54 CELESTIAL MECHANICS: LEUSCHNER 



(588) ACHILLES, 1906 TG. 



Discovered by Wolff 1 at Heidelberg 1906, February 22. 



From observations of February 22, and March 5, Berberich 2 com- 

 puted Elements A for a circular orbit (search ephemeris for April 

 1906). 



The first elliptic Elements B were published by Berberich. 3 They 

 are based on observations 1906, February 22, March 23, and April 

 22. An ephemeris for May and June, 1906, is also included. He also 

 points out that the aphelion point lies far beyond Jupiter's orbit, 

 and that the present orbit has had its form and position for a long 

 time. 



From Elements B, Charlier 4 finds that Achilles is approximately 

 55 ahead of Jupiter, consequently very close to one of Lagrange's 

 libration points. He also points out that we may have here a case 

 (as he has shown) 5 where the planet does not remain at the apex of 

 the equilateral triangle, as suggested by Lagrange, but oscillates about 

 it with a period of about 148 years. 



Comments on the character of the orbit of Achilles have been pub- 

 lished by Berberich, 6 Crommelin, 7 Ristenpart, 8 Stroobant. 9 



On the basis of observations extending from 1906, February 22, to 

 May 19, Bidschof 10 has published a set of Elements C and an 

 ephemeris for 1907. A continuation of the ephemeris with corrections 

 to the same was published by Bidschof. 11 An observation 1907, 

 February 12, gives Aa 51 s , and AS +6'.7. 



An ephemeris for 1909 based on Elements B. J. 1911 with special 

 perturbations due to Jupiter was published by Franz. 12 



For the following years, to 1919, the B. J. (Kleine Planeten) pub- 

 lishes elements by Bidschof brought forward to a new epoch and 

 mean equinox, 13 Elements D. 



As an application of Leuschner's 14 satellite method, Einarsson 15 

 computed a preliminary orbit, (Elements E), based on observations 

 1907, February 12, April 15, and June 2. Special perturbations, due 

 to Jupiter, computed by Encke's method, were included for the period 

 covering the observations. The maximum residuals, for nine observa- 

 tions of 1907, were Aa cos 8 3".6, AS 0".7. Elements E were 

 used, without perturbations, to represent an observation 1906, May 

 19, with the following results: Aa cos 8 +1'01".8, AS 39".2. 



The most recent work on Achilles was done by Julie M. Vinter- 

 Hansen. 16 With Bidschof's Elements D, the special perturbations, 

 due to Jupiter and Saturn, were computed from 1906 to 1914. All 

 observations of 1906 and 1907, two of 1913, and one of 1914, were 

 then represented and residuals determined. With these as a basis, 



