 202 189s 002 202 137 78 3J H.c.r, 12.19 1.80 1.28 10.38 2i;7 1910 051) 190 137 78 3:1 14.74 12.09 1..S0 1.27 1035 272 1922 057 190 137 78 34 14.82 11.99 1.8.;. 1.25 1031 277 lOiU 054 193 137 78 3.") 11.90 11.80 l.sl 1 24 1027 2S2 1940 052 no 137 77 ,'-2 1970 647 184 137 77 3S 15.12 11.59 1.82 1.20 1015 297 19«2 045 isl 137 70 39 15.19 11.49 1.82 1.18 1011 3tt2 1993 042 179 137 76 40 15. 2n 11.38 1.82 1.17 lOOS 30S 201 15 Olii 170 137 70 41 15.33 11.28 1.S2 1.15 1002 313 20 1<; 037 17 ! 137 70 42 15.39 11.17 1.82 1.11 1597 319 2028 035 171 137 70 43 15.40 11. oo 1.82 1. 13 1592 325 2039 032 los 137 75 44 15.52 10.95 1.82 1.11 1587 3.31 2051 030 160 137 74 4.'> 15.58 10. St 1.S2 1.10 1 582 337 2003 027 103 1.37 74 4(i 15.(;4 10.,.; 1.82 1.0S 1577 313 2073 2 4 100 137 74 47 15.70 10.02 1.82 1.07 1572 349 20S4 021 158 137 73 4S 15.7C 10.51 1.82 1.00 1500 355 2095 0, 1 155 137 73 49 15. Ml 10.40 1.82 1.04 1501 302 2100 1 153 137 73 r>o 15.86 10.28 1.S2 1.03 1550 308 2117 1 ■ 150 137 72 61 15.91 10.17 l.x-i 1.01 1550 375 2128 r,i(i 148 137 ! 72 5ii 15.90 10.05 1.82 1.00 1545 381 213S 007 145 137 72 r)3 10.01 9.94 l.sl 0.08 1539 388 21 19 00 4 H3 137 72 f)4 10.05 9.82 1.81 0.97 1533 394 2159 001 140 137 72 55 10.09 9.70 1.81 0. 90 1527 401 2170 598 13S 137 ■ 71 5n 10.13 9.59 1.81 0.94 1521 408 2180 595 135 137 71 57 10.17 9.47 1.80 0.93 1515 415 2191 592 133 137 71 58 10.20 9.35 1.80 0.92 150!) 422 2201 589 1.11 137 71 59 10.23 9.23 1.80 0.90 1502 430 2212 5, SO 129 137 71 1 fiO 10.20 9.11 1.80 0.89 1490 437 2222 683 127 137 1 '' 236 THE ORBIT OP URANUS. TAIJLE IX, Ami. 2.— Continued. \rK. '(w.c.O) Diir. (r.».l) Diir. Sccvur. (I'.f.l) DilT. Stcvur'! u.«.2) Diff. Sec.vnr, (il «2 f,3 C4 C5 Gl> (!7 68 69 70 71 7-2 73 74 75 7t! 77 78 7 it 80 81 82 83 84 85 8<; 87 8S 89 90 91 92 93 94 95 9i> 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 IIS 119 44.35 . 44.47+°'^ 44.59 44.71 44.83 44.95 45.07 45.19 45.31 45.44 45.56 45. CS 45.80 45.92 40.04 46.16 46.29 46.41 40.53 4U.65 46.77 46.,S9' 47.01 47.13 47.24 0.12 0. 12 0.12 0. 12 -1-0.12 O. 12 O. 12 O.I3 0. 12 -t-0.12 0.12 O. 12 O. 12 O.I2 + °-M O. 12 O. 12 O. 12 O. 12 ,+O.I2 O. 12 O. 12 O. II O.I2 4 7.. 16 , 47.48+°'^ 47.5!) 47.71 47.82 o. I I O. 12 O. II O.I 2 47.94 , 48.05+°-" 48.16 48.27 48.38 o. I I o. I I O.I I O.ll 48.49 , 48. ,59+°'° 4,S.70 °" 48.50 °'° 48.91 °" o.io 49.01 , 49.11+°'° 49.21 °'° 49.31 °'° O. ID 49.50 J „ 49.59+°-°'' 49.77 °?'^ *'J«5 .In 0.09 49.94 ,„„o •'•»" c"oS f-'-'-'fi col fiO.34 , 50.41 +°-°7 f.0.48 °-°7 50.,55 °°7 50.62 222.07 , 223.30 + 224.52 225.73 2a,:.. 94 228.14 229.33 + 230.51 231.69 232.86 234.02 , 235.17"^ 236.31 237.45 238.53 2.39.70 , 240.81''" 241.91 243.00 244.09 245.17 , 246.24"^ 247.. 30 248. ;{5 249.38 250.41 251.43 252.44 253.44 254.43 + .14 •14 •'3 .11 . 10 .09 .09 .08 .07 .06 ■°5 •03 ■°3 .02 .01 .00 0.99 0.98 S:^i+°-''r 257.34 °-'^^ 2.^8.29 °-'^5 259.23 °-'^-' 260.16 261.08 261.99 262.88 263.76 4.0.92 0.91 0.N9 0..S8 0.87 ^65l9+°-«'^ S'l"-"'* o.«4 268.00 0.81 268. SI 269.61 270.41 271.19 271.96 120 50.69 0.07 0.07 +0.S0 o.Xo 0.78 0-77 °-7S 272.71 , 273.45+°-7-» 274.18 274.90 275,60 , 0.09 276,29 , .„ 276.97+°'^'^ 277.63 278.28 278.!»2 0-73 0.72 0,70 279,55 o.M) 0,^15 o.().\ 0.63 1.78 1.76 1.75 1.73 1.72 1.70 1.69 1.67 1.66 1.64 1.63 1.62 1.60 1.59 1.57 1.56 1.55 1.53 1.52 1.50 1.49 1.18 1.46 1.45 1.43 1.42 1.40 1.39 1.38 1.36 1.35 1.33 1.32 1.30 1.29 1.28 1.26 1.25 1.23 1.22 1.20 1.19 1.17 1.16 1.14 1.13 1,11 1,10 1,09 1.07 1.06 1,05 1.03 1.02 1.00 0.99 0,98 0,97 0.96 0,94 270.69 269.89" 269.09 268.28 267.45 266.61 265.76" 264.89 264.02 263.13 262.23 261.32" 260.39 259.46 258.52 257. .56 256.59- 255.61 254.62 253.62 252.61 251.59- 250.56 249.51 248.45 247.38 246.30" 245.21 2.44.12 243.01 241.89 210.76- 239.62 238,48 237.33 236.16 234.98" 233.79 232.60 231.39 2.30.18 228.96- 227.73 226.50 225.25 223.90 222.73" 221.46 220.18 218.88 217.58 216.28" 214.97 213.65 212.33 210.99 209.65" 208.31 206.96 205.60 -0.80 0.80 0.81 0.83 0.S4 -0.85 0.87 0.87 0.89 0.90 -o 91 0-93 0-93 0.94 0.96 -0.97 0.98 0.99 00 0,93 204,23 0.14 0.13 0.13 0.13 0.13 0.12 0.13 0.12 12 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.13 0.13 0.13 0,13 0.13 0.13 0.14 0.14 0,1 t 0.15 0.15 0.16 0.16 0.16 0,17 0,17 0,18 0,18 0,1s (,19 0,19 0.20 0.20 0,21 0,21 0,22 0.22 238 239 240 240 241 241, 242, 243, 243. 244. 244. 245. 245. 246. 246. 247. 247. 248. 248. 249. 249. 249. 250. 250. 251. 251. 251. 252. 252. 252. 252. 253. 253. 1253. 253. 254. 254, 254. 254. 254, 255, 255, 255. 25.'-), ^g+0.64 12 °-^5 '. 0.0 1 0.60 53+°-58 10 °S7 "T til .y.2 °-55 0.54 79 °-5' 0.50 0.49 0.48 2fi , 7.) +0.47 18 "--tS (;., 9-44 05 °-»-5 0.42 JJ7+0.40 26 64 29 78 0.38 01 °-37 37 7[+o-34 04 °-'53 36 °-^' 67 °--^' 0.29 50 '' 0.23 0.23 98 0.26 0.2; 21 ^^«+°-; 63 82 99 o, 19 0.17 o. 16 255, 255, 255. 255. 256, 256. 256, 256. 256. 255, 255, 255, 255, 255, 0.13 O. 12 O. I! C.09 ^.,-l-o.oS 90 °°7 95 °-°S <)<) 004 0.03 02 , 04+°°' 04 °°° 03-°-°' 00 °°| 0.04 96 '^4 76 67 0,23 1255,56 I 0,07 o.oS 0.09 0.1 1 0.91 0.90 0.98 0.87 0.86 0.84 0.83 0.82 0.81 0.79 0.73 0.77 0.75 0.74 0.73 0.71 0.70 0.69 0.68 0.66 0.65 0.64 0.63 0.62 0.61 0.60 0.58 0.57 0.56 0,55 0,54 0,53 0,52 0.51 0.50 0,49 0,48 0,47 0,46 0.45 0,44 0,43 0.42 0.41 0,40 0,39 0,39 0,38 0.37 0.36 0,35 0,34 0,33 0.33 0.32 0.31 0.311 0,29 0,28 ' r.e.2) DilT. Scc.var, 192.67 191.55' 190.42 189.29 188.14 186.99 185.83' 184.67 183.51 182.34 181.16 r....97" 178.78 177.59 176.40 175.20 173.99- 172.78 171.. 56 170.34 169.11 167.88" 166.65 165.41 164.17 162.92 ici.rj- 160.42 159.16 157.90 156.63_ 155..'!6~ 154.09 152.82 151.55 150,27 I4s,!('.r 147,71 146,42 145,13 143.84 142.55" 141.26 139.97 13><.n7 1 37.. 37. 136,07" 134.77 133.47 132.17 130.87 129,57" 128,2(; 1 21;. 96 125,66 121 36 123,05" 121,75 120.45 119,15 12 '3 13 »S '5 16 16 16 '7 18 »9 «9 •9 >9 20 27 I17,s5 -1. 21 1. 21 1.22 1.22 >-23 -1.23 1.24 1.24 '•-'5 1.2O 1.26 1.27 -1.27 1.27 1.27 1.27 1.28 -1.28 1.2.S 1.29 1.29 1.29 -1.29 1.29 1.29 1.30 1.30 -1,30 1.30 1.30 1.50 1.30 -1.30 '•3> 1,30 1.30 1.30 -•■3' 1.30 1.30 1.30 '•30 0.22 0.23 0.23 0.24 0.25 0.25 0.26 0.27 0.28 0.28 0.29 0.30 0.31 0.32 0.33 0.34 0.34 0.35 0.36 0.37 0..38 0.39 0.40 0.41 0,42 0,43 0,43 0,44 0.45 0.46 0.47 0.48 0.49 0..50 0.51 0.52 0.54 0.55 0.56 0.57 0.58 59 0.60 0.61 0. 1!2 0.63 0.'!5 0.66 0.67 0.68 0,69 0,70 0,71 0.73 0.74 0.75 0.76 0.77 0.78 0.80 81 TUB ORBIT OP URANUS. 287 T.\I5LE IX, Abo. 2 . — Continued. 1 Avg. (r.,s.3) (..C.3) (V.sA) ((!.C.4> (p.e.O) (p.«.l) (p.C.l) (p.«.2) (p.c.2) (p.8.3) (h.c.3)| r,o If 10.20 It 9.11 n 1.80 It 0.89 1490 437 2222 583 127 137 71 fii lo.i;9 9.00 1.79 0.88 1489 445 2233 580 125 137 70 (12 10.32 8.88 1.79 0.87 1483 453 2243 577 123 137 70 (i:i 10.35 8.70 1.78 0.85 1477 400 2254 574 121 137 70 G4 10.38 8.04 1.78 0.84 1470 40S 2204 570 119 137 69 (15 10.41 8.52 1.7S 0.83 1404 470 2275 507 117 130 69 or, 1G.43 8.40 1.77 0.81 1457 484 2285 504 115 130 68 f.7 10.45 8.28 1.77 0.80 1451 492 2295 500 113 130 68 (IS 10.47 8.10 1.70 0.79 1444 501 2300 557 112 130 67 (19 10.49 8.04 1.75 0.78 1438 509 2310 553 110 130 67 70 10.50 7.91 1.75 0.70 1431 518 2320 550 108 130 66 71 10.51 7.79 1.74 0.75 1424 527 2330 640 107 130 60 72 10.52 7.07 1.74 0.74 1417 530 2340 543 105 130 60 73 10.53 7.55 1.7! 0.73 1410 545 2350 640 103 135 65 74 10.54 7.43 1,72 0.71 1403 554 2305 530 102 135 65 75 10.54 7.31 1.72 0.70 1390 603 2375 533 100 135 04 7(1 10.54 7.19 1.71 0.09 1389 572 23S4 630 99 135 64 77 10.54 7.07 1.70 0.08 1382 581 2394 520 VJS 135 64 78 10.54 0.95 1.70 0.07 1374 590 2403 523 90 135 63 79 10.53 0.83 1.09 0.05 1307 599 2413 519 95 135 63 80 10.53 0.71 1.08 0.04 1.300 008 24 2f! 510 94 135 02 81 10.52 0.00 1.07 0.03 13.53 018 2431 512 93 135 02 «2 10.52 0.48 1.00 0.02 1340 027 2440 608 92 134 01 83 10.51 0.30 1.05 0.01 1339 037 2449 505 91 134 01 84 10.50 0.24 1.04 0.00 1332 640 2457 501 89 134 00 85 10. 4S 0.13 1.03 0.59 1325 050 2400 498 88 134 CO 80 10.40 0.01 1.02 0,5S 1318 004 2475 494 87 134 59 87 10.44 5.89 1,01 0.57 1311 070 24 S3 490 80 134 59 88 10.42 5.78 1,00 0.50 1304 084 2492 4X7 85 134 58 89 10.40 5.00 1.59 0.55 1290 090 2500 483 84 133 58 00 10.38 5..55 1.5S 0.54 1289 700 2509 480 84 133 57 ill 10.35 5.44 1.57 0.53 12SI 717 2517 470 83 133 57 92 10.33 5.32 1.50 0,52 1273 727 2525 472 82 133 50 93 10.30 5.21 1,55 0,51 1 200 73S 25.34 409 82 132 50 94 10.27 5.09 1.51 0.50 1258 749 2542 405 81 132 55 !I5 10.21 4. 98 1.5:t 0,50 12.50 700 2550 401 81 132 55 ilCi 10.20 4.87 1.52 0,49 1242 771 255 S 457 80 131 64 !I7 10.17 4.70 1,51 0,4S 1234 782 2500 454 80 131 54 9S 10.13 4.04 1,50 0.48 1220 793 2574 45(1 80 131 63 99 10.09 4.54 1.49 0.47 121S 804 25M2 440 79 130 53 100 10.05 4.44 1.48 0.40 1210 815 2590 442 79 130 52 KiL 10.01 4.33 1,40 0.40 1202 S20 259X 43S 79 130 51 102 15.90 4.23 1.45 0.45 1194 83S 2005 435 79 129 51 103 15.92 4.13 1,41 0.44 1180 H49 2013 431 78 129 5(( 104 15.87 4.02 1.43 0.44 1178 801 2020 42S 78 129 60 105 15.83 3,92 1.42 0.43 1170 872 2027 424 78 128 49 100 15.78 3.82 1.41 0,43 1102 8><3 2034 420 7s 128 49 107 15.73 3,72 l..i9 0.42 1154 895 2041 410 78 128 48 108 15.08 3.02 1.3S 0,42 '140 907 20 47 413 78 127 4S 109 15.02 3.52 1.37 0,41 1138 919 2054 409 78 127 47 110 15,50 3.43 1.30 0,41 1130 931 2001 405 78 120 40 111 15.51 3,34 1,.!4 0, 10 1122 943 2(;os 402 78 120 40 112 15.45 3,24 1,33 0.10 lilt 955 2074 39S 79 125 45 113 15.39 3.15 1,32 0. !(l 11(10 907 20><1 394 125 45 114 15.33 3,(((> 1,31 0.39 109S 979 20S7 391 79 124 44 115 15.27 2.97 1.29 0,39 1090 991 2093 3S7 80 124 44 no 15.21 2.88 l.-.'S 0,3S 10H2 1003 2099 383 SO 123 44 117 15.) 4 2. HO 1,20 0,:!8 1074 1010 2705 380 80 1 22 43 lis 15.08 2.71 1,25 0.3S 1000 1028 2711 370 SI 122 43 119 15.01 2.02 1.24 0.3S 1059 1(141 2710 373 81 121 42 120 14.94 2.54 1.23 0.37 1051 1053 2722 370 82 120 42 238 THE ORBIT OP URANUS. i I "II TAIJLK IX, . Vro. 2. — Continued. Ai'ff. (v.c.O) Diff. ,11.8.1) Diff. Soc.var. KV.C.l) Diff. Sco.var. (V.g.2) Dill'. Scc'.var. '' 'I II (i'.c*.2) Dilf. Sccvar. 11 II tf ft tt tf It /' // tt II 120 60.09 279.55 , , 280.10+°-^' 0.93 204.23 0.23 255,50 C.27 117,85 0.81 121 50.70 +°-°J 50.82 °°^ 50.88 °°^ 0,92 202. 8(!' -> 37 1.38 1.38 0.24 255.44-°" 0.20 110.55-'-3° 0.82 122 0.90 201.48 0.24 2,55.31 °'^ 0.20 115,25 •-■'° 0.84 12:j 281.35 °59 0.89 200.10 0.25 255.10 °- S 254.00 °'^' 0.25 113,95 '••5° 0.85 124 50.93 °°5 281.92 °-5J 0.88 198.71 « ■ M) 0.25 0,25 112,05 '--5° 0,87 0.05 0.56 1..10 0.17 1.29 125 50.98^^ - 282.48 , 0.87 197.31 0.20 254.83 0,24 111.30 0.88 121! 51.03 +°°3 28,3. 02+°- 5* 0.85 195.91' -1.40 0.27 254.04-°- ^ 0,23 110.07-! -^5 0.89 127 51.08 °-°-^ 283 55 °-5.? 0.84 194.50 1. 41 0.27 254,44 °- ° 0,23 108.77 •^° 0,91 12H 51.13 °°-^ 2,84.07 °-5; 0.83 193.09 1.41 0.28 2.54,23 "• 0,22 107,48 ]:l 0.92 129 61.17 °°-* 0.05 284.57 °:^; 0.81 191.07 1.42 1.4J 0.28 254,00 l:ll 0.22 100,19 1:;^ 0.94 130 51.22 51.20 +°°-* 285.00 0.80 190.25 0.29 25.3.70 0.21 104.91 0.95 i;ii 285. 53+°- -'J 0.79 IS8.82' -'•43 0.30 253,51-°- -5 0.21 103.03-'-=^ 0.96 1.12 51.30 °°-* 285.99 °:'' 0.7 < 187.39 «.4,^ 0.31 253.24 °-7 0,20 102.35 '-^^ 0,98 133 51.34 °°-* 280.44 °-'5 0.70 1S5.95 1.44 0.32 252,90 °-'^ 0.20 101.07 '-^^ 0,99 134 51.37 °°' 0.03 280.87 °-'' 0.42 0.75 184.51 1.44 1.44 o.;i3 252,07 °'^ 0,31 0.19 99.79 '-^^ 1.27 1.01 135 51.40 287.29 0.74 183.07 0.33 252,30 0.19 98.52 1.02 130 51. 43+°°^ 287.09 +°-*° 0.73 181.02- -1-43 0.34 252,04 -°''^ 0.18 97.25 -'-7 1.03 137 51.45 °°' 2 •(8! 08 °-,59 0.72 180.17 1-45 0.35 251.71 °-" 0. 1 8 95.99 '-^ 1.05 138 61.47 °°-' 288.45 °--'7 0.70 178.72 1.45 0.30 251.30 °-35 0.17 94.73 '•-^' l.OG 139 51.49 °°- 0.02 288.81 °--5'' °j5 0.09 177.26 1.46 1.46 37 251.00 °i^ 0-37 0.17 93.47 '--^ 1.26 1.08 140 61.51 289.10 0.08 175.80 0.38 250.03 O.lfi 92.21 1.09 141 61.52+°°' 289.49+°.'-' 0.07 174.33- -1.47 0.39 250.24-°- 39 0.10 90.90— '--5 1.10 142 51.53 '^•°' 289.80 °->' 0.00 172.80 1.47 0.10 249.84 °-»° 0.15 89.71 '-^5 1.12 143 51.54 °°' 290.10 °-5° 0.05 171.39 1.47 0.41 249.43 °-»' 0.15 88.40 '--5 1,13 144 61.65 °°' 0.00 290.39 °-'> 0.27 0.03 109.92 1.47 1.48 0.42 249.01 °-^' 0.44 0.14 87.22 '-'•* 1.24 1.15 145 51.55 200.00 290.92 +°--'' 0.02 108.44 0.42 2-««-'''' „ ,r 0.14 85.98 1.16 140 61.55 °°° 0.01 100.90- -!..(,'! 0.43 248.12-°-'»5 J,. ,,,, 0.46 247.18 °-f 0.14 84.75-'-^3 1.17 147 51.55 °°° 291.10 °-^ 0.00 105.48 1. 48 0.44 0.13 83.52 '--'3 1.19 14< 51.54-°°' 2'.ll,38 °-" 0.58 104.00 1.48 0.45 0.13 K.. ...) "-23 1.20 149 51.53 °°" 0.01 291.59 °-' 0.20 0.57 ';;2.52 1.48 1.49 0.40 240.09 °-^'' 0.50 0.12 81,07 '■'' 1,22 1.22 150 61.52 201.79 0.50 101.03 0.47 -+'l'" 0.. 0.12 79.85 1.23 151 61.50-°°^ 291.97+°'° 0.55 159.54- -I.. 19 0.!8 245.08-°-5' 0.12 7.S.(i4-'--' 1.24 152 51.49 °°' 292.13 °"' 0.51 158.05 1.49 0.49 21.5.15 °-5-3 0. 1 2 77,44 ••^° 1.20 153 51.47 °°- 292.28 °'-i 0.53 150.50 1,49 0.50 244.01 °-5j 0.11 70.24 \l° 1.27 154 51.45 °°- 292.42 °'-* 0.52 155.07 1.49 0.51 244.00 °-5-! 0,50 0.11 75.04 '•^° 1.29 0.02 0. 12 I 49 1. 19 155 51.43 292.54 0.51 153.58 0.52 243,50 O.U 73.85 72.(;7-- '^ 71.49 • '; 1.30 150 51.40-°°' 292.04+°'° 0.50 152.09" -1.49 0.54 242,92 — °-S" 0.11 1.31 157 51.37 °°' 292.73 °°'^ 0.49 150.00 1.49 0..-.5 242 .■(3 ° ■-■''> 0.11 1.33 158 51.34 °°'> 292, KG °°7 O.H 149.11 1.49 0.50 241,73 °f° 0.10 70.;!2 - 1.34 159 61.30 °■°^ 292.80 °°'' 0.47 147.02 1.49 0.57 241,12 °'" 0.10 09.15 '-'J 1.16 1.30 0.04 0.04 '5° 0.63 100 51.20 292.90 0.40 140.12 0.58 240.49 0.10 07.99 l.:i7 101 61.22-°°» 292.9.3+°°,' 0.45 144.03- -1.49 0.59 239.,S)-°-^'4 0.10 0(1. S4 — '-'5 1.39 102 51.17 °°' 292.91 °°' 0.4t 143.13 1.50 0.00 239.20 °''S 0.10 05.09 '-'5 1.40 103 61.12 °°' 292.94 °°° 0. (3 141.03 1.50 0.01 238.54 °-^''^ 0.10 04.55 '-'4 1.42 104 61.07 °°' 0.06 292.92"°°" 0.42 140.14 1.49 0.02 237.87 °-^'7 0,10 0-.t.41 '•'■* 1.43 0.04 '5° 0.69 '•'3 105 61.01 5o.y5-°-°^' 50.89 °°'^ 292.88 0.41 138.04 0.04 237,18 0.09 02.28 1.45 loo 292 m;)— °°.^ 0.40 137.15- -1.49 0.05 230. 4 8-°- 7° 0.(19 01. 10-''^ 1.47 107 292^70 °■°^ 0.40 135.00 1.49 0.00 2,!5.77 °-'' 0. 0.09 57.84 '•'° 1.51 0.07 0. 1 1 1.48 0,74 1.09 170 50.09 292,47 0,37 131.20 0.09 233,58 232..S2— °7'' 0.09 50,75 1.53 171 60.02-"°7 292., 'U ~°-'-' 0.30 129.71- -1.49 0.70 0.09 ; .5.5.(;7-'°« 1.54 172 .'.0.54 °'°^ 2i|.) 10 °- '5 0.35 IJS.23 1.48 0.71 232.05 "■~l (1.(19 54.59 '•"« 1.50 173 50.40 °°'' 2!»2!o3 °'^' 0.:!5 1 ■-'•;. 75 1..18 0.73 ! i>;j| .>1 O.7.S 0.10 .53. -.2 '"^ 1.57 174 50.37 °°'J 0.09 291,80 °'7 0,19 0.,i4 125.27 I. ;« 1.4S ■ '■• i 23(».48 °i" 0,80 0,10 ; 52.45 '•°7 1,05 1.59 175 50.28 291.07 0..33 123.79 0.7.^ 229,08 228.s7-°-^' 228.05 °Z' 227,22 °-^' 220.37 °-'^P 0.8O 0.10 ' 51.40 1.00 170 .'.0.19-°°'' 291.40-°-' 0.32 122.32- -1.47 0.7<> 0. 10 50.,!5-'°-'5 I.Ol 177 •-"••O 000 291. 2» ° ■'■ 0.,!1 120.85 1.47 0.77 o.lo 49.31 '"^ 1.03 17S •'•"•»! n?o 291,01 °^' 0.,'!l 119.38 I. 17 0.79 0.11 4.S28 '•°' 1.04 179 ^■•••«l 0°;:: 290,70 °"'^ 0,27 0.,!0 U.S. 91 ...17 ..46 0.80 0.11 1 47,20 '°-' 1. 01 1.00 ISO 4'.I.,S1 290 49 0,29 110.45 O.Sl 225.51 0.11 ' 40.25 1.07 ,T^£tf ^aiwN*'**— ~ -^ -. . -T - ' THE ORBIT OP URANUS. 239 TABLE IX Alio. 2. — Continued. Arg. {V.S.3) (u.f.3) ((>.«. 4) {u.cA) (p.c.O) y^f.S.\){f.C.\) (p.«.2) (f.c.'l)\ (|,.».3) (p.c.3j 120 It 14.04 n 2.54 1.23 0.37 1 1051 1053 2722 370 82 120 42 121 14. «7 2.4(! 