SE&amp;lt;JT. 2.] OF WHICH TWO ONLY ARE coMrLETE. 243 



XI. ? &quot; cos $ &quot; = R &quot; cos B&quot; cos ( &quot; I &quot;} 



H 1+P &quot;^y (?&quot; cos 0&quot; cos ( &quot; &quot;)+ ,5&quot; cos ,5&quot; cos (a &quot; I&quot;]) 



**(!+$) 



- jL (&amp;lt;? cos cos (a &quot; a ) + # cos cos (a &quot; I )), 



XII. o &quot; sin |3 &quot; = #&quot; sin B&quot; -4- 1 + P &quot; =, fo&quot; sin 0&quot; 4- # sin B&quot;} 



*&quot;(! + ) 



-^(e sin/S -f JZ sin.ff ). 



The formulas IX.-XII. are derived without difficulty from equations 1, 2, 3, article 

 112, if, merely, the symbols there used are properly converted into those we here 

 employ. The formulas will evidently come out much more simple if B, B , B&quot; 

 vanish. Not only (), but also /? will follow from the combination of the formulas 

 IX. and X., and, in the same manner, besides /&quot;, also ft &quot; from XI. and XII. : the 

 values of these, compared with the observed latitudes (not entering into the 

 calculation), if they have been given, will show with what degree of accuracy 

 the extreme latitudes may be represented by elements adapted to the six remain 

 ing data. 



171. 



A suitable example for the illustration of this investigation is taken from Vesta, 

 which, of all the most recently discovered planets, has the least inclination to 

 the ecliptic.* We select the following observations made at Bremen, Paris, 

 Lilienthal, and Milan, by the illustrious astronomers OLBERS, BOUVARD, BESSEL, and 

 ORIANI : 



* Nevertheless this inclination is still great enough to admit of a sufficiently safe and accurate deter 

 mination of the orbit based upon three observations: in fact the first elements which had been derived 

 in this way from observations only 19 days distant from each other (see TON ZACH S Monatliche Cor- 

 respondenz, Vol. XV. p. 595), approach nearly to those which were here deduced from four observa 

 tions, removed from each other 162 days. 



