ASTRO N OM I SCHE NACH RI CHTEN. Nr. 3923. Band 164. I I.
On the Degree of Accuracy Attainable in Determining the Position of Laplace's Invariable Plane of the Planetary System. By 2". J. J.See.
I. Theory of the Invariable Plane. 1 attaching to the Invariable Plane was chiefly a theoretical 1 one. But as the masses of the planets are gradually be- In the year I 784 Laplace discovered that in every system of bodies *) revolving freely in space and subjected coming known with greater and greater precision, while the to the mutual gravitation of its parts, there exists an In- planetary elements are already quite satisfactory, it appears variable Plane, which is a plane of maximum areas. Under worth while to inquire into the degree of accuracy at present the action of the various physical causes which affect our attainable in the determination of the Invariable Plane of system, this plane is the only geometrical element which is our System. rigorously fixed in space. On that account considerable Suppose Ma, nr,, m, , m3 . . mn to be the n + I theoretical interest attaches to it, and several determinations bodies of the Solar System, and let gi, qi, gi denote their of its position have been made by astronomers. Unfortu- coordinates referred to the origin taken at any point in nately the elements of the planetary orbits for a long time space. Then it may be shown (Cf. Laplace's MCcanique were not known with extreme precision, and the masses of Celeste, Liv. I, Ch. V, 9 22) that the rigorous integrals of the planets remained still more uncertain, so that the interest the areas are:
Laplace observes that the second terms of the right We shall now put these expressions into a suitable members of these equations, when multiplied by di, express form for computing. the position of the Invariable Plane of the Sums of the projections of the elementary areas described the Solar System. n(n+ I) n(n+ I) by each of the right lines connecting the Any areal element of the expression 2 2 pairs of bodies in the system made up of n + I masses, namely, the Sun, and n planets; one of the bodies in each group being supposed to move about the other considered at rest, and each area being multiplied by the product of occuring in the first term of the second member of (I) may the two masses connected by the right line. The functions be given in terms of the elements of the orbit of the body represented by the two terms of the second members of about the Sun. Suppose the Semi axis major to be aj, the these equations are similar in character. The first term eccentricity to be ej, and the mass mi; then whatever be represents areas described by the radii vectores drawn from the position of the axes Og, O,,, we shall have the centre of the Sun; and the second, relative areas de- n(n I) scribed by the + right lines connecting the other 2 bodies in pairs, Thus when taken for the whole system If we multiply the second member of this equation by the two expressions have exactly analogous properties. cosxi, where xi is the inclination of the plane of the orbit
*) Oeuvres Completes de Laplace, Tome XI, pp. 548-551. II I 63 3923 I 64
to the fixed piane of reference (the ecliptic), we shall obtain the area projected on this plane.; and if the axes Oc, O,, be taken in this plane gi 'Ti - 7, '6, -____ dt - + mi) (I,(I - e,a) cos xt (3) --___ But v(Mo+ m,) a, = nza2a, where n, is the mean motion of the planet in a Julian year; and if we put ei = sin 9,,or I - eta = cosasp,, any element of the first term of the second member of (I) becomes
The elements projected on the other coordinate planes would be
where vj is the longitude of the ascending node of the planet's orbit on the fixed plane of reference at a particular epoch. The Sun's mass Mo is usually taken as unity, and hence when we neglect terms of the second order in the right member, our equations take the form
The terms of the second order depending on products I with the uncertainty ui attaching to the value adopted; and of the masses of the general type mz"j are here omitted, I also other required elements of the planetary orbits, taken because the largest of these terms (that depending on Jupiter I chiefly from the theoretical investigations of Professor G. and Saturn = I : 3665725) barely exceeds the uncertainty W.Hill. The improvements in these elements effected by still attaching to a term of the first order in the case of this illustrious geometer, along with similar work by Professor Neptune, where the uncertainty is I : 3862600 of the sun's 1 Newcomb, have rendered our knowledge of most of the mass. In all other cases the second order terms are exceeded planetary orbits quite satisfactory. Using these data the by the uncertainty attaching to terms of the first order. The constants of the areas which determine the Invariable plane following Table I gives the adopted masses of the planets m,, 1 are deduced in Table 11.
