of the Satellites from their Primaries. 181 



differ very little. Vesta is separated from them by an interval 

 much larger than those by which they are separated from each 

 other. The smallness of the difference may indicate, as in other 

 cases, a cause of derangement from which the distance corres- 

 ponding to it is exempt. 



The four planets Vesta, Juno, Ceres, and Pallas, appear to 

 be an instance of the law for the case in which r = i. 



A curious inference, which is equally certain with the reality 

 of the law, may be drawn from it : — There can be no planet 

 nearer the Sun than Mercury, and no satellite nearer the several 

 primaries than the nearest of those in each system, which have 

 been discovered. 



10. It will now be easy to express the distances of the satel- 

 lites by series as simple in form as that which Bode found out 

 for the planets. 



For Jupiter's satellites a = 63084, 6 = 35914, 7=^77 nearly, = - . 



Suppose the distance of the second satellite = 7 + 4 = 11. As there 

 is nothing to determine the distance to which a term of the series 

 should be equated, I have selected the second, because I observe 

 that the minus differences in Art. 6, are in excess, and it is i^ro- 

 bable that the terms of the series will more nearly express the 

 distances which would obtain were there no disturbances, if the 

 plus and minus differences more nearly counterbalance each other. 



Empirical Distances. True Distances. Diiferences. 



Hence 7 = 7 6.91 + .09 



7 + 4 = 11 11.00 



7+4x2i =17 17.54 - .54 



7+4x {2lf = 32 30.86 + 1.14 



For Saturn's .satellites a = 336, 6 = 82, - = —y-. = -■ nearly. Let 



a 33o 4 



the distance of the first satellite = 4. 