1.22 0.37 1043 1 0(;5 2727 300 82 120 42 122 14.H0 2.38 1.20 0.37 1035 1078 2732 302 83 120 41 12;$ 14.73 2.30 1.19 0.37 1028 1090 2737 359 83 119 41 124 14. (iO 2.22 1.18 0.37 102U 1103 2742 855 84 119 40 125 14.59 2.15 1.17 0.37 1012 1110 2747 352 85 119 40 12(5 14.52 2.07 1.15 0.37 1004 1129 2751 348 85 118 39 127 14.44 2.00 1.14 0.37 99(i 1141 2750 344 80 118 39 12S 14.37 1.93 1.13 0.37 989 1154 2700 341 87 117 38 12 'J 14.29 l.SC 1.11 0.37 9;1 1107 2705 337 88 117 • 38 l.!0 14.21 1.79 1.10 0,37 973 IHO 2709 334 89 110 37 l.il 14.13 1.72 1.09 0,37 9(;5 1193 2773 330 90 115 37 i;i2 U.OC) l.C.O 1.07 0,37 958 1 200 2777 327 91 115 30 i;j3 13.08 1.59 l.OCi 0,37 950 1219 2781 324 92 114 30 134 13.90 1.53 1.05 0.38 942 1232 2785 321 93 113 35 l;i-) 13. H2 1.47 1.04 0.38 035 1245 27S9 318 04 113 35 l:i(i 13.74 1.41 1.02 0.38 927 1258 2793 314 95 112 35 l.J7 13.(;(> l.:!5 1.01 0.38 920 1271 2790 311 90 111 34 l.JS 13.57 1.30 1.00 0.39 012 12S4 2^i00 308 97 110 34 131) 13.49 1.24 0.99 0.39 905 1298 2803 305 99 109 33 no 13.40 1.19 0.97 0.39 897 1311 2800 302 100 108 33 Ml 13.32 1.14 O.liC, 0.40 889 1325 2S(I9 298 )(il 108 33 142 13.24 1.09 0.!15 0.40 882 1338 2.sn 295 103 107 32 14:{ 13.15 1.04 0.94 0. II 875 13.-2 2X14 292 104 100 32 144 13.07 0.99 0.93 0.41 807 1300 2810 289 100 105 32 145 12.08 0.94 0.91 0.41 850 1379 2819 280 107 105 31 1411 12.90 0.90 0.90 0.42 852 1393 2821 282 109 104 31 147 12.81 0.8f, 0.X9 0.42 844 1407 2823 279 no 103 31 14S 12.73 0.82 O..SS 0.43 837 1420 2825 270 112 102 30 149 12.04 0.78 0.87 0.44 829 1133 2S20 273 113 101 30 150 12.55 0.74 O.SC, 0.44 822 144 7 2><28 270 115 100 30 151 12.40 0.70 0.S5 0.45 815 1401 2829 207 117 99 SO 152 12.38 o.(;7 0.84 0.40 807 1474 2831 204 118 98 29 15.i 12.20 0.04 0.83 0.40 800 14SS 2S32 201 120 97 29 154 12.20 0.01 0.82 0.47 793 1502 2X34 258 122 90 29 155 12,11 0.58 0.81 0.48 780 1515 2835 250 124 95 28 15(1 12.03 0.55 0.80 0.49 778 1529 28,3(1 253 120 04 28 157 11.94 0.52 0.79 0.49 771 1543 2X37 250 128 93 28 15(S 11. 8(; 0.50 0.78 0.50 704 155(; 2S37 247 129 92 28 151) 11.77 0.47 0.77 0.51 750 1509 2838 244 131 91 27 ICO 11.08 0.45 0.7(1 0.52 749 r^s.'J 2838 242 133 90 27 ICl 11.59 0.43 0.75 0.52 742 1597 2s;!H 2.39 135 90 27 102 11.51 0.41 0.74 0.53 735 1011 2838 2.30 137 89 27 KI.J 11.42 0.40 0.73 0.54 728 1024 2S37 233 139 88 27 KU 11.33 0.38 0.72 0.55 7:>1 10.38 2837 230 141 87 27 1(!5 11.24 0.37 0.7! 0.5(5 714 1052 2X30 228 143 80 27 Kid 11.10 0.3(> 0,71 0.57 707 1005 2x:!.-, 225 145 85 27 1(!7 11.07 0.35 0.70 0.58 700 1079 2x34 222 147 84 27 i(;,s 10.09 0.34 O.CiO 0.50 093 1093 2x.!3 219 149 83 27 Kil) 10.91 0.3il 0.C8 0.00 C80 1700 2833 210 JM 82 27 170 10.82 0.32 0.08 o.(;i f.79 1720 2X31 214 153 82 27 171 10.74 0.32 0.07 o.(;2 072 17.!4 2829 211 150 81 27 172 lO.O'i 0.3) O.CO 0.03 005 1717 2X28 208 158 80 27 173 10.57 r ■ O.C.C, 0.04 059 1701 2820 20(1 ' 1(10 79 27 174 10. 4!' ..31 0,05 o.cs G52 1775 2S24 204 102 78 28 175 10.41 0.31 0.05 0.00 045 1788 2822 202 105 77 28 I7(i 10.3:» 0.3! 0.04 0.07 038 1802 2820 199 107 70 28 177 10 2-) 0.31 (..03 0.08 032 1815 ::i7 197 170 75 28 17S 10. li 0.32 0.03 0.09 025 1 829 2X15 1 95 173 74 28 170 10.00 0.32 0.03 0.70 018 18J2 2812 192 175 73 28 180 10.01 0.33 0.02 0.71 Oil 1S55 2810 190 178 72 28 --CiJ d 940 THE ORBIT OF U 11 ANUS. I i •■ i I TAHLK IX, Alio. 2. — Continued. \Tg. (y.c.O) Diir. (w.s.l) Diir. Sfc.viir O'.C.I) Diir. Sfc'.var. (0.H.2) Dill'. Sir.var (,i'.c,2) Dill'. Sce.var. // II // ft /r It It II II It II It It ft 18U 49.81 '^""•*!' ^ ,„ 0.29 Ilfi.45 -1.46 1.46 0.81 225.51 224.C4-°-^7 0.11 4C.25 1.C7 181 49.71-°'° 290.20-°-"' 0.28 114.99 0.82 0.11 45.24-'°" 1 . (18 182 49. CO °]\ 289.90 °-^° 0.28 113.53 0.84 223 77 °-**' 222.89 °-"** 0.11 **-'^ ".Z 1.70 188 49.49 °" 289.59 °-^' 0.27 112.08 >-4S 0.M5 0.12 43.25 °'>'> 1.71 184 49.37 III 289. 2G ^^ 0'35 0.20 ' 1I0.C3 >-45 «4S 0.8C 221.99 °-9° 0.91 0.12 ^^-^« o::j; 1.73 185 *^'^^ n ,, 2*^8.91 „ , 288.55-°- 3^ o.2r, 109.18 0.88 ^-'-^"* on. 0.12 i;;!!--? 1.71 18(1 49.13-°''^ 0.25 107.74 —'•44 0.M9 220.IC-°-''^ 0.12 1.75 187 49.00 °'^ 0.24 101;. 30 1.44 0.90 219.23 °-"^3 0.12 39.40 °-'^5 1.77 188 48.87 l\] 287.78 111 0.24 104.87 '■43 0.91 21s. 29 °-'^-* 0.13 3S.4C °;l\ 1.78 189 ^«-^^ l:\l =^^^-" ix. 0.23 103.44 '•43 1.42 0.93 217.34 °'^^ 0.96 0.13 ■'^■^■^ V,\ l.so I'JO ^'^••'■l „,. 28C.95 „ ., 0.22 102.02 0.94 2'«-;^'^ „„, 0.13 •'"■'•'■'1 n n. 1.81 191 48.47-°- j 28C.05 °-;^^ 21 100.00 ■ —1.42 0.95 215.41 -°-9 0.13 •'!-'-7''~!oo 1.82 192 48.33 °' ;^ 0.21 99. 1 9 1. 41 1. 41 1.40 1.40 0.97 214.44 °'97 213.4C °-9« 0.14 •^^-^O o! 9 33.91 'Z 1.S4 19a ^^•>» o 1 285.58 °-]l 0.20 97.78 0.9S 0.14 1.S5 194 *«"^ lit 0.20 9C.38 1.00 212.4C ;:°° 0.14 l.,S7 195 47.00 2S4.no 0.19 94.98 1.01 211.45 0.15 ■ji;;;--f l.ss 19a 47. 75-°- '5 284.09—°--''' 0.19 93..59- -1-3') 1.02 210.43-'-°* 0.15 1.90 197 47. CO °'5 47.44 °'^ 283..5fi °-53 0.18 92.20 '•.?9 ..38 1.04 209.40 '-"3 0.15 S;S =t 1.91 198 283.01 °-S5 282.45 °-5'^ 0.18 90. .S2 1.05 20.S.37 '-°3 15 1 '13 199 47.28 °'f 0.17 89.44 1.38 1.07 207.33 •-°-* O.IC 28.77 °''^ 1.94 o. i6 0-57 '•37 '•OS 0.S2 200 47.12 281.88 0.17 88.07 -1.36 1.0s 2(»C.28 O.IC 27.95 „ 1.9c 201 4(!.95-°'7 2«tl.29-°-S9 2S().C8 °-^' 280.0C °-^' 279.43 °p 0.17 8C.7r 1.09 205.22-'-°'^ 0.17 27.14-°-^' 1.97 202 4C.7S °-'7 0.1c 85. ;m ' .55 1.11 204.15 ^■°l 0.17 2C.34 °'^° 1.99 20:i 4i;.Cl °'7 0. 1 C 84.01 ' 3,S 1.12 203.07 '-°'^ 0.1s 2-V55 °'7'> 2.00 204 4C.44 °'7 0.1c 82. CC '•.?5 1.14 201.98 '■°'> 0.18 24.78 °-77 2.U1 O.I 8 0.65 « 34 1.09 0.77 205 4C. 2fi 27'*. 78 .. '>7s 12-°-'''' ^77:44 °f 0.15 81.32 1.15 200. S9 0.19 24.01 2.03 20(i 4C.08-°;« 0.15 79.99" -'•.33 I.IC 199.79-'-'° O.-JO 2.'t.2C-°'" 2.04 207 45.90 °'^^ 0.15 7H.C7 '3^ LIS 19S.C8 '-" 0.20 22.52 ''■* 2.05 201 4.^.71 °"^ 27r,.75 °-'^9 0.14 77.3C '3' 1.19 197. 5C '-'^ 0.21 21.79 °'!-' 2. OC 209 45.52 °"^ 0.19 27C04 °-7' 0.72 0.14 70.05 '3' 1.30 1.21 19C.43 ■-'•' '•'3 0.21 ^'-0^ 0:;: 2.08 210 45..'53 ^ , 275.32 O.M 74.75 1.22 195.30 0.22 20 3C „ , 2.09 211 4.V14-°"' 274..58-°-^^ 0.13 73.40" -1.29 1.28 1.23 194. IC-'-'-* 0.23 19.CC-°'/° 2.10 212 4i.94 °" 273.83 °-7j 0.13 72,18 1.25 193.01 '-'5 191.85 '-'^ 190. CO '-'" 0.23 IS 97 °' "^ 17. C4 . , 2.12 2I:( 4474 III 273.0C °!I 0.13 70.91 1.27 1.2c 0.24 2.13 2U 44.54 °" 272.28 "•'" 0.13 r.'i i;4 1.27 1.28 0.25 2.14 0.20 0.79 1.27 1. 17 ' 0.64 21.-) 44,34 271 49 „ 0.12 n^ ,; 1.29 189.52 „ 0.2c 17.00 f 2.1c 21 li 44.13-°-^' 270.CS-°-^' 4.3.92 °'! 2'19.SC °-^^ 4!. 71 °-" 2-.9.03 °'^'' 0.12 C7.12 -1.25 1.31 18S.34 -'-'^ 0.27 15 13 , 2.17 217 0.12 fi5.S8 1.24 1..-S2 1S7.15 \\l 0.28 2.1s 21 S 0.12 C4.C5 '•23 1.34 1S5.9C '-'^ 0.28 2.19 219 43.50 III 2C8.18 °;«| 0.12 C3.42 '-23 1.22 1.35 184. 7C '•'° 1.20 0.29 ^•r.3 -° 2.21 320 43.28 , 2fi7..32 0.11 ' fi2.20 1.37 I83.5C . _ 0.29 1^05 „ ,, 2 2" 821 43.0c -°" 2CC.44-°'^''' 0.11 CO. 99 -1. 21 1.38 1S2.35-- 0.30 IV'r°: 1 v> xO -' 2.23 222 42.84 °ll 2C5.55 °>^'> 0.11 59. XO 1. 19 1.40 ' isi.u -; 0.31 2 25 223 42.C2 °" 2'U.C5 °''° on 5^.(11 1.19 i.>8 1.11 : 179.92 - 0.32 12:27 °-5-l 2.2c 224 42.40 °" 2C3.73 ° ^^ 0.11 57.43 1.43 1 17.S.C9 '■'•' 0.33 11.73 °-:| 2.27 0.23 0.93 1 1. 17 '•23 0.52 325 42.17 2C2.S0 0.11 ; 5C.2C -1.16 1.14 177.4C 0.34 11.21 2.29 32(! 41.94-°-*' 2'!1.85-°''.f 0.11 .55.10- 1.45 17 c. 22 •"'•*-♦ 0.34 10.70-°-^' 2..-i0 327 41.71 °-^' 2C0.M9 °'>^ Oil 53. 9C 1.14 K47 174.97 '--5 0.35 10.20 °-?° 2.31 328 41.48 °-^.? 259 92 °^' Oil 52.82 1. 14 1.48 173.72 '-*5 172.4C '-"'5 3C 9.72 °-»'^ 2.32 329 41.24 °'-* 2.58:91 "-'^'^ 0.11 51.C9 '•'3 1.50 0.37 9.25 °-»J 2.34 1 0.24 1. 00 1.12 '•27 0.46 230 41.00 257.94 0.11 .50.57 1.51 171.19 0.3s 8.79 2.35 2:tl 40.7C-°-^-+ 25'i.03-'°' 0.11 49.4C- -I.I I 1.52 1r,9.92-'--7 0.39 1 S.35-°-*-» 2.3I-, •2:ii 40.52 °='-* 25.5.92 ' °' 0.11 4S.37 1.09 1.54 1CS.C5 '--• 0.40 1 ^2 °--»3 2.37 2;i:l 40.28 °-t 251.89 ■■°' 0.11 47.28 1.09 1.08 I.5.'i IC7.37 '•*'„ 0.41 7:50 °"»^ 2.3S 2.U 40.03 °^-' 2.53.85 '"^ 0.11 ! 4C.20 1.57 ICC. 09 '--'" 0.42 7.00 °-»' 2. 39 0.25 1.06 1 1.07 '29 0-39 2:55 , 39 78 252.79 0.11 ' 45.13 1.58 Ifil.SO 0.12 fi.Tf 2.41 23fi ; 39.53-°- '5 251.72^ '"^ 0.11 44.08" -1.05 l.co ir,3 51 -'•''> 0.43 c..-t2-°--^;' 2.42 2:i7 1 39.28 °-^^ 250. C4 '"'^ 0.12 43.05 1.03 L'!l 1C2.22 '^^ 0.44 5.9C °--^'^ 2.4.! 2H 1 39.03 °l^ 249.55 '-""^ 0.12 42.02 1.03 1.03 ICO. 92 '■•''^ 0.45 5. CI °--« 2.14 2.9 1 3S.77 °-^ 248.45 j- 0.12 ; 41.00 1 .02 1^'54 ir.9.C2 '-■'° 0.4C 5.27 °--^-* 2.45 1.02 '3' I °32 240 38.51 247.33 0.12 1 39.98 l^Ofi 158.31 0.47 4.95 2.4C THE ORBIT OF URANUS. 241 TABLE IX Aua. i.— Continued. Art,'. (i>.«.3) (C.C.3) (0.8.4) (w.f.4; (p.c.OJ (p.8.1, (p.f.l, (p.8.2 > (p.c.2 (p.«..3; (p.c.3) 180 10.01 tt 0.33 // 0.02 tt 0.71 Oil 1855 2810 190 178 72 28 181 9.93 0.34 0.03 0.72 004 1869 2807 188 180 71 28 183 9.85 0.35 0.61 0.73 598 1882 2804 186 183 70 28 183 9.78 0.36 0.01 0.75 591 1895 2801 184 185 69 28 184 9.70 0.37 0.61 0.76 584 1908 2798 182 188 68 28 185 9.03 0.38 0.60 0.77 577 1921 2794 180 190 07 29 186 9.55 0.39 0.00 0.78 571 1935 2791 178 192 60 29 187 9.47 0.41 0.00 0.79 564 1948 2787 176 195 05 29 188 9.40 0.43 0.60 0.80 557 1901 2783 174 197 04 29 189 9.33 0,44 0.59 0.81 551 1974 2779 172 200 03 30 190 9.26 0.45 0.59 0.83 544 1987 2775 171 202 62 30 191 9.19 0.47 0.59 0.84 538 2000 2771 109 205 01 30 193 9.12 0.50 0.59 0.85 531 2013 2700 107 207 (,0 30 193 9.05 0.,'-.2 0.59 0.86 525 2025 2702 105 210 59 31 194 8.D8 0.64 0.59 0.87 519 2038 2757 103 212 58 31 195 8.91 0.56 0.59 0.88 512 2051 2752 161 215 57 32 196 8.84 0.59 0.59 0.90 500 2004 2747 159 218 50 32 19T 8.^8 0.01 0.59 0.91 600 2076 2742 157 220 65 32 198 8.71 0.04 0.59 0.92 494 20>^9 2737 1 50 223 54 33 199 8.65 0.06 0.59 0.93 488 2101 2731 154 225 63 33 200 8.59 0.69 0.59 0.94 482 2113 2726 153 228 52 34 201 8.53 0.72 0.59 0,95 470 2120 2720 151 231 51 34 202 8.47 0.75 0.59 0.97 470 2138 2714 150 233 50 35 20:j 8.41 0.78 0.59 0.98 404 2160 2708 148 23(! 49 35 204 8.36 0.81 0.00 0.99 459 2102 2702 147 239 4S 30 205 8.30 0..S4 0.60 1.00 453 2174 2695 140 242 47 30 206 S.25 0.«T 0.00 l.OI 447 2186 2089 H4 244 47 37 207 8.19 0. 90 0.01 1.02 441 2198 2082 142 247 46 37 20S 8.14 0.93 0.01 1.03 430 2210 2675 III 250 45 38 209 8.08 0.90 0.01 1.04 430 2222 20(;h 140 253 44 39 210 8.03 1.00 0.03 1.0.-) 425 2234 2061 138 250 43 39 211 7.98 1.03 0.02 1. 00 419 2215 2051 137 259 43 40 212 7.93 1.06 0.62 1.07 413 2257 2040 135 202 42 41 2l;i 7.88 1.10 0.03 1.08 408 2268 2039 134 205 41 42 214 7.84 1.13 0.03 1.09 402 2280 2031 133 208 40 43 215 7.79 1.17 o.ot 1.10 397 2291 2024 132 271 39 43 2ir. 7.74 1.21 0.04 1.11 392 2302 2616 130 274 39 44 217 7.70 1.24 0.05 l.U> 386 2313 2008 129 277 38 44 218 7.05 1.28 0.00 1.13 380 2324 2000 128 2S0 37 45 219 7.01 1.32 0.66 1.14 375 2335 2592 127 283 30 46 220 7.57 1.35 0.07 1.15 370 2340 2584 120 286 35 40 221 7.53 1.39 0.08 1.10 304 2357 2575 125 2S9 35 47 222 7.49 1.43 0.09 1.17 359 2307 2567 125 292 34 47 22{ 7.45 1.46 0.09 1.18 354 2378 2558 124 295 33 48 224 7.41 1.50 0.70 1.19 S49 2389 2550 123 298 32 49 225 7.38 1.54 ! 0.71 1.20 344 2399 2541 122 301 32 50 22(1 7.34 1.58 0.71 1.20 339 2409 1 2532 121 304 31 50 227 7.31 1.02 0.72 1.21 334 2420 ; 2523 121 307 30 51 228 7.28 1.06 0.73 1.22 329 2430 2513 120 310 30 52 229 7.25 1.70 0.74 1.23 324 1 2440 2504 119 313 29 53 230 7.22 1.74 0.75 1.24 819 2450 2494 lis 316 28 1 54 231 7.19 1.78 0.75 1.24 314 21i>0 ■ 2484 lis 318 28 I 54 2:12 7.17 1.81 0.76 1.25 sn 2470 1 2474 117 ?21 27 55 2:!:j ; 7.14 1.85 0.77 1.25 305 2480 i 2464 110 321 27 50 234 7.11 1.89 0.78 1.26 300 : 2489 ! 2454 110 327 26 ] 57 235 7.09 1.93 0.79 1.27 296 2499 j 2444 115 330 25 '' 58 23fi 7.07 1.97 80 1.27 291 2508 ; 2433 114 333 25 59 237 7.05 2.00 0.81 1.28 2«7 2518 1 2423 114 330 24 60 238 7.03 2.04 0.82 1.28 283 2527 ■ 2412 113 339 23 01 239 7.01 2.08 0.83 1.29 278 2536 2401 113 342 22 62 240 fi.99 2.12 0.84 1.30 274 2545 ■ 2390 113 345 21 53 31 Jul7. 1873 ^^^.m^t^ r 2ii THE OUBIT OF U RAN US. TABLE IX, Aho. 2 . — Continued. Arg. (u.c.O) Uiir. j(u.s.l) Diir. S or.vor. (u.c.l) Diir. Soc.vor. (.U.S. 2) 1) IT. 8 cc.vur. (V.C.I) Diir. S L'e.vor. tl H It / ti ft H It It It It /' /; // 240 38.51 f 247.33 38.25-°-^^ 24f..20-' 37.99 °- 7 245.00 ' 0.12 39.98 1.66 158.31 0.47 4.95 2.46 241 ■13 0.12 38.98-'-°° 3S.0O °'^l 1.67 157.00-' •3' 0.48 *■'•■* "o'lo 2.47 242 '4 0.12' 1.09 155.69 ' 3' 0.49 4.34 °-° 2.48 243 37.73 °- ^ 243.91 'S .16 18 0.13 ■J7 ni °97 1.70 154.37 ' •3* 0.50 4.00 °- '^ 2.49 244 3^-*« III 242.75 1 0.13 1.71 153.05 ■ ■i- •3» 0.51 3.79 °- 0.25 2.50 245 37.20 „ „ 241.57 .18 0.13 •''•''•12 „„, 1.73 151.73 , 0.53 3 •'•* „,, 2.51 24G 3r..93-°-^7 240.39- 0.13 34.l9-°-W 1.74 150.40" 33 0.54 3.;!0-°- 1 2.52 247 SC.Ofl °- 239.20 .19 0.13 33.27 °:i\ 1.75 149.07 •33 0.55 30T °- f 2.53 243 30.39 °- 237.99 ' . 21 0.14 "2.36 °'5' 1.76 147.74 ■ii 0.50 2-^« o' 2.54 249 3«-ia l\] 236.77 j . 23 .23 0.14 3l-*« °oH 1.78 140.41 J ■a •34 0.57 2"« l\l 2.55 250 35.85 . ,, 235.54 , 0.14 30.57 „ ,, 29.70-°-^?, ^;-?? °s ^'•'^ 0.S2 1.79 145.07 , 0.58 2^8 „,, 2.56 251 3.-..58-°-'J 35.31 l\l ^^ oM 3^-7^ 0.28 234.30" 24 ■ n 26 .28 0.14 1.80 143.73 •34 .34 •34 •34 •35 0.59 2'l~o ? 2.57 252 233.05 1 0.15 1.S2 142.39 0.00 2"'' i 2.58 253 231.80 0.15 1.83 141.05 0.01 2 02 °- 1 2.59 254 2.50.54 1 0.16 1.85 139.71 , 0.02 '■«» TA 2.60 255 .34.19-°*? 229.20 .28 0.10 2.5.52-°-^' 1.80 138.. 30 0.64 1.78 2.61 256 227.98-' 0.10 1.88 137.01-' •35 0.05 1.08-°-'° 2.61 257 33 91 °-^l 3.3.03 °-^^ 226.08 ' 30 0.17 24.73 °-" 2.3.95 °ll 1.89 135.00 ' •35 0.00 1.00 °°^ 2.62 258 225.37 ' 3' 0.17 1.91 134.31 ' •35 0.07 l.,53 °-°/ 1.47 °-°^ 2.03 259 33.34 l^l 224.06 ' •3« 0.18 23.18 °-" 1.92 132.90 \ •35 0.68 2.04 0.28 •3» 0-75 •35 0.04 200 33.00 32.78-°'^ 222.74 0.18 22.'»3_„,^ 1.94 131.01 0.09 '■^3 „„, 2.05 2ni 221.41-' ■M 0.18 21.69-°-74 1.95 130.20"' •35 •36 0.70 l-^O^oof 2. 00 2C2 32.49 °\l 220.07 ' .?4 0.19 20.97 °-^ 1.97 128.90 ' 0.71 1.397°-°' 2.07 2fi3 32.21 °'^ 218.72 ' •.?S 0.19 20.26 °- ' 1.98 127.. 55 ' •35 0.73 '-^0+°° 2.07 2U4 31.92 °-'9 0.29 217.36 ' -.0 •37 0.20 »«-S« 0^8 1.99 120.20 : •35 35 0.74 1 -"a °° 0.03 2.08 2r,5 SI"'' 215.99 0.20 18.88 „ , 18.21-°-^' n.56 °-^. 16.92 °;l\ 2.01 124.8.'> 0.75 '■4!^ ,„„- 2.69 2fir, 3!..34-°-''' 214.02—' ■.?8 0.21 202 123.50—' •35 0.-0 1..5u+°-°5 , -,. 0.06 2.70 2(;7 31.05 °-"' 213.24 ' 0.21 2.03 122.14 ' 0.77 2.71 2R8 30.70 °-\l 211.85 ' •39 0.22 2.04 120.79 •35 0.T9 '•"3 °'° 2.71 2U9 30.47 °:\i 210.45 ' .40 40 0.22 2.0G 119.44 \ ■35 •35 O.HO 1.72 ^-°^ '■° O.IO 2.72 270 ^^ 1" « ,„ 209.05 0.23 '^■50 ol6 2.07 118.09 O.SI 1.82 . 2.73 271 29.88-°-^° 207.04-' 41 0.24 2. OH 110.74-' •35 0..X2 '•'•'^"^o";! 2.74 272 29.59 °-;2 200.22 ' 4^ 0.24 2.10 115.39 ' •35 0.S4 2-07 i 2.74 273 29.29 °-° 204.79 ' 43 0.25 '•'•"* o'» 2.11 114.04 ' •35 0..S5 2 21 ,•* 23: °-"' 2.75 274 29.00 °:;;j 203.36 ' 43 44 0.25 •3.30 :\\ 2.13 112.70 1 •34 •35 0.87 2.75 275 SS-TI „ ,„ 201.92 0.20 •2.S5 „ ,, 2.14 111. .35 , O.ss 2. 55 ,„ , 2.70 270 2^-41 ~^'° 2IM).4S— ' 44 0.27 '233 -°-^^ 2.15 110.01-' •34 O.Sil 2-7*'*'oM 2.77 277 -'^•13 v^ 199.03 ' 46 47 0.27 10.86 Ti 2.17 108.07 •34 •34 ■34 ■ii O.'