S ECT . 1.] THREE COMPLETE OBSERVATIONS. 229 



160. 



In the exposition of the method to which the preceding investigations have 

 been devoted, we have come upon certain special cases to which it did not apply, 

 at least not in the form in which it has been exhibited by us. We have seen 

 that this defect occurs first, when any one of the three geocentric places coincides 

 either with the corresponding heliocentric place of the earth, or with the oppo 

 site point (the last case can evidently only happen when the heavenly body 

 passes between the sun and earth) : second, when the first geocentric place of the 

 heavenly body coincides with the third ; third, when all three of the geocentric 

 places together with the second heliocentric place of the earth are situated in the 

 same great circle. 



In the first case the position of one of the great circles AB, A B , A B&quot;, and in 

 the second and third the place of the point JB*, will remain indeterminate. In 

 these cases, therefore, the methods before explained, by means of which we have 

 shown how to determine the heliocentric from the geocentric places, if the quan 

 tities P, Q, are regarded as known, lose their efficacy : but an essential distinction 

 is here to be noted, which is, that in the first case the defect will be attributable 

 to the method alone, but in the second and third cases to the nature of the prob 

 lem; in the first case, accordingly, that determination can imdoubtedly be effected 

 if the method is suitably altered, but in the second and third it will be absolutely 

 impossible, and the heliocentric places will remain indeterminate. It will not be 

 uninteresting to develop these relations in a few words : but it would be out of 

 place to go through all that belongs to this subject, the more so, because in all 

 these special cases the exact determination of the orbit is impossible where it 

 would be greatly affected by the smallest errors of observation. The same defect 

 will also exist when the observations resemble, not exactly indeed, but nearly, 

 any one of these cases ; for which reason, in selecting observations this is to be 

 recollected, and properly guarded against, that no place be chosen where the 

 heavenly body is at the same time in the vicinity of the node and of opposition 

 or conjunction, nor such observations as where the heavenly body has nearly re 

 turned in the last to the geocentric place of the first observation, nor, finally, such 



