No. 3.] MINOR PLANETS— LEUSCHNER, GL.A.NCY, LEVY. U 



When Jupiter's mean motion and tliat of the planet are nearly commensurable, the inte- 

 gration of Hansen's difTerential equations becomes impracticable through the presence of large 

 integrating factors. The integrals are of the form: 



sm 



■('■-'^7 



' \(in-i'n')t 

 ' cos 



•OO <t< +00 

 0<i'< +00 



For the Tlecuba group the mean motion is approximately twice the mean motion of Jupiter. 

 Hence, for exact oommensurability, 



7l' 1 1 •/ o • 1 



j = 00 ; I = 2, a = 1 . 



'n 2' 



■(*-*¥)" 



By introducing the exponential in place of the sine and cosine, the indeterminateness can 

 be removed, for if in — i''n' = 0, then (.V-i('''-i'n')(= j. This ia one of Bohlin's modifications. 



For any given ])lanet the ratio is not exactly commensurable, and the developments are 

 originally made for the case of exact commensurabihty. They are then expressed, for a given 

 case, by Taylor's series in ascending powers of a small quantity w, which depends upon the 

 difference between the real ratio and exact commensurability. In addition to positive powers 

 of w there will occur negative powers. They are due to the following causes. An argument is 

 introduced (see p. 13), from which the mean anomaly of Jupiter is eliminated through the intro- 

 duction of w. It is a necessary consequence of the form of the partial differential equations in 



which -J- appears, that the integration of first-order terms shall contain w~^ and that higher 



order terms shall contain other negative powers. Hence the integrals are series in both posi- 

 tive and negative powers of w. 



In distinction to the method of Hansen the elements appear explicitly in the arguments 

 or as factors in the terms of the series. 



An important feature of v. Zeipel's theory is his treatment of the constants of integration. 

 Since the method is essentially Hansen's, the constants of integration must be determined con- 

 sistently with that method. Given osculating elements, the constants of integration are deter- 

 mined by the condition that, at the date of osculation, (t = 0), the perturbations and their 

 velocities shall be equal to zero. 



V. Zeipel ailopts osculating elements as his initial elements. With these elements and the 

 perturbations and their velocities at the date of osculation, he computes elements, designated 

 by the subscript unity, in which the constants of integration are absorbed. They are analogous 

 to Hansen's constant elements, i. e., the fundamental equations of Hansen are valid. 



Our transformations of the elements differ from v. Zeipel's in two respects. First, the 



constants in -., and in its velocity have not been introduced into the elements i, Si, but 



cos ^ 



are treated in the usual Hansen manner. Second, v. Zeipel introduces certain terms in the 



perturbations which have the same period as the planet (argument e), into the elements to 



form mean elements. This ha^> not been done. 



The general tables, XXXV, XXX\TII, XLIII, LIV, LVi, LVii, LVI, LVH, which are 

 required in computing the perturbations, are given at the conclusion of the formulae. The 

 formulae for any planet of the group ^ are given completely, and they are supplemented by 

 numerical values for the planet (10) Hygiea. 



The references to v. Zeipel's paper arc indicated briefly by Z, followed by the number of 

 the page. 



The osculating elements of the planet are taken from Z 139; the elements for Jupiter are 

 taken from Astronomical Papers of the I'nited States Nautical Almanac Office, Vol. VII, p. 23. 



