SECT. 1.] THREE COMPLETE OBSERVATIONS. 173 



have been assigned to x, y. If, accordingly, the geocentric place for the time of 

 the third observation is computed by means of these elements, its agreement or 

 disagreement with the observed place will determine whether the assumed values 

 are the true ones, or whether they differ from them ; whence, as a double com 

 parison will be obtained, one difference (in longitude or right ascension) can be 

 taken for Jf, and the other (in latitude or declination) for Y. Unless, therefore, 

 the values of these differences come out at once = 0, the true values of x, y, may 

 be got by the method given in 120 and the following articles. For the rest, it is 

 in itself arbitrary from which of the three observations we set out : still, it is betr 

 ter, in general, to choose the first and last, the special case of which we shall speak 

 directly, being excepted. 



This method is preferable to most of those to be explained hereafter, on this 

 account, that it admits of the most general application. The case must be ex 

 cepted, in which the two extreme observations embrace a heliocentric motion of 

 180, or 360, or 540, etc., degrees; for then the position of the plane of the orbit 

 cannot be determined, (article 110). It will be equally inconvenient to apply the 

 method, when the heliocentric motion between the two extreme observations 

 differs very little from 180 or 360, etc., because an accurate determination of 

 the position of the orbit cannot be obtained in this case, or rather, because the 

 slightest changes in the assumed values of the unknown quantities would cause 

 such great variations in the position of the orbit, and, therefore, in the values of 

 X, Y, that the variations of the latter could no longer be regarded as propor 

 tional to those of the former. But the proper remedy is at hand ; which is, that 

 we should not, in such an event, start from the two extreme observations, but from 

 the first and middle, or from the middle and last, and, therefore, should take for 

 -X, Y, the differences between calculation and observation in the third or first 

 place. But, if both the second place should be distant from the first, and the 

 third from the second nearly 180 degrees, the disadvantage could not be removed 

 in this way ; but it is better not to make use, in the computation of the elements, 

 of observations of this sort, from which, by the nature of the case, it is wholly 

 impossible to obtain an accurate determination of the position of the orbit. 



Moreover, this method derives value from the fact, that by it the amount of 



