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except for the change due to Jupiter's mass. The resulting Elements J represent the heliocentric longitude and latitude as follows:
Galle states these differences may be accounted for if the perturbations of Saturn and Mars were taken into account.
In A. N. No. 636 osculating Elements K are published for each year from 1839 to 1850. These were computed by Galle. The starting elements are those for epoch 1810, January 0. In computing the special perturbations, Encke and Galle used mass of Jupiter 1/1053.924.
From 1851 to 1870 Galle11 continues the special perturbations by Jupiter and later with the elements of Günther also those by Saturn. These were used in computing the ephemerides published in the Astronomiches Jahrbuch from 1862 to 1870. (See Elements L.)
Beginning with the year 1871 and continuing to 1919, the Jahrbuch published and used Farley's12 osculating elements for computing the ephemeris. (See Elements M and N). Farley's computation includes the perturbations by Venus, Earth, Mars, Jupiter and Saturn. His computations are also the basis for the ephemerides published in the British Nautical Almanac. With Farley's elements we have the following comparisons:
In Annales de l'Observatoire de Paris, Vol. I, Le Verrier publishes the results of his investigation on "Développement de la fonction perturbatrice relative à l'action de Jupiter sur Pallas. Calcul du terme dont dépend une inégalité à longue période du mouvement de cette dernière planète." Le Verrier states that the aphelion of Pallas is 54° from the intersection of the orbit with Jupiter. Consequently when Pallas is at aphelion the distance from Jupiter is increased on account of the great inclination of the orbit. This large inclination diminishes the effect due to the large eccentricity. Le Verrier gives the series for the reciprocal of the distance in a more convergent form and develops the equation in longitude depending on the argument 182-7 Pallas. The maximum of the term is 895". In his report before the Paris Academy,18 Cauchy compares his theory
with the results by Le Verrier; his value for the inequality is 906". Cauchy's investigation is more fully elaborated by M. Puiseux in Annales de l'Observatoire de Paris, Vol. VII. In Vol. VIII, ibid., Hoüel has recomputed the inequality.
The development of the reciprocal of the distance was later extended by Tisserand.14 He shows that the development depending upon the inclination and eccentricity is divergent in some parts of the orbit of Pallas and proceeds to give the analytical development and to apply it to the case of Pallas.
In Bulletin Astronomique Vol. XII, 1895, M. P. Bruck has published the results of his work on "The secular variations of the elliptic elements of Pallas due to the action of Jupiter." He used the method developed by Gauss and extended by Hill and Callandreau. He utilizes elements by Farley for the epoch 1878.
In 1910 George Struve1 published his results on "Die Darstellung der Pallasbahn durch die Gauss'sche Theorie für den Zeitraum 1803 bis 1910." The result of his work based on 63 normal places is a more accurate value for the mean motion of Pallas (769′′.1385). The new value for the annual motion of Pallas compared with Jupiter becomes 18n' 7n123". The deviation between observation and computation still amounts to ±4' which is attributed to the second order perturbations. These residuals are somewhat reduced by empirical terms.
In A. N. No. 205, p. 225, M. Viljev has published his "Recherches sur le mouvement de Pallas." He attempts to reduce the residuals from Struve's work (±4') by taking into account second order terms in the general perturbations, employing the method by Hill. He reports his results as negative.
Juno was discovered by Harding at Lilienthal near Bremen, September 1, 1804. Gauss computed several orbits successively correcting the elements by new observations. Elements VII1 corrected with Bessel's observations, 1807. Ephemeris for 1808, April-December, approximately given. (Elements A.) A number of additional orbits were computed by Gauss' students at Göttingen (Wachter, Möbius, etc.).
Wachter2: Elements, (eccentricity omitted) from the last four oppositions after Gauss' Method, (Neue Comment. der Göttingen K. Societät, Bd. 1) including the opposition 1812. Eccentricity supplied from Bode's Astronomische Jahrbuch, 1816, p. 233. (Elements B.) Möbius3: Oppositions used: 1810, 1811, 1812, 1813. Correcting mean longitude by +4′55′′, the representation of the observations 1815, March, is +8" in longitude, and 51" in latitude. (Elements C.)
