104: RELATIONS BETWEEN SEVERAL [BOOK 1. 



In this way we have 



_ 2 r/ sin |(.y JV) s i n (ijy _[_!# _ 77) 

 P a cos (A 77) 



/ r 

 t&amp;gt; -^ __ 



a cos (A 71) * 



These formulas are especially convenient when p and e are to be computed for 

 several values of H ; r, r, N, N continuing the same. Since for the calculation 

 of the auxiliary quantities a, A, the angle H may be taken at pleasure, it will be 

 of advantage to put II ~ J (-N-\- W)&amp;gt; by which means the formulas are changed 

 into these, 



(/ _ r ) cos l(N N) = a cos (A i N I N ) 



And so the angle A being determined by the equation 



tan (A i N i N ) = -^- tan i (N 

 we have immediately 



[& 



cos i (2V&quot; _ jy) C os ( A 77) 



r 1 -\-r 



The computation of the logarithm of the quantity -/-__- may be abridged by a 

 method already frequently explained. 



81. 



If the eccentricity c is given, the angle IT will be found by means of the 

 equation 



cos (A IT}- _&quot;(^-*^m 



ecos i(N N) 



afterwards the auxiliary angle A is determined by the equation 

 tan (A i N * N ) = ^ tan * (W N}. 



The ambiguity remaining in the determination of the angle A 77 by its cosine 

 is founded in the nature of the case, so that the problem can be satisfied by two 

 different solutions ; which of these is to be adopted, and which rejected, must be 

 decided in some other way ; and for this purpose the approximate value at least 



