SECT. 1.] THREE COMPLETE OBSERVATIONS. 223 



Since the differences of all these numbers from those which the second 

 hypothesis furnished are very small, it may be safely concluded that the third 

 hypothesis requires no further correction, and, therefore, that a new hypothesis 

 would be superfluous. Wherefore, it will now be proper to proceed to the calcu 

 lation of the elements from 2/ , $ , r, r&quot; : and since the processes comprised in 

 this calculation have been most fully explained above, it will be sufficient to add 

 here the resulting elements, for the benefit of those who may wish to perform the 

 computation themselves : 



Right ascension of the ascending node on the equator .... 158 40 38&quot;.93 



Inclination of the orbit to the equator 11 42 49 .13 



Distance of the perihelion from the ascending node 323 14 56 .92 



Mean anomaly for the epoch 1806 335 4 13 .05 



Mean daily (sidereal) motion 770&quot;.2662 



Angle of eccentricity, y 14 9 3 .91 



Logarithm of the semi-axis major 0.4422438 



158. 



The two preceding examples have not yet furnished occasion for using the 

 method of article 120 : for the successive hypotheses converged so rapidly that 

 we might have stopped at the second, and the third scarcely differed by a sensible 

 amount from the truth. We shall always enjoy this advantage, and be able to do 

 without the fourth hypothesis, when the heliocentric motion is not great and the 

 three radii vectores are not too unequal, particularly if, in addition to this, the 

 intervals of the times differ from each other but little. But the further the con 

 ditions of the problem depart from these, the more will the first assumed values 

 of P and Q differ from the correct ones, and the less rapidly will the subsequent 

 values converge to the truth. In such a case the first three hypotheses are to 

 be completed in the manner shown in the two preceding examples, (with this 

 difference only, that the elements themselves are not to be computed in the third 

 hypothesis, but, exactly as in the first and second hypotheses, the quantities 17, rj&quot;, 

 P , Q , X, Y) ; but then, the last values of P , Q are no longer to be taken as 



