2 MEADOWCROFT, Motion in Elliptic and Hyperbolic Orbits 



Let fx x - /u 2 = 2a, t cos ^ (fd l + fji 2 ) = cos fi 

 .'. k= 2a sin a sin fi ... 



r z + r a + £=2a [i -cos (/3 + a)] 

 r, + r 2 - £ = 2a [i - cos (fi - a)] 

 nt=2a — 2 sin a cos /3 . 



(4), 



(5), 



(6), 



(7), 

 (8). 



If now we put /3 + a = 0„ /3 - a = <f> 2 , the equations (6) and (7) lead to 

 the required expressions for sin \ n sin \ <f> 2 , whilst (8) gives 

 «/=|j3 + o-sin (/3 + a)] -[{fi-a) -sin (fi - a)] 

 = (0 I -sin (p,)-^ -sin <£ 2 ) 



In the figure Q ti Q 2 are the points on the auxiliary circle which 

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

 of the corresponding ordinates. Then | ACQ x *=p» \ ACQ 2 = /u 2 and 



2a= - I CxCO, 



Now or cos \ (fi z + /u 2 ) 



= <r. C7V, if TV is the foot of the ordinate drawn from Q, the 

 middle point of Q Y Q 2 . Let Pbe the corresponding point on the ellipse 

 and r its focal distance. Thus a cos /3 = *. C7V, by (4), and in order 

 to obtain a geometrical representation for fi it is necessary to tranform 