.ll 2.95 °- 2.77 27S -'^•«3 0" 197.57 ' 0,28 2.18 107.33 ' 0.92 "■'T o"t 2.78 279 i^7-5* ::2^ 190.11 J 0.28 2.20 105.99 0.94 3^*0 l\\ 2.78 280 2SI 27 25 194.64 193.17—' 47 48 0.29 0.30 ":«S-:S 2.21 2.22 104.00 103.33 ■ii (t.95 o.!m; ^tv^, 2.79 2.79 2S2 2'l"fi o'.o 191.09 ' 0.31 "■•■^a T\ 2.23 102.00 J ■ii 0.9.S 2.S0 2S3 20.30 „•^- 100.20 ' 49 0.32 "•'0 040 2.24 100.67 I ■ii 0.99 ^•^^ 00 2.80 284 2G.07 l\l 188.71 J 49 49 0.33 «-^» o'.S 2.25 99.34 , ■ii •32 1.01 4 78 ■^ *■"' 0.32 2.81 285 S'i.fS „ ,„ 1S7.23 0.34 831 2.27 98.03 1.02 5.10 , 2.81 2sr. 25. 48-°- 3° 185.72—' 50 0.34 7. 94-°- 37 2.28 96.70-' •3» 1.(13 .5.43 +°-« 2.82 287 25.19 °-^,^ 184.21 ' 5« 0.35 7.59 y,\ 2.29 95.39 ' •3« 1.05 5.78 °-3S 2.82 288 24.89 °-3° 182.70 " 5' 0.36 7.25 °-3^ 2.30 94.08 ' 3« 1.00 6.14 "--J" 2.83 289 24.00 °-2 0.29 181.19 ' 5' 5» 0.37 6-93 IM 0.30 2.31 92.77 ' •3' •3° 1.08 6.51 °--57 0.39 2.83 290 2*3' „,n 179.67 0.38 "•^ „ ,„ 2.32 91.47 1.09 6.90 ,„ . 2.84 291 24.02-°- ^9 178.15-' 52 0.39 6.34-°- 9 2.33 90. 1 7-' •30 1.10 7.30+°-'t° 2.84 292 23.73 °;\l 176.03 ' 5» 0.40 6.07 °-%7 5.81 °-f 2.34 88.S7 ' ■30 1.12 7-^2 TA 2.S5 293 23.44 °- 9 175.10 ' 53 0.41 2.36 87.58 ' .29 1.13 8.15 °-^-5 2. 85 294 '■^^■'5 °1^ 173.57 ' 53 53 0.42 •^•^^ :;:] 2.37 80.29 1 .29 .28 1.15 u r.n 3-44 **-•'" 0.46 2.86 295 22.80 172.04 0.43 •'■••3< „ „ 2.38 85. ni 1.10 9.05 , ,, 2.86 29.<'. 1) Diir. .s (U.K. 8) Diir. Soc.var. (U.C.3) Diir. Mvc.var. ft tl tt II $t tt tt 11 300 81. 4J 104. »:> 0,47 4.44 2.44 78.07 1.23 11.55 , 2.88 itiil 21.13-°'^ 102.78" -'■55 0,4 8 4.31-° ■] 2.45 77.42-"S 1.24 12.09 +°S-« 2.88 ■M-i 20.85 I'Jt 80.57 °''l Hi 1.23 •55 0,49 ^'» T.' 8.40 70.17 '-'5 1,20 12.04 °-5S 2.88 :ioa 159.08 •■55 1,56 0,50 8.47 74.93 '•■'■♦ 1.27 13.81 °57 8.89 au4 80.29 "••'^ 158.12 0,51 8.48 73.70 '•'•» 1.29 13.79 °S» 8.89 O.JS '•55 0.07 '•23 059 soft 80.01 I5fi,57 -1,56 0,53 •'•"* ««, 8.49 72.47 1.30 14.38 , , 2.80 :>()i> 10.73 -°-^« 15ft. or 0.54 3.80-°°5 8.50 71. 25-'" 1.31 14.99+°-^' 8.80 307 30 S 10.45 °' « 10.17 °'1 153.40 151.00 '■55 "S6 1,56 0.55 0.50 3.86 °°^ 8.51 2.52 70.03 \\\ 08.88 1.33 1.34 15.01 °^' 10.24 °-^i 8. MO 8.90 30'J 18.00 °'^ 150.34 0.57 a-«4l°° 8.53 07.02 •■^° 1.30 10.88 "ij 8.90 0.J7 '•55 +0.01 1.20 0.66 8IU '^••''■» „ „ U8.7!) -1.56 1,56 '•55 1.56 1.56 0.58 l^'-" 2.54 0(1. J 2 1.37 l''.54 , ,. 2.90 311 18.3,i-°' U7.23' 0.50 2. 55 05.23- -^^ 114.05 \\^^ 02.87 ■ •; 1.39 18.-21+°-?7 18.90 °^'^ 2.90 312 1,8.09 °' 145.07 0,00 2! 5*0 1.40 8.90 313 17,82 ° 144.12 0.01 2.57 1.42 19.00 °- ° 8.90 314 "•^^ o.:] 142.50 0.02 8.58 01.70 \\l 1.43 ^"••" ::;; 8.90 315 17,28 111,00 0,04 ■»'".«. 2.5ft 00.54 1.45 21.03 2.91 31(! 17.02-° -''^ 130.44" 1.56 0,05 4,27+°" 2.59 50.39 -'-'S 1.47 21. 70+°- 73 8.91 317 10.75 °'J 137.88 O.CiO 4.40 °'' 2.i;o 58.24 '-'S 1.48 22.51 °V5 2.91 31.S 1G.49 °'^' 10.23 °-^'^ 136.33 = •55 1,56 0.07 4.55 °'; 4.71 °'^ 2.01 57.10 ' '4 1.50 23.27 °-7'' 8.91 319 134.77 0.08 2.02 55.97 ' '» 1.51 24.04 ° 77 8.91 0.26 '55 0. 18 I. 12 0.78 320 15,07 , 15.7l-°-^<^ 133.22 0.00 4.89 , 2.03 54.85 1.53 24.82 2.91 321 131.07" -'•55 0.70 6.08 -♦-°'"> 2.04 53,74 -'•" 1.54 25.02 -r°'*° 2.91 322 15.46 "•'5 130.12 '■55 0.71 6.29 °" 8.05 52.04 ''o 1.56 20.42 °'^° 8.91 323 15.21 ° 'S 128.57 '•55 0.7:. 5.51 °'-» 8.05 61.54 ' '° 1.57 27.24 ° "^ 2.90 324 14.90 °-^5 127.02 '•55 '54 0.74 5.75 °-'-» 2.00 50.45 '•°'J 1.08 1.59 28.07 °",^ 0.8? i: 90 325 14.71 125.48 0.75 "■"O . „ 2.07 49.37 1.(10 28.92 2.90 32(1 14.47-° ^^ 123.04" -'54 0.70 „.0T+o- 7 2.08 48.30 — '°7 1.01 29.77+0-85 2.90 327 14.23 °'-^ 122.41 '•5.? 0,77 fi.50 °=«9 2.09 47.23 ' °7 1.03 30.63 °-^^ 2.90 328 13.00 °-^-« 120.88 '•53 0.79 fi.,87 °'-^' 8.09 40.17 ' °'' 1.04 31.51 o-^S 8.89 32» 13.75 °^-t 0.24 119.35 • 53 '53 0.80 '■'» :i; 2.70 45.13 '"-^ 1.03 1.00 32.40 °-89 0.90 8.89 330 13.51 117.82 O.xl !:--+o..„ 2.71 44.10 1.07 33.30 2.89 331 13.2,S-°*-^ 110.30" -1.52 0. 82 2.72 43.(»7— '■°-' 1.08 34.21 +°-9' 2.89 332 1J.04 °'* 114.78 • 5^ 0.84 8.24 °-" 2.72 42.00 ' °' 1.70 35.13 °-9» 2.89 333 12.81 °-"-' 113.20 '5^ 0,85 2.73 41.0fi * °'^ 1.71 30.00 °-93 2.88 334 12.5S °-» 0.23 111.74 '52 ' 5« 0.80 9.02 °;;;° 2.74 40.00 '•°° 0.99 1.73 37.01 °'''.5 0.96 2.88 33:) 12.35 110.23 0.8S 2.75 39.07 1.74 37.97 2.88 331! 12.13-°" 108.72" -1,51 0.80 2.75 38.10— "•'>7 1.75 38.94 +-°-97 2.88 337 11.01 °" 107,22 1.50 0.90 10.33 °-'5 11.20 °-',^ 2.70 37.14 °-'/' 1.77 39.91 °-''7 2.88 33S 11,110 °- 105.7 I '■49 0.0 1 2.7 7 30.19 °-9S 1.78 40.89 °'i^ 2.87 33'J 'in n : 1111,24 1.49 0.93 2.77 35.24 °-9S 1.80 41.89 ^■°° 2.87 0. 2 1 '49 0.49 0.94 1. 01 340 •'•-« ^ ,. 102,75 -.,4s 0.01 ii:ilii 2.78 34.30 1.81 42.90 2.87 341 11,05-° 101.27" 0.05 2.78 33.38 -«-9» 1.82 43 <,i2-t-'-°* 2.87 842 '"■^^ o 00.80 ' -t? 0.07 2.7!t 32.47 °'" 1.84 44.95 ••°3 2.80 343 10.04 °- ° 98.33 '47 0,08 lis ^ 2.-9 31.57 °''° 1.85 45.99 ^°* 2.80 344 0.20 90.80 '■47 1,46 1.00 2.80 ao.«8 °.«0 1.87 47.03 '"i 1.06 2.80 345 10.24 95. 4 r 1 01 14.40 , 2,80 20.80 1.88 48.09 2.85 3411 10.05-°"' 03.05" -'■45 1.02 15.05-t-°-59 15.05 °^° 10.20 °-^" 10.89 °-^'» 0.65 8,81 28.93— °-^7 1.90 4y.i5+'-°^ 2.85 347 9.85 °'^° 92.50 '45 1.04 -Lx.\ 28.08 "•p 1.91 .'>0.22 " °7 2.85 34S y.fiii °"> 91.00 '44 1.05 8.8-i 27.2.1 °«S 20.39 °-^-» 1.93 61.30 '"^ 2.85 34'J U.47 °"' 89.03 '43 1.07 8.82 1.94 52.40 ' '° 2.84 0.19 1.42 0.82 1. 10 350 9.28 88.21 1.08 17.54 , ., l8.20+°-^'^ 18.88 °-?'* 10.57 °-^"^ 2.83 25.57, _ 1.00 53.50 2.84 35 1 9,10-°'t 80.70" -1,42 1.00 2.83 i,^.,.:-o.8i 1.97 64.01 -t-'" 2.83 352 8,02 °'^ 85.38 1,41 1 11 8.84 83.97 °'^'> 1.!I9 55.73 ' '^ 2.83 353 8.74 °'** 83.98 1,40 1.12 2.84 83.18 °-79 2.00 50.80 '-'3 2.82 354 8.57 °'7 0.17 82.58 1.40 '•39 1.14 20,28 °;;; 8.84 22.40 °78 0.70 2.01 57.99 '■'•' « '5 2.88 355 8.40 81.10 1.15 21 00j^„ ,. 2.85 21-«* „„ 2.03 59.14 2.81 35(i 8.23-°- 'J 8.07 °'^' 7.01 °'^ 7.75 °-^ 70.81" -1,3s 1.10 21. .4+0. 4 2.85 20.8!|— °"S 2.04 00.29 +'-'5 2.80 357 78.44 '37 1.18 22.49 °" 23,2.5 °ll 84.03 °r 0.79 2,85 20.10 °" 2.05 f.1.45 '"5 2.80 35S 35 !l 77.07 75.71 '•37 1.35 '35 1.19 1.21 8.. so 19.43 °73 18.71 °'^ 0. 70 2.00 2.08 02.02 ''7 03.80 J J;^ 2.79 2.79 8(;o 7.fiO 74.30 1.22 34.82 a. 80 18.01 2.09 64. 9d 2.78 THE Oil HIT OP URANUS. 24S TAULK IX, A HO, S .^Cmtdnued. Arg. (».«,3) it 8.92 (U.C.8) 3.35 («.f.4) n 1.43 {v.oA) {(..c.O 110 (p.".!) 1581 (h.«.2 143 (^..•.2 (,,.».. 3) ( p.t'.3 133 1.08 2848 512 15 ;iui (1.92 8.35 1.43 1.07 1 09 2S4!I 1566 144 515 15 134 !1(I2 3° 0.64 1.22 3«!5 3(iri 1.20 1.31 20.00 ,^„„ 2.88 2.88 u:ol-°t 13.48 °-^' 12.88 °'° 2 111 2.17 72.23^' ;, 2.74 2.74 3tiT «■«» o ■ 1.32 30.78 °''° 2.88 2.18 73.4(1 :• 2.73 3C8 (!.4,S °'-^ 1.34 31.(11 y' 2.88 2. 1 74.70 \\\ 2.72 36 9 C.3(! °'' (;2.(;,i ''7 1.35 32.(11 °'>^ 2.80 12.20 °-5'^ 2.21 ''••'•'^ .-.2^ 2.71 O. 12 1.25 0.93 0-57 310 r> 24 f.l.I'.S . ,, 1.37 :<•■'•''•» 1 „ r 2.80 11.72 2.22 2.71 3-1 fi.l2-°'' «"H--^1 1.38 34.40+°'^5 35.45 °'>^ 2.8<.) "■'^~o 5 2! 23 2.70 372 «.00 °'' 5S.01 • 1.40 2.80 10.(13 °-^ 2.25 V.KU ! 2.(10 373 5.8» °" 57.f.H ^ 1.41 3(1.42 °'^7 37.40 °:^« 2. 8!( 10.10 °- ■ 2.2(1 ^"" :; 2.(18 374 ^•" :;;; 5(1.4(1 ,:,„ 1.43 2.80 0.50 °-5' 0.50 2.27 2.(17 37.") S.fiH 55.2fi 1.44 38 40 2.80 0.00 2.20 83.5B , 2. fid :t7(; 5.57-°" 54.07-''9 52 .SO -'^ 1.45 30.41+'°" 2.80 8.(10-°-l'' 2.30 84.85+'-^'> 2.(1(1 377 5.47 °'° 1.47 40.43 '°^ 2.8il 8.12 °-»;^ 7(,(l °-»'' 2,31 8(1.14 '■''> 2.(15 37S 5.:i,s """J 51.72 "7 1.4S 41.47 ■•°* 2,8!) 2.32 87.43 '•-" 2.(14 37» 6.30 °°''' 50.57 ''5 1.50 42.52 '"S 2,80 7.21 °-«S 2.34 88.73 •■•'° 2.(13 o.oS 1.15 I.Ob 0.43 '•3' 3S0 5.22 *'^*'^ , ., 1.51 43.68 , 2.80 f..78 2.35 00.04 , 2.(12 3H1 5.14-°°'^ 4S/.1)-''' 1.52 44.(15 + ' °7 2.80 (1.3(1 -°-«^ 2.3(1 01.35+'-'" 2.(11 3S'2 5.07 °°l 47.17 ' ;; 1.51 45 74 '°y 2.8 2.8,S 2 0(1-° -J 2 70 °;^ 2.47 101.(18 +'•■'■■' 2,52 3!l-2 CO., :t5.(!.-, °''l 1.(10 57.25 ' -' 2.8K 2.48 10(1.03 '-'5 2.51 3'.);! 4 45 °°-' 1.70 5s.4(l ' -' 2,87 2 15 °'' 2,40 107.:i8 'll 2,50 3i)4 4.41 °°-» 0.0 ; ••^■^•' o..;6 1.71 50.(18 J '1* 2.87 2,22 °-' 0.21 2,50 2.40 305 4..'tS 3:! 71 1.73 (1(1.01 2,87 2ni „ ,„ 2.51 110.10 , , , 2.47 vm; 4.:..-.-^^°-' 32.77 -°''^ 1.71 (12.15 + ' •'■♦ 2.S7 '■'*'"o ^ 2.52 111.4.1+ •• 2.4(1 3!I7 4.:t:J °-^- 31. S5 '^•'^ 1.75 (13,40 ' -5 2,S7 '•'•■=2 °^ 2.53 w>.^-> •' 2.45 3!iS 4.31 °°- 30.04 °'i 1.7(1 .14.(17 • .15.05 ;-« 2.8C, l:^» li 2.54 III.IS [f 2.44 asiy 4.:i0 °°; 0.01 ="•■"3 :i; 1.78 2.8(1 2.55 1.5.55 \-H 2.43 4on 4.20 S;;i--i 1.70 (17.23 , . , 2.8r, l'« n.. 2.5(1 '"■•"-.L. „ 2.42 401 4.2S-°°' 1.80 (ls..-,2+ -"> 2.8(1 l'"~o • 2.57 lls,-20"^ •'' 2.41 40i> 4.2s °°° .-.42 °;'' 1.S2 (10,82 ••° 2.85 "'•"• 00 2.58 lio,.;(i [ll 2.40 4ii;t 4U4 4.2,S °°° 4 2S °-°° ^5:7^ °:? 1.83 1.85 71.13 72.45 '•'' 2.85 2.84 2.50 2.(10 121,03 •■" 122,11 •', 2.30 2.38 '•-4-0.01 ' 0.82 1 .53 0.07 '■37 4 Of) 4 ''0 2».01 1.80 73,78 . 2.8» 1 "■.■'■'' 2.(10 123.78 , , ^ 2.311 4 (It; 4(17 40S 40!l :.^o+°°: 2:!.3:t ° !, 1.88 I.SO 1.01 1.02 2,81 2.8;| 2.83 2.82 0.(10 -:;°5 0.5.1 °°| 0.00 2. Ill 2.(12 2.(13 2.(14 .25,l.l+- 12(1,53 ■•" 120,28 •" '■37 2.35 2.34 2.33 2.32 410 4.:iO 21. or, 1.04 8(1,50. 2.82 "•'••2l„„, 2.(15 1:10.(15 2.31 411 4 4.) 4-00,^ 20.;t3-°7.? 1.05 81,04 + 'V^ 2,8-> 0,51+°° 2. '1(1 l:i2.02 + ''7 2.30 412 4:4(1 °-°-' 10.C.2 °J' 1.07 83.33 ' '■' 2,81 "•■'^ on- 2.(17 1:1:1,10 [■^>l 2.20 4l:l 4..'-.0 °°-» IS.ilH ° "> lS-<5 °'''* 0.O7 l.os 84.73 ' '° 2.81 0.(12 °°S 0.(18 °'°^' o.oH 2.(17 1:11.77 '••" 2.27 414 4.54 °°' o.oi; 1.00 8(1.14 ' '" 1.41 2.80 2.(l.s 1:11;, It '-'^ '37 2.20 415 4.50 1 17. 5S 2.01 87.55 2.80 "f'''j_o„> 2.(10 1:17 51 , 2.25 41(1 4.(14+°°.^ lC,.03-°'^ 2.02 88 .17+''" '.'.70 0,85+°°" 2,;o l.!S,8s + ''37 2.21 417 4.70 °°^' 1C..30 °M 15.(;7 °',' 2.03 00.30 ' '■' 2 70 0,0(1 °" 2,VI 110 25 '••" 2.22 4H 4.7(! °°f 2.01 01.8-2 '••■' 2.78 1,08 °'' 2 71 111,,.. '37 2.21 41'.) 4.,S2 °°' 0.06 , i:..o(: °'^" 2.011 i 0.!.2(1 ' ■" I 45 2.78 1 1.22 °'' o'5 2.72 11 -.',00 '■■'/ '■3'> 2 20 420 4.S8 14 47 2.0, 04.7! 2.77 1.37 2,7:1 lll.:!5 2.19 .J^ ^ THE OR BIT OP U 11 AN US. isn TAULK IX, Abo. 2. — Continued. AfR. (U.8.3) (U.C.3) 2.53 (i'.».4) (...r.4) (p.r.O) (p.«.l) 710 fp.«.2) 253 (p.c.2) 027 (,..H.3) 67 (P.c.3) 181 SCO It f.07 If 1.23 0.38 140 2029 aiii 0.02 2.51 1.22 0.38 148 2021 097 254 028 08 I8l .S(>2 5.98 2.50 1.21 0.37 151 2013 085 250 629 (.9 182 •MV.l 5.94 2.49 1.20 0.30 153 2005 072 259 031 70 183 ■Mi 5.90 2.48 1.19 0.35 156 2590 000 201 032 71 183 3fi5 5.85 2.46 1.17 0.35 !59 2588 047 203 033 72 183 aiiti 5. Hi 2.45 1 10 0.34 102 2579 035 205 034 74 183 3 (17 5.7r, 2.44 1.15 0.34 104 2570 023 207 035 75 183 3r)8 5.72 2.44 1.14 0.33 107 2502 010 270 630 70 183 3(i9 5.C7 2 43 1.12 0.32 170 2553 698 273 038 77 184 370 5.fi3 2.43 1.11 0.32 173 2544 580 274 639 78 184 371 5.58 2.41 1 10 0.31 170 25;i5 .•>74 270 040 79 184 37. 5.53 2.41 1.08 0.31 180 2520 502 279 041 80 184 373 5.48 2.40 1.07 0.30 183 2510 550 281 042 81 184 374 5.43 2.39 1.05 0.30 180 , 2507 539 283 043 h3 184 375 6.38 2.39 1.04 0.29 190 2498 528 285 645 83 181 370 5.33 2.39 1.03 0.2!. 194 2488 517 288 040 84 184 377 5.28 2.38 I.OI 0.29 197 2479 500 290 047 85 |8( 3:» 5 22 2.38 1.00 0.28 201 2409 495 292 048 80 184 379 5.17 2.38 0.98 0.28 205 2400 485 294 649 87 184 380 5.12 2.38 0.97 0.28 208 24.50 474 297 650 88 184 3S1 5.07 2.38 0.90 0.28 212 2440 403 299 051 89 184 3H2 5.01 2.38 0.94 0.27 210 2429 453 302 052 90 184 3S3 4.911 2.39 0.93 0.27 220 2419 443 304 053 91 184 3S4 4. '.to 2.39 0.91 0.27 224 2408 4.33 300 054 92 184 3S-, 4.84 2.39 0.90 0.27 228 2397 423 309 6.55 94 184 3sr, 4.79 2. 40 0.88 .i..27 232 o-- ., , 413 312 650 95 184 3S7 4.73 2.40 0.80 0.27 230 2374 40i : 1 4 0.57 90 184 3.SS 4.07 2.41 0.8'. 0.27 210 230:! 394 317 057 97 183 3.S.» 4. Ill 2.42 0.81 0.27 245 2:!5! 384 319 058 98 183 3!t0 4.55 2.43 0.83 0.27 249 2310 375 322 059 99 183 3!ll 4 49 2.44 0.81 0.27 253 2328 300 321 OOO 100 183 ■M-i 4.43 2.40 0.79 0. 27 258 2'ilO 357 327 001 102 183 3it:t 4.37 2.47 0.78 0.27 202 2.!05 318 329 002 103 18a 391 4.31 2.48 0.76 0.27 200 2293 339 332 662 104 183 39') 4.21 2.49 0.75 0.27 271 2281 330 334 603 105 182 :!9l n If 430 <-88 ,„„, 14.47 2.07 94.71 , 2.77 1.37 . 2.73 144.35 , 2.19 421 *-"5+oo 13.90-°" 2.08 90.10 + 45 46 2.70 l-'^"*"o'8 2.74 14.5.72+ -37 2.18 422 so'-J III 0.03 13.35 °-53 2.10 97.02 ' 2.70 2.74 147,09 -37 148.45 -36 2.10 423 12.81 °-54 2.11 99.09 47 2.75 l.!12 °'° 2.75 2.15 424 12.28 °S3 2.13 100.50 1 47 2.75 ai3 °'' 2.75 149.80 '••'S 2.13 0.51 48 0.22 >-35 425 5-2fi ,„„„ 11.77 2.14 102.04 48 2.74 2. •'?•"> , „ , 2.70 151.15 , , 2.12 4M 5-a'^"^ooo ii.28--°-;^9 10.80 °-»? 10.34 If^ 2.15 103.52 + ; 2.73 2.M+°- ? 2.77 I52,50 + '-3S 2.11 427 5.44 °°'^ 2.17 105.01 J 49 2,73 2«^ 2.77 1.53.84 \\\ 2.09 428 5-W °?^ 2.18 100.50 J 49 2.72 -'» o's 3.39 °: 2.78 155.10 35 2.08 429 sea °- ° 9.89 °-»S 2.20 108.00 : 5° 2.72 2.78 150.54 '-'5 2.06 o.io 0-43 5« 0.30 1-34 430 5. 73 ,„ ,„ 9.40 2.21 109.51 , 2.71 3.09 , 2.79 157.88 , 2.06 431 6.83 +°-° 9.05 -°-»' 2.22 111.02+; 5« 2.70 4.00+°-^ 2.79 150.22 + j^; 2.04 432 lit -" 8.00 °-]1 '■'' oil i.h 112..53 . 5' 2.09 ^■■''^ ot 2.80 100., 55 , •" 2.02 433 2.24 114.04 : 5« 2. 0-l 4 07 °-'^-' 2.80 101.88 ••» 103.20 i:^; 2.01 434 2.25 115.56 \ 5^ 53 2.07 2.81 1.99 435 0.29 IM 2.27 117.09 , 2.00 5.39 , . 2.81 104.52 1.98 43i; fi.41+O.I2 T,-2:i—°-3i 2.28 118.62 + ' 53 2.00 .5.77 +°-^'" 2.82 105.84 + ''3' 1.91 437 fi.54 °'3 c.h °'J' 2.29 120.16 ' 54 2.05 0.17 °-l° 2.82 107.15 '-31 1.95 438 0.07 °^' 0.02 °-3° 2.30 121.70 ' 54 2.04 0.58 °->' 2.83 108.40 '-3' 1.94 439 6.80 °'-5 0.14 6.34 °-^8 0.27 2.31 123.24 ' 54 54 2.03 7.00 °-'' 0.44 2.83 109.76 '-3° 1.30 1.92 440 0.94 0.07 2.32 124.78 ,, 2.02 ■J^* . .. 2.84 171.00 1.91 441 7.08 +°'4 5.H2-°='S 2.33 120.33 + ' 55 2.01 7 H!.+°-»5 2.84 172.35+'-'' 1.00 442 7.22 °"l 6.. 59 °'i 2.34 127.88 55 2,00 8.84 °-*" 2.85 173.04 '•-'' 1.88 443 7.37 °'5 5.38 "-' 2.30 129.43 " 55 2.50 2.85 174.113 '-'2 1.87 444 7.52 °'5 5.18 °-° 2.37 130.98 ■ 55 56 2.58 9.33 °-»'' 2.80 170.21 '•^'* 1.85 0.15 0.18 o-5« 1.27 445 8.31 °"^ 5.00 2.38 132.54 , 2.57 0.84 , ^ 2.80 177.48 1.84 44r. 4. 83-°- '7 2.39 I34.10 + ' 56 56 56 56 2.57 10. .30+°- 5' 2.80 17H.75 + '-7 180.01 '•^'' 1.83 447 4.08 °'5 2.40 135.00 ' 2.50 10.8!) °-5.i 2.87 1.81 448 4.55 °'3 2.42 137.22 ' 2.55 11.44 °-55 12.00 °S'^ 2.87 181.20 '-'5 1.80 449 4.43 °'^ 2.43 138.78 1 2.54 2.88 182.51 '-^5 1.78 0.17 0.10 57 0-57 1.24 450 8.48 , 4.M - 2.44 140.35 2.53 12.57 , , 2.88 183.75 1.77 451 8.05+°- "7 4.25-°°? 2,45 lll.!)2 + 57 2,52 i;uo+°" 2.88 184.1I0 + ''-'' 1.70 452 8.82 °\l 9.18 °'» 4.19 °°'^ 2,40 14.3.49 ' 57 2.51 13.70 ° '° 14.37 °-'' 2.88 180,:!2 '-3 1.74 453 4.14 °°S 2,47 115,00 ' 57 2.50 2.8{. 7.44 '•" 1 73 454 4.10 °°-^ 2.48 140.03 1 57 2.49 2.89 ..((.00 " 1.71 0.18 O.Ol 57 1. 21 455 9.54+°'" 4.09 2.49 148.20 2.47 I?;!!; o:^J ^^ oS 0. 70 19.02 , , 2.89 189.87 1.70 45(i 4.09 °°, 2.50 14'.».77+; 57 2.40 2.80 101.07+'" 1.09 457 !»" tin 4.11+°°- 2.51 1.M.34 ' 57 2.45 2.80 192.27 If 1.07 458 «3^ 00 <••* ooi 2.52 152.91 ; 57 2.44 2.90 .93.40 \W 194.04 ;:;■; 105.81 1.06 459 10.11 °^ 0. 19 10.30 , <18 toi 0.07 4.25 , 2.53 154.48 J 57 57 2.43 2.90 1.04 4 no 2.54 150.05 , 2 42 2.00 1.03 4fil 10.50+°-^° 4.33 + °°« 2,.'->5 157.02 + 1 57 57 2.41 19.73+°- 2.!i0 100.08 + ''J 108.14 \\'l 1.01 41'. 2 10.70 °' ° <•*=« 0! 2,50 159.19 2,10 20,45 °'^ 2.90 1.00 403 "'•'•'0 o'.o ^•^'•■^ 2,57 100.70 57 2,311 21.19 °-lt 2.!I0 100.20 -^ 1,58 4G4 'I'" 0:;: *•"« ::;j 4.83 , , 2,58 102.33 , 57 56 2.38 ^'••'^ o.?-, 2.90 200.43 \\\ 1.57 405 11.31 2.5!) 103.89 , 2.30 22,71 . . 