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% David R. Wilkins
% School of Mathematics, Trinity College, Dublin 2, Ireland
% (dwilkins@maths.tcd.ie)
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% Trinity College, 2000.

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\centerline{\Largebf ON THEOREMS OF HODOGRAPHIC AND}

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\centerline{\Largebf ANTHODOGRAPHIC ISOCHRONISM}

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\centerline{\Largebf By}

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\centerline{\Largebf William Rowan Hamilton}

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\centerline{\largerm (Proceedings of the Royal Irish Academy,
   3 (1847), pp.\ 417, 465--466.)}

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\centerline{\largerm Edited by David R. Wilkins}

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\centerline{\largerm 2000}

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\noindent
\centerline{\largeit On a Theorem of Hodographic Isochronism.}

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\centerline{\largerm William Rowan Hamilton}
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\centerline{[{\it Proceedings of the Royal Irish Academy},
vol.~3 (1847), p.~417.]}
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\centerline{Communicated March 16, 1847.}

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The following note by Sir W.~R. Hamilton, announcing a theorem of
hodographic isochronism, was read:

If two circular hodographs, having a common chord, which passes
through or tends towards a common centre of force, be cut
perpendicularly by a third circle, the times of hodographically
describing the intercepted arcs will be equal.

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\noindent
\centerline{\largeit On a Theorem of Anthodographic Isochronism.}

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\centerline{\largerm William Rowan Hamilton}
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\centerline{[{\it Proceedings of the Royal Irish Academy},
vol.~3 (1847), p.~465--466.]}
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\centerline{Communicated May 10, 1847.}

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Sir William Hamilton stated and illustrated a theorem of
anthodographic (or anthodic) isochronism, namely, that if two
circular anthodes, having a common chord, which passes through,
or tends towards a common centre of force, be both cut
perpendicularly by any third circle, the times of anthodically
describing the intercepted arcs will be
equal:---the {\it anthode} of a planet being the circular locus
of the extremities of its vectors of slowness, or of straight
lines representing, in length and direction, the reciprocals of its
velocities, and drawn from a common origin.

The theorem is intimately connected with the analogous theorem
respecting hodographic isochronism (or synchronism), which was
communicated to the Academy by Sir William Hamilton, in a note
read at the Meeting in last March.  He had been led to perceive
that former theorem by combining the principles of his first
paper on a General Method in Dynamics, published in the second
part of the Philosophical Transactions for 1834, with those of
his communication of last December, since published in the
Proceedings of the Academy, respecting the Law of the Circular
Hodograph.  This {\it Hodograph} was, for a planet or comet, the
circular locus of the extremities of its {\it vectors of
velocity}, as the Anthode is the locus of the extremities of the
vectors of slowness; so that the rectangular coordinates of the
Hodograph are $x'$,~$y'$,~$z'$, if
$$x' = {dx \over dt},\quad
  y' = {dy \over dt},\quad
  z' = {dz \over dt};$$
while those of the Anthode may be denoted as follows:
$$x_\prime = - v^{-2} x',\quad
  y_\prime = - v^{-2} y',\quad
  z_\prime = - v^{-2} z',$$
where $v^2 = x'^2 + y'^2 + z'^2$.

He had effected the passage from the theorem respecting
hodographic to that respecting anthodic isochronism, by the help
of his calculus of quaternions; but had since been able to prove
both theorems by means of certain elementary properties of the
circle.

For a hyperbolic comet, the Anthode is a circular arc
{\it convex\/} to the sun; for a parabolic comet, the Anthode is
a {\it straight line}.  And for comets of this latter class the
theorem of isochronism takes this curiously simple form: ``Any
two diameters of any one circle (or sphere) in space, are
anthodically described in equal times, with reference to any one
point, regarded as a common centre of force.''  By this last
theorem, the general problem of determining the time of
{\it orbital\/} description of a finite arc of a
{\it parabola\/}, is reduced to that of determining the time of
{\it anthodical\/} description of a finite {\it straight line\/}
directed to the sun; and thus it is found that ``the interval of
time between any two positions of a parabolic comet, divided by
the mass of the sun, is equal to the sixth part of the difference
of the cubes of the sum and difference of the diagonals of the
parallelogram,
constructed with the initial and final vectors of slowness as two
adjacent sides.''  Another very simple extension for the time of
description of a parabolic arc, to which Sir William Hamilton is
conducted by his own method, but which he sees to admit of easy
proof from known principles (though he does not remember meeting
the expression itself), is given by the following formula:
$$t = {\textstyle {1 \over 2}} {\sc t} \tan
         ( \theta - \tan^{-1} {\textstyle {1 \over 2}}
               \tan {\textstyle {1 \over 2}} \theta );$$
where $\theta$ is the true anomaly, and $t$ is the time from
perihelion, while ${\sc t}$ is the time of describing the first
quadrant of true anomaly.

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