252 DETERMINATION OF AN ORBIT FROM [BOOK II. 



would be cause to fear lest an observation made correctly in right ascension, 

 but badly in declination (or the opposite), should be vitiated in respect to both 

 longitude and latitude, and thus become Avholly useless. The degree of precision 

 to be assigned to the mean found as above will be, according to the principles to 

 be explained hereafter, 



^ ( ee 4. e e &amp;gt; + e &quot; e &quot; + /&quot;/&quot; + etc.) ; 



so that four or nine equally exact observations are required, if the mean is to 

 possess a double or triple accuracy. 



174. 



If the orbit of a heavenly body has been determined according to the methods 

 given in the preceding sections from three or four geocentric positions, each one 

 of which has been derived, according to the precepts of the preceding article, 

 from a great many observations, that orbit will hold a mean, as it were, among 

 all these observations ; and in the differences between the observed and computed 

 places there will remain no trace of any law, which it would be possible to re 

 move or sensibly diminish by a correction of the elements. Now, when the whole 

 number of observations does not embrace too great an interval of time, the best 

 agreement of the elements with all the observations can be obtained, if only 

 three or four normal positions are judiciously selected. How much advantage 

 we shall derive from this method in determining the orbits of new planets or 

 comets, the observations of which do not yet embrace a period of more than 

 one year, will depend on the nature of the case. When, accordingly, the orbit 

 to be determined is inclined at a considerable angle to the ecliptic, it will be 

 in general based upon three observations, which we shall take as remote from 

 each other as possible : but if in this way we should meet with any one of the 

 cases excluded above (articles 160-162), or if the inclination of the orbit should 

 seem too small, we shall prefer the determination from four positions, which, also, 

 we shall take as remote as possible from each other. 



But when we have a longer series of observations, embracing several years, 

 more normal positions can be derived from them ; on which account, we should 



