180 THE ORBIT OF URANUS. 



Substituting these values in the expression for d- ..^ and integrating twice, we 



find, putting h for the coefficient of the time in N, of which the value, taking tiie 

 century as the unit, is -|-0.1472, and putting T'for the time in centuries, 



bl='^ m'av' |(4H" + — |^ + 2".6r)sin iV + (1837" +-^:^_51."47') cos i\r 



— 109".3 sin 2N — 5".5 cos 2N \ 

 — l&'.^m'anH" -{-cT^c, 



c and c being the arbitrary constants of integration, which are to be chosen so that 

 both bl and its first differential coefficient shall vanish at the epoch, Eeducing to 

 numbers, we find 



hi = (140".70 + 0".327') sin N 

 + (232.60 — 6.37 7')cosi\r 



— 13 .60 sin 2N 



— .70 cos 2N 



— 0.03r- 

 + 34 .27 T 



— 46 .76, 

 the last two terms being arbitrary. 



When we carry the perturbations of the eccentricity and perihelion to quantities 

 of the second order, we are troubled by the introduction of large terms depending 

 on the square of the disturbing force, which disappear from the rigorous expres- 

 sions for the co-ordinates. These may be avoided by substituting for the eccen- 

 tricity and perihelion the quantities h and Jc determined by the condition 



7t = e sin n 

 Jc = e cos 71 



If, as before, we count the longitudes from the perihelion of Uranus at the epoch 

 1850, we sliould substitute hn for jt in these expressions. The values of h and k 

 will then be given by the integration of the equations 



(Ih , 



— - = man/:, cos N 



at 



dk 

 dt 



-^fT = — v)'a.iili\ sin N- 



