228 DETERMINATION OF AN ORBIT FROM [BOOK II. 



middle observation, the mean anomaly is found to be 326 19 25&quot;.72, the loga 

 rithm of the radius vector, 0.4132825, the true anomaly, 320 43 54&quot;.87 : this last 

 should differ from the true anomaly for the first place by the quantity If&quot;, or 

 from the true anomaly for the third place by the quantity 2/, and should, there 

 fore, be 320 43 54&quot;.92, as also the logarithm of the radius vector, 0.4132817 : 

 the difference 0&quot;.05 in the true anomaly, and of eight units in the logarithm, is 

 to be considered as of no consequence. 



If the fourth hypothesis should be conducted to the end in the same way as 

 the three preceding, we would have X= 0, Y= 0.0000168, whence the follow 

 ing corrected values of x and y would be obtained, 



x = logP = 0.0256331, (the same as in the fourth hypothesis,) 

 y = \og Q= 0.7508917. 



If the fifth hypothesis should be constructed on these values, the solution would 

 reach the utmost precision the tables allow: but the resulting elements would 

 not differ sensibly from those which the fourth hypothesis has furnished. 



Nothing remains now, to obtain the complete elements, except that the posi 

 tion of the plane of the orbit should be computed. By the precepts of article 

 149 we have 



From the first place. From the third place. 



g 354 9 44&quot;.22 / .... 57 5 0&quot;.91 



h 261 56 6 .94 A&quot; .... 161 1 .61 



i 10 37 33 .02 10 37 33 .00 



8 80 58 49 .06 80 58 49 .10 



Distance of the perihelion I ^ g ^ 65 g 4 ^ 



from the ascending node j 



Longitude of the perihelion 146 53 .53 146 53 .62 



The mean being taken, we shall put i= 10 37 33&quot;.01, Q = 80 58 49&quot;.08, the 

 longitude of the perihelion = 146 53&quot;.57. Lastly, the mean longitude for 

 the beginning of the year 1806 will be 108 36 46&quot;.08. 



