Manchester Memoirs, Vol. Ixiv (1920) No. 1 



I. — A Discussion of the Theorems of Lambert and Adams 

 on Motion in Elliptic and Hyperbolic Orbits. 



By L. V. Mkadowcroft, B.A., M.Sc. 



(Communicated by Professor Sydney Chapman, M.A., D.Sc, F.R.S.) 

 (Read Dec. 2nd, ipip. Received for Publication, Dec. 29th, ipip.) 



The theorem of Lambert* on the motion of a planet in an elliptic 

 orbit is usually enunciated as follows : — 



" If / is the time of describing any arc P T P 2 of an ellipse, and k is the 

 chord of the arc, then nt = ((p 1 — sin T ) - (<f> 2 - sin <f) 2 ), where 



smj0 I = i / LZ 2 ^ , sinJ^ 2 = J / -i-L- » 



\J a sj a 



r z , r 2 are the focal distances of the points P x , P 2 , a is the semi-major 

 axis of the ellipse and n the mean angular velocity about the focus. 



The most elegant proof is that due to J. C. Adams, f which will be 

 reproduced here in view of the interesting geometrical results to which 

 it gives rise. 



Let fx T , fi 2 be the eccentric anomalies of P z , P 2 , 



.'. k 2 ■■= a 2 (cos (j,! - cos ^ 2 ) 2 + a 2 (i - e 2 ) (sin ^ - sin ^ 2 ) 3 



= 4tf 2 sin 2 £ (^-^[1 -e 2 cos 2 h (yu x + /u 2 )] .... (1), 



r z + r 2 = 2a — ae cos fi 1 — ae cos jx 2 



= 2a[i -e cos \ (fi T >+fi 2 ) cos J fai-^)] • ■ ■ (2), 



nt = /u I - jui 2 — e (sin /j. z — sin jjl 2 ) 



= (/*i - /* 2 ) - 2 e cos £ (fx T + // 2 ) sin J (/^ - /* 2 ) (3). 



Since a, and therefore n, are given, it follows from (1), (2) and (3) 

 that r r -\-r 2 , k and / are functions of the two quantities /^ - /x 2 and 

 * cos J ( / u I + /x 2 ). 



* " Insigniores orbitse cometarum proprietates," 1761. 



f British Association Report, 1 877, or Collected Works, p. 410. 



July 30th, IQ20. 