Table I. Planetary Elements from the Theories of G. W. Hill.
Planet i mi xi tvi
~~ -- ~ I I I Mercury I I : 14868548f743421 7" 0' ?:./I Venus 2 1 :408134&8163 3 23 34.83 Earth 3 I: 328715f328 o.ooooooo 1 o.ooooooo 0.0 I 6 7 7 I I 0 0 0 0.0 Mars 4 I : 3089967 f 10300 9.7256538-1°1 0.182897I 0.09326113 I 51 2.28 Jupiter 5 I: 1047.35&0.10 8.9258504-~~10.7162375 0.048255511 I 18 42.10 Saturn 6 I : 3500&2.0 ,- 8.5308179-~~109794956 0.05606025 2 29 40.19 Uranus 7 I: 22780f 76 0 46 20.54 Neptune 8 1: 193'3f96 I 47 1.68 * 65 3923 I 66
Table 11. Deduction of Integrals of Areas from adopted Elements. (Units of the 6th decimal).
mi Q aiz cos (pi cos xi ' mi r:) ai2 cos Sp, sin ;si cos vi 1 mi (2) ai*cos spi sin xi sin tpi -- . --. -.I -...... - _.____ - .. - ! 0.040645 + 0.003433 ! 0.003624 2.080146 i + 0.031231 0.1 19309 3.04 172I - - 0.39 7 5 3 I i + 0.008528 I 0.009605 2'75.713000 - 7,740952 49.2 I4443 880.3306 10 -14.58'735 35.471 106 192.214430 ! + 0.747426 i 2.481 I57 283.991 I20 6.7 62 7 40 -. i - .- 5,699903- . .. ._ .- _-.- -. - - 2; = 3537.869203 = c, E2 = -27.231972 = cz & = 94.061984 = c, Ecliptic and Mean Equinox $2 = '060g' 461688; 7 = lo 35' 7'745 1 January o.o, Greenwich M. T. Table HI. Integrals of Areas as modified by altering the masses by the amount of their uncertainty, so as to displace the Invariable Plane. (see pag. 175). (Units of the 6th decimal). n. 0: cos (pi cos xz I mi ('1 a: cos (pi sin xi cos tpi i ??ti (::) ai2 cos (pi sin xi sin tpj
.~ ~ - -. -. . . . - . - -- 0.038710 1 + 0.003269 0.00345 I 2.039357 I + 0.030618 0.1 I 6969 3.044760 - I - 0.398860 I + o.ooa557 0.009637 2175.565000 - 7.740213 49.209740 8 79.8 2 7 800 j - 14.573406 35.450850 192.8j7a40 I + 0.749928 2.489463 285.409700 I - 5.728244 6.796506 - .__ .-- 211=3539.182027 =c,' I 2;.= - 27.136177 =la' I 23' = 94.076616 = c,' 0 = 106O 5' 24120; y = io35' 4'906; d&2= & 2021)49; dy = f2:'839 Let y denote the inclination of the Invariable Plane plane are given in Table IV. These several determinations and 0' the longitude of its ascending node on the fixed 1 are not referred to a common ecliptic and mean equinox, ecliptic of 1850.0:-, then we shall have and such a reduction is hardly worth while in view of the diversity of elements and masses employed by the several c, tan y sin 62 = cs ; c, tan y cos 0 = cz , (7) I investigators. which would also greatly-~ affect the resulting The values of c,*. , c,,". c,., found in Table I1 give- I position of the Invariable Plane. Moreover, Stockwell's in- ' vestigation is the only one in which account is taken of y = 1~35'7:'745; = 106~8'46:'688 . a I the cxistence of the planet Neptune, which was discovered The previous determinations of the position of this i subsequent to the researches of Laplace and Pont6coulant. Table IV. Previous Determinations of the Position of the Invariable Plane. Epoch Authority Source ... -~$2 Y _- . - - . . 102" 57' 29" '035' 31" Ecliptic and Mean Equinox Laplace 1802 Mecanique Celeste, Liv. VI, ch. XVII, 0 46 Of 1750.0 102 57 '5 ' 35 3' Ecliptic of 1750, but ele- Laplace 1802 MCcanique Celeste, Liv. VI, ch. XVII, 9 46 ments of Planets for 1950.0 '03 8 45 ' 34 16 Ecliptic and Mean Equinox Pontecoulant 1834 Theorie Analytique du systeme du Monde, 103 8 50 ' 34 '5 of I 800.0. In second value Tome III, Liv. VI, ch. XXII elements of Planets are for 2000.0 106 14 6.00 I 35 '9.316 Ecliptic and Mean Equinox Stockwell 1872 Smithsonian Contributions to Knowledge, No. of 1850.0 232, p. 166 I I= 3923 I 68
