186 DETERMINATION OF AN ORBIT FROM [BOOK II. 



be in opposition or conjunction and in the ecliptic at the same time, y would be 

 indeterminate. But we assume that this is not the case in either of the three 

 observations. 



If the equator is adopted as the fundamental plane, instead of the ecliptic, 

 then, for determining the positions of the three great circles with respect to the 

 equator, will be required the right ascensions of their intersections with the equa 

 tor, besides the inclinations ; and it will be necessary to compute, in addition to 

 the distances of the points B, B , B&quot;, from these intersections, the distances of the 

 points A, A , A&quot; also from the same intersections. Since these depend on the 

 problem discussed in article 110, we do not stop here to obtain the formulas. 



137. 



The second step will be the determination of the positions of these three great 

 circles relatively to each other, which depend on their inclinations and the places 

 of their mutual intersections. If we wish to bring these to depend upon clear 

 and general conceptions, without ambiguity, so as not to be obliged to use 

 special figures for different individual cases, it will be necessary to premise some 

 preliminary explanations. Firstly, in every great circle two opposite directions 

 are to be distinguished in some way, which will be done if we regard one of them 

 as direct or positive, and the other as retrograde or negative. This being wholly 

 arbitrary in itself, we shall always, for the sake of establishing a uniform rule, con 

 sider the directions from A, A , A&quot; towards B, B ,B&quot; as positive; thus, for example, 

 if the intersection of the first circle with the second is represented by a positive 

 distance from the point A, it will be understood that it is to be taken from A 

 towards B (as D&quot; in our figure) ; but if it should be negative, then the distance 

 is to be taken on the other side of A. And secondly, the two hemispheres, into 

 which every great circle divides the whole sphere, are to be distinguished by suit 

 able denominations ; accordingly, we shall call that the superior hemisphere, which, 

 to one walking on the inner surface of the sphere, in the positive direction along 

 the great circle, is on the right hand ; the other, the inferior . The superior hemi 

 sphere will be analogous to the northern hemisphere in regard to the ecliptic or 

 equator, the inferior to the southern. 



