382 Royal Astronomical Society. 



being hopeless in its present situation, the only data by which a 

 knowledge of the elements of their orbits can be obtained are their 

 angles of position with the meridian, which are susceptible of toler- 

 ably correct determination. For the investigation of the elements, 

 the author considers the best method to be that which he has used for 

 determining the orbits of revolving double stars from data of the same 

 nature. The application of this process to the case of the satellites, 

 however, is greatly facilitated by the approximate knowledge already 

 possessed of their periods, and the situations of the planes in which 

 they revolve. In that of the double stars, these elements are wholly 

 unknown. But in the case of the satellites, by using the approximate 

 node and inclination, the observed angles of position, as seen pro- 

 jected on the heavens, may be reduced to the plane of the orbit, 

 thereby simplifying the computations considerably. And a knowledge 

 of the periodic time enables positions, observed in different revolu- 

 tions, to be used as if they had been observed consecutively in a sin- 

 gle revolution. In order, however, to justify this mode of proceeding, 

 it is necessary, in the first instance, to show that a sufficient approxi- 

 mation to the values of the elements is already possessed. The au- 

 thor has at present limited himself to this preliminary verification, as 

 he had not sufficient time to investigate the subject more completely. 

 There are given 49 observed angles of position of the first of the 

 two satellites, and 59 of the second ; 31 of the first, and 32 of the 

 second, having been observed by Sir William Herschel between 1787 

 and 1798 ; and the remainder by the author, between 1828 and 1832. 

 Assuming the position of the nodes of the satellites' orbits, and their 

 inclinations to the orbit of the planet, as given by Sir W. Herschel 

 (viz. longitude of ascending node of each 165° 30', and inclination 

 101° 2'), the observed angles of position are reduced to the plane of 

 the satellites' orbits : and supposing that the orbits are circular, and 

 assuming the times of revolution as delivered by Sir VV. Herschel, 

 two epochs are determined at which the satellites pass through the 

 ascending node ; the first being in the year 1787, and the second in 

 the year 1828. And, the number of revolutions performed in the 

 interval being known from the approximate periods, the following 

 correct values of the times of revolution are obtained, viz. : 



d h m s 



First Satellite . . . . 8 16 56 31-3 



Second Satellite . . . 13 11 7 12-6 



The first being 26 s * 1 greater, and the second being l m 46 s *4 less, 



than the times given by Sir William Herschel in his paper above 



mentioned. 



On comparing the angles of position, computed from the corrected 

 elements, with those actually observed, the errors for the first satel- 

 lite seldom exceed 10°, and for the second 7°; and they are, for the 

 most part, inferior in amount. This the author considers a reason- 

 able degree of precision under the circumstances. There is reason to 

 suspect an ellipticity in the orbit of the first satellite, corresponding 

 to an excentricity of 0035. 



The. author gives formulae for determining the positions of the two 



