I. FORMULAE AND TABLES FOR THE HECUBA GROUP, ACCORDING TO THE 

 THEORY OF BOHLIN-v. ZEIPEL, AND AN EXAMPLE OF THEIR USE. 



DETERMINATION OF CONSTANT ELEMENTS AND OF PERTURBATIONS OF THE MEAN ANOMALY. 



The planet (10) Hygiea was selected by v. Zeipel as an example of the use of his tables for 

 the group ^. We have used it as a preliminary example for the application of our own tables, 

 so as to provide further comparison of our tables with those of v. Zeipel. 



This example is presented with the direct purpose of meeting the needs of the computer. 

 For this reason, no attempt is made to explain the significance of the functions involved, yet 

 their use will be less mechanical, if, in a general way, some of the essential principles under- 

 lying their development are understood. The theory of v. Zeipel is taken up in the second 

 section of this memoir. 



The method proposed by v. Zeipel is a practical adaptation of Bohhn's method of com- 

 puting the perturbations by Jupiter upon planets whose mean motions bear nearly commen- 

 surable ratios to that of Jupiter. In particular, the formulae are derived for the planets of the 

 Hecuba group. Tracing the history of this method one step further back, Bohhn's method is a 

 modification of the theory of Hansen for the indeterminate case of nearly commensurable mean 

 motions. Or, concisely, in \'. Zeipel's own words, ''Die benutzte Methode kann einfach dadurch 

 charakterisirt werden, dass die Differentialgleichungen von Hansen mittels des Integrations- 

 verfahrens des Herrn K. Bohlin gelost worden sind."' 



Certain principles of Hansen are fundamental to an understanding of some of the important 

 equations. Briefly, the perturbations are reckoned in the plane of the orbit and perpendicular 

 to it. In the plane of the orbit nbz signifies the displacement in the planet's mean anomaly 

 (52 is the perturbation in the time) ; v gives the disturbed radius vector through the relation 



r = f(l +v) 



u 

 and the displacement in the third coordinate is denoted by =. With Hansen's choice of ideal 



COS » 



coordinates, the fundamental analytical relations are: 



£ — e sin £ = nt-\-c-\- nSz 



f cos f=a (cos £ — e) (1) 



r sin f=a ■>Jl —e^ sin £ 



r = f(l +1^) 



dB = ^^.asm 1" 

 ^ cos % 



Jx = dp cos a (2) 



Ay = dp cos h 



Az = dp cos c 



x = r sva.asva. {A' -\-f)-\-Ax 



y = r sin b sin (B' +/) + Ay . (3) 



2 = r sin c sin (C +f) +Az 



where £,/, f are fictitiously disturbed coordinates, which, in connection with constant elements 



u ' 

 and the perturbations nhz, v, and = give the true position of the body. A', B', C, sin a, sin h, 



sin c are the constants for the equator. The notation for the eccentric anomaly and the true 

 anomaly is v. Zeipel's; in Hansen's notation they would be written e,f. 



1 Angenahertc Jupiterstorungen fiir die Hecuba-Grappe, p. 1. 

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