2.91 201.50 1.55 40(1 II r,o+0-2' 4 q,,+o.ir, 2.59 105.45+' ^6 56 2,35 23,4S+°" 2.01 202.08-t-''' 1,53 407 11.73 °-' b.V-J °'» 2.00 ir,7 01 ' 2,34 24,27 °i'^ 25,07 °-^° 25.88 °-^' 2.91 203,70 ' " 1,52 4 OS 11.94 °=' 5.37 °'° 2.01 108.57 2.33 2.91 204,00 '■" l.;.0 409 12.15 °^' 5.59 °" 2.02 170.13 1 2.32 2.91 200.00 '■'° 1,49 0.}2 0.2J 55 0.82 1.09 470 12.37 , 5. 82 2.03 171,08 2.31 20,70 ^ „ 2,01 207.00 1.47 47! |o r,<)+o.zi 0.07+°'.'! 2.04 173,23 + ' 55 2.30 27.53 +°«-» 28.37 °";^-» 2,01 208.17+'°" 1,40 472 12:81 °" 0.34 °»7 2.05 174.78 ' 55 2,29 2,01 200.21 ^■°l 1.44 473 13,03 °-'' 0.02 "-''^ 2.05 170.33 ' 55 2 27 29 22 °-2 2,00 210..iO '•°^ 1.43 474 13.20 °-^-' 0.91 °''' 2.00 17 7. 87 ' 54 2,20 30.0!) °-^7 2,90 211.30 •■°<* 1,41 0.2.5 °-3> I 54 0.88 1.04 475 13.49 , 7.22 , 2.07 179.41 Ho.ns-*"' 2.25 31.80+°'*'' 2,00 212,40 . 1 40 470 l.!.72+°-'-] 7.55 +°-^.^ 2. OH 54 2.21 2,!M) 213, 43 + '°' 1,30 477 13.95 °' 7.90 °-." 8.20 °-3'i 2.0!) 182.48 ' 53 2.23 32,70 °''° 2,00 214.45 '•°' 1.37 47H 14.18 °'^' 2.0!) 184.01 ' 53 2.21 33,07 °'^' 2,80 215.47 '•°' 1.30 479 14.41 °'- 8.03 °-" 2.70 185.54 ' 53 2.20 34,69 °'" 2,80 210.48 '■°' 1.34 480 o.a4 °J9 I 5» 0-93 0.99 14.05 o.oa 2.71 187.00 2.19 35.52 2.80 217.47 1.13 " TUB ORBIT OP UUANUS. 241) TABLE IX Alio. 2. — Continued. Arjj. (r.y.:i) (r.c.3) 3.29 (o.i..4) (v.c.i) (p.e.O) 407 ((..«. 1] (p.C.lj (f-x.2] (,.c.-> {,>..s.3 (p.f.3) 4 '20 ft 2.fi7 0.40 It 0.42 1957 100 390 072 128 172 421 2. 0.12 0.80 040 14 OS 88 480 005 149 149 455 0.94 5.75 0.12 O.SS 048 1453 89 480 004 150 148 45(! 0.91 5. SI 0.11 0.89 05(! 143S 90 i 492 0(i 1 150 148 4". 7 O..S.S 5.91 0.11 0.9(1 0(i t 1 423 92 ; 49t 00.3 15(» 147 4:.s 0.S5 0.03 0.11 0.92 071 1 409 93 ' 497 0(i2 151 140 459 D.H-i 0.13 0.10 0.94 , 079 1394 95 499 001 151 145 4(iO o.so 0.23 0.10 0.95 ' 087 1379 90 502 001 151 144 4 lit 0.77 0.32 i 0.10 0.97 (i95 1304 98 1 505 (ifiO 151 144 41; 2 0.75 O.tJ 0.10 0.99 703 1319 100 .507 (i(iO 152 143 4c;;j O.7.! r,..-,i 0.10 1.00 711 i:i:!5 j 102 1 51(1 (i59 152 142 4U4 1 0.71 1 C.lil 0.10 1.02 ! 719 1320 105 512 058 152 142 405 o.r,9 0.71 ' 0.(i;i 1.03 727 1305 1 lOS 515 057 ' 1.52 141 4lili O.tlS O.sl 0.(19 1.0.-. 735 1290 llo 517 (i57 l.V< 140 4(i7 1 o.tin 0.91 O.O!) 1.0(i 713 1275 113 52(( (i5ii 153 139 4(i8 j 0.(!4 7.01 0.09 1.08 751 1201 IKi 52! 055 153 139 4(!» 1 0.(!:l 7.11 0.09 1.10 759 1210 119 525 j 05 » 154 13S 470 O.C'^ 7.21 0.09 1.11 707 1232 123 ' 52S ! 653 154 137 471 0.fl2 7.31 0.10 1.13 ! 775 1217 127 ■ 5.!l Ii52 151 130 17 2 (Mil 7.42 0.10 1.15 784 1203 1.11 533 (i5l 155 l:t(i 47:1 ll.lill 7.52 0.10 1.10 792 II 89 1.35 .53ii Ot9 155 1. 15 47 « O.till 7.ii2 1 0.10 , 1.18 800 1175 139 639 j 048 155 134 475 O.)>0 7.72 0.10 1.19 809 IIOI 143 542 047 150 13.3 47(! O.liO 7. S3 0.11 1. 21 817 1117 I4S 545 i 010 150 1 3.! 477 O.tiO 7.9» o.lt 1.23 82(i 113.! , 15.t 548 t 044 1.50 , 132 47>< O.iiO N.dl 0.11 1.21 834 1119 15S .551 i 013 15'i III 479 O.til 8.15 12 1 . 25 843 1105 1 103 55 1 1 012 15ii 130 4St) O.lil 8.20 0.12 1.27 851 1091 108 ,55(1 040 150 130 2 July. lavs. 250 TIIK ORBIT OF URANUS. A TAULE IX, Aro. 2 . — Cunliinied. Arff. ' (u.c.O) IUff. yv.g.l) l»ilT. Sec.vnr iu.e.l) Diir. S cf.vur. (t'.».2) Diff. Scf.vur. (('.f.2) Diff. Si'c.vnr »/ // tf II ft II II '» // II // II If II 4«0 14.f.5 , 9.02 , 2.71 187.00 2.19 35.52 , 2.89 217.47 , 1.33 4K| 14.H9+°-^-» 9.4:1 +°-» 2.72 188.57 '*"''5' 2.18 30.4(1 + 3-94 2 89 2l8.40+°-W 1.32 4H3 15.13 11^, 9.S5 °-'' 2.72 190.08 ''5' 2.10 37.41 3.y6 3.y.8 2.89 219.44 °'^? 220.40 °'-'l' l.:io 4.S3 '''-*T n'^ '0.29 °-»^ 2.73 191.59 '-S' 2.15 38.37 2.88 1.29 4M 15.r,i °-^-t 10.74 °-»5 2.74 193.09 50 2.14 39.35 2.88 221. :i5 °'^5 1.27 o.-M 0.47 1.50 - y.S 0.94 4Hr. 15.S5 , 11.21 , 2.74 194.59 , 2.12 40.33 , 2. 88 000 01) 1.20 4MI! i(;.o',» +"•-'•♦ ii..;9+°f 2.75 19.1.08 + '-''' 2.11 41.32 + 3.99 2.88 .'.r( oJ+°W 1.25 4«7 ir,..i3 °;-» 12.19 °5° 2.7.1 197..57 '•■♦^ 2.1.1 42.32 1. 00 2.88 '"•"4 1.3 °'^' 1.2:1 4HS l(i.5S °-*3 12.71 °=' 2.77 I99.(»5 '■•♦•^ 2.09 43,:!3 I.OI 2.87 225;04 °'-" 1,22 489 l.i..3 °]l '"< 1.11 2.77 200.53 ;;i; 2.07 44.35 1.02 1.0 J 2.87 225.94 y° 0.09 1.20 4'.>'> 17.08 13.79 , ^ 2.7s 2.12,00 2.00 45.38 2.87 220.8.3 , „„ 1.19 4!li 17.;i3+o-^i 14. 35+°- 56 2.79 203.4.1 + '-«^^ 2.05 10.42 + 1.04 2.87 227.71 +°'!'^ lis 4:'2 1-.5S 0-25 n.93 "s** 2.79 204.92 "■-♦'' 2.03 47.40 1.04 2.8r, 228.58 °'^7 1.1(1 4!».t 17.H3 °-5 15.52 °;'^ 2 80 20(1.37 '•+'' 2.02 48.51 1.05 2.8(1 229.43 °'^5 1.15 4'.t4 is. 09 o^** 0-5 lfi.l3 °f' 0.62 2.80 •J07.81 '••♦•♦ >-4J 2.00 49.58 1.07 1.07 2.8(; 230.27 °Z^ o.«3 1.13 4!t5 IS.. '{4 lfi.75 , ^ 2.81 209,24 1.99 50.f,5 2.85 231.10, „ 1.12 4'.M! ls.t;o+°-^'' I7.39+°-^'* 2.81 210,07 + '•■♦■5 1.98 51.73' .oS 2.85 "31 +°''^ l.lo 4'.t7 l.s.«5 °--^ 1,S.04 °^^ 2.82 212.09 '• '- 1.90 52. SO ,09 2. 85 232:7:! °'*' 1.09 4 IIS lit. 11 °-^ 1S.71 °''l 2.82 213.50 '■■♦' 1,95 5:1.92 [. lo 2.85 2;i3.52 °" 1.07 4!)!) 19.3ti °^'' 0.26 19.39 °^"^ 0.70 2.83 214.90 '•■*° 1.40 1.93 .55.03 . 1 1 . 1 1 2.84 234. :iO °''^ 0.77 1.00 5(10 in.(i2 20.09 2.83 2If..30 1.92 .50.14 , 2.84 2.35,07 , , 1.04 501 l!t.8S+°-^" 20.,S0'^°'' 2.S3 217..;9 + '-''2 1.91 57.2(1 + .12 2.83 2.15.83 '--•■^'^ 1.03 ■M\-2 20.14 °-'^' 21.53 °"' 2.84 21 9.. 17 '■■' 1.89 58.38 . I 2 2.83 230.58 "■'■■^ 1 01 f.o:t 20.40 ° -'■' 00 on °- "4 2.84 220.44 '•'' 1.8S 59.52 .14 2.82 237. :'l °'-' I.(>0 .'•.04 20. CO °'^(' iiiioi °'; 2.84 221,81 '•■'/ 1.80 00.(17 •'.S 2.82 238.(13 °'^" 0.9;; 0.26 0.76 1.36 '.S 0.71 r.or. 20. (12 23.77 , u 2.85 223.17, 1.85 01.82 2.81 238.74 , 0.97 .')OI'i 2!.lS+°-'' 2t.55+°J''^ 2.85 224.52 + '''5 1 84 02.98 + r. 16 2.80 2:19.44 ^°?° 0,9.1 r.(»7 21.4; °-^' •25.:i4 °- '; 2.85 :'•'"). s.'-, ■•3.? 1.82 01. U . 16 2.80 240,12 °;;' 240,79 °-f/ 9 95 ^(tS 2 I.e.!) °'^-' 2.;. 15 °i' 2.85 227.18 '■■'•' 1.81 05.31 •'7 2.79 0,94 i')0!t 21, It.) °-^' 2.1.97 °'' 2.8.1 228.50 '••'" 1.79 00.49 . 18 2.79 241.45 °''' 0.9'' C.J 6 0.S4 '•3' 1.19 0.65 510 >)■) ■)) 30.40 \,„ 2.80 229.81 1.78 07.08 2.7s 242.10, , 0.91 r.i2 .')i,{ 2147 +°-''' 22.72 °%^ .>•> C(.^ 0.26 2.8(1 |23l.ll+'-'° 2.87 i232.:l9 '-'^ 2.87 2:i3..;7 '■" 1,77 1,75 1,74 fis 87 + 70,(i7 71,28 r.19 .20 .21 2.77 2,77 2,70 242.73+°- ;•' 243.30 ° '■ 213.98 °^' 0.90 0.S8 0.87 514 2^:21 °-''' 3 1.2s °'^ 2.87 : 234.94 ' •;/ 1.72 72 49 .21 2,75 241.58 °-'':^ 0.8(i 0-5 o.yo 1 , JO , 22 o.5,^ 515 2:1.49 32. H , 2.8S !2:iC..20 , ! 71 73 71 , 271 ?»5.1fi , 0.84 5ii; 2:1, 75+°'^^ ■•'•""•'"^00 2.88 2:!7.45"'" ■;^ 1,(19 71,9:! + .22 2,71 21.5,71 -^°- 0.83 517 24 00 °-j 34.01 °'>' 2.88 2;t8.(19 "■* 1,(18 7'1, K! -,1 2,73 940,20 °-^-1 0.82 51 H 2».2r, °-° «4.95 I'lJ 2.88 239. '12 '■"'■' 1,00 77 :J9 ■-,? 2,72 2i'0..8O ''■♦ 0.8] 51'J 24.:-.! "''^ =«-"" l.U 2.89 1241.13 '-■' 1.05 78,(1:1 .2.\ 2,72 2(7 .33 ° ■^■' 0.79 0.25 1 I.JO .24 05' 5-.>0 24 715 3r..sfi , 2,89 212,33 , . 1.(13 79.S7 2,71 2(7,84 , 0,7s 5J1 2.-..02+°-'* :i7.H3+°''7 2,89 '2I3,52 + '- ',J 2.89 1 214. 7.1 '' ■ 1.02 81,12 + •25 2.7.> 2(8.3 +°--';° (..77 5-.>-i 25 27 °-'-^ 3SS2 °"'> 1.00 82,:i7 •-'5 2.09 24 8,83 °-''' 0.75 52:! 521 2.-...-.2 °-^ 1 .00 40.83 ,.„, 2.89 ; 2(5.87 ; ;;, 2.89 247,03 . '■ '.1 1,59 1,57 8:1,(12 84.88 • Ji j6 .j6 2.08 2.07 219:10 ° 'J 0,71 0.73 525 2fi.0-i 41.85 . 2.89 248 18 1.50 80. 1 1 87.41 + 2,00 250,21 , 0.71 52ii 211.27 +°--?> 42.88 + '°' 2.89 219.31 +"'3 1 55 •-'7 .28 .j.S 2, 0(1 ■ 250,C,4+° •»3 0,70 527 2ii.52 ° -S 4:1.92 '°' 2,89 [250,43 '•'■' 1 53 88. CK 2,05 251, Of, °-^^ 0,09 52S 2i;.7ii °--* 44.97 '"^ 2.89 1251.53 ' '■' 1 52 S9.'.I7 ' 2.01 251,47 °-" o,ris 521> 27.00 ° ■'^ 40.04 '°7 2.89 252.. 13 '*° 1.50 91.25 .28 2.03 25l,8(i °-39 0.0(1 0.24 1.08 1.09 .29 0. 58 5:10 27.24 47.12 , 2.89 05:1.72 , 1 49 92.51 93.83"*" 2.02 252.24 , ' 0.05 5111 27. 4v: + '-'■» 48 21+' "^ 2.8-. ,254 79 + '-°; ".89 25 -..85 '■°'' 1,4S .29 2.01 252.(10+°3<5 0.04 5:t2 27.72 °''-« 49.31 ' ■° 1 4ft 95.12 29 2,00 252 95 °''''' o.i;:i 5:!:t 27. '.m °-'» 50.42 ' 2.89 25.1.90 '"S I 15 9.1.42 -.la 2,59 25.T29 °-'* 0. 02 5.14 2S.19 °-.' 51.53 ' ■' 2.89 ,257.93 ' "■' 1.43 97.72 30 2.58 2.53.(11 °-'-' 0.01 0. 2,i ' '.5 1 1.02 30 0-3' 5;r. 2S.42 52 ;,f, . 2.88 ,258,95 1 42 99.. t2 2,58 253. !)2 , 0.(10 MC, 2s (;5+°-^' ;,380 + ''^ 2.88 259. 9.; + '"°' 1.10 loo.:!2 + •30 2,57 o-,4 oo+°-30 2.54;.5('. °-''^ 0,5 s 5:!7 OS ss °-' 54.95 ' '5 5.1.11 '■'^' 57.27 "^' 2.88 ; 2.10.95 °''1 l.:i9 1.11.(13 3" 2.5t! 0,57 5:is 2'.U1 °-' 2.88 '2.11.93 °'^ i.:i7 102 91 3' 2.55 2.54.77 l\l 0, 5(1 r,;i9 29.:t:i °-'' 2.88 2.12.90 °''''' l.:!(l 101.25 3' 2.5 J 255.02 °^5 0,:-.5 0.22 1 . 1 .8 o<)5 '3' 0,24 540 •»;..55 58. »5 2,88 203,85 1.34 105.5(5 2. 53 2.55.20 0,54 THE ORBIT OF URANUS. 251 * TABLE IX, Am. S . — Continued. Arj? (r.«.3) (u.f.3) (W.8.4) (0.C.4) 1 (p. 8.1) (p.C.1) (p.».2) (p.c.2) 1 (p.«.3) (P.O. 3) 480 O.C.l 8.2fi ft O.ll ff 1.27 851 1001 108 550 640 156 130 481 0.02 8, 31) 0.12 1.28 8(10 1070 173 559 C39 156 129 482 0.(!4 8.47 0.13 1.30 808 1003 179 501 037 156 128 483 o.i;5 8.58 0.13 1.31 877 1040 184 504 030 156 128 484 0.«t! 8. GO 0.14 1.33 885 1035 190 500 634 156 127 4s:> 0.(!7 8.79 0.14 1.35 894 1021 190 509 033 156 120 4Hr. 0.1.0 8.00 0.15 1.3fi 902 1007 202 571 632 156 125 4S7 0.71 O.Ol 0.15 1.37 911 094 208 574 630 156 124 4t»S 073 0.11 0.1(1 1.39 919 980 214 570 629 166 124 480 0.75 9.22 0.17 1.40 927 900 220 579 627 156 123 400 0.77 0.33 0.17 1.42 930 953 227 581 620 156 122 401 0..80 0.43 0.18 '..43 945 040 234 584 024 156 121 402 0.82 0.54 0.10 1.44 954 920 241 580 023 150 120 '0:t 0.85 0.(15 0.10 1.4(1 903 913 249 580 021 150 120 ;i>4 88 9.75 0.20 1.47 972 000 250 501 619 150 119 4 or. 0.01 9.8fi 0.21 1.40 081 887 204 504 018 150 118 40(; 0.04 9.07 0.22 1 . 50 000 874 272 500 1 (1 150 n7 407 0.08 10,07 0.23 1.51 090 801 280 509 015 150 117 4 OS 1.02 10.18 0.23 1.53 1008 848 2HH 001 013 155 no 400 lot! 10.29 0.24 1.54 1017 835 290 0O4 on 155 115 500 1.10 10.39 0.25 1.55 1020 822 304 000 010 155 115 501 1.14 10.50 0.2(1 1.57 1035 810 312 008 008 1.55 114 502 1.10 10. CO 0.27 1.58 1(144 707 321 on 000 155 114 503 1.24 10.70 0.28 1..59 105:) 785 320 013 (104 154 113 504 1.20 10,81 0.29 1.(10 1003 772 338 615 002 154 113 50;-. 1.34 10,01 0,30 l.r>2 1072 700 340 f.l7 000 154 112 501', 1.30 11, (tl 0.31 1.(13 1081 748 355 020 598 154 112 607 1.44 11.12 0.32 1.(14 1 000 7.30 304 022 500 154 111 50S 1.50 11.22 0.33 1.(15 1009 724 37.{ 021 504 154 11(1 500 l.afi 11.32 0.34 1.(1(1 1108 712 382 020 502 153 llo 510 1.(12 11.42 0.35 1.07 1117 700 391 028 500 153 109 511 l.<;8 11.52 0.37 i.r.8 1120 088 401 031 588 153 109 512 1.74 ll.ti2 0.38 1.(19 1135 077 410 033 585 153 108 51.t 1.80 11.71 0.30 1,70 1114 000 420 035 583 153 108 514 1.87 11.81 0.40 1.71 11.53 054 430 037 581 153 107 515 1.04 11 01 0.41 1,72 Il(!2 043 440 030 578 1,53 100 5111 2.01 12,00 0.42 1,73 1171 032 450 042 570 153 100 517 2.08 12.00 0,43 1 74 IHO 021 40(1 014 573 152 100 518 2. Hi 12.10 0.45 1,75 1189 010 471 040 571 152 105 619 2.23 12.28 0.4« I.7fi 1197 599 481 048 500 152 105 520 2.31 12.37 0.47 1.77 1 20(1 588 403 050 .500 152 104 ■,il 2. 30 12.111 0.40 1.78 1215 577 503 052 504 152 104 522 'i■^^ 12.5.-. 0.50 1.70 1221 507 514 051 501 151 103 52;j 2.55 12.(13 0.51 1.70 1 2:!3 557 525 050 558 151 103 524 2.(!3 12.72 0.52 1.80 1242 540 530 058 550 151 103 525 271 12.81 0.5 « 1,81 1250 530 548 000 553 151 102 52(i 2.80 12,80 0.55 1,82 1 250 r,..,! 550 002 550 151 102 527 2.88 12.08 0.57 1,83 12117 ..17 571 004 518 150 101 528 2.07 1 1.0(1 0.58 1,83 1 270 5(17 583 000 515 150 lot 520 3.0ti l.t.14 0.(10 1.84 1281 408 595 008 5(2 ho 100 5110 3.15 13.22 O.CI 1.85 1203 488 fi07 07O 540 15(1 100 5,{1 3.24 13 :10 0.(12 1 85 1 . 479 018 (;-2 537 150 100 5,12 3.;' 4 13.37 n,(it 1,8(1 1310 470 O.'iO 073 535 U'.l 00 5;i:i 3.43 13.45 0.(15 1.8(1 1318 4111 042 075 532 140 00 634 3.53 13.52 0.(17 1.87 1327 452 064 077 520 149 08 535 3.03 13.59 0.fi8 1.87 1335 443 000 078 527 140 98 53r. 3.73 13.(1(1 0.(10 1.88 1313 43 4 078 080 521 148 98 537 3.83 13,73 0.71 1.88 1352 420 (100 0S-.< 521 14s 08 538 3.03 13,8(1 0.72 1.80 I.'IOO 417 7(12 083 5 1 !» lis 07 530 4.03 13,87 0.71 I.SO 13HS 400 711 osr. 510 I4S 07 510 4.13 13.o;i 0.75 1.00 1370 401 72'1 osci 513 117 07 35:; THE OUBIT OF URANUS. 11 TAHI.K IX, Ami i . — (Unwluili'd. Arj5. (D.c.O) Diff. (u.D.l) Diir. Sfc.var (c.p.l) Diir. !• M'c.viir. I i'.(i.2) Diff, Sec.vnr '(i^.c.2) Diff. Sic.vnr. 540 29.55 58.45 , , „ 2.88 203.85 1.34 105.50 If 2.53 2.55.20 , 0.54 541 L>9.77+°" 59. ('.3 + '''^ 2.S8 204.79 +°-9^ 1.33 100.88 + •3» 2.52 2.-.5.49+°-' 0.53 543 29,99 °'^' 00. X2 '•"' 2..S8 205.71 °'^' 1.32 108.19 ' ■,i' 2.r,l :2.-.5.70 °-' 0,52 543 30.-.>l in 02.02 ■•-° 2.87 200.02 °-'^' 1.30 109.51 ; •,»J 2.50 2.M.90 °;° ,250.08 °''^ 0.17 0,51 544 30.43 °- 03.24 ;•;- 2.87 207.52 l;^ 1.29 110.83 J ■32 •3^ 2.49 0.50 545 54fi 547 ••M.O.; ll\ 04.40 , 05.09 + ' -'3 00.93 ••-'♦ 2.87 2.87 2.87 208.40 , „ .,,;,, 07+O.S7 270.' ?2 °-':;5 1.27 1.20 1.25 112.15 ,, 113.47+1 114.80 , 3-' •33 2.47 2.40 2.45 250.25 , , 2.Ml,41+°"^ 250..55 °'-* 0.49 0.48 0.47 548 549 08.18 ;•-' 09.43 \]l 2.80 2.80 270.90 °-'^-* 271.79 °'^3 1.21 1.2 J 110.12 117,45 J ■3-s 32 2.44 2.43 2,-.0.08 °'3 250.79 I'' 0. 10 0.40 0.45 550 551 0. 19 71.90+ -' 73.23 \:i '-«o ,:3o 2.80 2.80 272.00 , „ „ 273.40+°-^° •--T4.I8 °! 271.95 °t^ 1.21 1.20 118,77 ,, 120.09'^, 32 2.42 2.41 2.J0.S9 , 2.V198+°°9 0.03 0.44 0.43 652 5.".3 554 2.85 2.85 2.84 1,18 1,17 1.15 121.42 , 122.74 124.07 , 32 32 2,40 2.39 2.38 0.42 0.41 0.40 555 32.f.3 , - 32.Sl+°''^ 33:17 °;« 77.10 , 2.S4 270.41 1.14 125,39 , 2. .30 257,18 , 0.40 550 7«.40 + '--'° 2.84 277.10+°-7^ 1 12 120.71+' 32 2.35 257,20+°°* 0.39 55T 79.71 •••'■ 2.83 277.80 °^^° 1.11 128.03 ' 3-' 2,31 2.-.7.2> °°° 0,38 558 81.03 '••5^ 2.83 27 s.. 55 °'"} 1.09 129.35 ' 3-' 2.33 257, 1 9 -'^■°' 0,37 559 33.35 °'^ 82.35 '-3- 2.82 279.23 °"'; 1.08 130.07 ' 3-' 2.32 257.10 °°3 0.30 0.17 >•.?,? 0.66 34 0.04 5f;o 33.r.-.> , 83.08 , 2.82 279. «9 1. 00 132.01 2.31 257,12 0.35 5i;i 33.(;9+°-'7 85.01+'-" 2.82 2s().54+°-''5 1.(15 1:13.33 + ' 32 2.30 2.--7,o7-°-°-; 0.34 5(!2 33.Hfi °-'7 80.-35 '•■'•* 2.81 2si.n °''' 1.03 131.05 32 2 29 257,00 °°l 0.33 6(;3 34.93 °'; 34.19 °'^' 0. :6 87.09 '■•'■* 2.81 2S1.78 °-^' 1.02 135.97 3- 2.27 0.33 5U4 89.04 '-3; i,,;6 2.80 2S2.38 °-^° °-59 1.01 137.28 \ 3* 32 2.20 2.-,o',s2 ° '° 0, 1 1 0.32 5r.5 34 .n5 , , 31.51 +°'f' 90.40 , , 91.70+'-"^ 2.80 282.97 0.99 13^,00 2.25 -•^'■•'' 0,, 0.31 5i;(i 2.79 2s3.54+°-5' 0,98 139.91 +, .1 ( 2 21 2.-,0 59-°" 30 5<.04+ 3 2.77 2S5.07 , 9.93 145.10 , 2.19 255,90 0.27 !)71 2.70 2S0.10+°I2 ■1.92 140.40 + ! 30 2.18 255,77-°:^ 0.20 572 3.V41 l[-\ 1(111.03 '••''^ 2.70 280.04 ^-l:' 0.90 147.70 J 30 2.10 2.V..57 °- ° 0.20 573 3:V5:, °- -^ 101.42 '••" 2.75 2S7.10 °-''^' 0.89 14!1.05 29 2.15 255.30 °- 0.25 574 O.I.^ I()2.S2 '■•'° 1.40 2.75 2S7.55 "•■'■^ -I,? 0.87 1^.0,35 , 30 29 2.13 ^^^•'^ :.]i 0.25 575 57(1 35.95+° '• 101.22 , l(l.-,.r,2 + ''° 2.74 2.73 287.98 , 28^.40+°'- 0.8fi 0.85 151.01 ,, l-.2.9::"^' 20 2.12 2.1] 25 1 , S9 r 254,03-°:; 0.24 0.23 577 3n.08 °'-^ 107.03 '•;'' 2.73 2'.!^ so °-^° 2S9.18 °-'''^ 0.S3 154,22 2') 2.09 2.-i4.30 ^-^l 0.23 578 3(;.2i I'l 10S.44 [^' 2.72 0.W2 155.50 2.08 0. 22 579 3(i.33 0.12 109.85 J-^J 2.72 289.55 °-'J o.j6 0,80 150,77 , -'7 28 2.00 ^^^'^" :.^ 0.22 5>*0 5S1 3(;.(;9 °'-' 3.;.s, °;-: '"■'-"4-1 12 112.08+ ■■' lU.lO ["[l 11.-..52 •;^ 110.95 ;;]j 2.71 2.70 289.91 , 290.25 +°-'^ 0.79 0,78 158.05 ,, 159.32''" 27 2.05 2.01 -•'"■••«_o V 0.21 21 5^2 5^3 2.09 2.08 290.57 °-;' 290. S8 °''' 0,77 0,70 100. .