Sun, namely, independent methods, yet the data employed is usually dif- tn" = I : 408134 (Tables du Soleil, p. I). ferent, or the method of procedure somewhat varied, so that the values found appear to be the best obtained recently. Several other determinations have been made since, but this The simple mean of these determinations (I : 408805) by value still remains about the mean of those determinations Leverrier,Bauschioger, and Newcomb, is so near which apparently are entitled to the most confidence. On the by Haosen and Olufsen 1853, that it
...... deduced from the motion of the node of Mercury Is72, I smaller; and, after all, Newcomb's value differs very littlk probably is quite uncertain. Somewhat greater confidence from Hansen's. We take the uncertainty in the mass of may perhaps be placed in the following determioations : i Venus to be one-fiftieth part of the whole, which appears to be a safe limit in the present state of Science. a) Leverrier's latest mass, I III) The Mass of the Earth and Moon. Here we nz" = I : 412150 (Mem. Paris Obs. VI. 309). use the value found by Professor Newconib Giving the mass depending on Deimos a weight of The value deduced by Professor G. W. Hill in his twice that depending on Phobos, these separate results inay heory of Jupiter and Saturn (Asronomical Papers of the be combined into the final value, Lmerican Ephemeris, vol. VII, part I, p. I 7) from the mutual mIv = I : (3089967 f 10300). lerturbations of the two planets is not likely to be capable The uncertainty is about one three-hundredth part of )f improvement for a long time to come, and must there- the whole. The values numbered 7 and 8 in the above table ore be assigned as high weight as any other determination depend on observations made at the Lick Observatory and ret made. This value of Hill tends to confirm the essential at the Naval Observatory, Washington ; these observations iccuracy of the Heliometer values found by Bessel and A. *all, jr.; and it becomes a question what value should be were discussed by Professor S. J. Brown, U.S.Navy, in A. J.467, but are not included in the material treated by Professor ised in the present state of Science. Struve. The adopted value of the mass of Mars therefore It seems probable that Professor Hall’s observations with rests on all the observations of the satellites up to the he micrometer were affected by some kind of systematic present time, and may be considered highly satisfactory. :nor. The same cause appears in Professor Brown’s micro- V) The Mass of Jupiter. This is taken from the neter work after the great telescope was remounted at the researches of Professor Newcomb, who has given much lJew Naval Observatory in I 893. Professor Brown has pointed attention to the subject for many years, especially in con )ut (A. J. 443) a correction depending on difference in magni- nection with the perturbations of Polyhymnia. His Summary of ude of Titan (8.5) and Iapetus (XI) amounting to 0!017. the process used is as follows (Astronomical Constants, p. 97): Under atmospheric conditions which made Iapetus very faint ie says this correction might affect the result by two or Reciprocal of mass of Jupiter from P wt. .hree times its amount. If we use a correction of twice All Observations of the Satellites 1047.82 I .his amount, which appears quite admissable, the result will Action on Faye’s Comet (Moller) 1047.79 1 ie ,u = 3491.7 + 8.5 = 3500.2, in close agreement with Action on Theniis (Krueger) 