96 THE ORBIT OF URANUS. 



riom the expressions for p and q we obtain 



cos ^ D,^ = sin D,p -\- cos A?; 

 sin (p Dfi = cos D^p — sin Dtq. 



And, neglecting (Z>(^)- X sin <p, we have farther, 



cos ^D\<p = sin ^ {Dfif + sin QD\iy + cos dD\q ; 



sin ^D\d = — 2 cos ^DfiDt^ + cos OX-^ j> — sin ^L^'^q. 



Since <^ is only 46' we may put cos ^ and cos /? both eqnal to unity in these 

 expressions, wliile we have, for 1850, 



sin = .9573 

 cos = .2890 

 D,p = — 10".12 — r'.14u 

 D,q = + 41 .54 — 0.52^ 

 L\p = — .38 

 L\q = — 0.12 

 logsin<^= 8.129606. 



The above formulae then give 



Z>,^= + 2-.31 — 1".24,7 

 Dfi = — 316T'.5 + 12".6;t 

 D\<p = + 0".26 

 D\d = + 5".6 



<^ = ^^, _^ (0".3i _ i".24u) r+ 0".13 T^ 

 = 0,— (3167".5 — U".6u)T-\- 2 .8 T\ 



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

 of date, 



= Oo + (1857".7 + 12".6,«) T -\- 3".9 T-. 



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

 variation included, will be given by the expression 



sin j3 = sin <p sin (r — 0). 



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

 ^0 sin (v — 0), the secular term to be added will be 



1(2". 31 — 1".24») T-\- 0".13 T'\ sin {v — 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 (h.s.l), (b.c.l), etc., the terms 



