194 DETERMINATION OF AN ORBIT FROM [BOOK II. 



by means of the formulas 11-16, depending on given quantities alone. The 

 quantities b, c, e, will not themselves be required, only their logarithms. 



There is a special case in which these precepts require some change. That 

 is, when the great circle BB&quot; coincides with A B&quot;, and thus the points B, B* 

 with jy, D, respectively, the quantities a, b would acquire infinite values. Put 

 ting, in this case, 



R sin d sin (A D S + a) _ 

 JJ sind sin^D&quot; rf) ~ 



in place of equation HI. we shall have 



,, n sin (z a) 



= nn t, 



smz 

 whence, making 



?rsin a 

 tan w =2 



p_j-(l_rtcos&amp;lt;i) 

 the same equation IV. is obtained. 



In the same manner, in the special case when a = 0, c becomes infinite, and 

 w = 0, on account of which the factor c sin w, in equation IV., seems to be inde 

 terminate ; nevertheless, it is in reality determinate, and its value is 



as a little attention will show. In this case, therefore, sm z becomes 



142. 



Equation IV., which being developed rises to the eighth degree, is solved by 

 trial very expeditiously in its unchanged form. But, from the theory of equa 

 tions, it can be easily shown, (which, for the sake of brevity, we shall dispense 

 with explaining more fully) that this equation admits of two or four solutions by 

 means of real values. In the former case, one value of sin z will be positive ; 

 and the other negative value must be rejected, because, by the nature of the 

 problem, it is impossible for r to become negative. In the latter case, among the 

 values of sin z one will be positive, and the remaining three negative, when, 



