16 REPORT — 1877. 



and 



tit 



Or 



= (^j) t=u— u' — e(simi— sin?*'). 

 -=1 — (ecos — — J cos— — 



— = sin- _±— sin 2 __ +(1 - e 2 ) cos 2 -L- sin- — — 



= sin — 



pi'{l-e 2 cos 2 ^[, 



and 



nt=u-u'—2 (ecos— I — Ism—- — . 



Hence we see that if a, and therefore also n, be given, then r+r', k, and t are 

 functions of the tioo quantities 



Let 



Then 



u — u' and ecos — ^ — . 



u-u' = 2a and e cos 9 -^~ = cos /3. 



H^L=1 — cos a cos 0, 

 2a 



— - = sin a sin /3 : 

 2a 



therefore r+r'+k =l _ CQS ( /3+a ) ) 



2a 

 and 



also 



!^^=l-COS(/3-a); 



?ji = 2a — 2 sin a cos /3, 

 = [/3+a- sin(/3+a)]-[/3-a- sin(/3-a)]. 



The first two of these equations give fi+a and /3— a in terms of r+r'+k and 

 r-|-r' — A, and the third equation is the expression of Lambert's Theorem. 

 An exactly similar proof may be given in the case of an hvperbolic orbit. 

 Let 



|(e«+f ~") be denoted by csh (w) 

 and 



i( e «— e-«) by snh (V), 



which quantities may be called the hyperbolic cosine and hyperbolic sine of w. 

 Then we have 



csh 2 (?<)— snh 2 (m) = 1, 

 csh O) + csh («') =2 csh "+"' csh w ~"', 



csli («) — csh (V) = 2 snli "j" w snh — ~^, 



snh («)- snh («')=2 csh ^±!i mh"~ w . 



— *. 