1047.54 5 .he Heliometer values. Professor Brown has also corrected Action on Saturn (Hill) 1047.38 7 :A. J. 443, p. 88) Hall’s work for difference in the value of Action on Polyhymnia (Newcomb) 1047.34 20 ;crew originally employd (9Y948) and that finally adopted Action on Winnecke’s Comet (von Haerdtl) 1047.I 7 10 :91936). As thus corrected Hall’s work becomes: 1047.35 f 0.065 IJ. P (mean err.) I) Titan J 1876.7 3496.3 f 1.84 Aa, Ad1 We have estimated the uncertainty here as not ex- 1 1878.7 3482.7 f 1.49 sandp 1 3489.5 ceeding one ten- thousandth part of the whole. 1876.7 3481.2 f 0.65 Aa, Ad) 2) Iapetus VI) The Mass of Saturn. The mass of Saturn found 1 1880.2 3493.8 & 0.97 sandp [ 3487*5 by Bessel from observations of Titan with the Konigsberg Heliometer was for a long time universally used by astro- The mean of I) and 2) is 3488.5, which is perhaps nomers. But during the last quarter of a century some doubt the best mass reciprocal derivable from Hall’s work. If we has arisen as to the true value of this mass, on account of should apply to this value a systematic correction similar the discordant values furnished by excellent series of micro- to that suggested for Brown’s work, amounting to +8.5, meter measures made with large refracting telescopes. The we should have p = 3497.0, which would make Hall’s observations by Professor Hall with the 26- inch Refractor result quite accordant with H. Struve’s (3496.7). As some at Washington in 1876-80 gave the first decidedly dis- exception might be taken to the validity of this introduction cordant values, and the fine series subsequently made at of corrections which-are more or less arbitrary, it seems Poulkowa by Professor H. Struve tended to confirm the best to use the simple mean values corrected for error of larger value found by Hall. A similar result was also de- screw, giving each a weight proportional to its probable duced by Professor S. J. Brown with the 26-inch Refractor importance. Professor Struve’s work is summarized thus : at Washington in 1894-96. Beob. d. Saturnstrabanten, Erste Abteilung 1888, p. I I 7. In 1886 Professor A. Hall, jc. undertook a redetermi- Iapetus 3500.2 0.82 Aa, Ad nation of the mass of Saturn by the method of Bessel, using Titan 3495.7 1.43 combined with Rhea the new Yale Heliometer, and exercising every known pre- f caution against systematic error. The probable error attached Mean value p = 3499.1 f 0.71 . to this mass is not so small as in the case of Bessel’s work, Beob. d. Saturnstrabanten, but on the other grounds this value appears to be entitled Vol. XI, Publ. de I’Observatoire Central Nicolas, 1898, p. 239. to the highest confidence. Hall’s result was I II tZM = I : (3500.5 f 1.44). Tethys - Rhea 349o.a 3493.5 On the other hand a rediscussion of Bessel’s work by Enceladus - Tethys 3494.1 3501.5 Professor H. Struve, who applied small systematic corrections Dione - Rhea 3489.4 3492.0 for slight error in the value of Bessel’s Heliometer scale Rhea - Saturn 349 1.3 3499.0 found by Auwers, and for mean distance of Saturn due to Titan - Saturn 3495.6 3496.4 defect in Bouvard’s Tables, led to the final value 3492.1 3496.5 ?lfM = I : (3502.5 f 0.77). Simple mean p = 3494.3. By combining this value with 3499.1 given above, we (3498 and 3495.3) adopted by Struve in his two memoirs, get p = 3496.7, which utilizes all of Struve’s material, and and is quite free from any arbitrary corrections. Under the is probably the best value afforded by his observations