-o 101,S5 27 2,02 2.01 0.20 0.20 684 3.1.93 °;; 2.07 291.18 :-^ 0,75 103,11 , 26 1.99 0.19 585 37.04 118.38 , 2,07 291.40 0,73 104,37 1.98 251,75 0.19 58(5 37.1:. +°" 119.81 +'•'»•' 2.00 291.72+°-'^ 72 105,02 + ' 2,i 1.97 251, 30 -°-''' 0.18 5.S7 37 2fi °- ' ' 121.24 '•■'■? 2.05 291.90 °-* 0,71 100,v7 ' 25 1.95 250,90 °-'° 0.18 5S8 37.;;7 °" 122.07 ' '■' 2.04 292.19 ° -■' 0,70 KWII ' ■4 1.94 2.-.0.55 °-t' 0.17 589 87.48 °" 0. 10 121.11 '^-^ >-43 2.03 292.41 °-' 0. 20 0.09 109,35 ' -M 1.92 2.-10,13 °-" 0.44 0.17 5'10 37.59 , 125.54 , 2.02 292.01 O.OS 170.58 1.91 219.09 0.10 591 37.(;>s+°'° 120. 98 + '-'-* 2.(11 292.79 +°''^ 0,07 171, H1+' 23 1.90 249.24-°-;^ 248,78 °-t'^ 0.10 592 37.78 ^'^ 128.41 '•« 2.00 292.90 ° 'J 293.12 °' 0,00 173,03 ' , 22 1.88 0.15 593 37.88 °' 129.85 ' •♦^ 2.59 0.04 171,25 ' ,22 1.87 248,31 °-*l 217.83 °-»** 0.15 594 37.98 °'° 131.29 '••♦■♦ 2.58 293.20 °''* 0.03 175.40 ' .21 1.85 0.14 0. 10 1.44 0.12 .21 o.";o 595 3S.08 , 132.73, ,, 2.58 293.38 0,02 170.07 , 1.84 217,33 0.14 59A 3S.18+°''" ,,.,.,7 + 1.44 2.57 293.49+°" 0,01 177. X7+' .20 l.«3 240 S2- 5 0.14 697 3>< 27 ° °'' 135.01 '•••♦ 2.50 I293.5X °°'> 0.00 17".t.(t7 . ,;o l.Sl 210,30 °-''' 0.13 598 38.30 °"'' 1. •17.05 ' '* 2.55 293 00 °°'^ 0,5 s 1HO,20 ' II) 1,S0 21577 ° •'^•' 0.13 699 3S.45 "■'"' i;is..-,o ' •'^ 2 51 293.73 °"' 0.57 1>4l 4.23 13.99 0.70 ! 1.90 I3H1 393 738 088 510 147 97 542 4.34 14.05 0.78 1.90 1392 380 751 089 507 147 90 54:{ 4.45 14.11 0.79 1 . 90 1400 a78 703 091 503 140 90 544 4.50 14.17 0.81 1.91 1408 371 770 092 500 140 95 545 4.07 14.23 0.83 1.91 1410 303 7S8 093 497 140 95 54ti 4.79 14.28 0.84 1.91 1124 350 801 095 491 140 95 547 4.90 14.34 0.85 1 1.91 1431 319 813 090 491 140 94 54S 5.01 14.39 0.87 1.92 1439 343 820 098 4S7 145 94 549 5.13 14.44 0.89 1.92 144(; 330 t'38 099 484 145 94 550 5.24 14.49 0.90 1.92 1454 329 851 700 4SI 145 93 551 5.35 14.54 0.92 1.92 1401 323 804 702 47s 145 93 552 5.47 14.58 0.93 1.92 1408 310 877 703 475 145 93 55;{ 1 5.58 14.03 0.95 1.92 1475 310 890 704 471 144 93 554 5.70 14.07 0.90 1.92 14S2 304 903 705 408 144 92 555 5.82 14.71 0.98 1.92 14S9 298 910 7ni; 405 144 92 55r> 5.94 11.75 0.99 1.92 1 490 292 929 70s 402 I4t 92 55 T 0.00 14.79 1.01 1.92 1502 2H7 913 709 45S 141 92 558 0.18 14.82 1.02 1.92 1509 2S1 950 710 455 143 92 559 0.30 14.85 1.04 1.92 1515 270 970 711 452 143 91 500 (5.42 14.88 1.05 1.92 1522 270 983 712 449 143 91 5r,l 0.54 H.91 1.07 1.92 1528 205 w.h; 71.i 445 143 91 502 0.07 11.93 1 08 1.92 15.J4 200 1010 711 4 42 143 91 5G:J C.79 It. 90 1.10 1.91 1541 255 1(23 715 438 142 91 5G4 6.91 It. 99 1.11 1.91 1547 250 M:!i; 710 435 142 90 505 7.04 15. ni 1.13 1.91 1.553 240 10(1) 710 431 142 90 500 7.10 15.(i:l 1.14 1 91 15.-.9 242 lOi;:! 717 428 142 90 507 7.29 15.05 1.10 1.90 15r,4 2.3 s lOTO 717 421 142 90 508 7.41 I5.0fi 1.17 1.90 1570 231 lOSO 717 421 HI 90 509 7.54 15.08 1.18 1.90 1575 230 1102 718 417 141 89 570 7.00 15.09 1.20 1.S9 1581 220 1110 7is 413 141 89 571 7.79 15.09 1.21 1.89 I58(i 22.3 112:1 718 410 141 89 572 7.91 15.10 1 . 23 1.89 I5:il 219 1112 71S 400 Itl 89 573 8,04 15.10 1.24 1.88 I59i; 210 1155 719 4o:t 140 89 574 8.10 15.11 1.20 1. 88 1001 213 1109 719 399 140 89 5T5 8.29 15.11 1.27 1.S7 1005 210 1182 719 395 140 88 570 8.42 15.12 1.29 l.Sfi 1010 207 1195 719 392 no 88 577 8.54 15.12 1 30 1.80 i<;i4 205 1208 719 388 HI) 88 57 S 8.07 15.11 1.32 1.8.T ' 1018 202 1221 719 3S.^, 140 88 679 8.80 15.10 1.33 1.84 1022 200 123 4 719 381 139 88 5 SO 8.93 15.09 1.34 1 84 1020 19S 1218 719 37 s 139 88 5SI 9.05 15.08 j 1.35 I. S3 j 1 030 19i; 1 21; 1 719 371 139 87 5S2 9.18 15.07 1.30 1.8.1 10:!4 19 1 1271 719 371 l;!9 87 5s;} 9.31 15.05 1.38 1.82 ir,:{7 I9i JOsIS 719 307 139 87 584 i 9.43 15.04 1.39 1.81 1041 191 1301 719 304 139 87 585 9.50 15.02 1.40 1.91 1045 ISO 1315 718 300 139 87 680 9.08 15.00 1.42 1.80 HUS 188 : 1328 718 357 139 87 587 9.81 14.98 1.43 1.79 1051 IM.; 1342 718 35:1 139 87 588 9.93 14.90 1.44 1.79 1054 185 1350 717 350 138 80 589 10.00 14.93 1.45 1.73 1057 184 1309 717 340 138 80 590 10.18 14.91 1.40 1.77 1000 18.3 1382 717 343 138 sr, 591 10.31 14.88 1.48 1.70 |i;(;2 1S2 1390 j 710 339 138 80 592 10.43 1 t.s5 1.49 1.75 1005 1S2 1409 1 710 335 138 80 593 10.55 14.81 : 1.50 1 74 1007 isi , 142i 710 332 138 80 594 10.08 14.78 ; 1.51 1.73 1009 1 181 1430 715 328 138 85 595 10.80 14.74 1.52 1.72 1071 ISl 1448 715 324 138 85 590 10.92 14.71 1.53 1.72 1073 181 HOI 711 321 138 85 597 11.04 14.07 i 1.55 1.71 1071 181 1t7t i 711 317 137 85 598 11.10 14.03 1.50 1.70 ICTO ls| 14S7 713 313 137 85 599 11.28 14.58 i 1.57 1.09 11,77 182 ! 1500 712 310 137 85 fiOO 11.40 14.53 1.58 1.08 1078 182 1513 711 300 137 85 t 254 TIIK ORBIT OF URANUS. [':; TAlil.K X, Alio. 3. — Aitupn otf Nkiti nk. Ar-. (f.r.O) Diir. (I'.-i.l) Diir. (r.c.l) Diir. (r.«.2) I)iir.(,iv.2) Dill'. (r.«.3) (I'.r.W) (r.^.i) , (•••'••4) „ #f tl /» // M tt tt ft tf It tf tl 02. Of. 0.87 , 31.22 , 7.42 , 3.05 0.80 1.15 —0.85 — 1.13 1 ii2.r.7-°"^ 0.08 +°'" 31.77 +°-^S 7.30-°°^' 3.82-°-'-' O.sl I.IO O.SO 1.14 i 02 38 °-'^ 1.10 °'^ 35.31 °-S-» 7.;iO °°'^ 3.70 °'^ 0.X3 lis 0.S7 1.15 3 02.07 °-'' 1.23 °'-l 35.S5 °-5« 7.23 °°7 3.58 °-'- C.S4 1.20 SO 1.10 4 »'•" I:]: 0.15 30.39 ::M 7.10 °°7 3.47 °" 0.07 O.II U.80 1.22 O.OI 1.17 5 01. 4n 1.53 , 30.02 , 7.00 .. 3.30 0.S8 1.23 —0,03 — 1.18 (> 01.15-°'' 1.70+°-'7 37,41 +°--^=' 7.01 -°°'' 3.L'5-°" U. 0(1 1.24 (», !I5 1.18 7 0O.S4 °''' l.ss °'"' 37.00 °-5-' 0.02 °°'' 3.15 °'° 0.01 1.25 0.07 1.10 s 00.52 ° •^'' -'•"■^ 0'.' 3.48 °-5^ 0.S2 °'° 3.05 °'"^ 0.03 1 . 20 1.00 1.19 9 00.19 :^ 2 29 °'" 30.00 °-"' 0.51 0.72 °'° 2.05 °'° O.II o.oy 0.05 1.27 1.02 1.20 10 so.sfi 2.51 , 30.51 0.01 2. SO 0.07 1.28 —1.04 — 1 20 11 H0.53— °--^' 2.75 +"•-''» 4(l.(i2+°-5' 0.51 -°"° 2 77-o-oy 1.00 1.28 1.00 1.19 12 SO. 18 °-3S 2. CO °-t 40.52 °-5° 0.40 °" 2.00 °-°''* 1.02 1.2:. 1.08 1 10 l.'t HA.H3 °'^'' 3.20 °-7 41.01 °-W ,. ojj 0. I J ., ,;| 0.08 1.05 1,29 1.00 1 18 14 88.47 °-''' 3.54 °-''' 41..,0 °-l'> oilO °'-l2;54 °-°7 1.07 1.29 1.11 1.18 0.3S 0.J9 o..»y 0. 12 0.07 1") 8*<.00 3. S3 , „ 41.00, 0.04 2.47 1.00 1.29 — 1.12 — 1.17 ir> K7,7,-°-3^ 4.13+°-*° 4.J .(7+0.4S 5.01 -°'-^ 2.41-°°" 1.12 1.20 1.13 1.15 17 H7.31 °-«° 4.15 °i' 4-1 04 °-»7 5.78 °'3 2.;i,i °°5 1.14 1.2s 1.15 1.13 IS 8r.,oo °-*' 4.7s °-53 4.!. 40 °-tf 5.04 °'-« 2.31 °'°5 1.17 1.27 1.10 1.12 1!) 80.49 °»' °-4.5 6.13 °y= 0.3O 43.80 °-*^ 0.45 5 50 °'t 2 "7 °°'' 0. 141 "'" 0.04 1.19 1.20 1 17 1.10 20 8(1.0(5 5.40 , 44.31 , 5.30 !2.23 1 21 1.25 — I. is — l.OS k>| s.-,.r,2-°'',' 5.s7+°-''^ 4,.;, +0.4.1 5 22—°''|2 •.'0-°°-J 1.23 1.23 1.10 1.00 '2-2 85.10 °-t'' 0.2(1 °-''^ 45.10 °-*^ 5.08 °" 2.18 °°' 1.25 1.22 1.10 1.03 'J.l 84.00 °-'7 0.(;7 °-i' 45 58 °-i- 4.03 °'-^ 2.10-°°* 1.27 1.20 1.20 1.00 a 4 84.22 °-'7 0.50 7.00 °''-' 0.4.) 45.0s °--l° 0.40 4.7s °'-^ 2.10 °°° 0.15 0.00 1.29 1.10 1.20 0.00 2r) 83.72 7.53 , 40.3Sj^ 4 03 2.10 1.31 1.10 — 1.20 —0.07 •jc. 8:!.22-°-5'' 7 !is+°-»5 4C,.-T+°-'^'' 4.48-°- "S 2.10 °'°° 1.33 1.11 l.lOi 0.05 27 82.70 °-^' S.I5 °-i' 4T.lt "■•7 4 n °'^ '» 17+°'°' 1.35 1 1.12 1.101 0.0;! 2S 82.17 °-^' s.oi °-'" 47.10 °-y> lilS °'-V2:iO °°- 1.30 1.10 lis O.itl 2'.» 81. 02 °-^-^ 0-57 0. ^0 ''■'' 0.5. 47. S3 °-'t 033 j „.) oif' ., .,0 0.03 ' ' o.iSi"" 0.04 1.37 1.07 I.17j 0.00 SO 81.05 0.05 , 48.10 , 3.S7 ■ I2.20 1.38 1.05 -1.17 —0.89 :!l 80.48—^''' ,„ ,^+°o,5 4s.i7+°-'' .; r.i-o'.i ., .,„4-o°4 1.3s ' 1,02 1.15 0.88 »4 70. SO °P'' 70.20 ° '° 11 5s °-:'.' 12.15 °:' 0.5.S 4..7,; "•=:' 4:1.. 11 °-'; 40.. 10 °-''^ o.2j 3:.5S °'-t 2;35 °°s 3.13 °'-^ 2.11 °°'^' 3.2s °'-=' 2.4S °°7 0.14 0.07 1.30 1.40 1.40 1,00 07 0.01 1.13 111 1 00 ; O.SO 0. S5 0.S4 35 7S.05 , 12.73 , , 40.55 . 3.14 2 .55 , 1.40 O.OI — 1.00 — 0.83 :<)> (7.41 y l:i.:!3+°?" 4'..77+°-=-' 3.01-°'' 2.0l+°-°9 1.40 o.ss 1.07 0.S3 .•a 3'J ■Hi Hi 0.65 40. OS °-^' 50.10 °';^ 50.32 °'^ 0.14 0.14 „-., 0.09 •"74 °'-' ■>'s:i °'° 201 °"' :';o4 °-" 0.12 O.II 1.40 1.39 ■ 1.38 0.S5 O.S-J 0.70 1.05 1.03 0.00 0. 82 0.S2 0.81 40 41 4-2 43 44 >^-s" C.70 S^s+°- 2.10 3.05 , 03s -°" 3 17+°" 2.27 °" 3 30 °'' 2 17 °'° 3 14 °'-' •..37 1.35 1.34 1.32 1.30 0.70 0.73 0.70 0.07 0.05 —0.00 07 0.05 0.03 0.01 —0.81 O.sl 0.S2 0.S2 0.83 45 71.14 10.20 , 50. S3 1.07 !3'^,„,. 1.28 0.02 —0.01 —0.83 4ri 70.39 -°-75 10.07+°- 7' 50. S3 °-°° 1 ss-°°'' 3.H7+°-'5 isl °°7 4.o:t o-*(> 1 -. 0.00 , |i) 0. 10 ,;.;; 0.06;^;.,;. 0.17 1.25 0.00 0.91 0.84 47 00.03 °''' 2(1. OS °"' 50.sl-°-°;' 1.23 0.58 0.S7 O.SO 4S OH. 87 °'7*' 21.40 ^^-7-' 50.75 °°'^ 1.20 0.50 0.80 0.S7 40 08.10 °" 22.12 °'- 50.07 °°'^ 1.17 0.54 0.80 0.88 0.78 07.5 o.io 0.0^ 0.17 50 07.32 22 S5 50.57 1.04 ■ 4. .53 1.14 0.52 —0.84 —0.00 51 (•,0.54 -°''^ 23.5S+°-7' .50.11-° '.' 1.01 -°°-^ 4.71+^-'!^ 1.10 0.51 0.84 02 52 05.75 °" oj j.j 0-74 50.20 °'5 1.5S °°-^ 4. so °''^ 1.07 0.40 0.83 0.04 53 04.00 °-7" 25.(10 °"> 50.11 °''^ 1.50 °°^ 5.(»7 ° '■;; 1.03 0,4 s 0.82 0.00 54 04. IP, °''° o.Si 25.70 °-7.' 0.74 40.00 °" 1..55 °°' 5.25 °''^ 0.24 0.00 0. iS 1.00 0.47 0.81 0.98 55 03.. 35 20.53 , 49.00 1.55 . 5.43 . 0,00 0.47 —0.81 — 1.00 5fi 02.54 -°-^' 27.27+°- 7' 40.:io-°-^7 1 57+°°' 5.02+°"' 0.02 0.47 0.81 1.01 5t 01.7.3 °f 27.00 °"- 40 11 °-'^ 1.50 °°- 5.SI °") ix.i °-°-' 5.00 °'« 1.00 °°,',0.I7 °''' O.sO 47 0. S2 1.03 5S 00.02 °''' 1 OS 7-2 °-7,? 4S.70 °-^: 0.S5 (•48 0.82 1.05 50 00.11 °-^' , 20:45 °-7-' 4s.ll °-^^ O.sl 0,48 0.83 1.07 o.^l 1 - •' 0.7J 0-37 0.06 0. i<; (10 50.30 1 30.17 4S.07 11.72 0.30 0.77 I) 40 —0.83 ,— 1.09 TIIK OK II IT OF U KAN US. 205 TAItl.R X, AlUJ. n.— Continued. 1 Arff. (I'.c.oj Diir. (i).«.i) uiir. ( ' ) Diir. (f.N.a) Diir. ((•..•.2) Diir. (...«.3) (u.cM) (.-.8.4) It If tt II /» /' " II j // " II It tf If no r.9.;!0 „ 80.17 , ^;:2-°n l:;^o.o. ^;Jl+°- 0.77 0.49 —0.83 —1.09 ni r.s.49-°-^' :ii(.H,H+°-7' 0.73 0..50 0.84 1.10 (\i r.7.07 °-'' 31.59 °1' 47.24 °-'-' l.HO °°'^ 0.71 °'' u.i;9 0.52 0. 85 111 (\:i M.HC, °'^' ;;....., 070 40.79 °-'j 1.95 °°'J 0.H9 °"' 0.00 0.51 0.87 1.13 i;4 50.05 °'^' o.Si 32:97 If^ 0.67 40.31 °-'''' 2.05 °;° 0.50 0.1 1 ^•"« :.\] 0.02 0.5(; 0.8S 1.14 05 r>5.'_>4 3.3.04 45.81 2.10 , 7.23 , , 0.59 0.59 —0.89 — 1.15 (',(! 54.44-°^° 3..30+°^''' 4-, .,i-°S.l ., .,s+°'^ 7.39+°;' 0.55 0.02 0.91 l.KS fi7 (is 5:!..;4 °-° 31.95 °'5 :5.,5S °^ Jiiis °P 0; s^ii 7.09 ° = 0.53 0.50 0.05 0.08 0.93 0.95 1.10 1.17 f.9 52.05 ° '^ 0.79 <»■'■« IZ ■"' -s ^•«^ o.\] 0.47 0.71 0.97 I.lH 70 51.2fi 30.82 . 42.93 , 2.85 . ,, T-f^^L^ ,, 0.45 0.75 —0.99 -1.19 71 50.47-°- J'' .37,41+°" 4.. .>H-°^i •Mc.+° '7 •^•"^"^o' 0.13 0.78 1.00 1.19 7 J 7:» li?,;; - .37.98 °' 3H.53 °--''' .8,20 °': H.31 °" 0.41 0.39 0.H2 0.S7 1.02 1.04 1.19 1.18 71 '••■' :t 39.00 °-2^ 0. ,0 ^"-'^^ ::;i ='.^^" ::;:; «•"• ::::] 0.38 0.91 1.00 1.18 75 47.43 39.50 39.51 3,75 , '^•■*!' . 0.37 0.90 —1.08 — 1.18 70 4(;.(;o-° 74 4(1.05+° ■•'' 3s,77-°-7« 3,90 +°-'' 8.57+°-°'' 0.30 1.00 1.10 1.17 77 4:.. or, °7' 40,52 °t7 3S,01 °7" .j.K; °-'° 8.01 °°7 0.30 1.05 111 1.10 7S 4.-.. 2 4 °7' 40,90 °l> 37,23 °T^ 4,37 °-' 8.09 °°-=! O..30 1.10 1.12 1.15 7 'J 44.52 °7^ 0.70 41.38 °-l^ 0,40 30.41 °T> 4.59 °-' o.Si ■ o.Ji 8.73 °°4 O.OJ 0.30 1.14 1.13 1.13 HO 43.83 41.78 35.03 „ 4,80 , 8.70 , 0.37 1.18 —1.14 —1.12 Kl 4:1.14-°^'^ 42,10+° '« 31,81 -°'^- 5,02+° " H.77+°°' 0.38 1.23 1.15 1.10 S3 42.40 °^'? 41. HO °^;^ 42,51 °.'5 33,98 °^^5,2t °" 8.78+°°' 0.40 1.28 1.10 1.08 h:! 42.83 °-'- 33,13 °'^> 5,17 °-'-^ 8.77-°°' 0.11 1.32 1.17 1.00 H4 41.15 °f' 43.12 °-7 0. 26 32.28 °-^; 5.09 °-'' 0..SO 0.22 8,75 °°^ 0.05 0.43 1.30 1.17 1.04 H5 40.52 43,38 , 31,12 5.91 , 8.72 0.10 1.41 — I.IH — 1.02 HI! .■!;i.90-°f^ .•i;t.:!0 °'*'' 4:!,01+°■-^' 30..55-°«7 ,i.l:{+°-'^ 8.08 -°°,t 8.02 °°'^' 0.19 1.15 I.IH 1.01 X7 4.3.82 °-' 29.(18 °-^7 ,;.:i5 °-'^ 0.52 1.49 I.lH 0,99 «S .M.Tl "^''^ 41.00 ° ''^ 2H.H0 °f 0,57 °-''' S.55 °-°7 0.55 1.52 1.17 0,97 SO 38.14 °:i 4 4.10 °"^ 0.12 27.91 °'^'' 0.79 °-- o.yoi 0.21 8.47 °°'' 0.0(> 0.59 1.50 1 10 0.95 00 :!7.58 41.28 . 27. ni „ 17.00 , 8.38 0.02 1.50 —1.10 —0.94 ill .•!7.04-°-5> 41,38+°'° 20.12-°«'' 7.21 +°'^' 8.27-°" 0.00 1.01 1.15 0.92 !)2 :!:.52 °'^' 41,14 °°^ 25.23 °'^'' 7.11 °;° 8.15 °'-' 0.71 1.01 1.14 0.91 !i;j ;!(;.02 °=^° 41,l8+°°-« 21.33 °''° 7,00 °- '^ 8,02 °'-' 0.7(> l.OC, 1.13 0.90 y» 0. .(6 44.48 °°° —0.02 23,13 °''° 7,79 ° 2 o.Sy 0. IS 7.89 °'} o'5 O.SO l.OH 1.12 0.89 !)5 .'tr>.07 41.40 , 22,51 „ 17,97 , 7.71 . 0.H5 1.70 —1.11 —0.88 !)(! ;!j.02-°'3 41,10-° °^' .,, ,;-,-o,80 >, 1,4-0.17 in °f h:3,1 ° -^ 7.58-°"' 0.90 1.71 1.10 0.H7 !»7 :!1,20 ° »* 41.32 °°'' 7.11 °;^ 0.95 1.72 l.OH O.H(i its xuo °-'i 41.20 °'^ 19.90 °i^ H.45 °'' 7,21 °'7 1.00 1.72 1.00 0.H5 09 -■" ::;;■; ^'•"^ :::7' -■' ::';;;;«"o i:\i 7,05 °"^ ' 0,20 1.05 1.73 1.01 0.84 100 :!:?.(»! 43,90 H.IO a. 8.73 . „ ,, •■'^•"' „ „ MO 1.72 — 1.03 — 0.H3 tol :i2.70-°^* 43.70-°-'° 17.31-°^-^ H.s.;+°- ■ —0, 20 1.15 1.72 1.02 0.H3 102 .T.'.:t8 "•'' :;;:;! "" 10.10 °''^ H,97 °" ... 0,21 (11 1.21 1.70 1.(10 0.H3 lO.'l .-12.08 °'° 1.-..02 ° ■'» 9.00 °°'' . ... 0.21 0.23 1.20 1.09 0.9H o.H:i 104 =">^« :J6 42.92 °-"' •^•- ::«; '■• '-^ Vol "•••' III 1.31 1.07 0.90 0.83 105 ni.54 42,01 13.99 9.23 , „ . 5.79 1..35 1.05 —0.95 -0.83 IOC, 107 los -m1 ,.j.,„-°-7') ., •!0+°°^ 5.57~°" ■ 12 ° ^'^ ; - °°-^ 534 °'^ ?: ° 7^ -9 °°! ': ^ :'\ 1.40 1.44 1.48 1.03 1.00 1.50 0.94 0.92 0.90 (t.84 OH,") O.HC, 109 ^■•-^ oi; ■'•■"^ :;?] "'^ :::V'^^ -3 1.53 1.53 O.SO 0.87 no 30.03 40. r, I 10.19 9.13 , 4.01 1..50 1.49 — 0.8H —0.88 111 ;to.5l-°" 40.l7-°-'7 ,, ,,,-0.70 ,, ,,+0.01 ^ ,1,-0 2.» 1.59 1.45 0.87 0.89 113 :io.43 °°'> 39,08 ° f H.SO °'!'> 9.42-°°- 4.17 ° -'■' 1.02 1.41 O.HO 0.91 It:} .•!0.:!5 °°7 39.10 °-'i' HII °'j'^ 9.10 °°' 3.91 °-' 1.05 1.37 0.80 0.93 114 ;jo.;to °°5 3H.01 °5? 7. .50 ° ^'* 9.30 °°4 3.71 °-' 1.07 1.32 0.85 0.95 — O.OJ 0.56 0.62 1 005 1 o- 23 il5 .10.28 3H.fl5 _ .37.17-°S« 0.S8 , 9.31 3.48 1.70 1.27 —0.84 —0.90 lir. ;io.2n+°°' C, .)s— '•o ,-o 07 3.20-°-" 1.72 1.22 0.84 (I.!t7 117 :to.:i2 °°' 3.;,.Hi; °-^" 5.71 °-^7 9 17 °°7 3,01 °-^ 1.73 1.17 OS 4 0.99 118 .•to.;j7 °°^ 30,21 °^' 35,00 °^'t 5.17 °-5-* 9.08 °°'' 2.S2 ° -^ 1.74 1.12 0.H4 1,01 119 30.10 °"'^ 4.05 °S- 8.99 °°'^ 2.01 °-' 1.74 1.07 0.H4 1,02 O.IO 0,65 0..49 0. II 0.20 120 .30. .^0 34.95 4.10 8.88 2.41 1.74 1.02 — 0.H4 —1.04 25G T II K O 11 It 1 T O F U K A SVA. T.VIlLK X, .\[ui. 3,— ^»/l^/lH.7/. 1 Arg. (y.c.O) Diir. ,(».».l) Diir. (o.i'.l; Diir. i».«.2;Diir. (.u.c.aiDill (('.8.3) i|).r.3 (i'.».4 1 " 1 " »f »/ #/ 1) II »r »» " n It 130 30. 5C , 3;;,5,s °'"^ 4.10 H.NS 2.41 ■ 1.74 1.02 —0.84 l_ 1.0 1 1 iL'l ii0.70+°'» 3.(;9-°'<7 8.70-°'» 2.31-°-^° 1.74 0.97 0.8-1 l.OC 133 30.,s5 °'-^, 3.2(1 °-'» ,S.03 °'-' 2.02 °"^ 1.73 0.91 O.8.- 1.08 i^a 81.03 ° ''^ 33. ss °- ° 2.S5 °-',', 8.49 °''» 1.84 °''^ 1.73 0.^(1 0.8( 1.09 124 31.33 °-'° 33.15 °- • a. 47 °'!; 8.34 °:5 1.C7 °''; 1.71 0.81 0.87 l.lu o^.} 0-7J 0.34 0. tO 0. i() li'S 31.4(1 . •■"•^•■2 „ ,. 2.13 S.18 , s.<.2-°"* 1.51 1.70 '•■ 0.77 —0.88 _1.11 llT. 31. 71+°='-^ 30.C7-°!-! l.«l-°-'-' l.:io-°;5 l.n7 0.72 0.9( ' 1.13 137 13S 81. lis °-7 3.) 0^ °-.lo 30.1.1 °lt i..5a °-;;' 7.S5 "■;' 1.30 °- ^7.07 °- •' 1.21 ° '3 1.08 °'' 1.05 1.(13 0.C8 o.(;3 0.93 1 1.13 1.13 139 33! (Tl °-'3 3«.39 °1' 1.03 °:''-7.48 °"> 0.90 °''- 1.59 0.59 0.94 1.14 °-35 o.-.S 0.30 0.19 0. 12 I.'IO 33.!)fi , _ ! 27.00 „ , 0.S3 , 7.29 „ 0.84 1.5(1 0.55 0.95 — 1.15 l.'U s:i.:n+°-3^ 3C.s3-°- " 7 09-°° 0.74 -°''° 1.53 0.51 0.9(1 1.15 1.13 33.74 °f 2C.03 °-." "••* n^ (;.,S8 °- 0.05 °°'' 1.49 0.48 0.98 1.15 l:t3 l:!4 31.1(1 °-»- 3 4. CO °-«/ "5 -'l "'J 24:i] If 0.44 °"° 0.40 °-' :.i'i " 1.45 1.41 0.45 0.43 0.99 1.01 1.15 1.10 0.4O 0.79 0.04 O.