SECT. 1.] THREE COMPLETE OBSERVATIONS. 199 



The two former will give i (n&quot;-\- u) and sin/ , the two latter (u&quot; 11) and cos/ ; 

 from sin/ and cos/ we shall have/ . It will be proper to neglect in the first 

 hypotheses the angles I (&quot;-)-??) and \ (&quot;--?&amp;lt;), which will be used in the last 

 hypothesis only for determining the position of the plane of the orbit. 



hi the same way, exactly,/ can be derived from a, C D and C&quot;D; also/&quot; 

 from t&quot;, CD&quot; and C D&quot; ; but the following formulas are used much more con 

 veniently for this purpose : 



in which the logarithms of the quantities ^ , ^-, are already given by the pre 

 ceding calculations. Finally, the whole calculation finds a new verification in 

 this, that we must have 



if by chance any difference shows itself, it will not certainly be of any impor 

 tance, if all the processes have been performed as accurately as possible. Never 

 theless, occasionally, the calculation being conducted throughout with seven 

 places of decimals, it may amount to some tenths of a second, which, if it appear 

 worth while, we may with the utmost facility so distribute between 2 /and 2f&quot; 

 that the logarithms of the sines may be equally either increased or diminished, 

 by which means the equation 



p _ r sin 2/&quot; _ n&quot; 

 r&quot;sin2/ n 



will be satisfied with all the precision that the tables admit. When /and/&quot; differ a 

 little, it will be sufficient to distribute that difference equally between 2/ and 2/&quot;. 



145. 



After the positions of the heavenly body in the orbit have been determined in 

 this manner, the double calculation of the elements will be commenced, both by 

 the combination of the second place with the third, and the combination of the 

 first with the second, together with the corresponding intervals of the times. 



