SECT. 1.] THREE COMPLETE OBSERVATIONS. 203 



terms log?}, log?/ , but less complete, for the former greatly exceed the latter. In 



r r 



general, it is not possible to determine any thing concerning the sign of log r,. 



Now, as often as the heliocentric motion between the observations is small, it 

 will rarely be necessary to proceed to the fourth hypothesis ; most frequently the 

 third, often the second, will afford sufficient precision, and we may sometimes be 

 satisfied with the numbers resulting from even the first hypothesis. It will be 

 advantageous always to have a regard to the greater or less degree of precision 

 belonging to the observations; it would be an ungrateful task to aim at a pre 

 cision in the calculation a hundred or a thousand times greater than that which 

 the observations themselves allow. In these matters, however, the judgment is 

 sharpened more by frequent practical exercise than by rules, and the skilful 

 readily acquire a certain faculty of deciding where it is expedient to stop. 



149. 



Lastly, the elements themselves will be compiited in the final hypothesis, 

 either from/, r, r&quot;, or from/&quot;, r, /, carrying one or the other of the calculations 

 through to the end, which in the previous hypotheses it had only been requisite 

 to continue as far as t], r&quot; ; if it should be thought proper to finish both, the 

 agreement of the resulting numbers will furnish a new verification of the whole 

 work. It is best, nevertheless, as soon as /,/ ,/&quot;, are got, to obtain the elements 

 from the single combination of the first place with the third, that is, from f,r, r&quot;. 

 and the interval of the time, and finally, for the better confirmation of the com 

 putation, to determine the middle place in the orbit by means of the elements 

 found. 



In this way, therefore, the dimensions of the conic section are made known, 

 that is, the eccentricity, the semi-axis major or the semi-parameter, the place 

 of the perihelion with respect to the heliocentric places C, , C&quot;, the mean 

 motion, and the mean anomaly for the arbitrary epoch if the orbit is elliptical, or 

 the time of perihelion passage if the orbit is hyperbolic or parabolic. It only 

 remains, therefore, to determine the positions of the heliocentric places in the 

 orbit with respect to the ascending node, the position of this node with reference 

 to the equinoctial point, and the inclination of the orbit to the ecliptic (or the 



