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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 THE HODOGRAPH, OR A NEW METHOD OF}
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\centerline{\Largebf EXPRESSING IN SYMBOLICAL LANGUAGE}
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\centerline{\Largebf THE NEWTONIAN LAW OF ATTRACTION}
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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.\ 344-353.)}
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\centerline{\largerm Edited by David R. Wilkins}
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\centerline{\largerm 2000}
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\noindent
{\largerm THE HODOGRAPH, OR A NEW METHOD OF EXPRESSING IN
SYMBOLICAL LANGUAGE THE NEWTONIAN LAW OF ATTRACTION\par}
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\centerline{\largerm William Rowan Hamilton}
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\centerline{[{\it Proceedings of the Royal Irish Academy},
vol.~3 (1847), pp.\ 344--353.]}
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\centerline{Communicated December 14, 1846.}
\bigskip
Sir William R. Hamilton made a communication respecting a new
mode of geometrically conceiving, and of expressing in symbolical
language, the Newtonian law of attraction, and the mathematical
problem of determining the orbits and perturbations of bodies
which are governed in their motions by that law.
Whatever may be the complication of the accelerating forces which
act on any moving body, regarded as a moving point, and,
therefore, however complex may be its {\it orbit}, we may always
imagine a succession of straight lines, or vectors, to be drawn
from some one point, as from a common origin, in such a manner as
to represent, by their directions and lengths, the varying
directions and degrees (or quantities) of the velocity of the
moving point: and the curve which is the locus of the ends of the
straight lines so drawn may be called the {\it hodograph\/} of the
body, or of its motion, by a combination of the two Greek words,
$\overbq{o} \delta \acute{o} \varsigma$, a {\it way}, and
$\gamma \rho \acute{\alpha} \phi \omega$, {\it to write\/}
or {\it describe\/}; because the vector of this hodograph,
which may also be said to be the {\it vector of velocity\/} of
the body, and which is always parallel to the tangent at the
corresponding point of the orbit, marks out or indicates at
once the direction of the momentary path or way in which the body
is moving, and the rapidity with which the body, at that moment,
is moving in that path or way. This hodographic curve is even
more immediately connected than the orbit with the {\it forces\/}
which act upon the body, or with the varying resultant of those
forces, for the tangent to the hodograph is always parallel to
the direction of this resultant; and if the element of the
hodograph be divided by the element of the time, the quotient of
this division represents (to the usual units) the intensity of the
same resultant force; so that the whole accelerating force which
acts on the body at any one instant is represented, both in
direction and magnitude, by the element of the hodograph, divided
by the element of the time. We have also the general proportion,
that the {\it force is to the velocity}, in any varied motion of
a point, {\it as the element of the hodograph is to the
corresponding element of the orbit}.
These general remarks respecting varied motion, under the
influence of {\it any\/} accelerating forces whatever, having
been premised, let it be now supposed that the force is
constantly directed towards some one {\it fixed point\/} or
{\it centre}, which it will then be natural to choose for the
origin of the vectors of the hodograph. The straight lines drawn
to the moving body from the centre of force being called, in like
manner, the vectors of the orbit, or the {\it vectors of
position\/} of the body, we see that each such vector of position
is now parallel to the tangent of the hodograph drawn at the
extremity of the vector of velocity, as the latter vector was
seen to be parallel to the tangent of the orbit, drawn at the
extremity of the vector of position; so that the two vectors, and
the two tangents drawn at their extremities, enclose at each
moment a {\it parallelogram}, of which it is easily seen that the
{\it plane and area are constant}, although its position and its
shape are generally variable from one moment to another, in the
motion thus performed under the influence of a central force.
If, therefore, this constant area be given, and if either the
hodograph or the orbit be known, the other of these two curves
can be deduced, by a simple and uniform process, of which account
the two curves themselves may be called {\it reciprocal
hodographs}.