as circumstances the following values may, I think, be safely a whole. This is within 0.05 of the mean of the values used in deducing the probable mass of Saturn: P wt. 1) 3502.2 25 Hill, Theory of Jupiter and Saturn 2) 3502.5 2 5 Bessel, Heliometer Observations of .Titan 3) 3500.5 25 A. Hall, jr., Heliometer Observations of Titan 4) 3496.7 I 7 H. Struve, Micrometer Observations of Satellites 5) 3488.5 3 A. Hall, Micrometer Observations of Titan and Iapetus 6) 3491.7 -5 Brown, da, Ad of Titan and Iapetus %an 3499.98 It thus appears probable that the mass of Saturn is applied in the table for errors of screw, and the assignment the reciprocal of 3500.0 f 2.0, so that the uncertainty will of weights is tolerably obvious. We have utilized as many not exceed one part in I 750. independent methods of finding this mass as possible, with VII) The Mass of Uranus. The deduction of this a view of reducing the effect of systematic error depending mass is concisely shown in Table V. Corrections have been on any one method. - - Table V. Determinations of the Mass of Uranus. I No Epoct Authority p=- Method Source
I The earliest determination of mass. - 2. Used formulae of Laplace. - 3. Same Observations, worked by Laplace. - 4. Same Observations as in No. I. - 5. Bouvard’s data rediscussed by Hill gives No. 18. - 6. Obs. of Oberon and Titania, Munich Io1/2-inch Refractor. - 7. Adams prefers mass of I : 2 1000 to any heretofore used. - 8. Careful discussion, but Obs. not very good. - 9. Observations with 6 feet Reflector, Birr Castle, 1872-3-4. - 10. The first really fine Observations. All the Washington determinations have been corrected for effect of erroneous value of Micrometer screw (91,948) originally used instead of 9Y936, which Hall showed to be the true value. - I I. Corrected for error of screw. - 12: Hall’s final value as corrected. - 13. Hill finally adopted this value. - 14. Obs. treated by Bouvard more thoroughly discussed by Hill. - 15. Hill thinks difference between 18 and 19 due to constant cause, as yet unknown. - 16. Newcomb’s latest value. '73 3923 ' 74
VIII) The Mass of Neptune. The mass of this planet is treated in Table VI, where the method is similar to that followed in the case of Uranus. Table VI. Determinations of the Mass of Neptune. - ~ ~~~~~ ~~ ~ ~ II I No. Spoch ' Authority p = -$ViE I wt. Method I Source (Rem. I - .- - I .- - -__ -. ._ ____ -- - - - . I I 0. Struve Obs. of Satellite i C. K. XXV, 813 1 2 1848 Peirce Perturbations of Uranus A. N. XXVII, 203, No. 637 iz 3 18471848 I Peirce Obs. of Sat. by Lassell and Bond M. N. R. A. S. VIII, 128 3 4 1849 I Hind Obs. ofsat. by Lassell, Bond, 0.Struve ~ M. N. R. A. S. IX, 203 i 5 1850 A. Struve Obs. of Sat. by 0. Struve Dorpat. Heob. XIII, 1856, Anh. 24 ! 4 6 1851 I G.P.Bon8 Obs. of Sat. by G.P.Bond, W. C.Bond 1 A. N. XXXI, 38, No. 7 23 (5 7 1855 Hind Obs. of Sat. by Lassell, at Malta 1852 M.N. R. A. S. XV, 47 '6 8 1862 Safford Perturbations of Uranus ' M. N. R. A. S. XXII, 144 !7 9 1874 Newcomt Perturbations of Uranus 1 Smith.Contrib. to Knowledge XIX, I 7 3 ' 8 10 I875 Newcoml Obs. Sat. Wash. 1873-4 ~ Wash. Obs. 1873, App. I 19 'I 1876 Holden Obs. of Satellite 1 A. N. LXXXVIII, 186 10 I2 1876 Hall Obs. of Sat. 1875-6 j Wash. Obs. I 88 I, App.II; cf. H. Struve 1 I I Beob. d. Neptunstr., Mem. St. Pet., Acad. XLII, 4, p. 65 I '3 1882 1 Hall Obs. of Sat. 1881--2 Wash. Obs. 1881,App. I1 I2 Obs. of Sat. 1883-4 Wash.Obs. 1881,App.II;cf.H.Struve 1 13 loc. cit. 15 1892 I A.Halljr