SECT. 1.] THREE COMPLETE OBSERVATIONS. 189 



Thence are derived the following formulas, better suited to numerical calculations. 

 Putting, 



[7] tan ft sin (a.&quot; I } tan ft&quot; sin (a / ) = S, 

 [8] tan ft cos (a&quot; I ) tan ft&quot; cos (a I ) = Tsint, 

 [9] sin(a&quot; a) = Tcost, 

 we shall have (article 14, II.) 

 [10] tan((T o)=*r 



7 sin (&amp;lt; -{- /) 



The uncertainty in the determination of the arc (&amp;lt;? - cr) by means of the 

 tangent arises from the fact that the great circles AB , BE&quot;, cut each other in 

 two points ; we shall always adopt for B* the intersection nearest the point B , so 

 that may always fall between the limits of 90 and -f- 90, by which means 

 the uncertainty is removed. 



For the most part, then, the value of the arc a (which depends upon the 

 curvature of the geocentric motion) will be quite a small quantity, and even, gen 

 erally speaking, of the second order, if the intervals of the times are regarded 

 as of the first order. 



It will readily appear, from the remark in the preceding article, what are the 

 modifications to be applied fo the computation, if the equator should be chosen 

 as the fundamental plane instead of the ecliptic. It is, moreover, manifest that 

 the place of the point B* will remain indeterminate, if the circles BB&quot;, AB&quot; 

 should be wholly coincident; this case, in which the four points A,B,B ,B&quot; lie in 

 the same great circle, we exclude from our investigation. It is proper in the 

 selection of observations to avoid that case, also, where the locus of these four 

 points differs but little from a great circle ; for then the place of the point B *, 

 which is of great importance in the subsequent operations, would be too much 

 affected by the slightest errors of observation, and could not be determined with 

 the requisite precision. In the same manner the point B*, evidently, remains 

 indeterminate when the points B, B&quot; coincide,f in which case the position of the 



fOr when they are opposite to each other; but we do not speak of this case, because our method ia 

 not extended to observations embracing so great an interval. 



