250 DETERMINATION OF AN ORBIT FROM [BOOK II. 



173. 



In the first place, it is of the greatest importance, that the several positions of 

 the heavenly body on which it is proposed to base the orbit, should not be 

 taken from single observations, but, if possible, from several so combined that the 

 accidental errors might, as far as may be, mutually destroy each other. Obser 

 vations, for example, such as are distant from each other by an interval of a few 

 days, or by so much, in some cases, as an interval of fifteen or twenty days, 

 are not to be used in the calculation as so many different positions, but it would 

 be better to derive from them a single place, which would be, as it were, a mean 

 among all, admitting, therefore, much greater accuracy than single observations 

 considered separately. This process is based on the following principles. 



The geocentric places of a heavenly body computed from approximate ele 

 ments ought to differ very little from the true places, and the differences between 

 the former and latter should change very slowly, so that for an interval of a 

 few days they can be regarded as nearly constant, or, at least, the changes may 

 be regarded as proportional to the times. If, accordingly, the observations should 

 be regarded as free from all error, the differences between the observed places 

 corresponding to the times t, t , f, t &quot;, and those which have been computed from 

 the elements, that is, the differences between the observed and the computed 

 longitudes and latitudes, or right ascensions and declinations, would be quanti 

 ties either sensibly equal, or, at least, uniformly and very slowly increasing or de 

 creasing. Let, for example, the observed right ascensions a, , a&quot;, a&quot;, etc., cor 

 respond to those times, and let a -\- $, a -\- &amp;lt;$ , a&quot; -\- d&quot;, a &quot; -\- d &quot;, etc., be the 

 computed ones ; then the differences d, 8 , 8&quot;, 8 &quot;, etc. will differ from the true 

 deviations of the elements so far only as the observations themselves are errone 

 ous : if, therefore, these deviations can be regarded as constant for all these ob 

 servations, the quantities d, d , d&quot;, 8 &quot;, etc. will furnish as many different determi 

 nations of the same quantity, for the correct value of which it will be proper to 

 take the arithmetical mean between those determinations, so far, of course, as 

 there is no reason for preferring one to the other. But if it seems that the same 

 degree of accuracy cannot be attributed to the several observations, let us assume 



