SECT. 1.] THREE COMPLETE OBSERVATIONS. 169 



It is evidently admissible, also, to interchange in these formulas the quantities 

 a, b, A, B, with , V, A , B , or with a&quot;, b&quot;, A&quot;, B&quot;. 



The common denominator of all these expressions, which may be put under 



the form (A A) (B&quot; B} (A&quot; A) (ff B), becomes 



whence it appears that a, a, a&quot;, b, b , b&quot; must be so taken as not to make 



y 5 gdj&amp;gt; 



otherwise, this method would not be applicable, but would furnish, for the values 

 of and vj, fractions of which the numerators and denominators would vanish at 

 the same time. It is evident also that, if it should happen that ad tiy = 0, the 

 same defect wholly destroys the use of the method, in whatever way a, a, a&quot;, 

 I, b , b&quot;, may be taken. In such a case it would be necessary to assume for the 

 values of X the form 



and a similar one for the values of F, which being done, analysis would supply 

 methods, analogous to the preceding, of obtaining from values of X, Y, computed 

 for four systems of values of x, y, true values of the latter. But the computation 

 in this way would be very troublesome, and, moreover, it can be shown that, in 

 such a case, the determination of the orbit does not, from the nature of the ques 

 tion, admit of the requisite precision : as this disadvantage can only be avoided 

 by the introduction of new and more suitable observations, we do not here dwell 

 upon the subject. 



121. 



When, therefore, the approximate values of the unknown quantities are ob 

 tained, the true values can be derived from them, in the manner just now ex 

 plained, with all the accuracy that is needed. First, that is, the values of X, T, 

 corresponding to the approximate values (a, b) will be computed : if they do not 

 vanish for these, the calculation will be repeated with two other values (a, b ) 

 differing but little from the former, and afterwards with a third system (a&quot;, b&quot;) 



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