272 



DETERMINATION OF AN ORBIT FROM 



[BOOK II. 



tion is considerably abridged, especially when the observations do not embrace a 

 &amp;lt;Teat interval of time ; here the final determination of the orbit is not yet 

 proposed. In such cases the following method may be employed with great 

 advantage. 



Let complete places L and L be selected from the whole number of observa 

 tions, and let the distances of the heavenly body from the earth be computed 

 from the approximate elements for the corresponding times. Let three hypothe 

 ses then be framed with respect to these distances, the computed values being 

 retained in the first, the first distance being changed in the second hypothesis, 

 and the second in the third hypothesis ; these changes can be made in proportion 

 to the uncertainty presumed to remain in the distances. According to these 

 three hypotheses, which we present in the following table, 



let three sets of elements be computed from the two places I/, L , by the methods 

 explained in the first book, and afterwards from each one of these sets the geo 

 centric places of the heavenly body corresponding to the times of all the remain 

 ing observations. Let these be (the several longitudes and latitudes, or right 

 ascensions and declinations, being denoted separately), 



in the first set .... M, M , M&quot;, etc. 



in the second set . . . M-\-a, M -(- , M&quot;-\-a&quot;, etc. 



in the third set . . . . M + /?, M +/? , M&quot;-}- p&quot;, etc. 

 Let, moreover, the observed 



places be respectively N, N , N&quot;, etc. 



Now, so far as proportional variations of the individual elements correspond 



* It will be still more convenient to use, instead of the distances themselves, the logarithms of the 

 curtate distances. 



