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[209
209.
A DEMONSTR ATION OF SIR W. R. HAMILTON’S THEOREM OF
THE ISOCHRONISM OF THE CIRCULAR HODOGRAPH.
[From the Philosophical Magazine, vol. xiv. (1857), pp. 427—430.]
Imagine a body moving in piano under the action of a central force, and let h
denote, as usual, the double of the area described in a unit of time; let P be any
point of the orbit, then measuring off, on the perpendicular let fall from the centre
of force 0 on the tangent at P to the orbit, a distance OQ equal or proportional
to li into the reciprocal of the perpendicular on the tangent, the locus of Q is the
hodograph, and the points P, Q are corresponding points of the orbit and hodograph.
It is easy to see that the hodograph is the polar reciprocal of the orbit with
respect to a circle having 0 for its centre, and having its radius equal or proportional
to VA. And it follows at once that Q is the pole, with respect to this circle, of the
tangent at P to the orbit.
In the particular case where the force varies inversely as the square of the
distance, the hodograph is a circle. And if we consider two elliptic orbits described
about the same centre, under the action of the same central force, and such that the
major axes are equal, then (as will be presently seen) the common chord or radical
axis of the two hodographs passes through the centre of force.
Imagine an orthotomic circle of the two hodographs (the centre of this circle is
of course on the common chord or radical axis of the two hodographs), and consider
the arcs intercepted on the two hodographs respectively by the orthotomic circle; then
the theorem of the isochronism of the circular hodograph is as follows, viz. the times
of hodographic description of the intercepted arcs are equal; in other words, the times
of description in the orbits, of the arcs which correspond to the intercepted arcs of
the hodographs, are equal. It was remarked by Sir W. R. Hamilton, that the theorem
is in fact equivalent to Lambert’s theorem, that the time depends only on the chord
