SECT. 1.] 



THREE COMPLETE OBSERVATIONS. 



215 



for the middle observation is continued to the geocentric place, the results dif 

 fer from observation only by a few hundredths of a second, (article 63 ;) these 

 differences are absorbed, as it were, in the unavoidable errors arising from the 

 want of strict accuracy in the tables. 



We have worked out the preceding example with the utmost precision, to 

 show how easily the most exact solution possible can be obtained by our method. 

 In actual practice it will rarely be necessary to adhere scrupulously to this 

 type. It will generally be sufficient to use six places of decimals throughout; 

 and in our example the second hypothesis would have given results not less accu 

 rate than the third, and even the first would have been entirely satisfactory. We 

 imagine that it will not be unacceptable to our readers to have a comparison of 

 the elements derived from the third hypothesis with those which would result 

 from the use of the second or first hypothesis for the same object. We exhibit 

 the three systems of elements in the following table : 



By computing the heliocentric place in orbit for the middle observation from 

 the second system of elements, the error of the logarithm of the radius vector is 

 found equal to zero, the error of the longitude in orbit, 0&quot;.03 ; and in comput 

 ing the same place by the system derived from the first hypothesis, the error of 

 the logarithm of the radius Vector is 0.0000002, the error of the longitude in 

 orbit, 1&quot;.31. And by continuing the calculation to the geocentric place we have, 