-M 0.05 l,!.') 35.0(i , „ „ 35.54 +°-»'^ 23. C5 0.33 0.25 O.40 1.37 0.40 —1.03 — l.Ifl mr, 33.H(J-°-7y 0.32-°°' 0.03-°- --• 0.41 -°°5 1.33 0.38 1.05 1.15 l:!7 8C.05 °5' 33.0(5 °''° 0.:{5+°°3 5.SI °-- 0.38 °°.? 1.38 0.37 l.OC 1.14 i;)s 3C..5S °-5' 31.37 °'" 0.40 °°5!5.5» ° -- 0.30-°-°^ 1.33 0.35 1.07 1.13 l:i<.) 37.13 °55 20.4S °-7'^ 0.49 °°'^ 15.38 °-' O.30 °°° 1.18 0.34 1.08 1.12 °-57 0.79 0.12 0. J2 0.00 140 37.70 , , 10.00 O.OI , 5.10 0.30 , 1.13 0.34 —1.09 —1.11 141 3s.:io+°-^° H.'iO— °''' 0.70+°';' 4.95-^-' 0.38+°°' 1.09 0.33 1.09 1.09 143 3><.ii3 °-^'- ls.13 °"7 0.04 °'" 4.73 °-- 0.41 °°3 1.01 0.31 i.lo 1.07 143 3'.t.55 "l'' 17.30 °'J 1.15 °-^' 4.52 °'-' 0.45 °°» 1.00 0.31 1.11 1.00 144 40.31 °-^''' 0.O7 IC.fiO ° ''' °-75 1.39 ° -^' 4.31 °-' 0.J7 0.21 0.50 °°-^ C.07 0.95 0.35 1.13 1.05 145 «>•■'*'* , ^0 15.,S5 1.00 . 4.10 0.57 0.91 0.30 —1.13 —1.04 it<; 41.5(J+°- 3.71 °;'^ 0.f<3 0.40 1.13 1 03 14V( 43.00 °"-' 13.07 °- -' 2. ("3 °-'i 3.53 ° "-' 0.82 °°? 0.7S 0.41 1.13 1.01 1,9 43.73 °-7^ 13.97 °J° 3.01 °-''^ 3.33 °''' 0.93 °" 0.74 0.44 1.14 0.99 0.76 0. 69 0.41 0. IS 0. 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(- Hso '.>,"i'.'0 2S."iO 107 — :,•>:, — 101 — IlKl r.iiio Mil 111+° 5° s,;is 7.'js s70 'Jdll 207(1 :'.(i.! :>.;:\ lot inn ri7ii lj!i:>.-. °-»'' 111.0:1 " ■- S..M 7 :il sr.s •.Till.'. ;i!iii-.> fiiiii .Ml 10.-. 1(17 I'.ISd ljo.0.1 °-»'' M-.'..Ml " '•' li;t.-jo °fi S ."lO 7 :il n;.s •J7"^0 :!-Jii7 .M.". .'■..'.(I 1 1(1.". 1117 I'.l'.lll l-.-s.,-,7 ° '•' .s 111 1 . -> 1 SIS -jsi;; :i:;:..| :,j| .'..•.s 1 1(1(1 111.. o.4,S o.^5 ■JIM III — IJS 0!) -in s:, — s {'2 -7 I'l I s;is__-."i.-,i, -.;i(:;i ,-,j: .Mir, — Kill _|iiH / ^%. %> t^ IMAGE EVALUATION TEST TARGET (MT-S) 1.0 I.I 1.25 m m IIM IIIIL2.5 '- IIM IIIIIZ2 2.0 lAO 1 1.8 U III 1.6 Photographic Sciences Corporation 23 WEST MAIN bTRflE' WEBSTER, N.Y. )45t:0 (716) 873-4503 4 4< .4^ ^^ \\ ^v O^ #.^ '^^^ ^ ^o^ f/j i \\ o'^ 27 28 29 32 33 34 35 3(i 37 38 39 40 41 42 43 44 45 190 191 192 193 194 195 19(> 197 198 19 199 200 201 202 203 204 205 20() 207 208 209 30 210 31 21 1 212 213 214 215 2 III 217 218 219 220 221 222 223 224 225 R 10.00 9.(>7' 9.35 9.02 8.70 8.37 8.05' 7.73 7.42 7.10 0.80 C.49- 0.19 5.89 6.00 5.32 5.04" 4.70 4.49 4.23 3.98 3.73- 3.49 3.2(; 3.04 2.82 2.(;2- 2.42 2.23 2.05 1.8S 1.73- 1.58 1,44 1.31 1.20 l.Olt 99 0.91 0.»4 -°.33 0.32 °-33 0.32 °-33 -0.32 0.32 0.31 0.32 0.30 .0.31 0.30 0.30 0.29 0.28 -0.28 0.2S 0.27 0.26 0.25 -0.2s 0.24 0.23 0.22 0.22 -0.20 0.20 0.19 0.18 0.17 -0.1 5 0.15 0.14 o 13 O. II — O.II O.IO 0.08 0.07 0.07 0.77 0.72 -°°5 0.08 °'°-^ 0.03 O.OI 0.01 0.05 0.04 0.03 50 51 52 53 55 50 57 58 59 00 01 02 03 04 05 00 07 08 09 70 71 72 73 74 75 70 77 78 45 225 40 I 220 47 ; 227 48 I 228 49 229 54 234 1.09 230 231 232 233 235 230 2371 238 j 239 240 241 242 243 244 245 240 247 248 249 250 251 252 253 254 255 250 257 258 I 2591 200 201 I 202 i R 0.03 , 0.04+°°' O.OI 0.05 0.08 0.72 0.03 0.04 0.0; 2:n+°°v 0.99 °°8 1.20 1.31+°" 1.44 °'-5 1.58 °'-* 1.73 °'5 0.15 1.88 2.05+°- '7 „ .,., 0.18 240 °'9 2.02 °-° 0.20 2.82 , 3.04+°-^-' 3.20 3.49 3.73 0.23 0.24 3.98 , 4.23+° 4.49 ° 4.70 ° 5.04 0.28 5.32 5.00 5.89 0.19 0.49 +0 o o o o S4 203 204 85 205 80 200 87 207 88 208 89 209 90 270 0,80 7.10+°- 7,42 °- 7.73 °- 8.05 °- o. 8.37 , 8.70+°- 9.02 °- 9.35 °- 9.07 °' 90 91 92 93 94 95 90 97 98 99 270 271 272 273 274 j 275' 270 \ 2771 278 279 I 110 111 112 113 114 115 110 117 118 119 120 121 122 123 124 125 3° 120 32 127 31 128 3-! 129 32 130 33 131 3^ 132 ,>,.•, 133 3-! 134 33 R 10.00 , 10.33+°-" 10.05 °-3^ 10.98 °'" 11.30 °-3^ 0-33 11.03 , 11.95+°- 3^ 12.27 12,58 12.90 0.32 0.31 0.32 0.30 1001 280 : 101 i 281 102: 282 103 1 283 104 I 284, 105 285: 100! 280 ! 107^ 2S7 lOS 2S8 109 289 13.20 13.51 +°-3' 13.81 14.11 14,40 0.30 0.30 o, 29 0,28 290 291 I 292 293 294: 295 2911 297 i 298 299 300 301 302 303 304 I 305 3(10 307 30S 309 310 311 312 313 314 14.08 , „ 14,90+°-=^ 15,24 °-^'^ 15.51 °%7 15.77 °'^ 0.25 10.02 , 10.27 +°-'5 10.51 °-^^ 10.74 °-'3 10.90 °" 0.22 17.18 n,38 17.58 17.77 17.95 4-0.20 0.20 o. 19 0.18 0.17 18.12 , 18 27+°-'5 19.42 °'5 1>^<'« o.M 18,80 18.91+°" 19,01 °-'° 19.09 °-°'^ 19.10 °°7 0.07 19.23 ]5 7-65 7,04 7-f',3 7.O2 7.61 7.O0 40 20 21 16.70 40 30 2.92 8.C2 S.o_' 30 30 13 56.11 30 30 £1 23. S8 7-17 7.1O 30 40 (i 10.94 20 40 114 3.S2 20 40 21 31.05 20 50 IS. 95 8. 01 8.01 10 5') '\ 14 11.52 10 50 21 38.21 1 7.14 10 8° 10' 2l!.9(! 31.97 S.oi 172° 50' 18' 10' 14 19.21 It 26.90 162' 50' 28' 10' :;l 45.35 21 52.49 7.14 ■? I 2 162° 50' 20 (i 42.97 8.00 40 20 14 34.58 40 20 21 59.61 ■7 13 40 30 1) 50.97 8.00 30 30 ' 14 42.25 30 30 22 0.73 /• '-^ 30 40 (i 5S.97 8.00 20 40 14 49.92 20 40 22 13.83 7.10 7,09 7,08 20 50 7 (•,.90 7-9'; 8.00 10 50 14 57.58 10 50 22 20.92- 10 9° 7 14.90 171° 19' 15 5.23 161° 29° 22 28.00 7,07 "7 0^ 151° 10' 7 22.95 7-99 50' 10' 15 12.S8 50' 10' 22 35.07 5(i' 20 7 .'i(l.9;i 7.9S 40 20 15 20.52 40 20 22 42.12 7-05 7-°3 701 40 30 7 3S.91 7.98 30 30 15 28.15 30 30 22 49.17 30 40 7 4(1. S8 7-97 20 40 15 35.77 20 40 22 56.20 20 50 7 54.85 7.97 10 50 15 43.38 10 50 23 3.23 / -^.1 7.01 ! 10 10° 8 2.S2 7-97 170° 20° 15 50.98 160' 30° 23 10.24 150° 190° !i 350° ■u 200° /5 340' u 210° a 830° u 35 July, 1873. 271 THE ORBIT OF TTRaNUS, TABLE XIX, Alio. M. — Cuiilin ued. u ' 3 u P u 3 210^ / 1/ 330" 220" / »/ 320" 230° / V 310° 30" 2;3 10.24 150" 40" 29 47.27 6 t9 6., 7 6.16 6.14 6.13 140 50" 35 29.99 130° 10' 23 17.24 7.00 6.99 6.1,7 6.97 (>')5 50' 10' 29 53.40 50' 10' 35 35.18 5- 19 50' 20 23 24.23 23 31.20 40 30 20 30 29 59.03 30 5.79 40 30 20 3(' 35 40.35 35 45.50 5'7 5-'5 40 30 40 23 3S.17 20 40 30 11.93 20 4j 35 50.03 5- '3 20 r)0 23 45 12 10 50 30 13.06 10 60 35 55.75 512 10 6.94 6. II 5.10 31° 23 52.00 6-93 6.91 0.90 6.89 6.88 0.86 149" 41" 30 24.11 6.10 6.08 6.06 6.05 6.03 6.02 139° 51° 30 0.85 5.08 129° 10' 23 5S.!)i) 50' 10' 30 30.27 50' 10' 3(! 5.93 50' 20 ao 40 50 24 5.!»0 24 12.80 24 ll).(i!) 24 20.57 40 30 20 10 20 30 40 50 30 3f;.35 30 42.41 30 48.40 30 54.49 40 30 20 10 20 30 40 60 30 11.00 30 10.05 30 21.07 30 20.08 5-07 5-°5 5.02 5.01 4-99 40 30 20 10 32' 24 33.43 6.85 6.84 6.83 148° 42° 31 0.51 138° 52" 30 31.07 128° 10' 21 40.28 50' 10' 31 0.51 6.00 50' 10' 30 30.04 4-97 50' 20 24 47.12 40 20 31 12.50 5-99 40 20 30 40.99 4-95 40 30 24 53.05 30 80 31 18.47 5-97 30 30 30 45.93 4.94 30 40 2.-I 0.77 6.82 20 40 31 24.43 5-96 20 40 30 50.84 4.91 20 M 25 7.57 6.80 10 50 31 30.37 5-94 10 50 30 55.73 4. 89 10 33= 25 14.3r, 0.79 6.7s 6,76 6-75 6.74 6.7.' 147" 43" 31 30.29 5-92 137° 53" 37 0.01 4.88 127° 10' 25 21.14 50' 10' 31 42.20 5-91 50' 10' 37 6.47 4.86 50' 20 25 27.90 40 20 31 48.09 S-«9 40 20 37 10.31 4-84 40 ;{0 25 34.05 30 30 31 53.90 S-«7 30 30 37 15.13 4. 82 30 40 25 41.39 20 40 31 59.82 5. 86 20 40 37 19.93 4.80 20 CO 25 48.11 10 50 32 6.00 5-84 10 60 37 24.72 4-79 10 34" 25 54.82 6.71 6.70 6.68 6.68 6.66 6.65 6.6,5 6.62 146° 44" 32 11.49 5.81 136" 54" 37 29.48 4.76 126° 10' ■2C, 1.52 50' 10' 32 17.30 50' 10' 37 34.22 4-74 50' 20 ;io 2() 8.20 .'..S0 40 39.09 40 43.50 40 47.41 40 51.24 40 55.05 40 5.><.84 2.00 0.35 10.07 13.78 17.40 21.13 24.77 2K.39 31.99 35. 5G 39.12 42.05 40.17 49.00 5.!. 13 50. 5S O.Ol 3.42 o.so 10.17 13.51 10.83 20.13 23.41 20.07 29.91 42 33.12 42 30.32 42 39.49 42 42.04 42 4."). 77 42 4S.S7 4 2 51.90 42 55.02 42 58.00 43 1.08 4.08 7.05 10.01 12.94 15.85 18.74 21.00 24.45 27.27 30.00 32.84 3 4- 03 4.01 4.00 3-97 3-95 3-93 391 3«9 3-«7 3-85 383 3.S1 3-79 3. 76 3-75 3-7^ 371 3.6S 367 3(>4 3.62 3.60 3-57 356 3-53 3-5^ 3-49 3-47 3'4.=; 3-43 3- 4 1 3-3« 3-37 3-34 332 3i° 3,28 3.26 3-^4 3-2I 3.20 317 43 43 43 43 43 43 43 43 43 43 43 31.=; 3- '.I 3.10 3°9 3.06 3°4 3.02 3.00 2.97 2.96 2-93 2.91 2.89 2.86 2.85 2.82 2.79 2.78 300° 120" 50' 40 30 20 10 U9° 50' 40 30 20 10 118° 50' 40 30 20 10 U7° 50' 40 30 20 10 116° 50' 40 30 20 10 115° 50' 40 30 20 10 114" 50' 40 30 20 10 113° 50' 40 30 20 10 112° 50' 40 30 20 10 111° 50' 40 30 20 10 110° 290° u 250° 70° 10' 20 30 40 50 •71° 10' 20 30 40 50 72° 10' 20 30 40 50 73° 10' 20 30 40 50 74° 10' 20 30 40 50 75° 10' 20 30 40 50 76° 10' 20 30 40 50 77° 10' 20 30 40 50 78° 10' 20 30 40 50 79° 10' 20 30 40 50 80^ 260° 43 32.84 43 35.00 43 38.33 43 41.04 43 43.73 43 40.40 43 49.04 43 51.00 43 r4.20 50.84 59.40 1.93 4.44 0.93 !>.39 11.83 14.25 10.05 19.03 21.38 23.71 44 20.02 44 28.30 44 30.57 44 32.82 44 35.04 44 37.23 44 39.40 44 41.55 43.08 45.79 47.87 49.93 51.97 53.98 55.97 57.94 44 59.89 45 1.81 43 43 44 41 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 l45 45 45 45 45 45 3.71 5.59 7.44 9.27 11.08 12.80 14.02 10.30 18.08 19.77 21.44 23.09 24.71 20.31 27.89 29.45 3it.98 32.49 33.97 35.43 30.87 38.29 2.76 2-73 2.71 2.69 i 2.67 2.64 2.O2 2.60 2.58 2.55 2-53 2.51 2.49 2.46 2.44 2.42 2.40 2.38 2-35 2-33 2-3' 2.28 2.27 2.25 2.22 2.19 2.17 215 2-13 2. t 1 2.0S 2.06 2.04 2.01 1.99 1.97 1-95 1.92 1.90 1. 88 1,85 1.S3 i.Ri 1.78 1.7O 1.74 1.72 1.69 1,67 ,.65 1.62 1.60 1.58 1.56 1-53 1. 48 1.46 J. 44 1.42 290° 110° 50' 40 30 20 10 109° 50' 40 30 20 10 108° 50' 40 30 20 10 107° 50' 40 30 20 10 106° 50' 40 30 20 10 105° 50' 40 30 20 10 104 5(1' 40 30 20 10 103° 50' 40 30 20 10 102° 50' 40 30 20 10 101' 50' 40 30 20 10 100' 280° n u 260° 80° 10' 20 30 40 50 81° 10' 20 30 40 50 82° 10' 20 30 40 50 83° 10' 20 30 40 50 84° 10' 20 30 40 50 85° 10' 20 30 40 50 8G° 10' 20 30 40 50 87' 111' 20 30 40 50 88° 10' 20 45 38.29 45 39.08 45 41.05 45 42.40 45 43.73 45 45.03 45 40.30 45 47.50 45 48.79 45 50.00 45 51.18 45 52.34 45 53.48 54.59 55.08 50.75 57.79 58.81 59.80 0.78 1.73 2.00 3.50 4.44 30 40 50 89° 10' 20 30 40 50 90° 270° 45 45 45 45 45 45 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 41; 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 5.30 0.14 0.95 7.74 8.50 9.24 9.90 10.05 11.32 11.97 12.59 13.19 13.70 14.31 1 1 . 84 15.35 15.83 10.29 10.72 17.13 17.52 17. S!) 18.23 18.55 18.84 19.11 19.30 19.59 19.79 19.97 40 20.12 40 20.25 20.30 20.44 20.50 20.53 20.54 3 1-39 1-37 1-35 1-33 1.30 1.27 1.2O 1-23 1. 21 i.iS 1. 16 1.14 I. II 1.09 1.07 1.04 1.02 0.99 0.98 0-95 0-93 0.90 0.8.S 0.86 0.S4 0.81 0.79 0.76 0.2: 0.09 o.r,7i 0.05 0.62 0.60 °-S7 °o5 0-53 0.51 0.48 0.46 0-43 0.4T °-39 °-37 0-34 0.32 0.29 280° 100° 50' 40 30 20 10 99° 50' 40 30 20 10 98° 50' 40 30 20 10 97° 50' 40 30 20 10 96° 50' 40 30 20 10 95' 50' 40 30 20 10 94° 511' 40 30 20 10 93° 50' 40 30 20 10 92° 40 40 4li 40 40 0.27 50' 0.25 40 0.23 30 0.20 20 o.iS 10 0.15 91° 013 50' 0. 1 1 40 0.08 30 0.06 20 0.03 10 O.OI 90° 270° u f! 276 THE ORBIT OF URANUS. Tablk XX. XXI. XXII. Aim. 1 2 3 (6.S.1) (6.C..1) (6.C.0) (6.«.1) (6.C.1) (//.S.2) (6.e.2) (6.C.0) (b.S.l) (h.c.V (6.M.2I (6.C.2) // // tt " tt // If II U n 1.20 1.12 0.04 5.99 5.42 0.21 0.17 0.0" 1.58 1.34 0.07 o.n 10 1.15 1.17 0.03 6.13 5.23 0.19 0.18 0. 06 1.63 1.14 0.08 0.23 20 1. 10 1.21 0.03 6.25 6.04 r.18 0.18 0.(15 1.67 0.93 0.11 0.29 30 1.04 1.25 0.(12 6.34 4.84 (,.17 0.18 0.04 1.68 0.72 0.15 0.32 40 0.98 1.28 0.02 6.41 4.62 0.16 0.18 0.03 1.67 0.51 0.21 0.33 50 0.92 1.31 0.02 6.45 4.40 0.15 0.18 0.03 1.60 0.33 0.27 0.32 GO 0.8(i 1.33 0.(12 6.45 4.18 0.16 0.18 0.(12 1.48 0.18 0.32 0.28 70 0.79 1.34 0.02 6.40 3.94 O.lfi 0.18 O.ol 1.32 0.09 0.35 0.23 80 0.72 1.35 0.02 6.32 3.69 0.16 0.17 0.(!0 1.13 0.07 0.36 0.18 90 O.GIl 1.35 0.(13 6.19 3.43 0.17 0.16 0.00 0.94 0.13 0.35 0.13 100 0.59 1.34 0.03 G.02 3.15 0.17 0.15 O.OI 0.77 0.25 0.32 0.09 110 0.52 1.32 0.04 5.81 2.87 0.18 0.14 0.02 0.66 0.41 0.29 0.07 120 0.4f. !.30 0.04 5.56 2.57 0.19 0.12 0.04 0.62 •.1.60 0.25 0,(1(1 130 0.40 1.27 0.04 6.29 2.26 0.20 0.11 0.06 0.64 0.7 V 0.22 O.Ofi 140 0.34 1.24 0.05 4.99 1.96 0.20 0.10 0.08 0.72 0.91 0.20 0.07 mo 0.28 1.20 0.05 4 (,'i 1.66 0.20 0.09 0.09 0.83 1.00 0.18 0.08 1(10 0.23 1.15 0. 05 4.36 1..S8 0.20 0.08 0.10 0.95 1.04 0.17 0.09 no 0.19 1.10 O.OG 4.03 1.11 0.20 0.08 O.U 1.04 1.04 0.17 0.10 ISO 0.15 1.04 O.OG 3.69 0.86 0.19 0.07 0,11 1.11 l.OI 0.16 0.10 190 0.12 0.98 0.06 3.36 0.65 0.17 0.07 0.11 1.14 0.97 0.14 0.11 200 0.09 0.92 O.OG 3.03 0.56 0.16 0.08 O.ll 1.14 0.93 0,13 0.12 210 0.07 0.8*'" O.Ofi 2.71 0.32 0.14 0.08 0.11 1.11 0.91 0,11 0.13 220 o.or, 0.79 O.Ofi 2.30 0.20 0.12 0.09 0.10 1.07 0.91 0.10 0.15 230 0.05 0.72 O.OG 2.08 0.12 0.1 1 O.U 0.10 1.02 0.92 0.10 0.18 240 0.05 O.GG 0.07 1.77 0.08 0.09 0.13 0.09 0.98 0.9C 0.10 0.21 250 o.on 0.59 0.07 1.48 0.08 0.07 0.15 0.09 0.96 1.00 0.12 0.23 2fiO 0.08 0.52 0.07 1.21 0.10 0.05 0.17 0.08 0. 95 1.05 0.14 0.25 270 0.10 0.4G 0.08 1.05 0.16 0.03 0.20 0.08 0.94 1.09 0.16 26 2S0 0.13 0.40 0.08 0.72 0.26 0.02 0.22 0.07 0.95 1.13 0.19 0.26 290 0.1 fi 0.34 0.09 0.52 0.38 0.01 0.25 O.Ofi 0.95 1.16 0.22 0.25 300 0.20 0.28 0.10 0.35 0.54 0.01 0.27 O.Ofi 0.94 1.18 0.23 0.23 310 0.25 0.23 0.11 0.22 (1.73 0.01 0.29 O.Ofi 0.93 1.20 0.24 0.21 320 0.30 0.19 0.12 0.12 0.94 0.02 0.31 0.07 0.91 1.22 0,23 0.19 330 0.3rt 0. 15 0.13 0.0(1 1.18 0.03 0,33 0.08 0.88 1.25 0,22 0.18 340 0.42 0.12 0.14 0.04 1.44 0.05 0.35 0.09 0.80 1.28 0.20 0.17 350 0.48 o.no 0.15 O.Ofi 1.72 0.07 0.36 0.09 0.84 1.34 0.18 0.17 3r,o 0.54 0.(17 O.lfi 0.12 2.01 0.10 0.37 0.10 0.85 1.40 0.16 0.18 370 O.fil O.Ul) 0.17 0.21 2.31 0.13 0.37 0.11 0.88 1.46 0.15 0.19 380 0.G8 0.05 0.18 0.34 2.62 0.16 0.37 0.12 0.92 1.51 0.14 0.20 390 0.74 0.05 0.19 0.50 2.92 0.19 0.36 0.12 0.98 1.54 0.13 0.21 400 0.81 O.Ofi 0.20 0.70 3.21 0.23 0.35 0.11 1.03 1.54 0.13 0.22 410 0.88 0.08 0.20 0.92 3.54 0.26 0.31 0.10 1.07 1.50 0.12 0.23 420 0.94 0.10 0.21 1.17 3.84 0.80 0.33 0.08 l.Ofi 1.44 0.12 0.25 430 1.00 0.13 0.21 1.45 4.14 0.33 0.31 0.06 1.01 1.37 0.12 0.27 440 1.06 O.IG 0.21 1.75 4.42 0.36 0.29 0.04 0.91 1.30 0.12 0.29 450 1.12 0.20 0.21 2.06 4.70 0.38 0.27 0.03 0.77 1.28 0.12 0.32 4 CO 1.17 0.25 0.21 2 39 4.96 0.40 0.25 0.02 O.fil 1.31 0.14 0.35 470 1.21 0.30 0.20 2.72 5.20 0.41 0.23 0,01 0.47 1.40 0.17 0.38 4 SO 1.25 0.3G 0.19 3.06 5.42 0.42 0.21 0.(11 0.36 1.56 0.21 0.39 490 1.2H 0.42 0.18 3.39 5.62 0.42 0.20 0.01 0.32 1.76 0.27 0.39 500 1.31 0.48 0.17 3.71 5.79 n.42 0.18 0.01 0.36 1.96 0.32 0.37 510 1.33 0.54 O.lfi 4.02 5.92 0.41 0.17 0.01 0.46 2.15 0.37 0.33 520 1.34 O.Gl 0.14 4.31 6.01 0.39 0.16 0.02 0.62 2.21 0.40 0.27 530 1.35 0.G8 0.13 4.58 6.06 0.38 0.16 0,02 0.81 2.35 0.40 5.20 540 1.33 0.74 O.U 4.84 6.08 0.35 0.15 0.03 1.00 2.34 0.38 0.14 550 1.34 0.81 0.10 5.07 G.Ot 0.33 0.15 0.03 1.18 2.2fi 0.33 0.08 5i;o 1.32 0.88 0.09 5.28 5.98 0.30 0.15 0.04 1.31 2.11 0.27 0.05 570 1.30 0.94 0.07 5.48 5.88 0.28 0.16 0.(14 1.41 1.94 0.20 0.04 5 so 1.27 1.(10 O.Ofi 5.67 5.75 0.2fi 0.16 0.(15 1.48 1.74 0.14 0.07 590 1.24 l.dCi 05 6.81 5.59 0.23 0.17 0.06 1.53 1.51 0,1(1 0.11 000 1.20 1,12 0.04 5.99 5.42 0.21 0.17 0.(1(1 l..^S 1.34 0.07 0,17 THE ORBIT OF URANUS. 8TT T.vm.KXXlH. Year. (6.C.0) (fc.ti.l), {b.c.l) 1300 +0 fifil— 3.S5'_12.74 llilO ; O.flSl 3.81 13-20 0.04, .78 1330 ' 0.03 3.74 1340 I 0.01 3.70 12.0,0 12.:)0 1-2.40 12.37 (t.».2) (6.C.2) .0,341—0.07 0.34 i 0.00 0.341 0.05 Year. (6.c,0) (ft.s.l) (b.c.l) V>-i-'^)\ (l'-'-'-'^) 1350 +0.00—3.00—12.27 1300 0.59; 3.03 1370 1 0.58 1380 0.57 1390 0.50 1400 +0.55 0.54 I 0.53, 0.52 i 3.00 3.57 3.54 12.17 12.07 11.07 11.80 —3.51 —11.70 1410 1420 1430 1440 3.48 3.40 3.44 0.51 1 3.43 1450 +0.5" —3.42 11.05 11.54 11.43 11.32 0.34 0.34 -0.34 0.34 0.34 0.34 0.34 0.04 0.03 '—0.02 0.00 0.80 0.88 0.87 1800 +0.15 1810 , 0.14 1820 ! 