SECT. 3.] PLACES IN ORBIT. 103 



r 77) 



from which, if one of the quantities p, e, IT, is also given, it will be possible to 

 determine the two remaining ones. 



Let us first suppose the semi-parameter p to be given, and it is evident that 

 the determination of the quantities e and 77&quot; from the equations 



ecos(N 77)=:^ 1, 



can be performed by the rule of lemma III. in the preceding article. We have 

 accordingly 



tan ( N 77) = cotan ( N f N} -^ ~ ( f^2,- 



r (p r) sin (N N) 



r 4-r 



P 

 80. 



If the angle 77 is given, p and e will be determined by means of the equations 



_ rr (cos (N 77) cos (N II)) 

 ^n) r cos (N 1 77) 

 r r 



_ _ 



~ r cos (2T^~if) r cos (N f 77) * 



It is possible to reduce the common denominator in these formulas to the form 

 a cos (A 77), so that a and A may be independent of 77. Thus letting H de 

 note an arbitrary angle, we have 



rcos(jy 77) r cos(N 77)=(rcos(^ H) /cos(JY 7J))cos(7J 77) 



(r sin(^ 7J) /sin (N H}) sin (7777) 



and so 



= a cos (A 77), 



if a and A are determined by the equations 



r cos (N 77) / cos (N l 77) = a cos (4 77) 

 r sin (^ 77) / sin (Jf 77) = a sin (4 77) . 



