174 DETERMINATION OF AN ORBIT FROM [BOOK II. 



the variations which the elements experience, if the middle place changes while 

 the extreme places remain fixed, can be estimated without difficulty : in this way, 

 therefore, some judgment may be formed as to the degree of precision to be 

 attributed to the elements found. 



125. 



We shall derive the second from the preceding method by applying a slight 

 change. Starting from the distances in two observations, we shall determine all 

 the elements in the same manner as before ; we shall not, however, compute 

 from these the geocentric place for the third observation, but will only proceed 

 as far as the heliocentric place in the orbit ; on the other hand we will obtain the 

 same heliocentric place, by means of the problem treated in articles 74, 75, from 

 the observed geocentric place and the position of the plane of the orbit; these 

 two determinations, different from each other (unless, perchance, the true values 

 of x, y, should be the assumed ones), will furnish us X and Y, the difference be 

 tween the two values of the longitude in orbit being taken for X, and the differ 

 ence between the two values of the radius vector, or rather its logarithm, for T. 

 This method is subject to the same cautions we have touched upon in the -pre 

 ceding article : another is to be added, namely, that the heliocentric place in orbit 

 cannot be deduced from the geocentric place, when the place of the earth happens 

 to be in either of the nodes of the orbit ; when that is the case, accordingly, this 

 method cannot be applied. But it will also be proper to avoid the use of this 

 method in the case where the place of the earth is very near either of the nodes, 

 since the assumption that, to small variations of x, y, correspond proportional 

 variations of X, Y, would be too much in error, for a reason similar to that which 

 we have mentioned in the preceding article. But here, also, may be a remedy 

 sought in the interchange of the mean place with one of the extremes, to which 

 may correspond a place of the earth more remote from the nodes, except, per 

 chance, the earth, in all three of the observations, should be in the vicinity of the 

 nodes. 



