■i'i, y,,j, ü,j, Ui, —^ as functions of the masses, or rather, as f unc- 



( 657 ) 



Here y/ and ^i are the oum inclinations and nodes. The angles 

 ^i vary proportionally witli ihe time and the coefticients ^y depend 

 on the masses. We have again Oa = 1. to and <9 are the inclination 

 and node of the mean equator of the planet on the orbital plane 

 for 1900.0. In the discussions we have used the abbreviations : 



.^•p = — to si)i (if^o — (9) 



y^ — — CO cos (ip^ — d) — to„. 



x^ and //o thus determine the position of the equator. The adopted 

 fundamental plane nearly co-incides with the equator, and the node 

 tpo has nearly the theoretical motion of the node 6. ^) The angle 

 i\\ — <9 is therefore constant and very nearly equal to 180°. 



In Gron, Puhl. 17, Chapter IV, I have given the quantities (2/, /^% r^ 

 d(J)i dOi 



~di' '''■' ""'' ""''' '''■' Vt 



tions of the small quantities x' and iv, w^hich are defined by 

 Jb' — 0.0219087 (1 4- >«) (6 = 1 for d = 39".0) 



=: 0.0000 0000 530042 (I + x) (astronomical units). 



x' = x + 0.055 ■ 

 m^ — 0.0000 40 (1 -I- r J m^ — 0.0000 80 (1 + v,) 



m, = 0.0000 22 (1 + rj m, = 0.0000 424 751 (1 + v,). 



Of >c', i\ and I's only the first power was kept, of v^ and i\ all 

 powers, which could be derived from Souillart's formulas were 

 taken into account. 



For the reciprocal of the mass of the system the value 



^^ = 1047.40. 

 was adopted. 



The data to be derived from the observations can be divided into 

 three groups. 



A. The inclinations and nodes, represented by the quantities p 

 and qi, i.e. the quantities determining the latitudes. 



B. The data determining the longitudes and radii-vectores. These 

 are the mean longitudes, the equations of the centre and the coeffi- 

 cients of the great inequalities of the three inner satellites (//, hi,ki,Xi) 



C. The mean distances ai. 



A. In determining the elements from eclipse-observations, the 

 elements of group B are derived from the observed epochs of the 

 middle of the eclipse, those of group A from the duration of the 



1) The motion of J/q actually used by Marth is not exactly the theoretical 

 motion of L The difference is however negligible. 



45 



Proceedings Pioyal Acad. Amsterdam. Vol. X. 