0.13 1830 0.12 1840 0.11 1850 +0.10 1800 0.09 -4.00 —0.00 1870 0.08 _0.35'— 0.80 0.35 j 0.85 0.35 0.84 0.35 1 0.83 0.35; 0.8- 1880 : 0.07 1890 0.00 4.08 4.75 4.83 4.92 -5.00 5.09 5.18 5.27 5.30 5.80 5.01 5.41 5.21 ff ft —0.44 —0.40 0.45 0.45 0.45 0.44 0.45 0.43 0.46 n .it\ 0.42 n ii —5.00 —0.40 —I 4.79! 0.47 1 4.58 1 0.47 1900 '+0.05! —5.45 1010 I 0.04 1020 0.03 1030 0.02 1400 1470 1480 1490 1500 1510 0.49 0.48 0.40 0.45 +0.44 0.43 -11.21 —0.35 3.41! 11.10 3.40 1 10.08 3.40 1 10.80 3.40 10.74 _:?,40 —10.02 .10 10.50 1520 0.42 1530 I 0.41 1540 i 0.40 3.41 3.42 10.38 10.20 3.43 10.14 1000 '+0.34 1010 ' 0.33 1C20 i 0.32 1030 0.32 1040 1050 1 000 1070 1080 1090 1700 1710 1550 +0.39—3.45 —10.01 1.500 ' 0.38: 3.47 0.88 1570 0.371 3.40 0.75 1580 0.30! 3.51 1590 0.351 3.r.3 0.35 0.30 0.30 0.30 -0.30 0.30 0.30 0.3- 0.37 -0.80 0.70 0.78 0.77 0.70 — 0. 0.7t 0.73 (1.72 0.71 1040 t + 0.01 1950 1 000 1070 1080 1000 2000 0.00 —0.01 0.02 0.04 0.05 5.55 5.04 5.74 5.84 —5.94 0.04 6.14 6.25 6.30 0.40 0.,39 4.30; 0.48 i 0.38 4.141 0.48i 0.37 I _3.02 — 0.49 —0.30 3.09 3.40 3.22 2.97 0.40 0.35 0.50 0.34 0.50 j 0.33 0.50 1 0.32 _') 73—0.51 —0.30 2.48' 0.51; 0.29 2.24 0.52! 0.28 1.99 0.52 1 0.27 1.73 1 0.521 0.20 _0.00 —0.47 —1.471— 0.531-0.25 _n.37 — O.TO 0.70 9.01 0.47 —3.50 — 0..33 0.31 +0.30 0.20 0.28 0.27 0.20 +0.25 0.24 1720 ' 0.23 3.50 3.02 3.05 3.09 0.19 0.05 8.00 0.37' 0.37; 0.37 I 0.38 0.00 0.08 0.07 2010 I 0.07, 0.58 2020 ! 0.0S| 2030 2040 0.00 0.00 0.80 0.10 i 0.92 1.211 0.5:i 0.04 0.00 0.54 0.54 1 0.24 0.23 0.22 0.38, 0.541 0.20 -0.11' —7.03 L-0.10'-0..55'— 0.19 2050 •2000 1 0.12! 7.14 :+o. IS 0.55; 0.18 _0.3S — 0.00 0.381 0.05 0.30! 0.04 0.30 0.03 0.22 0.21 1730 1740 1750 1700 1770 1780 1790 1800 +0.15 — 3.77 S.liU - 8.44 3.81 8.20 3.85 8 13 3.00 1.00 —3.05 — 7.79 ' 4.00 7.03 4.00 7 40 4.12 7.20 4.18 t.l2 i o.\ — 0.05 0.10 4.32 0.18 4.38 0.17 4.45 0.10 1 n 1 ^ 4.53 I r.n' 0.30, 0.02 _0.39'— 0.01 0.40 1 0.00 0.40] 0.50 0.40 0.58 0.41 -0.41 0.41 0.42 0.42 0.42 0.43 2070 2080 2090 0.13; 7.20 0.15 1 7.37 0.10 2100 '—0.17 2110 ! 0.18 2120 • 0.19 2130 i 0.21 2140 O.22I 7.48 0.47 0.50 0.70 0.50 1.05 0.50 j 0.17 0.10 0.15 _7 00 +1.35 —0.57' —0.14 7.72 1 1.05 0.571 "■''* 7.84 7.07 8.09 2150 —0.23 —8.21 2100 i 0.241 2170 ' 0.251 2180 0.27! 8.33 8.44 1.00 0.57; 0.12 2.27 0.58 0.10 2.58 1 0.58 +2.90—0.58 3.22i 0.50 3.541 0.50 3.87 i 0.50 0.09 —0.08 0.07 0.00 0.05 2100 0.281 2200 —0.20 — S 210 2220 2230 0.301 0.311 0.33! 0.77 0.43 0.50 0.58 0.43 0.49 0.39 0.44 0.48 0.20 0.44 0.47 - 0.00 —0.44 —0.40 2240 , 0.34 2250 —0.35 —9 2200 2270 2280 2290 2300 —0.42 —10 0.30 0.38 0.30 0.40 1 08 4.20 0.00 0.03 SO + 1.54 —0.00 —0.02 92 4.88 0.00 —0.01 04 5.23 0.01 0.00 10 5 58 0.01 +0.01 .28 5.94 0.01 0.03 40 + 0.29 —0.02 +0.04 52 0.05 0.02 0.05 04 7.01 0.03 0.07 70 7.37 0. 03 0.08 .88 t.74 0.03 0.10 .00 +8.10 -0.04 + 0.11 ( ( ! ii in 'i TABLE FOR FORMING THE PRODUCTS OF GIVEN NUMBERS BY THE SINE OR COSINE OF A GIVEN ANGLE. This tab. 3 is formed for thn especial purpose of facilitating the formation of the products («.«.3) siu 3r,, (r.c.3) cos JJ./, etc., (,.«.!) sin g, (p.cl) cos j, for entire degree^ tf q It is so arranged that the required products can be taken out at siglit. Supposing the number to be given in seconds and decimal fractions ot a second, we first seek the given angle at the top or bottom of the page, and then enter one of the first nine lines of the table with the fraction part of the second, niterpolatu.g for the hundredths. We then add the result mentally to the number corrcspondnig to the entire seconds. The algebraic signs at the sides of the angles are those of the sines or cosines corresponding to the angle and to the column above or below. If the number does not exceed 3" we can enter the table as if it were ten times greater, and remove the decimal pouit one place to the left ni the result. For exanii>le, to find the value of 2l".67 sin 280' + 2".25 cos 280° we find the angle 280° at the bottom of a pair of columns, the right hand one being the sine column. Entering this column with 0.67 as the argument, we find O.CUi. Entering with 2.1, we find 20.68, to which adding 0.66, we have 21".:U as the sine product. Entering the other column with 22.5, and moving the decimal point, we find 0".39 for the cosine product. Noticing the algebraic signs on each side of 280°, we find the result to be - 21" M + 0'.39 = - 20".7 + cos — 356 + —355 + sin cos sin cos xin cos sin cos 0.1 0.00 0.10 0.00 0.10 0.01 0.10 0.01 0.10 0.01 0.10 0.1 0.2 0.00 0.20 0.01 0.20 0.01 0.20 0.01 0,20 0.02 0.20 0.2 0.3 0.01 0.30 0.01 0.30 0.02 0.30 0.02 0,30 0.03 0.30 0.3 0.4 0.01 0.40 0.01 0.40 0.02 0.40 0.03 0.40 0.03 0.40 0.4 0.5 0.01 0.50 0.02 0.50 0,03 0,50 0.03 0.50 0.04 0.50 0.5 O.fi 0.01 o.(;o 0.02 0.(10 0.03 0,(10 0.04 O.CO 0.05 0.(10 0.6 0.7 0.01 0.70 0.02 0.70 0,04 0.70 0.05 0.70 0.0(1 0.70 0.7 0.8 0.01 o.so 0.03 0.80 0.04 0.80 O.Ofi 0.80 0.07 0.80 0.8 0.9 0.02 0.90 0.03 0.00 0.05 0.90 O.Ofi 0.90 O.OS 0.00 0.9 1.0 0.02 1 00 0.03 1.00 0.05 1.00 0.07 1.00 0.00 1.00 1.0 2.0 0.03 2.00 0.07 2.00 0. 10 2.00 0.14 2.00 0.17 1 . 09 2.0 3.0 0.0,) 3.00 0.10 3.00 O.lfi 3.00 0.21 2.00 0.2(1 2.09 3.0 4.0 0.07 4.00 0.14 4.00 0.21 3.99 0.28 3.99 0.35 3.08 4.0 6.0 0.09 5.00 0.17 5.00 0.20 4.99 0..35 4,09 0.44 4.08 5.0 6.0 0.10 fi.OO 0.21 0.00 0.31 5,99 0.42 5,09 0.52 5.08 6.0 7.0 0.12 7.00 0.24 7.00 0.37 fi,09 0.49 fi.08 0.61 6.07 7.0 8.0 0.14 8.00 0.28 8.00 0.42 7 09 0..'>fi 7.08 0.70 7.07 8.0 9.0 O.K! 9.00 0.31 8.09 0.47 8.09 0.03 8.98 0.78 8.07 9.0 10.0 0.17 10.00 0,35 0.90 0.52 0.99 0.70 0.98 0.87 O.Ofi 10.0 11.0 0.1!) 11.00 0.38 10.90 0.58 10.08 0.77 10.07 0.96 1 O.Ofi 11.0 12.0 0.21 12.00 0.42 11.00 O.fi 3 11.98 0.S4 11.07 1,05 11.05 12.0 1.3.0 0.23 13.00 0.45 12.00 O.fiS 12.98 0.01 12.07 1,13 12.05 13.0 14.0 0.24 14.00 0.49 13.99 0.73 13.98 0.98 13.97 1.22 13.05 14.0 15.0 0.2fi 15.00 0.52 14.99 0.70 14.98 1.05 14.06 1.31 14.94 150 IG.O 0.28 Ifi.OO 0.5fi 15.99 0..S4 15.98 1.12 15.96 1.30 15.04 16.0 17.0 0.30 17.00 0.50 Ifi.OO O.SO lfi.98 1.10 16.96 1.4S lfi.04 17.0 18.0 0.31 IS. 00 O.C.S 17.09 0.04 17.98 1.2(; 17.06 1.57 17.03 18.0 19.0 0.33 10.00 O.fi (5 IS. 99 0.00 18.97 1.33 18.05 l.Gfi 18.93 19.0 20.0 0.3:) 20.00 0.70 10.09 1.05 19.07 1.40 10.05 1.74 19.92 20.0 21.0 0.37 21.00 0,73 20.09 1.10 20.07 1.4fi 20.05 1.S3 20.02 21.0 22.0 38 22, Ou 0.77 21.99 1.15 21.07 1.53 21.95 1.02 2 1 . 02 22.0 23.0 0.40 23.00 0,sO 22.09 1.20 22.97 1 . (10 22,04 2.00 22.01 23.0 24.0 0.42 24.00 0.84 23.09 1.2fi 23.97 1.(17 23,94 2.00 23.01 24.0 25.0 0.44 25.00 0..^7 24.98 1.31 24.97 1.74 24 94 2.18 24.00 25.0 2ti.0 0.4n 2(!.00 », ' 25.98 1.3fi 25.90 l.sl 25,94 2.27 25.00 2(1.0 27.0 0.47 27.00 0.04 2fi.98 1.41 2fi.96 1.S8 2(1.03 2.35 2fi.90 27.0 28.0 0.40 28.00 0.08 27.98 1.47 27. 00 1,05 27.93 2.44 27. SO 2S.0 29.0 0.51 20.00 1.01 2S.98 1.52 28.90 2.02 28.93 2.53 28.80 20.0 30.0 0.52 30.00 1.05 29.98 1.57 29.96 2.00 20.03 2.fil 20.80 20.0 cos Bin CO.S sin cos sin cos .sill cos sin + 2- 71°— + 27 2 — +27 3 — + 2- 14 — + 2T 5 — — 2e 9 — —26 8 — — 2£ n — — 2( 56 — —26 5 — — J n + — £ (2 + — £ 3 + — S 4 + — £ •5 + + i J9 + + £ 18 + + £ 17 + + i 36 + + £ 15 + ' J TABLE OP PRODUCTS OF SINES AND COSINES. 281 J + 6 + + -7^ + + 8"+ + r+ + 10^ + 174 — + 173 — + 172- + 171 — ^ 170 — — 186 — — 187 — — 188 — — 189 — — 190 — — 354 + — 358 + —352 + — 351 + — 350 + Kill CD.i ttin C(JH sin cos sill tus Hin cos 0,1 0.01 0,10 0.01 0,10 0.01 0.10 0.02 0.10 0.02 O.IO 0,1 0.2 0.02 0,20 0,02 0.20 0.03 0.20 0.03 0.20 0.03 0.20 0.2 0,3 0.03 0..30 0.01 0.30 0.04 0.30 0.05 0.30 0,05 0.30 (1.3 0,4 0.04 0.40 0.05 0.40 0.00 0.40 0.06 0.40 0.07 0.39 0.4 0.5 0.05 0.50 0.00 0,50 0,07 0,50 0.08 0.49 0.09 0.49 0,5 0. li 0.00 0.00 0.07 0,00 0.08 0.59 0.09 0.59 0,10 0.59 0.0 0.7 0,07 0.70 0.09 0,09 0.10 0.09 0.11 0.09 0,12 0.09 0.7 O.S O.OS 0.80 0,10 0.79 0.11 0.79 0.13 0.79 0,14 0,79 0.8 0.9 O.OD 0.90 O.lt 0.89 0.13 0.80 O.U 0.«9 0.10 0,89 0.9 1.0 O.lO 0,99 0.12 0.99 0.14 0.99 0,10 0.99 0.17 0,98 1.0 2.0 0.21 1.99 0.21 1.99 0.28 1.98 0.31 1.98 35 1,97 2.0 3.0 0,31 2.98 0.37 2.98 0.42 2.97 0.47 2.90 0.52 2,95 3.0 4.0 42 3.98 0,49 3.97 O.uO 3,90 0.03 3.95 0.09 3,94 4.0 5,0 0.52 4.97 0.01 4.90 0.70 4.95 0.78 4.94 0.87 4,92 5.0 fi,0 0.03 5.97 0.73 5.90 0.84 5,94 0.94 5.93 1.04 5.91 0,0 7.0 0.73 0.90 0.85 0.95 0.97 0.93 1,10 0.91 1.22 0.89 7.0 8,0 0.H4 7.90 0.97 7.94 1,11 7.92 1,25 7,90 l..i9 7.88 8.0 9,0 0.94 8.95 l.IO 8,93 1.25 8.91 1,41 8.89 1.50 8.80 9.0 10.0 1,05 9.95 1.22 9,93 1.39 9.90 1.50 9.88 1.74 9.85 10.0 11,0 1.15 10,94 1.34 10.92 1.53 10.89 1,72 10.80 1.91 10.83 11.0 12.0 1.25 11.93 1,40 11.91 1.07 11,88 1,88 11.85 2.08 11.82 12,0 13.0 1.30 12,93 1,58 12.90 1.81 12,87 2,03 12.84 2.20 12,80 13.0 14.0 1.40 13.92 1.71 13.90 1.95 13,80 2.19 13.83 2.43 13,79 14.0 15.0 1.57 14.92 1.83 14.89 2.09 14.85 2,35 14.82 2.00 14.77 15,0 10.0 1.07 15.91 1.95 15.88 2.23 15.84 2.50 15.80 2.78 15.70 10. 17. D 1.7S 10.91 2.07 10.87 1 2.37 10.83 2.00 10.79 2.95 10.74 17.0 1S,0 l.ss 17.90 2.19 17.87 2,51 17.82 2,82 17.78 3.13 17.73 18,0 19.0 1.99 18.90 2.32 18.80 2.04 18.82 2.97 18.7T 3.30 18.71 19,0 20.0 2.09 19.89 2.44 10.85 2.78 19,81 3.13 19.75 3.47 19.70 20.0 21.0 2.20 20.88 2.50 20.84 2.92 20.80 3.29 20.74 3.05 20.08 21.0 22,0 2.30 21.88 2.08 21.84 3.00 21.79 3,44 21.73 3.82 21.07 22.0 23.0 2.40 22.87 2.80 22.83 3.20 22,78 3.00 22.72 3.99 22.05 23.0 24.0 2.51 23.87 2.92 23.82 3.34 23,77 3.75 23.70 4.17 23.04 24.0 25.0 2.01 24. SO 3.05 24,81 3.48 24,70 3.91 24.09 4.31 24.02 25.0 20,0 2,72 25.80 3.17 25,81 3,02 25.75 4.07 25.08 4.51 25.01 20.0 27.0 2,82 20.85 3.29 20,80 3.70 20.74 4.22 20.07 4.09 20.59 27,0 2S.0 2,93 27.85 3,41 27.79 3,90 27.73 4.38 27.00 4.80 27.57 28,0 29.0 3,03 28.84 3.53 28.78 4.04 28.72 4.54 28.04 5.04 28.56 29.0 30.0 3.14 29.84 3.00 29.78 4.18 29.71 4.09 29.03 5.21 29.54 30.0 cos sin COS siti cos sin cos sin cos sin + 2' 76 — + 277 — + 2' IB — + 279 — + 2- — — 2( 84 — — 263 — — 2( 32 — — 281 — — 2( 30 — — { :6 + - 97 + — i )3 + — 89 + — 1( )0 + + i J4 + + 83 + + J 32 + + 81 + + i JO + 3 a Aug ust. 1873 . 282 TABLE OF PRODUCTS OF SINKS AM) COS INKS, n |i! ; + 11° + + 12° + + 13°+ + 14°+ + 1B° + + 169 — + 168 - + 167 - + 166 — + 16B — — 191 — — 192 — — 193 — — 194 — — 105 — — 349 + — 348 + — 847 + — 346 + — 846 + sin t'08 «in (•(IH xln cos RJn cos Rin cog 0.1 0.02 0.10 0.02 0,10 0.02 0,10 0,(12 0,10 0.03 0.10 0.1 0.2 O.lit 0.20 O.dt 0,20 cot 0.19 0,05 0,19 0.05 0.19 0.2 0.3 O.OC, 0.2'J (l,0(> 0,29 0.(17 0.29 0.07 0,29 0.08 0.29 0.3 0.4 O.OH 0.39 0.08 0,39 0.09 0.39 0,10 0,39 0.10 0.39 0.4 O.ft O.IO 0.49 0.10 0,(9 0.11 0.49 0.12 0,49 0.13 0.48 0.5 O.C) 0. II 0.59 0.12 0.59 0.13 0,58 0.15 0,58 0,10 0.58 O.fi 0.7 0. l:! O.CO 0,15 0.08 0. 1 0,08 0,17 0.08 0.18 0,08 0.7 0.8 o.i:. 0.79 0.17 0.78 O.IS 0.78 0,19 0,-8 0,21 0,77 0,8 0.9 0.17 0.88 0.19 0.88 0.20 0.88 0.22 0,87 0,23 0.87 0.9 1,0 fl.IO 0.9S 0,21 0.98 0.22 0.97 0.24 0.97 0.20 0.97 1,0 2.0 o.;!s i.9(; 0.42 l.9(; 0,45 1,95 0,48 1.94 0.52 1,93 2.0 3.0 0..M 2.91 o.(;2 2.93 0,07 2.92 0.73 2,91 0,78 2.90 3.0 4.0 0.7(1 3.93 0.83 3.91 0.90 3.90 0.97 3.88 1.04 3.80 4.0 f).0 O.Of) 4.91 1,04 4.89 1.12 4,87 1,21 4,85 1.29 4,83 5.0 fi.O 1.14 5.89 1,25 5.87 1.35 5.85 1,45 5,s2 1.55 5,80 6.(1 7.0 l.:!4 n.87 1,4(1 fi.85 1.57 0.82 1.09 0.79 1.81 0,70 7,0 H.O i.r>.{ 7.85 i,(;r, 7 83 1.80 7,79 1.94 7.70 2.07 7,73 8.0 0.0 1.72 8.83 1.87 8.80 2.02 8.77 2.18 8.73 2.33 8.09 9.0 10.0 i.ni 9.82 2,08 9.78 2,25 9.74 2,42 9,70 2,59 9.06 10.0 11.0 2.10 l((.80 2.29 10. 7c. 2^47 10.72 2,00 10.07 2,s5 1(1,03 11.0 12.0 ') .)() 11.78 2,49 11.74 2,70 11,09 2,90 11.04 3,11 11,59 12,0 l.i.O 2'.4S 12.7(! 2,70 12,72 2,92 12,07 3,14 12.01 3.;io 12.50 13,0 14.0 2.(17 13.74 2.91 13.09 3.15 13.G4 3.39 13.58 3.02 13.52 14.0 in.o 2.Sfi 14.72 3,12 14.07 3.37 14,02 3.03 14,55 s.y-H 14.49 15.0 lil.O .•J.O,') 15.71 3.33 15.05 3,00 15.59 3.87 15.52 4.14 15.45 10.0 17.0 :i.24 l(i.(!9 3.5:i 10.03 3,82 10.50 4,11 10.50 4.10 10.42 17.0 18.0 .'i.4;! 17.07 3,7t 17.01 4,(15 17.54 4,35 17.47 ■<,00 17.39 18.0 ly.o •J. an 18. C5 3.95 18.58 4,27 18.51 4.00 18.44 4,92 18.35 19.0 20.0 3.S2 19.03 4.10 19.50 4,50 19.49 4.84 19.41 5,18 19.32 20.0 21.0 4.01 20.01 4,37 20.54 4,72 20.40 5.08 20.38 5.44 20.28 21.0 22.0 4.2(1 2I.(;0 4,57 21.52 4.95 21.44 5.32 21.35 5.09 21.25 22.0 2:t.0 4.:i9 22.58 4,78 22.50 5.17 22.41 5,50 22.32 5.95 22.22 23.0 24.0 4.58 23.50 4.99 23.48 5.40 23.38 5.81 23.20 0.21 23.18 24.0 25.0 4.77 24.54 5.20 24.45 5.02 24.30 0.05 24.20 0.47 24.15 25.0 2(i.O 4.!Mi 25,52 5.41 25.43 5.85 25, .",3 0.29 25.23 0.73 25.11 20.0 27.0 5.15 2(i.50 5,(;i 20.41 0.07 20,31 0.53 20.20 0.99 20,08 27.0 2.S.0 5.:!4 27.49 5,82 27.39 0.3() 27.28 0,77 27.17 7.25 27.05 28.0 29.0 5.53 28.47 6.03 28.37 0.52 28.20 7.02 28.14 7.51 28.01 29.0 30.0 5.72 29.45 0,24 29.34 G.75 29.23 7.20 29.11 7.70 2S.98 30.0 cos sin cos sill cos sin OIKS sin t'O.S sin + 281 — + 2£ J2 — + 2f }3 — + 2e 14 — + 2f J5 — — 259 — — 2E >8 — — 257 — — 2E •8 — — 2£ >5 — — 101 + — IC >2 + — 103 + — 10 4 + — IC )5 + + "79 + + - (8 + + 77 + + '■ 16 + + ^ '5 + TAin ,K OF PRODUCTS OF 8 N !•: S A Nl) C08IT s'BS. 283 + 10° + + ir+ -t 18°+ 4 19°+ + 20°+ + 164 — + 163 — + 162 — + 161 — 1 160 — — 196 — — 197 — ._ 198 — — 199 — — 200 — — 344 + — 343 + — 342 + — 341 + — 340 + Bill CHS Hin cos hIii C'OH KIM 1 j fos sill cog O.l 0.03 0.10 0.03 0.10 0.03 0,10 0.03 ; 0.09 0.03 0.09 0.1 o.a 0.li)> 0.1!) O.OO 0.19 O.OO (I 1!) 0.07 0. 1 9 0.07 0,19 0.2 0.3 o.os 0.2!) 0.09 , 0.29 0.09 0.2!) 0.10 ! 28 0.10 0,28 0.3 0.4 0.1 1 0.38 0.12 0.38 0.12 0.38 0.13 , 0.38 0.14 0,38 4 0.5 0.14 0.48 0.15 0.48 0.15 0.48 0.10 0.47 0,17 0,47 0.5 o.c, 0.17 0.58 0.18 0.57 0.19 057 0.20 0.57 0,21 0,50 O.li 0.7 0.1!) 0.07 0.20 0.07 0.22 0,07 0,23 0.06 0,24 0,06 0.7 0.8 0.22 0.77 0.23 0.77 0.25 0.76 0,20 0.76 0,27 0,75 0.8 0.9 0.25 0.87 0.26 0.86 0.28 0.80 0.29 0.85 0.31 0,^)0 0.9 1.0 0.28 0.96 0.29 O.OO 0.31 0.95 0.33 0.95 0.34 0,94 1.0 L'.O 0.55 1.92 0.58 1.01 0.02 l.!)0 0,r,5 1.89 0,08 1,88 2.0 ;i.o 0.H3 2.88 0.S8 2.87 0.93 2.S5 0.'.' 2.S4 1,03 2.S2 3.0 4.0 1.10 3.85 1.17 3.83 1.24 3.80 1.30 :;.78 1,37 3'70 4.0 5.0 1.38 4.81 1.40 4.78 1.55 4.76 1.03 4.73 1.71 4.70 5.0 r,,o l.(;5 5.77 1.75 6.74 1.85 5.71 1 95 5.67 2.05 5.04 6.0 V.o 1.113 0.73 2.05 6.09 2.10 0.06 ,.28 f..02 2.3 9 0.58 7.0 8.0 2.21 7.09 2.34 7.05 2.47 7.01 2.00 7.56 2.74 7.52 8.0 n.,; 2.48 8.65 2.03 8.01 2.78 8.56 2.93 8.51 3.08 8.40 9.0 10.0 2.7(5 0.01 2.92 9.56 3.09 0.51 3.0') 9.40 3.42 9.40 lO.O 11.0 3.03 10.57 3.22 10.52 3.40 10.46 3,5X 10.40 3.76 10.34 n.o 12.0 3.31 11.54 3.51 11.48 3,71 11.41 3,91 11.35 4.10 11.28 12.0 13.0 3.58 12.50 3.80 12.43 4.02 12.36 4,23 1::.29 4.45 12.22 13.0 14.0 3.80 13.46 4.09 13.39 4.33 13.31 4.50 13.24 4.7!) 13.10 14.0 15.0 4.13 14.42 4.39 14.34 4.04 14.27 4.88 14. IS 5.13 14.10 15.0 ICO 4.41 15.38 4.08 15.30 4.!I4 15.22 5.21 15.13 5.47 1 J. 04 10.0 n.o 4. CO 10.34 4.97 10.26 5.25 10.17 5.53 16.07 5.81 i5.!»7 17.0 is.O 4.',)li 17.30 5.26 17.21 5.56 17.12 5.80 17.02 0.10 10.91 18,0 10.0 5.24 18.26 5.56 18.17 1 5.87 18.07 0.19 17.96 6.50 17.85 19.0 20.0 5.51 19.23 5. 85 19.13 0.18 19.00 0.51 IS. 91 O.fvl 18.79 20.0 21.0 5. 79 1 20.19 0.14 20.08 0.4 9 1 1!1.'.I7 6.84 19. MO T.18 ; 19.73 21.0 22.0 fi.Oll 21.15 0.43 21.04 0.80 2().!t2 7.10 20.80 7.52 20.07 22.0 23.0 0.34 22.11 0.72 ! 