SECT. 1.] THREE COMPLETE OBSERVATIONS. 175 



126. 



The preceding method prepares the way directly for the third. In the same 

 manner as before, by means of the distances of the heavenly body from the earth 

 in the extreme observations, the corresponding longitudes in orbit together with 

 the radii vectores may be determined. With the position of the plane of the 

 orbit, which this calculation will have furnished, the longitude in orbit and the 

 radius vector will be got from the middle observation. The remaining elements 

 may be computed from these three heliocentric places, by the problem treated in 

 articles 82, 83, which process will be independent of the times of the observa 

 tions. In this way, three mean anomalies and the diurnal motion will be known, 

 whence may be computed the intervals of the times between the first and second, 

 and between the second and third observations. The differences between these 

 and the true intervals will be taken for X and Y. 



This method is less advantageous when the heliocentric motion includes a 

 small arc only. For in such a case this determination of the orbit (as we have 

 already shown in article 82) depends on quantities of the third order, and does 

 not, therefore, admit of sufficient exactness. The slightest changes in the values 

 of x,y, might cause very great changes in the elements and, therefore, in the val 

 ues of X, Y, also, nor would it be allowable to suppose the latter proportional to 

 the former. But when the three places embrace a considerable heliocentric mo 

 tion, the use of the method will undoubtedly succeed best, unless, indeed, it is 

 thrown into confusion by the exceptions explained in the preceding articles, 

 which are evidently in this method too, to be taken into consideration. 



127. 



After the three heliocentric places have been obtained in the way we have 

 described in the preceding article, we can go forward in the following manner. 

 The remaining elements may be determined by the problem treated in articles 

 85-105, first, from the first and second places with the corresponding interval of 

 time, and, afterwards, in the same manner, from the second and third places and 



