Full text: The collected mathematical papers of Arthur Cayley, Sc.D., F.R.S., late sadlerian professor of pure mathematics in the University of Cambridge (Vol. 9)

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581] 
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symmetrical 
he process 
581. 
ON A THEOREM IN ELLIPTIC MOTION. 
[From the Monthly Notices of the Royal Astronomical Society, vol. xxxv. (1874—1875), 
pp. 337—339.] 
Let a body move through apocentre between two opposite points of its orbit, say 
from the point P, eccentric anomaly u, to the point P', eccentric anomaly v!, where 
u, v! are each positive, u <ir, v! > ir. Taking the origin at the focus, and the axis 
of x in the direction through apocentre, then— 
Coordinates of P are x — a (— cos u + e), y = a v 1 — e- sin u, 
„ P' „ x = a (— cos v! 4- e), y = a. Vl — e 2 sin v!; 
whence, expressing that the points P, P' are in a line with the focus, 
sin u (— cos u + e) — sin u (— cos u' + e) — 0, 
that is, 
sin (u —u) = e (sin u' — sin u), 
which is negative, viz. u’ — u is >7r.
	        
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