The opposite angles of a parallelogram being equal, it is
evident, that if the central force be attractive, any one vector
of position is inclined to the next following element of the
orbit as the same angle as that at which the corresponding vector
of velocity is inclined to the next preceding element of the
hodograph. Also, if from either extremity of any small element
of the curve, a chord of the circle which osculates to that curve
along that element be drawn and bisected, the element subtends,
at the middle point of this chord, an angle equal to the angle
between the two tangents drawn at the two extremities of the
element; that is, here, if the curve be the hodograph, to the
angle between the two near vectors of position, which are
parallel to the two extreme tangents of its element. We have,
therefore, two small and similar triangles, from which results
the following proportion, that {\it the half chord of curvature
of the hodograph\/} (passing through, or tending towards the fixed
centre of force) {\it is to the radius vector of the orbit\/} as
the element of the hodograph is to the element of the orbit, that
is, by what was lately seen, {\it as the force is to the
velocity}.\footnote*{By an exactly similar reasoning, the
following known proportion may be proved anew, namely, that the
force is to the velocity as that velocity is to the half chord of
curvature of the orbit, whatever the law of central force may
be.}
But also, the radius of curvature of the hodograph is to the half
chord of curvature of the same curve, as the radius vector of the
orbit is to the perpendicular let fall from the fixed centre on
the tangent to the same orbit; therefore, by compounding equal
ratios, we obtain this other proportion: {\it the radius of
curvature of the hodograph is to the radius vector of the orbit,
as the rectangle under the same radius vector and the force is to
the rectangle under the velocity and the perpendicular}, or to the
constant parallelogram under the vectors of position and
velocity. If, therefore, the law of the inverse square hold
good, so that the second and third terms of this last proportion
vary inversely as each other, while the fourth term remains
unchanged, the first term must be also constant; that is, {\it
with Newton's law of force\/} (supposed here to act towards a
fixed centre) {\it the curvature of the hodograph is constant\/}:
and, consequently, this curve, having been already seen to be
{\it plane}, is now perceived to be a {\it circle}, of which the
radius is equal to the attracting mass divided by the constant
double areal velocity of the orbit. Reciprocally, we see that
{\it no other law\/} of force would conduct to the same result:
so that the Newtonian law may be {\it characterized\/} as being
the {\it Law of the Circular Hodograph}.
Another mode of arriving at the same simple but important result,
is to observe, that because the radius of curvature of the
hodograph is equal to the element of that curve, divided by the
angle between the tangents at its extremities, or (in the case
of a central force) by the angle between two corresponding
vectors of the orbit, which angle is equal to the double of the
elementary area divided by the square of the distance (of the
body from the centre of force), while the element of the
hodograph has been seen to be equal to the force multiplied by
the element of time, or multiplied by the same double element of
orbital area, and divided by the constant of double areal
velocity, therefore this radius of curvature of the hodograph
must, for any central force, be equal to the force multiplied by
the square of the distance, and divided by the double areal
velocity.
The point on the hodograph which is the termination of any one
vector of velocity may be called the {\it hodographic
representative\/} of the moving body, and the foregoing
principles show, that with a central force varying as the inverse
square of the distance, this representative point describes, in
any proposed interval of time, a {\it circular arc}, which
contains the same number of degrees, minutes and seconds, as the
angle contemporaneously described round the centre of force by the
body itself in its orbit, or by the revolving vector of position;
because, whatever that angle may be, an equal angle is described
in the same time by the revolving tangent to the hodograph.
Thus, with the law of Newton, {\it the angular motion of a body
in its orbit is exactly represented, with all its variations, by
the circular motion on the hodograph\/}; and this remarkable
result may be accepted, perhaps, as an additional motive for the
use of the new term which it is here proposed to introduce.
Whatever the law of central force may be, if the square of the
velocity in the orbit be subtracted from the double rectangle
under the force and distance, which has been seen to be equal to
the rectangle under the same velocity and the chord of curvature
of the hodograph, the remainder is the rectangle under the
segments into which that chord is cut by the centre of force,
being positive when this section takes place internally, but
negative when the section is external, that is,when the centre of
force is outside the osculating circle of the hodograph. In the
case of the law of the inverse square, this latter curve is its
own osculating circle, and the rectangle under the segments of
the chord is, therefore, constant, by an elementary theorem of
geometry contained in the third book of Euclid; if, then, the
square of the velocity be subtracted from the double of the
attracting mass, divided by the distance of the body from the
centre of force, at which that mass is conceived to be placed,
the remainder is a constant quantity, which is positive if the
orbit be a closed curve, that is, if the centre of force be
situated in the {\it interior\/} of the circular hodograph.
