SECT. 1.] THREE COMPLETE OBSERVATIONS. 227 



If we designate the three values of X by A, A , A&quot;; the three values of Y by 

 B, B, B&quot;; the quotients arising from the division of the quantities A B&quot;-A&quot;B. 

 A B AB&quot;, AB 1 A B, by. the sum of these quantities, by k, k , k&quot;, respectively, 

 so that we have R-{-tf-\-tf r =l; and, finally, the values of log P f and log Q in the 

 third hypothesis, by M and N, (which would become new values of x and y if it 

 should be expedient to derive the fourth hypothesis from the third, as the third 

 had been derived from the second) : it is easily ascertained from the formulas of 

 article 120, that the corrected value of x is M k (A -\- A&quot;) /c A&quot;, and the cor 

 rected value of y, N k (B -j- B ) k B&quot;. The calculation being made, the 

 former becomes 0.0256331, the latter, 0.7509143. Upon these corrected values 

 we construct the fourth hypothesis, the chief results of which are the following : 



w + a . . . . 714 45&quot;.247 

 log Qc sin CD . . 1.2094284 B 



7 2 12 .736 



log/ 0.4132817 



log/ .... 0.4062033 



*(&quot; + ) . . . 26257 38&quot;.78 



-M) . . . 273 29 20 .73 



2/ 62 55 16 .64 



2/ 31 19 1 .49 



2/&quot; 31 36 15 .20 



160 22 45 .38 



&quot; 262 15 3 .90 



logr 0.4282792 



The difference between 2/ and 2/-J- 2/&quot; proves to be 0&quot;.05, which we shall 

 distribute in such a manner as to make 2/= 31 19 1&quot;.47, 2/&quot;= 31 36 15&quot;.17. 

 If now the elements are determined from the two extreme places, the following 

 values result : 



True anomaly for the first place 289 7 39&quot;.75 



True anomaly for the third place 352 2 56 .39 



Mean anomaly for the first place 297 41 35 .65 



Mean anomaly for the third place 353 15 22 .49 



Mean daily sidereal motion 769&quot;.6755 



Mean anomaly for the beginning of the year 1806 . . 322 35 52 .51 



Angle of eccentricity, y 4 37 57 .78 



Logarithm of the semi-axis major 0.4424661 



By computing from these elements the heliocentric place for the time of the 