22.00 7.11 i 'Jl.87 7.49 21.75 7.87 21.01 23.0 24.0 G.02 23.07 7.02 22.05 1 7.42 22.83 7.81 22.60 8.21 22.55 1 24.0 25.0 O.SO 24.03 7.31 ] 23.91 7.73 23.78 8.14 2 ".04 8.55 23.49 25 20. 7.17 24.99 7.00 24.80 S.(K! 2t.73 8.40 'J4.58 8.8!) 24.43 20.0 27.0 7.44 25.05 7.89 25.82 8.34 25.08 S.79 25.53 9.23 25.37 27.0 2S.0 7.72 ' 20.92 8,19 20.78 8.05 20.03 9.12 i 20.47 9.58 20.31 28,0 2'J.O T.it'J 27.88 8.48 27.73 8.90 27.58 j 9.44 27.42 9.92 27.25 29.0 30.0 8.27 28.84 8.77 28.69 9.27 , 28.53 9 7T 28.37 10.26 28.19 30.0 con sin cos sin co.s sia cus BlU C08 I sin + 286 — + 287 — + 288 — + 2 89 — + 290 — — 254 — — 253 — — 252 - - — 2 51 — — 250 — — 106 + — 107 f — 108 + — 1 09 + — 110 + + "74 + + 73 + + ^2 + + 71 + + 70 + ■dB 284 TABLE OF PKODUCTS OF SINES AND COSINES, + 21 + + 22° + + 23' + + 24°+ + 25°+ ■i 159 — + 158 — + 157 — + 156 — + 155 — — 201 — —202 — — 203 — — 204 — — 205 — — 839 + — 338 + — 337 + — 336 + — 335 + sin cos Hin cos sin cos sin COS sin COS 0.1 0.04 0.09 0.04 0.09 0.04 0.09 0.04 0.09 0.04 0.09 0.1 0.2 0.07 0.19 0.07 0.19 0.08 0.18 0.08 0.18 0.08 0.18 0.2 0.3 O.ll 0.28 0.11 0.28 0.12 0.28 0.12 0.27 0.13 0.27 0.3 0.4 0.14 0.31 0.15 0.37 0.10 0.37 0.16 0.37 0.17 0.36 0.4 0.5 O.IS 0.4t 0.19 0.46 0.20 0.46 0.20 0.46 0.21 n.45 0.5 O.G 0.22 0.56 0.22 0.56 0.23 0.55 0.24 0.55 0.25 0.54 0.6 0.7 0.25 0.05 0.20 0.05 0.27 0.04 0.28 0.04 0.30 0.63 0.7 O.S 0.2!) 0.75 0.30 0.74 0.31 0.74 0.33 0.73 0.34 0.73 0.8 0.9 0.32 0.84 0.34 0.83 0.35 0.83 0.37 0.82 0.38 0.82 0.9 1.0 o..3r. 0.93 0..37 0.93 0..S9 0.92 0.41 0.91 0.42 0.91 1.0 2.0 0.72 1.87 0.75 1.85 0.78 1.S4 0.81 1.83 0.85 1.81 2.0 3.0 l.OS 2.80 1.12 2.78 1.17 2.76 1.22 2.74 1.27 2.72 3.0 4.0 1.43 3.73 1.50 3.71 1.56 3.08 1.03 3.65 1.69 3.63 4.0 5.0 1.70 4.67 1.S7 4.04 1.95 4.60 2.03 4.57 2.11 4.63 5.0 ().0 2.15 5.60 2.25 5.56 2.34 5.52 2.44 5.48 2.54 5.44 0.0 7.0 2.51 0.54 2.1 ' 0.49 2.74 6.44 2.85 6.39 2.96 6.34 7.0 8.0 2.H7 7.47 3.00 7.42 3.13 7.36 3.25 7.31 3.38 7.25 8.0 U.O 3.23 8.40 3. 37 8.34 3.52 8.28 3.06 8.22 3.80 8.16 9.0 10.0 3.58 9.34 3.75 9.27 3.91 9.21 4.07 9.14 4.23 9.00 10.0 U.O 3.94 10.27 4.12 10.20 4.30 10.13 4.47 10.05 4.05 9.97 11.0 12.0 4.30 11.20 4.50 11.13 4.09 11.05 4.88 10.96 6.07 10.88 12.0 13.0 4.(ii> 12.14 4.87 12.05 5.08 11.97 5.29 11.88 5.49 11.78 13.0 14.0 5.02 13.07 5.24 12.98 5.47 12.89 5.69 12.79 6.92 12.69 14.0 15.0 5.3S 14.00 5.02 13.91 5.80 13.81 6.16 13.70 6.34 13.59 15.0 Ki.O 5.73 14.94 5.99 14.83 0.2;. 14.73 6.51 14.02 6.76 14.50 10.0 n.o (l.O!) 15.S7 0.37 15.70 6.04 15.65 0.91 15.53 7.18 15.41 17.0 IH.O fi.45 10.80 0.74 10.09 7.03 10.57 7.32 10.44 7.61 10.31 18.0 I'J.O 6.x I 17.74 7.12 17.02 7.42 17.49 7.73 17.36 8.03 17.22 19.0 20.0 7.17 18.07 7.19 18.54 7.81 18.41 8.13 18.27 8.45 18.13 20.0 21.0 7.53 19.01 7.S7 19.47 8.21 19.33 8.54 19.18 8.S7 19.03 21.0 22.0 7.S.S 20.54 8.24 20.40 8. CO 20.25 8.95 20.10 9.30 19.94 22.0 23.0 8.24 21.47 8.li2 2 '..33 8.99 21.17 9.35 21.(»1 9.72 20.85 23.0 24.0 8.00 22.41 8.99 22.25 9.38 22.09 9.70 21.93 10.14 21.75 24.0 25.0 8. or, 23.34 9.37 23.18 9.77 23.01 10.!7 22.84 10.57 22.66 25.0 2(1.0 9.32 24.27 9.74 24.11 10.10 23.93 10.58 23 75 10.99 23.58 20.0 27.0 O.C.S 25.21 10.11 25.03 10.55 24.85 10.98 24.07 11.41 24.47 27.0 2S.0 10.03 20.14 10.49 25.96 10.94 25.77 11.39 25.58 11.83 25.38 28.0 2i».0 10.39 27.07 10.8(1 26.89 11.33 20.69 11.80 20.49 12.26 26.28 29.0 30.0 10.75 ) 28.01 11.24 27.82 11.73 27.62 12.20 27.41 12.68 27.19 30.0 cos .sin co.s sin cos sin cos sin COS sin 6 — + 2 91 — + 2! ?2 — + 293 — + 29 4 — + 2£ — 2- 49 — -2' 18 — — OAl — — 24 [B — — 24 t5 - — 1 11 + — 1 12 + — 113 + — 11 4 + — 11 6 + + 89 + + i J8 + + 67 + + e 8 + -r € 15 + TABLE OP PRODUCTS ( OF SINES AND COSINES. 285 + 26"+ + 27" + + 28°+ + 29" + + 30'+ + 154 — + 153 — + 152 — + 151 — + 150 — — 206 — — 207 — — 208 — — 209 — — 210 — — 334 + — 333 + — 332 + — 331 + — 330 + sin cos sin cos sin cos sin cos sin cos 0.1 0.04 0.00 0.05 0,00 0.05 0.00 0.05 0.09 0.05 0.09 0.1 0.2 0.09 : 0.18 0.00 0.18 0.00 0.18 0.10 0.17 0.10 0.17 0.2 0.:^ 0.13 : 0.27 0.14 0.27 0.14 0.26 0.15 0.26 0.15 0.26 0.3 0.4 O.IS 0.36 0.18 0.36 0.10 0.35 0.10 0.35 0.20 0.35 0.4 0.5 0.22 0.45 0.23 0,45 0.23 0.44 i 0.24 ! 0.44 0.25 0.43 0.5 O.fi 0.26 1 0.54 0,27 0,53 0.28 0.53 0.20 0.52 0.30 0.52 0.6 0.7 0.!J1 0.63 0.32 0.62 0.33 0.62 0.34 0.61 0.35 O.Cl 0.7 0.8 0.35 0.72 0.36 0.7i 0.38 0.71 0.30 0.70 0.40 0.69 0.8 0.9 0.39 O.sl 0.41 0.80 0.42 0.79 0.44 0.70 0.45 0.78 0.0 1.0 0.44 0.00 0.45 0.80 0.47 0.88 0.48 0.87 0.50 0.87 1.0 2.0 O.SH 1.80 0.01 1.78 o.ot 1.77 0.07 1.75 1.00 1.73 2.0 y.o 1.32 2.70 1.36 2.67 1.41 2.65 1.45 2.62 1.50 2.60 3.0 4.0 1.75 3.60 1.82 3.56 1.88 3.53 1.04 3.50 2.00 3.46 4.0 5.0 2.10 4.40 2.27 4.46 2.35 4.41 2.42 4.37 2.50 4.. 33 50 c.o 2.(53 5.30 2.72 5.35 2..S2 5.. 30 2.01 5.25 3.00 5.20 6.0 t.o 3.07 6.20 3.18 6.24 3.20 6.18 3.30 6.12 3.50 6.06 7.0 s.u 3.51 7.10 3 63 7.13 3.711 7.06 3.88 7.00 4.00 6.03 8.0 9.0 3.0,". 8.09 4.00 8.02 4.23 7.05 4.36 7.87 4.50 7.70 0.0 10.0 4.3^^ 8.00 4.51 8.01 4.60 8.83 4.85 8.75 5.00 8.66 10.0 11.0 4.S2 0.80 4.00 0.80 5.16 0.71 5.33 9.62 5.50 0.53 11.0 12.0 5.2(1 10.79 5.45 10.69 5.63 10.60 5.82 10.50 6.00 10.30 12.0 13.0 5.70 • 11.68 5.00 11. 5S 6.10 11.48 6.30 11.37 6.50 11.26 13.0 14.0 6.14 12.58 6.36 12.47 6.57 12.36 C.79 12.24 7.00 12.12 14.0 l.'i.O fi.58 13.48 fi.Sl 13.37 7.04 13.24 7.27 13.12 7.50 12.09 15.0 )(>.0 7.01 14.38 7.26 14.26 7.51 14.13 7.76 13.00 8.00 13.86 16.0 n.o 7.45 15.28 7.72 15.15 7.08 15.01 8.24 14.87 8.50 14.72 17.0 IH.O 7. SO 16.18 8.17 16.04 8.45 15.89 8.73 1.5.74 9.00 15.50 18.0 19.0 8.33 17.08 8.63 16.03 8.92 16.78 9.21 16.62 9.50 16.45 10.0 20.0 8.77 17.08 O.OS 17.82 0.30 17.66 0.70 17.40 10.00 17.32 20.0 21.0 0.21 : 18.87 0,53 18.71 0..'S(1 is.,'^4 10.18 18.37 10.50 18.10 21.0 22.0 0.(14 10.77 0.00 10.60 10.33 10.42 10.67 10.24 11.00 10.05 22.0 2:1.0 10. OS 20.67 10.44 ■ 20.40 10.80 20.31 11.15 20.12 11.50 10.02 23.0 24.0 10.52 21.57 10.00 21.38 11.27 ; 21.10 11.64 20.09 12.00 20.78 24.0 25.0 10.06 22.47 11.35 22. 2S 11.74 22.07 12.12 21.87 12.50 21.65 25 2li.0 11.40 23.37 11.80 23,17 12.21 22.06 12.61 \ 22.74 13.00 22.52 26.0 27.0 11. S4 24.27 12.26 24.06 12.68 23.84 13.00 23.61 13.50 23.38 27,0 2H.0 12.27 25,17 12.71 24.05 13.15 24,72 13.57 24.40 14.00 24.25 2S.0 2'.).0 12.71 26.07 13.17 25.84 13.61 25,61 1 14.06 25.36 14.50 25.11 20.0 30.0 13.15 26.06 13.62 i ' 26.73 14.08 ! 26.49 14. .M 26.24 15.00 25.08 30.0 CO..! .sin 96 — cos sin cos sin cos sin cos sin + 2 + 297 — + 298 — + 299 — + 300 — —3 44 — — 243 — — 242 — — 241 — — 240 — — ] Lie + — 117 + — 118 + -119 + — 120 + + 64 + + 63 + + 62 + + 61 + + 60 + 286 TABLE OF niODUCTS OF SIXES AND COSINES. t 1 1 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2.0 8.0 4.0 5.0 (i.O 7.0 «.0 !».0 10.0 11.0 12.0 l.f.O 14.0 15.0 in.o 17.0 IH.O 19.0 20.0 21.0 22.0 2:t.O 24.0 25.0 2tl.O 27 2S.0 29.0 30.0 + 31° + +i4r — — 211 — —329 + sm 0.05 0.10 0.15 0.21 0.20 0.31 0.30 0.41 0.40 0.52 1.03 1.55 2.01! 2.5S 3.09 3.01 4.12 4. 04 5.15 5.(;7 fi.lS 0.70 7.21 7.73 8.24 8.70 9.27 9.79 10.30 I li).X2 11.33 il.85 12.30 12.88 13.39 13.91 14.42 14.94 15.45 (■C).S + 32°+ +148 — —212 — —328 + cos 0.09 0.17 0.26 0.34 0.43 0.51 0.00 0.09 0.77 O.Sfi 1.71 2.57 3.43 4.29 5.14 6.00 0.86 7.71 17.14 18.00 18.86 19.71 20.57 21.43 22.29 23.14 24.00 24.86 25.71 sin + 301°— —239 — — 121 + + 59 + sm 0.05 0.11 0.16 0.21 0.26 0.32 0.37 0.42 0.48 0.53 1.06 1.59 2.12 2.65 3.18 3.71 4.24 4.77 5.30 5. S3 6.36 89 42 95 18 01 I 0.08 0.17 0.25 0.34 0.42 0.51 0.59 0.68 0.76 0.85 1.70 2.54 3.39 4.24 5.09 5.94 6.78 7.63 8.48 9.33 10.18 11.02 11.87 12.72 13.57 14.42 15.26 9.54 10.07 16.11 10.60 11.13 11.66 12.19 12.72 13.25 13.78 I4.:!i 14.84 15.37 16.96 17.81 18 lu; 19.51 20.35 21.20 22.05 22.90 23.75 24.59 15.90 25.44 co.s sin + 302 — —238 — — 122 + + 58 + + 33°+ +147 — —213 — —327 + sill 0.05 0.11 0.16 0.22 0.27 0.33 0.38 0.44 0.49 CU9 I 0.08 0.17 0.25 0.34 0.42 0.50 0.59 0.67 0.75 0.84 1.68 2.52 3.35 4.19 5.03 5.87 6.71 7.55 H.39 9.23 10.06 10.90 11.74 12.58 13.42 14.26 15.10 10.35 : 15.93 5.45 5.99 6.54 7.08 7.62 8.17 H.71 9 26 9.80 10.89 11.44 11.98 12.53 13.07 13.62 14.16 14.71 15.25 15.79 16.77 17.61 1.S.45 19.29 20.13 20.97 21.81 22.64 23.48 24.32 16.31 25.16 C(is sill +303 — —237 — — 123 + + 57 + + 34°+ + 148 — —214 — — 326 + 0.06 0.11 0.17 0.22 0.28 0.34 0.39 0.45 0.50 0.56 1.12 1.68 2.24 2.80 3.36 3.91 4.47 5 03 5. 6. 6 7. 7. 8. 8. 9. 10. 10 59 15 71 ,27 83 39 95 51 07 62 11.18 11.74 12.30 12.86 13.42 ' 19.90 cos 0.08 0.17 25 0.33 0.41 0.50 0.58 0.66 0.75 0.83 1 . 66 2.49 3.32 4.15 4.97 5.. 80 6.63 7.46 8,29 9. 1 2 9.95 10.78 11.61 12.44 13.26 1(.09 14.92 15.75 16.58 17.41 18.24 19.07 13.98 14.54 15.10 15. (',6 16.22 20.73 21.55 22.38 23.21 24.04 16.78 1 24.87 COS sin + 304 — — 236 — — 124 + + 58 + + 35°+ + 145 — — 215 — — 325 + 0.06 0.11 0.17 0.23 0.29 0.34 0.40 0.46 0.52 0.57 1.15 1.72 2.29 cos 0.08 0. 1 6 0.25 0.33 0.41 0.49 0..57 0.66 0.74 0.82 1.64 2.46 3.28 2.87 '■ 4.10 3.44 i 1.91 4.02 i 5.73 4.59 ! 6.55 6.16 ' 7.37 5.74 6.31 6.S8 7.46 8.03 8.60 9.18 9.75 10.32 10.90 11.47 12.05 12.62 13.19 13.77 8.19 9.01 9. S3 10.65 11.47 12.29 13.11 13.93 14.74 15.56 16.38 17.20 is. 02 1S.,S4 19.66 14.34 I 20.48 14.91 i 21.30 15.49 22.12 16.06 22.94 16.63 23.76 17.21 24.57 sin + 305 — —235 — — 125 + + 55 + 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 IS.O 19.0 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28. 29.0 30.0 TABLE OF PK0DUCT8 OF SINES AND COSINES. 287 + 36°+ + 37°+ + 38°+ + 39°+ + 40°+ + 144 — + 143 — + 142 — + 141 — + 140 — — 216 — — 217 - — 218 — — 219 — — 220 — — 324 + — 323 + — 322 + — 321 + — 320 + sin j cos sin cos sin cos sin cos sin co.s 0.1 0.00 0.08 0.00 0,08 0.00 0.08 0.00 0,08 0.00 0.08 0.1 0.2 0.12 0.10 0.12 1 0,10 12 0.10 0.13 ! 0.10 0.13 0.15 0.2 0.3 0.18 0.24 0.18 ! 0,24 0.18 0.24 0.19 0.23 0.19 0.23 0.3 0.4 0.24 0.32 0.24 0.32 0.25 0.32 0.25 0.31 0.20 0.31 0.4 0.5 0.20 0.40 0.30 0.40 0,31 0.39 0.31 0.30 0.32 0.38 0.5 0.0 o.^;'^ ; 0.49 0.30 0.48 0.37 0.47 0.38 0.47 0.39 0.40 0.0 0.7 0.41 > 0,57 .0.42 0.50 0.43 i 55 0.44 ■ 0.54 0.45 0.54 0.7 0.8 0.47 0.05 0.48 i 0,04 0.40 0.03 0.50 0,02 0,51 0.01 0.8 O.U o.5:j . 0.73 0.54 0.72 0.55 0.71 0.57 0.70 0.58 0.09 0.9 1.0 0.59 i 0.81 0.00 0.80 0.02 0.79 0.03 0.78 0.04 0.77 1.0 2.0 1.18 : 1.02 1.20 1.00 1.23 1.58 1.20 ; 1.55 1.29 1.53 2.0 ;{.o 1.70 2.43 1.81 2.40 1.85 2.30 1,80 2,33 1.93 2.30 3.0 4.0 2.^5 3.24 2.41 3.19 2.40 3.15 2,52 3.11 2.57 3.06 4.0 5.0 2.04 4.05 3.01 3.09 3.08 3.04 3.15 3,80 3.21 3.83 5.0 0.0 ;{.:):! 4.85 3.01 4.79 3.(;o 4,73 3.78 4.00 3.86 4.00 6.0 7.0 4.11 5.00 4.21 5.59 4.31 5,52 4.41 5.44 4.50 5.30 7.0 8.0 4.70 0.47 4.81 0,39 4.93 0,30 5.03 0.22 5.14 6.13 8.0 9.0 5.29 7.28 5.42 7.10 5.54 7,09 5.00 0.90 5.79 0.89 9.0 10.0 5. 88 8.00 0.02 7.99 0.10 7.88 6.29 7.77 6.43 7.00 10.0 11.0 0.47 8.90 0.02 8.78 0.77 8.07 0,92 8.55 7.07 8.43 11.0 li'.O 7.05 9.71 7.22 9.58 7.39 9.40 7.55 9.33 7.71 0.10 12.0 l.i.O 7.04 10.52 7.82 1(1.38 8.00 10.24 8.18 10.10 8.30 9,90 13.0 14.0 8.23 11.33 8.43 11.18 8.02 11.03 8.81 10.88 9.00 10,72 14.0 l.'i.O 8.82 12.14 9.03 11,98 0.23 11.82 0.44 11.00 9.04 11.49 15.0 10.(1 9.40 12.94 9.03 12.78 9.85 12.01 10.07 12.43 10.28 12.20 10.0 n.o 9.09 13.75 10,23 13.58 10,47 13.40 10.70 13.21 10.93 13.02 17.0 IH.O 10. 5S 14.50 10.83 14.38 11.08 14.18 11.33 13.99 11.57 13.79 18.0 19.0 11.17 15.37 11.43 15.17 11.70 14.97 11.90 14.77 12.21 14.55 19.0 20.0 11.70 10.18 12.04 15.07 12,31 15.70 12.50 15.54 12.80 15.32 20.0 21.0 12.34 10.99 12.04 10.77 12.93 1(;.55 13.22 10.32 13.50 10.09 21.0 22.0 12.0.{ 17.80 13,24 17.57 13.54 17.34 13.85 17.10 14.14 10.85 22.0 2.'!.0 1.!.52 18.(il 1.3.84 ' 18.37 14.10 18.12 14.47 17.87 14,78 17.02 23.0 24.0 14.11 19.42 14.44 10.17 14.78 ; 18.91 15.10 18.(i5 15.43 18.39 24.0 2:").0 14.09 20.23 15.05 10,97 15.30 19.70 15.73 19.43 10.07 19.15 25.0 21). 15.28 21.03 15.05 2(1.70 IC.dl 20.49 10.30 20.21 10.11 19.92 20.0 27.0 15.87 21.84 10.25 21.50 10.02 21. 2« 10.09 i 20.08 17.30 20.08 27.0 2^.0 1(').40 22.05 10.85 22,30 17.24 22,(10 17.02 21.70 18.00 21.45 28.0 29.0 17.05 23.40 17.45 23.10 17.85 22,85 \ 18.25 22.54 18.04 22.22 20.0 30.0 17.03 24.27 18.05 23.90 18.47 23.04 18.88 23.31 19.28 22.98 30.0 cog sin cos sin (,'os sin cos sin cos 1 sin + 308 — + 307 — + 308 — + 309 — + 310 — — 234 — — -233 — — 232 — — 231 — — 230 — — 126 + — 127 + — 128 + — 129 + — 130 + + 54 + + 53 + + 52 + + 51 + + 50 + 388 fir :! ■ ^ :' TABLE OF PRODUCTS OF SINES AND COSINES. + 41°+ + 42°+ + 43°+ + 44°+ + 4B°+ + 139 — + 138 — + 137 — + 136 — + 135 — 221 — —222 — — 223 — —224 — —225 — — 819 + — 318 + — 817 + — 318 + — 315 + sin C03 sill cos sill cos sin cos sill cos 0.1 0.07 0.08 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.1 0.2 0.13 0.15 0.13 0.15 0.14 0.15 0.14 0.14 0.14 0.14 0.2 0.3 0.20 0.23 0.20 0.22 0.20 0.22 0.21 0.22 0.21 0.21 0.3 0.4 0.20 0.30 0.27 0.30 0.27 0.29 0.28 0.29 0.28 0.28 0.4 0.5 0:33 0.38 0.33 0.37 0.34 0.37 0.35 0.30 0.35 0.35 0.5 O.fi 0.39 0.45 0.40 0.45 0.41 0.44 0.42 0.43 0.42 0.42 0.6 0.7 0.46 0.53 0.47 0.52 0.48 0.51 0.49 0.50 0.49 0.49 0.7 0.8 0.52 0.00 0.54 0.59 0.55 0.59 0.50 0.58 0.57 0.57 0.8 0.9 0.59 0.68 00 0.67 0.01 0.06 0.63 0.65 0.04 0.64 0.9 1.0 O.fifi 0.75 0.67 0.74 0.08 0.73 0.69 0.72 0.71 0.71 1.0 2.0 1.31 1.51 1.34 1.49 1.30 1.40 1..39 1.44 1.41 1.41 2.0 3.0 1.97 2.26 2.01 2.23 2.05 2.19 2.08 2.16 2.12 2.12 3.0 4.0 2.C2 3.02 2.08 2.97 2.73 2.93 2.78 2.88 2.83 2.83 4.0 5.0 3.28 3.77 3.35 3.72 3.41 3.00 3.47 3.60 3.54 3.54 5.0 fi.O 3.94 4.53 4.01 4.46 4.09 4.39 4.17 4.32 4.24 4.24 6.0 7.0 4.5'.) 5.28 4.08 5.20 4.77 5.12 4.80 5.04 4.95 4.95 7.0 8.0 5.25 6.04 5.35 5.95 5.40 5.85 5.56 6.75 5.60 5.66 8.0 9.0 5.90 0.79 0.02 0.69 6.14 6.58 6.25 (;.47 0.30 0.36 9.0 10.0 6.50 7.55 6.09 7.43 6.82 7.31 6.95 7.19 7.07 7.07 10.0 11.0 7.22 8.30 7.36 8.17 7.50 8.04 7.64 7.91 7.78 7.78 11.0 12.0 7.87 9.06 8.03 8.92 8.18 8.78 8.34 8.63 8.49 8.49 12.0 l:i.0 8.53 9.81 8.70 9.60 8.87 9.51 9.03 9.35 9.19 9.19 13.0 14.0 9.18 10.57 9.37 10.40 9.55 10.24 9.73 10.07 9.90 9.90 14.0 l.i.O 9.84 11.32 10.04 11.15 10.23 10.97 10.42 10.79 10.61 10.61 15.0 1(!.0 10.50 12.08 10.71 11.89 10.91 11.70 11.11 11.51 11.31 11.31 16.0 17.0 11.15 12.83 11. 3S 12.03 11.59 12.43 11.81 12.23 12.02 12.02 17.0 18.0 11.81 13.58 12.04 13.38 12.28 13.16 12.50 12.95 12.73 12.73 18.0 19.0 12.47 14.34 12.71 14.12 12.90 13.90 13.20 13.67 13.44 13.44 19.0 20.0 13.12 15.09 13.38 14.86 13.04 14.03 13.89 14.39 14.14 14.14 20.0 21.0 13 78 15.85 14.05 15.01 14.32 15.30 14.59 15.11 14.85 14.85 21.0 22.0 14.43 10.60 14.72 10.35 15.00 16.09 15.28 15., S3 15.50 15.56 22.0 23.0 15.09 17.36 15.39 17.09 15.09 10.82 15.98 16.54 10.20 16.26 23.0 24.0 15 75 18.11 16.06 17.84 16.37 17.55 16.67 17.20 10.97 16.97 24.0 25.0 16.40 18.87 16.73 18.58 17.05 18.28 17.37 17.98 17.08 17.68 25.0 2fi.0 17.06 19.62 17.40 19.32 17.73 19.02 18.00 18.70 18.38 IS. 38 20.0 27.0 17.71 20.38 18.07 20.06 18.41 19.75 18.70 19.42 19.00 19.09 27.0 28. 18.37 21.13 18.74 20.81 19.10 20.48 19.45 20.14 19. SO 19.80 28.0 29.0 19.03 21.89 19.40 21.55 19.78 21.21 20.15 20.80 20.51 20.51 29.0 30.0 19.08 22.64 20.07 22.29 20.40 21.94 20.84 21.58 21.21 21.21 80.0 cos sin cos sin cos sin cos Kin cos sin + 3] LI — + 31 2 — + 313 — 4 31 4 — + 3] 5 — — 22 3 — — 22 8 — — 227 — — 22 6 — — 2S 5 — — 15 J1 + -13 2 + — 133 + — 13 4 + — 13 5 + + 4 Q + + 4 8 + + 47 + + 4 6 + + 4 5 +