SECT. 1.] THREE COMPLETE OBSERVATIONS. 183 



the extreme observations are equidistant from the middle ; or, of the first order in 

 other cases. But this new form of that equation is not suited to the determina 

 tion of d , because it involves the quantities r, r , r&quot;, still unknown. 



Now, generally speaking, the quantities ^,-^, differ from unity by a quantity 

 of the first order, and in the same manner also the product ^: it is readily 

 perceived that in the special case frequently mentioned, this product differs 

 from unity by a quantity of the second order only. And even when the orbit 

 of the ellipse is slightly eccentric, so that the eccentricity may be regarded as a 

 quantity of the first order, the difference of T ~ f - } can be referred to an order one 

 degree higher. It is manifest, therefore, that this error remains of the same order 



fl fl// a off 



as before if, in our equation, 2rrV / is substituted for ^, whence is obtained the 

 following form, 



In fact, this equation still contains the unknown quantity /, which, it is evident 

 nevertheless, can be eliminated, since it depends only on d and known quantities. 

 If now the equation should be afterwards properly arranged, it would ascend to 

 the eighth degree. 



134. 



From the preceding it will be understood why, in our method, we are about 

 to take for x, y, respectively, the quantities 



W, and 2 -1 / ==&amp;lt;?. 



For, in the first place, it is evident that if P and Q are regarded as known quanti 

 ties, d can be determined from them by means of the equation 



A 7, I c + dP (-[ Q 



= b + T+^( l + 2?-* 

 and afterwards $,d&quot;, by equations 4, 6, article 114, since we have 



-- -Wl-L.-^ n &quot;- P (l\ Q\ 

 n &amp;gt; 1-f-PV J -~T2r 8 / ri~~ \-\-P\ 2r V 



In the second place, it is manifest that -j , 66&quot; are, in the first hypothesis, the 



