T1IK OK HIT OF U K A N I H9 



of a lower order of magnitude. 'I In- perturbations to be added will be those of* 

 /, f, ','', 7i, and -d. 



The point from which the longitudes are counted being aibitrary, we shall take 

 the position of tin- perihelion of I 'nnms for 1850.0 as the origin, and put, as before, 

 </ for the mean longitude of Uranus counted from this point, and let F represent 

 the mean longitude "of Neptune counted from the same point The terms of 



.,, within the brackets will thus become 



e&i sin (Wgtm) -f- e'k t sin (2f y (rf *)) 



or, if we put 



W-g = N 



J Bin (71 n) = h' 



<f cos (d 7i ) = V 



and notice, that to terms of the first order we have, sin tin = tat, cos in = 1, we 



*liall have 



- 3m'un 2 \(eki + &'/.-,) sin N (ek^ + h'kj cos N\ 



differentiating the quantities, of which the perturbations are to be considered with 

 rei-pcct to the sign 5, we find for the terms of the second order. 



sn 

 kJ IN ek^n k^h^cos N\. 



We have now to substitute in this expression the numerical values of the quanti- 

 ties within parentheses. Those of the perturbations of Uranus have already been 

 given in Chapter III, but it is necessary to diminish them by the factor 0.145* for 

 the altered mass of Neptune. Those of Neptune are taken from my investigation 

 of the orbit of that planet (p. 38). The mass of Uranus there adopted is TT ^ V , 

 while the investigation of Dr. Von Asten,t from the observations of Struve and 

 others, shows it to be ^iinr' Tllc perturbations are therefore diminished by ^j. 

 In accordance with the system adopted throughout both investigations, constants 

 are added to all the perturbations to make them vanish at the epoch 1850.0. A 



term is also added to make c also vanish at the epoch ; this corresponds to the 



at 



constant which ought to be added to a. The numerical values thus obtained, are : 



* This factor was adopted before the mass of Neptune to be employed had been finally decided 

 upon. Hence the difference between it and that in the preceding chapter. 

 f Me'moires de TAcade'inie de St. Pe'terebourg, tome xviii, vii s^rie. 



