SECT. 3.] PLACES IN ORBIT. 127 



96. 



The equations in the preceding article possess so much neatness, that there 

 may seem nothing more to be desired. Nevertheless, we can obtain certain 

 other formulas, by which the elements of the orbit are determined much more 

 elegantly and conveniently ; but the development of these formulas is a little 

 more abstruse. 



We resume the following equations from article 8, which, for convenience, we 

 distinguish by new numbers : 



L sin i v V/- = sin lH^(~L-\-e) 



IL cos i vu - cos i.Ey(l e) 



IH. si 



W. 



We multiply I. by sin i (F-\- g], II. by cos I (F-\-g], whence, the products being 

 added, we obtain 



cos * (f+g) \^ = sin i^sin * (F-\-g)^(l + ) + cos i^cos l^+ff) V(l e) 

 or, because 



\/ (1 -f- e) = cos i 9 -(- sin i 9, y/ (1 e) = cos J y sin 9, 



cos %(f -\-g}U - = cos i 9003(1^ kG-\-g] sin Jycos %(F -\-G}. 



In exactly the same way, by multiplying HL by sin i (F g\ IV. by cos i ( F g\ 

 the products being added, appears 



i-Z T % G g) sin (p cos 

 The subtraction of the preceding from this equation gives 



cos J (f-\-g] (\l -- V / ^ cos ^ 9 sul ^ sm ^ (^ 

 or, by introducing the auxiliary angle to, 



[27] cos (f-\-ff) tan2cu=:sin i ( F G)cos$(psmg if. 



