S ECTt 2.] OF WHICH TWO ONLY ARE COMPLETE. 247 



Hence are found 



whence will result newly corrected values of the quantities f, P&quot;, Q , Q&quot;, 



log P 0.059415, log Q = 9.604782, 

 logP&quot;= 9.972253, log Q&quot; = 9.697687. 



Finally, if the fourth hypothesis is formed with these new values, the following 

 numbers are obtained : 



c = 7.678116, log cf = 0.045723 



c&quot;= + 2.210773, log rf&quot;= 0.126084 



of = 2.032473, / = 23 48 16&quot;. 7, log / = 0.346638 



x&quot;= 1.942281, 0&quot;= 27 12 51 .7, log / = 0.339263 



i/ i/ = ir 8 5&quot;.l, i (&quot;+ iO == 176 7 50&quot;.5, * (&quot; w ) = 4 33 23&quot;.6 



v v= 14 21 51 .9, log r = 0.354503 



v &quot; t/ =18 51 9 .5, log/&quot;= 0.334263 



These numbers differ so little from those which the third hypothesis furnished, 

 that we may now safely proceed to the determination of the elements. In 

 the first place we get out the position of the plane of the orbit. The inclina 

 tion of the orbit 7 8 14&quot;.8 is found by the precepts of article 149 from /, u , 

 and A C = d z, also the longitude of the ascending node 103 16 37&quot;.2, the 

 argument of the latitude in the second observation 94 36 4&quot;. 9, and, there 

 fore, the longitude in orbit 197 52 42&quot;.l ; in the same manner, from y&quot;, u&quot;, and 

 A&quot;C&quot; = 3&quot;J , are derived the inclination of the orbit = 7 8 14&quot;.8, the longi 

 tude of the ascending node 103 16 37&quot;.5, the argument of the latitude in the 

 third observation 11144 9&quot;.7, and therefore the longitude in orbit 215 47&quot;.2. 

 Hence the longitude in orbit for the first observation will be 183 30 50&quot;.2, for 

 the fourth 233 51 56&quot;.7. If now the dimensions of the orbit are determined 

 from f&quot; t, r, r&quot;, and v &quot; v = 50 21 6&quot;.5, we shall have, 



