5 
If IS 
I 
192 ON A THEOREM IN ELLIPTIC MOTION. [581 
The time of passage from P to P' is 
nt = (u' — e sin u') — (u — e sin u), 
— u' — u — e (sin u’ — sin u), 
= v! — u — sin {u' — u), 
which, v! — u being greater than 7r and — sin (u' — u) positive, is greater than 7r ; viz. 
the time of passage is greater than one-half the periodic time. Of course, if P and P' 
are at pericentre and apocentre, the time of passage is equal one-half the periodic time. 
The time of passage from P' to P through the pericentre is 
nt — 2tt— (u — u) + sin (u' — a), 
which is 
= 27r — (11 — u) — sin {27t — (u 1 — u)}, 
where Ztt — (u' — u), =ct suppose, is an angle <7r. Writing, then 
nt = a — sin a, 
and comparing with the known expression for the time in the case of a body falling 
directly towards the centre of force, we see that the time of passage from P' to P 
through the pericentre, is equal to the time of falling directly towards the same centre 
of force from rest at the distance 2a to the distance a (1 4- cos a), where, as above 
a = 27t — (u' — u), u' — u being the difference of the eccentric anomalies at the two 
opposite points P, P'. If a = 7r, the times of passage are each = ^, that is, one-half 
the periodic time. 
The foregoing equation sin (11 —u) = e (sin v! - sin u) gives obviously 
cos ^ (u' — u) = e cos £ (u + u); 
that is, 
or, 
1 + tan b u tan \ u = e (1 — tan £u tan \u’), 
— tan | u tan ^ v! 
1 — e 
l+e ] 
(in the figure tan^it is positive, tan^-ti' negative); and we thence obtain further 
sin \ (u' — u) = cos ^ v! cos \ u (tan b u — tan ^ u), 
sin ^ (u' + u) = cos £ v! cos ^ u (tan ^ u' + tan ^ u), 
cos | (u' — u) = cos ^ u cos ^ u . Y^T e ’ 
2 
cos \ (u' + u) — cos i 11 cos \ u . — ;