In the case of a closed orbit, the positive constant, which is
thus equal to the product of the segments of a hodographic chord,
or {\it the constant product of any two opposite velocities of
the body\/} is easily seen, by the foregoing principles, to be
equal to {\it the attracting mass divided by the semisum of the
two corresponding distances of the body}, which semisum is
therefore seen to be {\it constant} and may be called (as in fact
it is) the {\it mean distance}. The {\it law of living force},
involving this mean distance, may therefore be deduced as an
elementary consequence of this mode of hodographic
representation, for the case of a closed orbit; together with the
corresponding forms of this law, involving a null or a negative
constant, instead of the reciprocal of the mean distance, for the
two cases of an orbit which is not closed, namely, when the
centre of force is on, or is outside the circumference of the
hodographic circle.
Whichever of these situations the centre of force may have, we
may call the straight {\it line\/} drawn from it to the centre of
the hodograph, the hodographic {\it vector of eccentricity\/};
and the {\it number\/} which expresses the ratio of the length of
this vector to the radius of the hodograph will represent, if the
orbit be closed, the ratio of the semidifference to the semisum of
the two extreme distances of the body from the centre of force,
and may be called generally the {\it numerical eccentricity\/} of
the hodograph, or of the orbit (without violating the received
meaning of the term).
Whatever the value of this numerical eccentricity may be, the
constant area of the parallelogram under the vectors of position
and velocity may always be treated as the sum or difference of
two other parallelograms, of which one is equal to the rectangle
under the constant radius of the hodographic circle and the
varying radius vector of the orbit, while the other is equal to
the parallelogram under the vectors of position and eccentricity;
and hence it is not difficult to infer that the length of the
vector of position, or of the radius vector of the orbit, varies
in a constant ratio, expressed by the numerical eccentricity, to
the perpendicular let fall from its extremity, that is, from the
position of the body, on a constant straight line or
{\it directrix}, which is situated in the plane of the orbit, and
is parallel to the hodographic vector of eccentricity. The
{\it orbit}, therefore, whether it be closed or not, is always
(with the law of the inverse square) a {\it conic section},
having the centre of force for a {\it focus}---a theorem which
has indeed been known since the time of Newton, but has not
perhaps been proved before from principles so very
elementary.\footnote*{The {\it hodograph of the earth's annual
motion\/} may be considered to be exhibited to
{\it observation\/} in astronomy as the {\it curve of aberration
of a star\/}; and it is known that this aberratic curve is a
{\it circle}, notwithstanding the eccentricity of the earth's
orbit; but the author is not aware that this circularity of the
aberratic curve (for a star near the pole of the ecliptic) has
ever been shown before to be a {\it consequence of the law of the
inverse square}, except by the help of the properties of the
{\it elliptic orbit\/}; whereas the spirit of the present
communication is the {\it derive that orbit from the circle}, and
to regard that circle itself as a sort of geometrical {\it
picture of Newton's law}, instead of being only one of many
corollaries from the laws of Kepler.}
Conceive a diameter of the hodograph to be drawn in a direction
perpendicular to the vector of eccentricity; the extremities of
this diameter correspond to the extremities of that chord of the
orbit which is perpendicular to the shortest radius vector, and
which is called the {\it parameter\/}; from which it follows that
the {\it semiparameter of the orbit is equal to the constant area
of the parallelogram under distance and velocity, divided by the
radius of the hodograph}, and, consequently, that it is equal to
the {\it square of the constant double areal velocity, divided by
the attracting mass}.
