( 106 ) 



in all. For eacli of' these corrections U, to (he assumed longitudes 



of the satellites were derived. These direct results from the obser- 

 vations can, however, nut lie used as they stand. There are, as has 

 heen mentioned above, in the longitude of each satellite four unequali- 

 ties, whose periods are between 400 and 500 days, and whose 

 coefficients are of the same order of' magnitude as the libration. 

 These inequalities therefore, during the few months over which each 

 of the ten series of' observations extends, are practically constant, 

 and the correction Alt derived from the observations consequently 

 contains, in addition to the correction Ae,- to the mean longitude, 

 and the libration, also the correction to the assumed values of these 

 inequalities. 



Now the coefficients of these inequalities are proportional to the 

 excentricities and depend on the massef, and are therefore incertain 

 to the same extent as these, i.e. to a very large extent. The periods 

 of' the four inequalities are so nearly equal, that they cannot be 

 separated from each other. Further the period of the most important 

 of' them — important both by its magnitude and by its uncertainty 

 — differs just so much from the average interval of one opposition 

 to the next that, when we consider only the values at the epochs of 

 opposition, the inequality presents itself as one having approximately 

 the period of' the libration, ami can therefore not be separated from 

 the libration itself'. For all these reasons it was impossible to 

 determine the libration (nul he long-periodic inequalities from these 

 observations alone. 



Kor the determination of the masses, leaving for the moment the 

 mass of IV out of consideration, we have (he following data: 



1. the large inequalities in the longitudes of the satellites I, II 

 and III, 



2. the motion of the perijove of satellite IV, 



3. the period of the libration. 



The motion of the perijove of IV also depends on the compression 

 of the planet, which must thus also be investigated, and is deter- 

 mined by 



4. the motion of the node of satellite II. 



The data mentioned under 1, 2 and 4 are those used by Laplack, 

 3 has for the first time been pointed out by me in the communi- 

 cation to the "Nederlandsch Natuur- en Geneeskundig Congres", 

 quoted above. 



The method by which the approximations have been conducted is 

 the following. Certain values of the masses, approximately verifying 

 lie conditions 1, 2, and 4, are assumed, and the corresponding 



