96 THE ORBIT OF URANUS. 



From the expressions for p and q we obtain 



cos <p D t $ = sin 6 D t p + cos D t q; 

 sin <j> D t Q = cos 6 D$ sin 6 D t q, 



And, neglecting {D t ^f X sin <>, we have farther, 



cos $Z>> = sin <p (D t Oy + sin QD\p + cos 6D\g; 



sin <pD\6 = 2 cos $D t QD$ + cos OD\p sin 6L z t q. 



Since <j> is only 46' we may put cos <p and cos /3 both equal to unity in these 

 expressions, while we have, for 1850, 



sin = .9573 

 cos 6 = .2890 



D t q = + 41 .54 . 

 L\p = .38 

 D\q = 0.12 

 logsin<^>= 8.129606. 



The above formulae then give 



D& = + 2". 3 1 - l'.24/ 



= 3167".5 

 2 ,$> = + 0".26 

 3 ,fl = -j- 5".6 



T+ 0".13 T 72 

 = (3167".5 - 12".6 lM ) T+ 2 .8 T 72 , 



or, adding Struve's precession, we have when is counted from the mean equinox 

 of date, 



6 = + (1857". 7 + 12".6p) T+ 3".9 T 72 . 



Using the values of <p and given by these expressions, the latitude, secular 

 variation included, will be given by the expression 



sin (3 = sin $ sin (v 0}. 



If we take from a table, as the principal term of the latitude, the value of sin 

 $> sin (v 0), the secular term to be added will be 



|(2'.31 - l'.24p) T+ 0".13 T 2 \ sin (* - 0). 



If we represent, as before, by o the variable distance of the perihelion from the 

 node, this term will be allowed for by adding to (S.s.l), (i.c.l), etc., the terms 



