THEORBITOFURAXUS. I79 



of a lower order of magnitude. The perturbations to be added will be those of 



I, I', e, e', 7t, and 71'. 



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



the position of the perihelion of Uranus for 1850.0 as tlie origin, and put, as before, 



g for the mean longitude of Uranus counted from tliis point, and let I' represent 



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



d-l 



-jw within the brackets will thus become 



ek, sin (;2l'—g—hn) + e'k. sin (^'2/'— ^— (.i'— 7t)) 



or, if we put 



e' sin {n — n) = h' 

 e' cos (71 — 7t) = 7/ 



and notice, that to terms of the first order wc have, sin ?,n = hn, cos ^71 ^ 1, we 

 shall have 



-^ = — •im'au- \ {el; -f A'/-^) sin N — {ek^ + h'k.) cos iV^| 



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

 respect to the sign h, we find for the terms of the second order. 



^ -^2 = — 3??i'a«- I {l\k + l',!iIJ+h'UN) sin .V 



+( (eA"i 4- ^-V.-^) hN— el\h7t — \>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 Trtio o» 

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

 others, shows it to be 220^0^- The 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- also vanish at the epocli ; this corresponds to the 

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



* This factor was adopted before the mass of Neptune to be employed had been 6iia]Iy decided 

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

 f M^moires de I'Acad^mie de St. P^tersbourg, tome xviii, vii s^rie. 



