52. 



ON A SIMPLE PROOF OF LAMBERT'S THEOREM. 



[From the British Association Report (1877).] 



THE following proof of Lambert's Theorem, which I find among my old 

 papers, appears to be as simple and direct as can be desired. 



Let a denote the semiaxis major and e the eccentricity of an elliptic 

 orbit, n the mean motion, and p the absolute force. 



Also let r, r' denote the radii vectores, and u, u' the eccentric anomalies 

 at the extremities of any arc, k the chord, and t the time of describing the 

 arc. 



Then r = a(lecosu), r' = a (I e cos u'), 



k* a 2 (cos u cos u')' 2 + a 2 ( 1 e 2 ) (sin u sin u'f, 



v 



/*/ 



r + r' I u + u'\ 



and nt = I . , I t = u u' e (sin u sin u'). 



a 



Or 





.u u' 



. 2 u - u' I -, 



= sin - 41 e cos 2 - 



2 { 2 



, / u + u'\ . u u 

 and nt = u u 2 ( e cos I sin . 



