52] ON A SIMPLE PROOF OF LAMBERT'S THEOREM. 413 



Also t = (~} t 2 csh O 8 ) sn M a ) - 2a ]> 



/a 3 \ * 

 = [snh (ft + a) - (/3 + a) - snh (/3 - a) + (/3 - a)]. 



As before, the first two of these equations give ft + a and /B a in 

 terms of r + r' + k and r + r' k, and the last equation is the expression 

 of Lambert's theorem in the case of the hyperbola. 



When the orbit is parabolic, a becomes infinite ; and since r + r' and 

 k are finite, the quantities a and /3 become indefinitely small. 



Hence 



= 1 - cos (/8 + a) = - (/3 + a) 2 ultimately, 

 2 a 2 



r+ r' /; x 1 . , n v> -,, . , T 



1 cos (p a) = ~(p ay ultimately ; 



- 

 also * = ( {^ + a-sin(/3 + a)-(/3-a)+sin(/3-a)} 



= (^) Ji( j g + a )_I(^_ a )i\ ultimately 



\ J 



-( -) (- -] I ultimately 



l\ / V / J 



== _ ) 

 which is Lambert's theorem in the case of the parabola. 



