2 MEADOWCROFT, Motion in Elliptic and Hyperbolic Orbits 



Let yu x - fj, 2 = 2a, e cos J (^ + ^u 2 ) = cos j3 

 .' . k= 2a sin a sin /3 ... 



r x + r 2 + k = 2a [i -cos (/3-f a)] 

 7*i + r 2 -/6= 2<z [i -cos (/3-a)] 

 «/= 2a — 2 sin a cos /3 . 



(4), 

 (5), 

 (6), 

 (7), 

 (8). 



If now we put /3 + a = ^, /} - a = 2 , the equations (6) and (7) lead to 

 the required expressions for sin \ <^, sin \ <j> 2 , whilst (8) gives 

 nt=[fi + a- sm (/3 + «)]-[(/3-a)-sm (/3 - aV| 

 = (<^-sin 0!)-(0 2 -sin <£ 2 ) 



In the figure <2 I3 (2 2 are the points on the auxiliary circle which 

 correspond to the points P z , P 2 on the ellipse, and N„ N 2 are the feet 

 of the corresponding ordinates. Then | ACQ T = ^ \ ACQ 2 = /x 2 and 

 2a= - I (2,67(2, 



NOW <3£ COS J (fij. + ^ 2 ) 



= (?. 67V, if JV\s the foot of the ordinate drawn from Q, the 

 middle point of <2i<2 2 . Let P be the corresponding point on the ellipse 

 and r its focal distance. Thus a cos (j = e. CN, by (4), and in order 

 to obtain a geometrical representation for /3 it is necessary to tranform 



