jgg THE ORBIT OF UK AX US. 



= + 7260" sin N 6658" + 4". 26* 

 = - 414 sinJV+ 380 

 & = 414 cosN- 165 

 M' = -4- 120 sinN- 110 

 M = 4- 12 cosJV-f 48 

 7* = - - 1.234 

 h = + 0.452 

 7/ = + 0.00695 

 7;' = 0.00486 



(PI 

 Substituting these values in the expression for 5-^ and integrating twice, we 



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

 century as the unit, is -[-0.1472, and putting T for the time in centuries, 



- 109". 3 sin 2N 5". 5 cos 



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

 both <5Z and its first differential coefficient shall vanish at the epoch. Reducing to 

 numbers, we find 



37 = (140".70 -4- 0".327 7 ) sin N 

 + (232.60 6.37 T) cos N 

 13 .60 sin 2iV 

 0.70 cos 2 N 

 .03 T 72 

 + 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 7t and k determined by the condition 



h = e sin 7t 

 lc = e cos ?t 



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

 1850, we should substitute &TI for n in these expressions. The values of 7t and Jc 

 will then be given by the integration of the equations 



dh 



dk , . 



-rr = m an*, sm 