It is evident that these results agree with and illustrate those
by which Newton shewed that Kepler's laws were mathematical
consequences of his own great law of attraction. In applying
them to the undisturbed motion of any binary system of bodies,
attracting each other according to that law, we have only to
substitute the sum of the two masses for the single attracting
mass already considered, and to treat one of the two bodies as if
it were the fixed origin of the vector of a {\it relative
hodograph}, which will still be circular as before. And even if
we consider a ternary, or a {\it multiple system}, we may still
regard {\it each\/} body as {\it tending\/} by its attraction, to
cause every {\it other\/} to describe an orbit of which the
hodographic representative would be a perfect circle.
When there is {\it one predominant mass}, as in the case of the
solar system, we may in general regard each other body of the
system as moving in an orbit about it, which is, on the same
plan, {\it represented\/} by a {\it varying circular hodograph}.
For if, at any one moment, we know {\it the two heliocentric
vectors\/} of position and velocity of a planet, we know the
plane and area of the parallelogram under those two vectors, and
can conceive a parellelepiped constructed, of which this
momentary parallelogram shall be the base, while the volume of
the solid shall represent the sum of the masses of the sun and
planet; and then the height of the same solid will be equal to the
radius of the momentary hodograph; so that, in order to construct
this hodograph, we shall only have to describe, in the plane and
with the radius determined as above, a circle which shall touch
the side parallel to the heliocentric vector of position, at the
extremity of the vector of velocity, and shall have its
concavity, at the point of contact, turned towards the sun. The
moon, or any other satellite, may also be regarded as describing,
about its primary, an orbit of which the hodographic
representative shall still be a varying circle.
As formul{\ae} which may assist in symbolically tracing out the
consequences of this geometrical conception, Sir William Hamilton
offers the following transformations of certain general equations
for the motion of a system of bodies attracting each other
according to Newton's law, which he communicated to the Royal
Irish Academy in July, 1845. (See Proceedings, vol.~III, part 2,
Appendix III and V.)
The new forms of the equations are these:
$$\rho
= \int \tau \, dt;\quad
\sigma
= {m' \over {\rm V} \mathbin{.}
(\rho' - \rho) (\tau' - \tau)};\quad
\tau
= {\textstyle\sum} \int \sigma \, d{\rm U} (\rho' - \rho);$$
in which $\rho$ and $\tau$ are the vectors of position and
velocity of the mass $m$ at the time $t$; $\rho'$ and $\tau'$
are the two corresponding vectors of another mass $m'$ at the
same time; $\sigma$ is another vector, perpendicular to the
plane, and equal in length to the radius of the {\it momentary
relative hodograph, representing the momentary relative orbit},
which the attraction of the mass $m'$ tends to cause the body $m$
to describe; $d$, $\int$, $\sum$ are marks of differentiation,
integration and summation, and ${\rm V}$, ${\rm U}$ are the
characterics of operations of taking respectively the vector and
versor of a quaternion. Or, eliminating $\rho$ and $\sigma$, but
retaining the {\it hodographic vector\/} $\tau$, and using
$\Delta$ as the mark of differencing, the conditions of the
question may be included in the following formula, which the
author hopes on a future occasion to develope:
$$\tau = {\textstyle\sum} \int
{ (m + \Delta m) \, d{\rm U} \,
(\int \Delta \tau \, dt) \over
{\rm V} (\Delta \tau \mathbin{.}
\int \Delta \tau \, dt) }.$$
Meanwhile it is conceived that any such attempt as the foregoing,
to simplify or even to transform the important and difficult
problem of investigating the mathematical consequences of the
Newtonian law of attraction, is likely to be received at the
present time with peculiar indulgence and interest, in
consequence not only of the brilliant deductive discovery lately
made of the new planet exterior to Uranus, but also of the
extraordinary and exciting intelligence which has just arrived
from Dorpat of the presumed discovery by Professor M\"{a}dler of
a {\it central cluster\/} (the Pleiades), and of a {\it central
sun\/} (Alcinoe, also called Eta Tauri): around which cluster,
and which sun or star, it is believed by M\"{a}dler that our own
sun and all the other stars of our sidereal system, including the
milky way, but exclusive of the more distant nebul{\ae}, are
moving in enormous orbits, under the combined influences of their
own mutual attractions, all regulated by the same great law.
\bye