52] ON A SIMPLE PROOF OF LAMBERT'S THEOREM. 411 



Hence we see that if a, and therefore also n, be given, then r + r', k, 

 and t are functions of the two quantities 



, u + u' 



u u and e cos - . 



2 



Let u u' = 2a and e cos - - = cos ft. 



r + ^ 

 Then = 1 cos a cos ft, 



= sin a sin ft ; 



therefore = 1 cos (/3 + a), 



and ~ = 1 cos (ft a); 



also nt = 2a 2 sin a cos ft, 



= [ft + a - sin (ft + a)] - [/? - a - sin (ft - a)]. 



The first two of these equations give ft + a and ft a in terms of 

 r + r' + k and r + 'r' k, and the third equation is the expression of Lambert's 

 Theorem. 



An exactly similar proof may be given in the case of an hyperbolic 

 orbit. 



Let -(e K + e~") be denoted by csh (u), 



and o( "~ e ~") by snh (u), 



which quantities may be called the hyperbolic cosine and hyperbolic sine of u 



Then we have 



csh 2 (u) snh" (u) = l, 



i f 



ni I Or* <1 1 fl I 



csh (u) + csh (u'} = 2 csh csh - , 



I / A , , 



csh (t*) csh (u') = 2 snh - snh 



, 



,u + u' ,u -u' 

 snh (u) - snh (') = 2 csh - - snh - , 



522 



