SECT. 2.] DETERMINATION OF AN ORBIT. 235 



from these elements the geocentric longitudes or right ascensions for the two 

 remaining observations may be computed. If these happen to agree with the 

 observations, the elements will require no further correction: but if not, the 

 differences X, T, will be noted, and the same calculation will be repeated twice, 

 the values of x, y being a little changed. Thus will be obtained three systems 

 of values of the quantities x, y, and of the differences X, Y, whence, according 

 to the precepts of article 120, will be obtained the corrected values of the quan 

 tities x, y, to which will correspond the values X= 0, Y= 0. From a similar 

 calculation based on this fourth system elements will be found, by which all four 

 observations will be correctly represented. 



If it is in your power to choose, it will be best to retain those observations 

 complete from which the situation of the orbit can be determined with the great 

 est precision, therefore the two extreme observations, when they embrace a helio 

 centric motion of 90 or less. But if they do not possess equal accuracy, you 

 will set aside the latitudes or declinations of those you may suspect to be the 

 less accurate. 



166. 



Such places will necessarily be used for the first determination of an entirely 

 unknown orbit from four observations, as include a heliocentric motion not too 

 great ; for otherwise we should be without the aids for forming conveniently the 

 first approximation. The method which we shall give directly admits of such 

 extensive application, that observations comprehending a heliocentric motion of 

 30 or 40 may be used without hesitation, provided, only, the distances from the 

 sun are not too unequal : where there is a choice, it will be best to take the 

 intervals of the times between the first and second, the second and third, the 

 third and fourth but little removed from equality. But it will not be necessary 

 to be very particular in regard to this, as the annexed example will show, in 

 which the intervals of the times are 48, 55, and 59 days, and the heliocentric 

 motion more than 50. 



Moreover, our solution requires that the second and third observations be 

 complete, and, therefore, the latitudes or declinations in the extreme observations 