Obs. of Sat. I 89 1-2 A. J. 267 I 16 1894 H. Struvt Obs. of Sat. Poulkowa 3o-inch Re- Beob. d. Neptunstr., Mern. St. Pet. 15 I fractor, 1887-1892 Acad., Vol. XLII 4, p. 6j I 17 I 1894-5' Barnard '9565 !2 Obs. of Sat. at Lick, 36.inch Re- A. J. 441 I l6 I1 ! fractor, discussed by Hall 18 I 189j I Newcoml 19540 Value adopted in Sec. Var. of the I Astronomical Constants, p. I 07 ; Ast. I I 7 I i2 Planets I Pap. Am. Ephem., Vol. VI, p. 12 I 19 I 1897 I Newcoml 19261 l3 Perturbations of Uranus Cf. Brown, A. J. 473 20 1 1899 ' Brown '9399 '3 Obs. of Sat. 1897-8 A. J. 473 21 I 1900 1 See 1844s ; ._ iI Obs. of Sat. 1899-1900 1 A. N. 3663 I9
I. Sat. very faint in Poulkowa I 5 -inch Kefr. - 2. First mass from theory of Perturbations. - 3. First suitable obs. of sat. for finding mass. - 4. Mass affected by systematic error. - 5. Excellent obs. with I 5 -inch Harvard Refractor. The first really good mass. - 6. He thinks I : I 7500 nearer the truth. - 7. An early mass deduced from Perturbations. - 8. First good theory of Neptune. - 9. Newcoinb's value is here corrected for error of Micrometer screw. 91948 was used iostead of 97936. - 10. Corrected for error of screw. Cf. H. Struve, Beob. d. Neptunstrabanten, Mem. St. Pet. Acad. XLII, 4, p. 65. - I I. Corrected for error of screw. - I 2. Good observations. Cor. for error of screw. - 13. Ex- cellent series. Cor. for error of screw. - 14. He considers his mass uncertain. - I 5. Fine series of observations, carefully discussed. - I 6. Good observations, well discussed. - I 7. Probably based on Perturbations and on obs. of Satellite. - 18. Excellent observations well discussed. - 19. See's first obs. of Satellite, conditions not very favorable. 111. Uncertainty Attaching to the Position of the Invariable Plane. In concluding this paper it seems very desirable to as- ' computed position of the Invariable Plane due to the al- certain the degree of probable uncertainty still attaching to the ' terations in the masses of the planets. For any other in- Inclination and Node of the Invariable Plane in the present i crease or decrease of the masses, within the limits of uncer- state of our knowledge respecting the masses and elements. 1 tainty assigned, will effect the position of the plane less Poisson has observed (Trait6 de Mecanique, Tome I, I than that here suggested, where the increments are all positive Paris 181 I, p. 281) that the Invariable Plane is a plane of 1 on one side of the plane, and negative on the other. Ac- moments; and hence if the several masses on one side of i cordingly we have computed the position of each orbit with this plane at a given epoch be increased by the aniount I reference to the Invariable plane by means of the formulae: of the uncertainty attaching to each mass, while those on 1 the other .side are-decreased by the a.mount of their uncer- I sinXt sin pio = sin Xi sin 0) (vi'- (9) tainty, the effect will be a maximum displacement of the I sin Xp COS lu? = COS sin Xi Cos (vj - 6;3) - sin y cos xi ] - .- .- - -- - ,' After this paper is finished and ready to go to the printer I am greatly surprised to find that Professor Newcomb finally adopted for Neptune a mass of I : 19314 (Tables of the Heliocentric Motion of Uranus, Ast. Pap. Am. Ephemeris, Vol. V11, p. 293). - Notc, Aug. 4, 1903. ' 75 3923 ' 76 Table VII. Longitudes of Asc. Nodes and Inclinations of orbits on the Invariable Plane, the Longitudes being reckoned from Descending Node of Ecliptic of 1850 on Invariable Plane.
