SKOT. 2.] OF WHICH TWO ONLY ARE COMPLETE. 241 



which we have given in article 144, not only C C&quot; = v&quot; v , but also the angles 

 (u, u&quot;,) at which the great circles Alt , A B&quot;, cut the great circle C C&quot;. 



After the arc v&quot; v has been found, v v, and r will be obtained from a 

 combination of the equations 



P /sin(j/ 



~ 



, i // /x l + P 



rsm(v v + v&quot; v }= ^~ 



14- 



-TT* 



and in the same manner, /&quot; and v &quot; v&quot; from a combination of these : 



sn 



All the numbers found in this manner would be accurate if we could set out in 

 the beginning from true values of P , Q , I*&quot;, Q&quot; : and then the position of the 

 plane of the orbit might be determined in the same manner as in article 149, 

 either from A C, u and /, or from A&quot;C&quot;, u&quot; and y&quot;; and the dimensions of the 

 orbit either from r, r&quot;, t , t&quot;, and v&quot; v, or, which is more exact, from r, r &quot;, t, 

 f, v &quot; v. But in the first calculation we will pass by all these things, and will 

 direct our attention chiefly to obtaining the most approximate values of P , P&quot;, 

 (X, Q&quot;. We shall reach this end, if by the method explained in 88 and the fol 

 lowing articles, 



from r, r, v v, f t we obtain (rj 01) 



r ,r&quot;,v&quot; v ,t&quot; t (ij!2) 



r&quot;,r &quot;,v&quot; v&quot;,t &quot; t&quot; * (17 23). 



We shall substitute these quantities, and also the values of r, /, r&quot;, /&quot;, cos k (v ? ), 

 etc., in formulas III.- VI., whence the values of P 1 , Q , P&quot;, Q&quot; will result much 

 more exact than those on which the first hypothesis had been constructed. With 

 these, accordingly, the second hypothesis will be formed, which, if it is carried to 

 a conclusion exactly in the same manner as the first, will furnish much more 

 exact values of P 1 , Q , P&quot;, Q&quot;, and thus lead to the third hypothesis. These 

 processes will continue to be repeated, until the values of P , Q , P&quot;, Q&quot; seem to 



31 