Nicolai at Seeberg, near Gotha, compared Gauss' observations with the orbit of Möbius, (empirically correcting L), determined the oppositions and derived new elements. Oppositions used: 1811, 1812, 1813, 1815. "Juno is nearly in conjunction with Jupiter and the perturbations may be large." (Elements D.)
Taking up the determination of the large perturbations by Jupiter by the method of special perturbations, Nicolai derived a new set of elements. Oppositions used: 1811, 1812, 1813, 1815, 1816, 1817, 1818. Representation of observations in 1819, Aa+2′.6 A8 —0′.2. (Elements E.)
Not satisfied with the representation of the observations by his last set of elements, Nicolai extended the computation and determined new elements, which represented the observations of the "Atom" well in 1820. Oppositions used: 1805-1819. Special perturbations by Jupiter. Representation of the observations 1820, May, Aa -7", A8-2". (Elements F.) This computation of the special perturbations was continued for some years.
In 1823, Nicolai' derived his final set of elements, including the determination of the Jupiter mass, for which he found 1/1053.924, in agreement with the value Gauss had found from his theory of the motion of Pallas. The representation of the observations cannot be improved by taking Saturn or Mars into consideration, but Nicolai considers the possibility of the active mass of Jupiter changing with the body acted upon. From these elements osculating elements for 1826 were computed, taking account of the special perturbations by
Jupiter. Berliner Jahrbuch uses the elements by Nicolai until 1830. Fifteen oppositions used: 1804-1823. Special perturbations by Jupiter, (Saturn, Mars, negligible). Residuals in longitude -23′′ to +27′′, still show a run with the period of Jupiter. (Elements G.)
In 1832 new elements by Enckes were introduced, and are carried forward with special perturbations to 1865 by Bremiker and Powalky, for the ephemerides published 1832-1865. Perturbations by Jupiter with mass, 1/1053.924. (Elements H.)
Damoiseau has published general perturbations in the Connaissance des Temps.
Hind 10 took over the work started by Nicolai, Encke, and Bremiker, to compute osculating elements for each opposition by special perturbations. The ephemerides are published in the Nautical Almanac. and the Berliner Jahrbuch. As a basis for this work he derived new elements. (Elements I.)
An attempt to apply Hansen's method of determination of the general perturbations was made by Berkiewicz,11 starting with Hind's elements. The perturbations of the first order with regard to Jupiter, Mars, and Saturn were determined, also the constants of integration leading to a mean motion, 814".090. No comparison with the observations is attempted.
Being aware that the corrections to the ephemerides computed according to Hind had increased to 3' in 1887, Downing 12 undertook to correct Hind's elements. The errors of the tabular heliocentric places published in Greenwich Observations, 1864-1887, are discussed. Equations for the longitude and latitude corrections were set up expressed in terms of corrections to the elements and combined to eliminate the corrections to the radius vector. The mean motion is included in the solution and receives by far the greatest weight. The representation of the oppositions show a pronounced run in Aa. These elements were used for the computation of the annual ephemerides to 1913 in the Nautical Almanac and to 1917 in the Berliner Jahrbuch. (Elements J). Oppositions used: 1864-1887. Special perturbations by Venus, the Earth, Mars, Jupiter, Saturn. Representation varies from 3" to 4" in Aa cos 8, and 1" to 2" in AS. Large residual (-11") for 1874. Representation for 1890 is then in a+3", in 8±0′′ against -65′′ and -6", according to Hind's computations. Since 1917 Ephemeriden der Kleinen Planeten gives mean elements by Boda13 derived by the method of Brendel. (Elements K.) Mean elements. Perturbations by Jupiter according to Brendel, A. N., Vol. 195, p. 417. Expected representation ±0°.5 to year 2000. Oppositions not stated.