II. TABLES FOR THE DETERMINATION OF THE PERTURBATIONS OF THE 



HECUBA GROUP OF MINOR PLANETS. 



DEVELOPMENT OF THE DIFFERENTIAL EQUATIONS FOR W AND FOR THE THIRD COORDINATE. 



It would be futile to attempt to give a brief but comprehensive outline of the fundamental 

 developments in the theory of Bohlin-v. Zeipcl which would assist the reader to an imderstanding 

 of the construction of the tables. In broad outlines, the problem is the integration of Hansen's 



differential equations for vZz, v, and -•> by means of the method developed by Bohlin and 



COS V 



according to the modifications introduced by v. Zeipel for purposes of numerical computation. 

 The first division of the problem is the development of functions of the partial derivatives of 

 the perturbative function; the second division of the problem is the integration of the Hansen 

 equations in the form of infinite series. 



For the theory the reader is referred to the original works of Hansen', Bohlin^, and v. Zeipel'. 

 As indicated in the introduction to the first section, unless otherwise stated, the references to 

 Bohlin refer to the French edition and arc designated by B; references to v. Zeipel are desig- 

 nated by Z. Although dupHcation of material which can be found in either reference is to be 

 avoided, our experience in attempting to reproduce v. Zeipel's tables led us to fill in certain 

 gaps which are troublesome to the reader and the computer. 



The first section of v. Zeipel's theory is concerned with an independent development of 

 Hansen's differential equations for ndz and v and a repetition of the differential equation for 



-t and the introduction of Bohlin's argument 0. In passing, it is well to emphasize two 



COS % 



facts: First, the variables e and /"used throughout the theory are analogous to Hansen's s andy- 

 the dash is unnecessary, for the physically real values do not appear. Second, the constant 

 elements a, e,Tz, c, Q,,i are neither osculating nor mean elements; they are defined in the section 

 on constants of integration. 



The perturbative function and its partial derivatives are developed in Fourier's series, in 

 which the arguments depend upon the relative positions of the disturbed and disturbing 

 bodies and in which the coefficients are infinite series in ascending powers of the eccentricities 

 and the inclination of tlie orbits. The coefficients in the latter are elliptic integrals depending 

 upon the ratio of the semi-major axes. 



Since these elliptic integrals are functions of the ratio of the semi-major axes, or of the 

 mean daily motions, they can be developed in Taylor's series, in which the given function and 

 its successive partial derivatives are expressed for exact conamensurabiht}' and the series pro- 

 ceeds according to a small quantity w, defined by w=l— 2//, where /i is the ratio of Jupiter's 



mean motion to that of the planet and where fi differs but little from ^ • These elliptic integrals 



enter the coefficients in all of the subsequent trigonometric series. Hence all the coefficients are 

 series in w. With some exceptions the terms in w°, w, and v/ have been used. The develop- 

 ment of all functions in powers of vj is the essential principle underlying the group method of 

 determining perturbations. 



Tlie following pages contain the tables which are, in general, parallel to those of v. Zeipel. 

 At the end of sections 2, 3, 4, 5 there are brief written comparisons. To facilitate comparisons 



' Auseinandersetzunc einer zweckmassiRen Methoilo zur Bereehnung der absoluten Storungen dcr klcinen Planeten. 



' Formeln und Tafeln zur gruppenweisen Bereehnung der allgemeincn Storungen bcnachbarter I'lanetcn. Nova Acta Reg. Soc. So. UpsalienslJ, 

 Scr. Ill, Band XVII, U'96. 



Bur Ic Di'vcloppcmcnt dcs Perturbations Plan^taires. Application aui Petites Planfttes. Stockholm, 1902. 

 ••■VngeniilieneJupiterstoningenfurdie Hecuba-Gruppe. St. PStersbourg, 1902. 



41 