(Stockwell) I Mercury 323" I I' 23:53 75" 7' 13:78 288" I' 33:'03 6O 20' 52:67 287'54' 5Y12 6" 20' 58Y08 Venus 243 57 44.34 129 17 14.36 307 24 41.17 2 11 15.11 307 14 8.10 2 11 13,57 Earth 99 48 18.67 100 21 21.34 I80 0 0.00 I 35 7.745 I80 0 0.00 I. 35 19.316 Mars 83. 9 16.93 333 17 53.49 249 6 13.50 1 40 30.85 248 56 21.45 1 40 43.70 Jupiter 159 56 24.98 II 54 31.67 210 4 32.84 o 19 42.09 210 7 35.44 0 I9 59.674 Saturn 14 49 38.09 90 6 41.37 16 45 35.15 o 56 2.79 16 34 26.66 0 55 30.924 Uranus 28 25 17.05 168 15 6.7 204 7 30.31 I I 36.19 204 12 33.78 1 1 45.27 Neptune 335 5 38.91 43 17 30.3 86 29 3.26 10 43 34.35 I 286 39 55.10* 0 43 24.845
The values of xjO and tpp are given in Table VII, and fixed stars are referred. If the inclination of the In- which also contains the ordinary elements L and rr omitted variable Plane were certainly known within the limits of from Table I. From this it will be seen that at the epoch fo!'2o, a degree of precision would be attained which would 1850.0, the Earth, Mars, Uranus, and Neptune were north leave very little to be desired. Considering the progress of the Invariable Plane, while Mercury, Venus, Jupiter and of Practical Astronomy since the time of Bessel, this im- Saturn were south of it. To shift the computed position of provement might be easily effected during the present cen- the Invariable Plane by a maximum amount for the uncer- tury. In that case Laplace's expectations of finding an im- tainty attaching to the planetary masses, we take ui with movable plane of reference equally good for all ages and potitive sign for the planets Earth, Mars, Uranus and Nep- serviceable alike for the Planets, Comets and Fixed Stars, eventually would be realized. tune, and negative sign for the planets Mercury, Venus, Jupiter and Saturn. The result of this computation is given It is hardly necessary to state that the Satellites, in Table III. Hence it appears that the variations in the Asteroids and Comets are too small and too symmetrically position of the plane are distributed to exert a sensible influence on the position of this plane. Nor is it likely that any other planet as yet do = +202:49 dy = f2Y839. undiscovered would need to be taken into account. Laplace has shown (cf. Mecanique Celeste, Liv. VI, ch. XVIII, 9 47) This shows that the Invariable Plane may now be determined that the great distance of the fixed stars renders their per- with very considerable accuracy. The actual shifting of the turbing action upon the planets so very minute that it seems plane due to improvements in the masses is likely to be unlikely ever to give this fundamental plane of our system about one third of the maximum variation here computed, a precessional motion among the stars which could be per- and hence we may, I think, conclude that the inclination ceived by the inhabitants of the terrestrial globe. Accordingly is uncertain to the extent of about I" and the Node by when the elements of the principal planets and their masses about 1'. are known with the required accuracy, the resulting position If serious effort were made during the next half cen- of the plane may be regarded as rigidly invariable. tury, probably the masses of the planets and their elements As transformation to such a plane in practice would could be found with such precision, that the Invariable plane be somewhat troublesome, it is not likely to be so useful would become known with all the accuracy required in Prac- in the theory of the fixed stars, as in the theory of the tical Astronomy. Already the inclination of the Plane is planets and comets, where the orbits undergo great periodic known with a degree of accuracy approximating that of our oscillations depending on the Secular Variations of the knowledge of the ecliptic and equator, to which the planets Elements tinder the action of Universal Gravitation.
Washington, D. C., 1903 Aug. I. T. 7. 3: See.
Beobachtung des Planeten (156) Xanthippe. 1904 Genn. 9 9h37m17~ t. m. Roma Coll. Rom. da = -0~46917 66 = +0'32:8 Cfr. 12.4 micr. filare ingr. 240 Gr. 1rm9 aapp. = 8h5m4919~(9.5~4~) dapp. = +7"51'35:3 (0.716). Stella di confront0 (1904.0): gh6'"34?66 +1f42 +7"51' 14:s -12:o AG. Leipz. XI 4415.
Roma, 1904 Genn. 10. B. Millsevich.
Inhalt zu Nr. 3923. T.y. 7.See. On the Degree of Accuracy Attainable in Determining the Position of Laplace's Invariable Plane of the Planetary System. 161. - E. Millosmich. Beobachtung des Planeten (156) Xanthippe. 175. - Cteichlosren zp04 Jan. 18. Herausgeber: H. Kreuts. Druck ron C. Schoidt. Expedition: Riel, Niemannsweg 103.