24 CELESTIAL MECHANICS: LEVSCHNER 



From 1871 to 1910 the Berliner Jahrbuch publishes elements (see 

 Elements G), by Farley. 10 His work is based on twelve oppositions, 

 1840 to 1855. This work forms the basis for later investigations of 

 the general perturbations. 



Probably the most extensive work on a minor planet are' the tables 

 of Vesta by Leveau published in Annales de FObservatoire de Paris 

 Memoires, XV, XVII, XX, XXII, XXV. The method applied by 

 Leveau is that of Hansen "Auseinandersetzung einer Zweckmassigen 

 Methode zur Berechnung der Absoluten Storungen der Kleinen 

 Planeten, I, II, III." The explanation for the choice of this method 

 is that the application to Vesta is a preparatory study to the motion 

 of Pallas, as Gauss' theory of Ceres was a preliminary study to his 

 theory of Pallas. Memoir XV contains the perturbations of the 

 first order of the masses of Venus, Earth, Mars, Jupiter, Saturn, 

 Uranus, and Neptune. The memoir concludes with the determina- 

 tion of the constants of integration and the expressions for nSz, v, 

 u seci and a representation of an observation 1858, April, 23.5 as 

 follows: Aa= s .! and AS= 0".4. Memoir XVII contains the 

 terms depending upon the square of the mass of Jupiter in n8z 

 and 2i/. The effect on u seci becomes noticeable after 100 years. 

 Memoir XX contains the terms depending upon the product of the 

 masses and concludes with a set of mean elements and corresponding 

 expressions for nSz, v, and u seci. The mean elements given in memoir 

 XX are slightly changed in memoir XXII (see Elements H), on ac- 

 count of some perturbations of the second order that Farley had in- 

 cluded and which form the basis of Leveau's work. Memoir XXII 

 contains the comparisons with 215 normal places founded on 5000 

 observations extending from 1807 to 1889. The computation has been 

 changed to conform to the solar tables by Newcomb. The correction 

 to the mean motion is zero. The new Elements I represent the obser- 

 vations in right ascension between 2" and +4" and in declination 

 between V and +4". A new value of Jupiter 1/1046, and Mars 

 1/3648000. The perturbations are collected in tables introducing the 

 mean anomaly and corrective terms for the eccentric anomaly. Mem- 

 oir XXV contains some supplementary terms depending upon the 

 product of the masses. One of the larger terms has a period of 3000 

 years. The effect of the critical terms, which are mentioned in the 

 beginning of his work, shows itself by comparing these terms before 

 and after integration. The coefficients of corresponding terms are 

 ten or twenty times larger, whereas the other terms mostly decrease. 



Leveau has made a comparison between observations and calcu- 

 lation of later positions 11 showing the residuals as obtained from 



