196 DETERMINATION OF AN ORBIT FROM [BOOK II. 



not then be doubtful which is to be adopted. But yet it certainly can happen, 

 that the equation may admit of two distinct and proper solutions, and thus that 

 our problem may be satisfied by two wholly different orbits. But in such an 

 event, the true orbit is easily distinguished from the false as soon as it is possible 

 to bring to the test other and more remote observations. 



143. 



As soon as the angle z is got, / is immediately had by means of the equation 



, _ K sins 



Further, from the equations P = and III. we obtain, 



nY _ (P+a)J?smff 

 n b sin (z a) 



/_ JL_ nY 



&quot; -~P ~7T 



Now, in order that we may treat the formulas, according to which the posi 

 tions of the points O, C&quot;, are determined from the position of the point C , in such 

 a manner that their general truth in those cases not shown in figure 4 may 

 immediately be apparent, we remark, that the sine of the distance of the point 

 C from the great circle CB (taken positively in the superior hemisphere, nega 

 tively in the inferior) is equal to the product of sin e&quot; into the sine of the distance 

 of the point C from D&quot;, measured in the positive direction, and therefore to 



- sin e&quot; sin C D&quot; = sin e&quot; sin (0 + A D&quot; d ) ; 



in the same manner, the sine of the distance of the point C&quot; from the same great 

 circle is sin t, sin C&quot;D . But, evidently, those sines are as sin CO to sin CO&quot;, or 

 as ^-, to ^p,, or as ri r&quot; to n r . Putting, therefore, C&quot;D C&quot;, we have 



Vff j-// n r sin . , \ &f TV/ w\ 

 r sin &quot; = .-. - sm (z 4- A o ) . 



if sm e 



Precisely in the same way, putting (7ZX = t, is obtained 



TTT !- ft Sin . / i Af T\ fc/\ 



VI. r sin !, = . -r 7 sin (z 4- A D 8} . 



n sin e 



VH. 



