[581 
581] 
ON A THEOREM IN ELLIPTIC MOTION. 
193 
C. IX. 
25 
than 7T; viz. 
if P and P' 
)eriodic time. 
t body falling 
rom P' to P 
3 same centre 
ere, as above 
at the two 
it is, one-half 
and thence also 
cos u + cos v! = 2 cos \ (u' + u) cos \ (v! — u), 
8e 
= cos 2 £ u' cos 2 \ u . 
But we have 
(l+e) 2 ' 
1 + cos (v! — u) = 2 cos 2 \ (u' — u) = cos 2 1- v! cos 2 1 
8e 2 
(1 + e) 2 ’ 
or, comparing with the last equation, 
1 + cos (u —u) = e (cos u + cos u), 
or, what is the same thing, 
1 — cos (v! — u) = (1 — e cos u') + (1 — e cos u); 
and in like manner, 
1 + cos (u' + m) = 2 cos 2 \ (u 1 -4- u) = cos 2 J u'. cos 2 ^ u 
or, comparing with the same equation, 
1 + cos (u' + u) = - (cos u + cos u'): 
c 
which are formulse corresponding with the original equation 
sin (u r — u) = e (sin v! — sin u). 
8 
(1 + e) 2 ’ 
urther