% This paper has been transcribed in Plain TeX by
% 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 RESEARCHES ON THE DYNAMICS OF LIGHT}
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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,
1 (1841), pp.\ 267--270.)}
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\centerline{\largerm Edited by David R. Wilkins}
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\centerline{\largerm 2000}
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\centerline{\largeit Researches on the Dynamics of Light.}
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\centerline{{\largeit By\/}
{\largerm Sir} {\largesc William R. Hamilton.}}
\bigskip
\centerline{Communicated January~14 and February~11, 1839.}
\bigskip
\centerline{[{\it Proceedings of the Royal Irish Academy},
vol.~1 (1841), pp.\ 267--270.]}
\bigskip
The Chair having been taken, {\it pro tempore}, by his Grace
the Archbishop of Dublin, V.~P., the President continued his
account of his researches in the theory of light.
As a specimen of the problems which he had lately considered and
resolved, the following question was stated:---An indefinite
series of equal and equally distant particles,
$\ldots \, m_{-1}, m_0, m_1,\ldots$,
situated in the axis of $x$, at the points
$\ldots \, -1, 0, +1,\ldots$,
being supposed to receive, at the time~$0$, any very small
transversal displacements
$\ldots \, y_{-1,0}, y_{0,0}, y_{1,0},\ldots$,
and any very small transversal velocities
$\ldots \, y_{-1,0}', y_{0,0}', y_{1,0}',\ldots$,
it is required to determine their displacements
$\ldots \, y_{-1,t}, y_{0,t}, y_{1,t},\ldots$
for any other time~$t$; each particle being supposed to attract
the one which immediately precedes or follows it in the series,
with an energy~$= a^2$, and to have no sensible influence on any
of the more distant particles. This problem may be considered as
equivalent to that of integrating generally the equation in mixed
differences,
$$y_{x,y}'' = a^2 ( y_{x+1,t} - 2 y_{x,t} - y_{x-1,t} );
\eqno (1)$$
which may also be thus written:
$$\left( {d \over dt} \right)^2 y_{x,t}
= {(a \Delta_x)^2 \over 1 + \Delta_x} y_{x,t}.
\eqno (1)'$$
The general integral required, may be thus written:
$$y_{x,t}
= \left\{
1 - {a^2 \Delta_x^2 \over 1 + \Delta_x}
\left( \int_0^t dt \right)^2
\right\}^{-1} (y_{x,0} + t y_{x,0}');
\eqno (2)$$
an expression which may be developed into the sum of two series,
as follows,
$$\eqalignno{
y_{x,t}
&= y_{x,0}
+ {a^2 t^2 \over 1 \mathbin{.} 2}
\Delta_x^2 y_{x-1,0}
+ {a^4 t^4 \over 1 \mathbin{.} 2 \mathbin{.} 3
\mathbin{.} 4}
\Delta_x^4 y_{x-2,0} + \hbox{\&c.} \cr
&\mathrel{\phantom{=}}
+ t y_{x,0}'
+ {a^3 t^3 \over 1 \mathbin{.} 2 \mathbin{.} 3}
\Delta_x^2 y_{x-1,0}'
+ {a^5 t^5 \over 1 \mathbin{.} 2 \mathbin{.} 3
\mathbin{.} 4 \mathbin{.} 5}
\Delta_x^4 y_{x-2,0}' + \hbox{\&c.};
&(2)'\cr}$$
and may be put under this other form,
$$\eqalignno{
y_{x,t}
&= {2 \over \pi} \Sigma_{(l) \,}{}_{-\infty}^\infty
y_{x + l,0}
\int_0^{\pi \over 2} d \theta \,
\cos (2l \theta) \cos (2at \sin \theta) \cr
&\mathrel{\phantom{=}}
+ {1 \over a \pi} \Sigma_{(l) \,}{}_{-\infty}^\infty
y_{x + l,0}'
\int_0^{\pi \over 2} d \theta \,
\cos (2l \theta) \mathop{\rm cosec} \theta
\sin (2at \sin \theta);
&(2)''\cr}$$
the first line of (2)${}'$ or (2)${}''$ expressing the effect of
the initial displacements, and the second line expressing the
effect of the initial velocities, for all possible suppositions
respecting these initial data, or for all possible forms of the
two arbitrary functions $y_{x,0}$ and $y_{x,0}'$.
Supposing now that these arbitrary forms or initial conditions
are such, that
$$y_{x,0}
= \eta \mathop{\rm vers} 2x {\pi \over n},
\quad\hbox{and}\quad
y_{x,0}'
= - 2 a \eta \sin {\pi \over n} \sin 2x {\pi \over n},
\eqno (3)$$
for all values of the integer~$x$ between the limits $0$ and
$-in$, $n$ and $i$ being positive and large, but finite integer
numbers, and that for all other values of $x$ the functions
$y_{x,0}$ and $y_{x,0}'$ vanish: which is equivalent to supposing
that at at the origin of $t$, and for a large number~$i$ of
wave-lengths (each $= n$) behind the origin of $x$, the
displacements and velocities of the particles are such as to
agree with the following law of undulatory vibration,
$$y_{x,t}
= \eta \mathop{\rm vers}
\left(
2x {\pi \over n} - 2at \sin {\pi \over n}
\right),
\eqno (3)'$$
but that all the other particles are, at that moment, at rest: it
is required to determine the motion which will ensue, as a
consequence of these initial conditions. The solution is
expressed by the following formula, which is a rigorous deduction
from the equation in mixed differences (1):
$$y_{x,t}
= {\eta \over \pi}
\left( \sin {\pi \over n} \right)^2
\int_0^\pi {\sin i n \theta \over \sin \theta}
{\cos (2x \theta + i n \theta - 2at \sin \theta)
\over \displaystyle
\cos \theta - \cos {\pi \over n}}'
\, d \theta;
\eqno (4)$$
an expression which tends indefinitely to become
$$y_{x,t}
= {\eta \over 2} \mathop{\rm vers}
\left( 2x {\pi \over n} - 2at \sin {\pi \over n} \right)
- {\eta \over 2 \pi} \left( \sin {\pi \over n} \right)^2
\int_0^\pi {\sin (2x \theta - 2at \sin \theta)
\over \displaystyle \sin \theta
\left( \cos \theta - \cos {\pi \over n} \right)}
\, d \theta,
\eqno (4)'$$
as the number~$i$ increases without limit. The approximate
values are discussed, which these rigorous integrals acquire,
when the value of $t$ is large. It is found that a vibration, of
which the phase and the amplitude agree with the law (3)${}'$, is
propagated forward, but not backward, so as to agitate
successively new and more distant particles, (and to leave
successively others at rest, if $i$ be finite,) with a velocity
of progress which is expressed by
$\displaystyle a \cos {\pi \over n}$,
and which is therefore less, by a finite though small amount,
than the velocity of passage
$\displaystyle a {n \over \pi} \sin {\pi \over n}$
of any given phase, from one vibrating particle to another within
that extent of the series which is already fully agitated. In
other words, the communicated vibration does not attain a
sensible amplitude, until a finite interval of time has elapsed
from the moment when one should expect it to begin, judging only
by the law of the propagation of phase through an indefinite
series of particles, which are all in vibration already. A small
disturbance, distinct from the vibration~(3)${}'$, is also
propagated, backward as well as forward, with a velocity $= a$,
independent of the length of the wave. And all these
propagations are accompanied with a small degree of terminal
diffusion, which, after a very long time, renders all the
displacements insensible, if the number~$i$, however large, be
finite, that is, if the vibration be originally limited to any
finite number of particles.
\bye