SECT. 1.] THREE COMPLETE OBSERVATIONS. 195 



accordingly, it will not be doubtful which must be adopted, or three positive 

 with one negative ; in this case, from among the positive values those, if there are 

 any, are to be rejected which give z greater than d , since, by another essential 

 condition of the problem, (&amp;gt; and, therefore, sin (d z\ must be a positive quantity. 

 When the observations are distant from each other by moderate intervals of 

 time, the last case will most frequently occur, in which three positive values of 

 sin z satisfy the equation. Among these solutions, besides that which is true ; 

 some one will be found making z differ but little from d , cither in excess or 

 in defect; this is to be accounted for as follows. The analytical treatment of 

 our problem is based upon the condition, simply, that the three places of the heav 

 enly body in space must fall in right lines, the positions of which are determined 

 by the absolute places of the earth, and the observed places of the body. Now, 

 from the very nature of the case, these places must, in fact, fall in those parts of 

 the right lines whence the light descends to the earth. But the analytical equa 

 tions do not recognize this restriction, and every system of places, harmonizing of 

 course with the laws of KEPLER, is embraced, whether they lie in these right lines 

 on this side of the earth, or on that, or, in fine, whether they coincide with the 

 earth itself. Now, this last case will undoubtedly satisfy our problem, since the 

 earth moves in accordance with these laws. Thence it is manifest, that the equa 

 tions must include the solution in which the points C. C , C&quot; coincide with A, A , A&quot; 

 (so long as we neglect the very small variations in the elliptical places of the earth 

 produced by the perturbations and the parallax). Equation IV., therefore, must 

 always admit the solution z = d , if true values answering to the places of the 

 earth are adopted for P and Q. So long as values not differing much from these 

 are assigned to those quantities (which is always an admissible supposition, when 

 the intervals of the times are moderate), among the solutions of equation IV., 

 some one will necessarily be found which approaches very nearly to the value 



z c r. 



For the most part, indeed, in that case where equation IV. admits of three 

 solutions by means of positive values of sin 2, the third of these (besides the true 

 one, and that of which we have just spoken) makes the value of z greater than 

 d , and thus is only analytically possible, but physically impossible ; so that it can- 



