SECT. 1.] THREE COMPLETE OBSERVATIONS. 177 



lastly, let the product of the constant quantity It (article 2) into the intervals of 

 the time. from the second observation to the third, from the first to the third, and 

 from the first to the second be respectively, 6, 6 &&quot;. The double computation of 

 the elements is begun, just as in the preceding article, both from rr f&quot; and 6&quot;, 

 and from r r&quot;,f, 6: but neither computation will be continued to the determina 

 tion of the elements, but will stop as soon as that quantity has been obtained 

 which expresses the ratio of the elliptical sector to the triangle, and which is de 

 noted above (article 91) by y or -- Y. Let the value of this quantity be, in the 

 first calculation, r&quot;, in the second, t]. Accordingly, by means of formula 18, arti 

 cle 95, we shall have for the semi-parameter^ the two values: 



if n&quot; 



But we have, besides, by article 82, a third value, 



4 rr r&quot; sin /sin/ sin/&quot; 



v\ - ___ J J __ ^L_ 



P- n n +ri 



which three values would evidently be identical if true values could have been 

 taken in the beginning for x and y. For which reason we should have 



(P_ _ /V 



rjn 



_]_ &quot; 4 g 0&quot;rrV sin/sin/ sin/ 7 _ n dff 



tjif nn&quot; ~ 2 n n rr i&quot; cos/cos/ cos/&quot; 



Unless, therefore, these equations are fully satisfied in the first calculation, we 

 can put 



2 rfrW cos/cos/ cos/&quot; 



This method admits of an application equally general with the second ex 

 plained in article 125, but it is a great advantage, that in this fifth method the 

 first hypotheses do not require the determination of the elements themselves, but 

 stop, as it were, half way. It appears, also, that in this process we find that, as it 

 can be foreseen that the new hypothesis will not differ sensibly from the truth, it 

 will be sufficient to determine the elements either from r,r ,f&quot;,6&quot;, alone, or from 

 r , r&quot;,f, 6, or, which is better, from r, r&quot; f, ff. 



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