SECT. 3.] ANY NUMBER OF OBSERVATIONS. 251 



that the degree of accuracy in each may be considered proportional to the num 

 bers e, e, e&quot;, e&quot;, etc. respectively, that is, that errors reciprocally proportional to 

 these numbers could have been made in the observations with equal facility; 

 then, according to the principles to be propounded below, the most probable 

 mean value will no longer be the simple arithmetical mean, but 



ee 8 -\- e e 8 -f e&quot;e&quot;8 -\- e &quot;e &quot;d &quot; -f- etc. 

 ~ee+ ~e e ~+ e&quot;e&quot; -\-e &quot;e &quot; -f- etc. 



Putting now this mean value equal to //, we can assume for the true right ascen 

 sions, a -\- d J, of -j- () - _/, ft&quot;- j- d&quot; - .J, a &quot;- j- d &quot;- A, respectively, and then 

 it will be arbitrary, which we use in the calculation. But if either the observa 

 tions are distant from each other by too great an interval of tune, or if suffi 

 ciently approximate elements of the orbit are not yet known, so that it would 

 not be admissible to regard their deviations as constant for all the observations, it 

 will readily be perceived, that no other difference arises from this except that the 

 mean deviation thus found cannot be regarded as common to all the observa 

 tions, but is to be referred to some intermediate time, which must be derived from 

 the individual times in the same manner as A from the corresponding deviations, 

 and therefore generally to the time 



eet + e eY + e W -}- e &quot;e &quot;t&quot;&amp;gt; + etc. 

 ~~e e -f e e + e V + e &quot;e &quot; + etc. 



Consequently, if we desire the greatest accuracy, it will be necessary to compute 

 the geocentric place from the elements for the same time, and afterwards to free 

 it from the mean error A, in order that the most accurate position may be ob 

 tained. But it will in general be abundantly sufficient if the mean error is 

 referred to the observation nearest to the mean time. What we have said here 

 of right ascensions, applies equally to declinations, or, if it is desired, to longitudes 

 and latitudes : however, it will always be better to compare the right ascensions 

 and declinations computed from the elements immediately with those observed ; 

 for thus we not only gain a much more expeditious calculation, especially if we 

 make use of the methods explained in articles 53-60, but this method has the 

 additional advantage, that the incomplete observations can also be made use of; 

 and besides, if every thing should be referred to longitudes and latitudes, there 



