236 DETERMINATION OF AN OKBIT FROM FOUR OBSERVATIONS, [BuoK II. 



are neglected. We have, indeed, shown above that, for the sake of accuracy, it is 

 generally better that the elements be adapted to two extreme complete observa 

 tions, and to the longitudes or right ascensions of the intermediate ones ; never 

 theless, we shall not regret having lost this advantage in the first determination 

 of the orbit, because the most rapid approximation is by far the most important, 

 and the loss, which affects chiefly the longitude of the node and the inclina 

 tion of the orbit, and hardly, in a sensible degree, the other elements, can after 

 wards easily be remedied. 



We will, for the sake of brevity, so arrange the explanation of the method, 

 as to refer all the places to the ecliptic, and, therefore, we will suppose four longi 

 tudes and two latitudes to be given : but yet, as we take into account the latitude 

 of the earth in our formulas, they can easily be transferred to the case in which 

 the equator is taken as the fundamental plane, provided that right ascensions and 

 declinations are substituted in the place of longitudes and latitudes. 



Finally, all that we have stated in the preceding section with respect to nuta 

 tion, precession, and parallax, and also aberration, applies as well here : unless, 

 therefore, the approximate distances from the earth are otherwise known, so that 

 method I., article 118, can be employed, the observed places will in the beginning 

 be freed from the aberration of the fixed stars only, and the times will be cor 

 rected as soon as the approximate determination of the distances is obtained in 

 the course of the calculation, as will appear more clearly in the sequel. 



167. 



We preface the explanation of the solution with a list of the principal sym 

 bols. We will make 



t, t , t&quot;, t &quot;, the times of the four observations, 



a, a , a&quot;, a&quot; , the geocentric longitudes of the heavenly body, 



(1, /? , ft&quot;, p &quot;, their latitudes, 



r, r, r&quot;, r&quot;, the distances from the sun, 



(), (&amp;gt; , (/ , (&amp;gt; &quot;, the distances from the earth, 



/, I , I&quot;, I &quot;, the heliocentric longitudes of the earth, 



