of Prof. Encke's Ephemerisfur 1831. 175 



The manner in which the perturbations of the small planets 

 are calculated, — viz. by applying the corrections to the ele- 

 ments themselves, and not to the places calculated by the 

 mean elements, — renders a generally true exhibition of their 

 orbits impossible, and the data given in astronomical books 

 neither refer to mean elements generally, nor even to variable 

 elements taken for a certain moment of time. As however for 

 all the four new planets, the effect of Jupiter at least, although 

 perhaps not on the same hypothesis of its mass, has been ap- 

 plied, it may perhaps be interesting to exhibit the form of the 

 four orbits for the same moment of time. The following ele- 

 ments of Pallas, Juno, and Vesta, refer to the moment of the 

 opposition of Pallas ; those of Ceres properly for the moment 

 of her opposition. The distance being however small, the 

 change of the latter, in order to reduce them accurately to the 

 moment to which the others belong, would be very small. 



Elements of the small Planets. 



Epoch of the mean longitude 1831. July 23. 



O*" mean Time of Berlin. 



Vesta. 



Mean longitude 84.°47' 3"-2 



Mean anomaly 195 35 26 '2 



Longitude of the perihelion 249 11 37 '0 



Longitude of the node 103 20 28-0 



Liclination 7 7 57 *3 



Angle of eccentricity 5 4 50 "8 



Mean daily sidereal motion 977*7554'0 



Log. of the semi-axis major 0*373185 



Juno. 



Mean longitude 74^39' 43"-6 



Mean anomaly 20 22 30 '9 



Longitude of the perihelion 54 17 12*7 



Longitude of the node 170 52 34*5 



Inclination 13 2 10-0 



Angle of eccentricity 14 48 24*2 



Mean daily sidereal motion 813*52533 



Log. of the semi-axis major 0*426424 



Pallas. 



Mean longitude 290°38' ll"-8 



Mean anomaly 169 33 1 1 '3 



Longitude of the perihelion 121 5 0*5 



Longitude of the node 172 38 29-8 



Inclination 34 35 49-1 



Angle of eccentricity 14 16*3 



Mean daily sidereal motion 768*54421 



L«>g. of the semi-axis major 0*412892 



Ceres. 



