46] ROYAL ASTRONOMICAL SOCIETY TO M. LE VERRIER. 367 



Moreover, in each of these series the equations are subdivided into eight 

 groups, corresponding to the distances of the planet from its perihelion, 

 to 45, 45 to 90, and so on. 



From these are formed four final equations, the solution of which gives 

 the corrections of the epoch, of the mean motion, of the eccentricity, and of 

 the longitude of the perihelion, in terms of the correction required by the 

 mass of Saturn, which is left in an indeterminate form. 



The substitution of these expressions in the thirty-two normal equations 

 corresponding to the several groups above mentioned gives the residual differ- 

 ences between theory and observation in terms of the correction of the mass 

 of Saturn. 



No conclusion can be drawn from the ancient observations; but from the 

 modern observations M. Le Verrier finds that the mass of Saturn assumed 

 which is that of Bouvard should be diminished by about its -g-^th part. 

 This correction is very small, but M. Le Verrier regards it as well established. 



On the other hand, Bessel's value of the mass of Saturn, founded on 

 his observations of the Huyghenian satellite, exceeds Bouvard's by about its 



sio tn P art - 



The equations of condition in latitude are treated in a similar manner, 

 being grouped according to the distances of the planet from its ascending 

 node. 



From these equations the corrections of the inclination of the orbit and 

 longitude of the node are found separately from the ancient and from the 

 modern observations. The results differ very little, but the second solution 

 is employed in the construction of the tables. 



After the application of these corrections to the elements, the agreement 

 between theory and observation may be considered perfect; so that the action 

 of the minor planets on Jupiter appears to be insensible, and there is no 

 indication of any unknown disturbing causes. 



There are some peculiarities in the mode of tabulating the perturbations 

 caused by the action of Saturn. The perturbations of longitude and of 

 radius vector are not, as usual, exhibited directly, but instead of them 

 M. Le Verrier gives the perturbations, both secular and periodic, of the mean 

 longitude, of the longitude of the perihelion, of the eccentricity, and of the 

 semi-axis major of the orbit, and then from the elements corrected by these 



