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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.