% This paper has been transcribed in Plain TeX by
% David R. Wilkins
% School of Mathematics, Trinity College, Dublin 2, Ireland
% (dwilkins@maths.tcd.ie)
%
% Trinity College, 2000.

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\centerline{\Largebf ON THE PROPAGATION OF LIGHT}

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\centerline{\Largebf IN VACUO AND IN CRYSTALS}

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\centerline{\Largebf By}

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\centerline{\Largebf William Rowan Hamilton}

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\centerline{\largerm (British Association Report, 1838,
   Part~II, pp.\ 2--6.)}

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\vfill

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

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\centerline{\largerm 2000}

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{\largeit\noindent
On the Propagation of Light\/} {\largerm in vacuo.}
{\largeit By Professor\/}
{\largerm Sir} {\largesc W.~R. Hamilton}.

\bigbreak

\centerline{[{\it Report of the Eighth Meeting of the British
Association for the Advancement of}}
\centerline{{\it Science; held at Newcastle in August 1838}.}
\centerline{(John Murray, London, 1839), Part~II, pp.\ 2--6.]}

\bigbreak

The object of this communication was to advance the state of our
knowledge respecting the law which regulates the attractions or
repulsions of the particles of the ether on each other.  The
general differential equations of motion of any system of
attracting or repelling points being reducible to the form
$${d^2 x \over dt^2} = {\rm S} \mathbin{.} m_\prime \, \Delta x f(r),
   \eqno (1.)$$
the equations of minute vibration are of the form
$${d^2 \delta x \over dt^2}
   =  {\rm S} \mathbin{.} m_\prime
      \left(
         \Delta \delta x \mathbin{.} f(r)
       + \Delta x \mathbin{.} \delta f(r)
      \right),
   \eqno (2.)$$
in which
$$\delta f(r) = f'(r) \, \delta r,
   \eqno (3.)$$
and
$$\delta r
   =  {\Delta x \over r} \, \Delta \delta x
    + {\Delta y \over r} \, \Delta \delta y
    + {\Delta z \over r} \, \Delta \delta z.
   \eqno (4.)$$
A mode of satisfying the differential equations (2), and at the
same time of representing a large class of the phenomena of
light, is to assume
$$    {\delta x \over \xi^\backprime}
   =  {\delta y \over \eta^\backprime}
   =  {\delta z \over \zeta^\backprime}
   =  \hbox{const.}
    + \cos {2\pi (vy - ax - by -cz) \over \lambda},
   \eqno (5.)$$
in which $\xi^\backprime$, $\eta^\backprime$, $\zeta^\backprime$
are constants, depending on the extent and direction of
vibration: $a$, $b$, $c$, are the cosines of the inclinations of
the direction of propagation of a plane wave to the positive
semi-axes of $x$,~$y$,~$z$; $v$ is the velocity of propagation of
that wave, and $\lambda$ is the length of an undulation; and
$\pi$ is the semicircumference of a circle, of which the radius
is unity. With this assumption (5.), and with a natural and
obvious supposition respecting a certain symmetry of arrangement
in the ether, causing the sums of odd powers to vanish, it is
permitted to substitute in (2.) the expressions
$${d^2 \delta x \over \delta t^2}
   =  - \left( {2 \pi v \over \lambda} \right)^2
      \mathbin{.} \delta x,
   \eqno (6.)$$
$$\Delta \delta x
   = - \vers \Delta \theta \mathbin{.} \delta x,
   \eqno (7.)$$
in which
$$\Delta \theta
  =  - {2\pi \over \lambda}
         ( a \, \Delta x + b \, \Delta y + c \, \Delta z);
   \eqno (8.)$$
and thus arises a system of conditions of the form
$$\eqalignno{
\xi^\backprime  \left( {2\pi v \over \lambda} \right)^2
   &= \xi^\backprime m {\rm S} \mathbin{.}
         \left\{ f(r) + {\Delta x^2 \over r} f'(r) \right\}
         \vers \Delta \theta \cr
   &\mathrel{\phantom{=}}
    + \eta^\backprime m {\rm S} \mathbin{.}
         {\Delta x \, \Delta y \over r} f'(r)
         \vers \Delta \theta \cr
   &\mathrel{\phantom{=}}
    + \zeta^\backprime m {\rm S} \mathbin{.}
         {\Delta x \, \Delta z \over r} f'(r)
         \vers \Delta \theta
   &(9.)\cr}$$
the masses $m^\backprime$ of the etherial particles, being
supposed each $= m$.  Three conditions of this form (9.) exist
for every particle, and determine, in general, for any given
values of $a$, $b$, $c$, $\lambda$, that is, for any given
direction of propagation, and any given length of wave, the value
of $v$, and the ratios of $\xi^\backprime$, $\eta^\backprime$,
$\zeta^\backprime$, that is, the velocity of propagation of the
wave and the direction of vibration of the particle. Accordingly,
with some slight differences of notation, they have been proposed
for this purpose by Cauchy, and adopted by other mathematicians.
Suppose now, for simplicity, that the plane wave is vertical, so
that $c = 0$, and let, at first, the direction of its propagation
coincide with the positive semi-axis of $x$, so that $b$ also
vanishes, and $a$ is $=1$.  Then, for transversal vibrations, the
expression for the square of the velocity of propagation is
$$v^2
   =  \left( {\lambda \over 2\pi} \right)^2 m {\rm S}
         \left\{ f(r) + {r^2 - \Delta x^2 \over 2r} f'(r) \right\}
         \vers {2\pi \, \Delta x \over \lambda};
   \eqno (10.)$$
which appears to extend not only to interplanetary spaces, but
also to all ordinary transparent media, and contains, for them,
the theoretical law of dispersion, which was first discovered by
Cauchy, namely, the expression
$$v^2 = A_0 + A_1 \lambda^{-2} + A_2 \lambda^{-4}
         \enspace \hbox{\&c.}
   \eqno (11.)$$
in which
$$A_i
   =  {(2\pi)^{2i} m
         \over 1 \mathbin{.} 2 \mathbin{.} 3 \mathbin{.} 4
         \, \ldots \, (2i + 2)}
      {\rm S}
         \left\{ f(r) + {r^2 - \Delta x^2 \over 2r} f'(r) \right\}
         \Delta x^{2i+2}.
   \eqno (12.)$$

But, in order that this law may agree with the phenomena, it is
essential that the series (11.) should be convergent, even in its
earliest terms; and this consideration enables us to exclude the
supposition which has occurred to some mathematicians, that the
particles of the ether attract each other with forces which are
inversely as the squares of the distances between them.  For if
we suppose $r f(r) = r^{-2}$, and therefore
$f(r) =r^{-3}$, $f'(r) = - 3 r^{-4}$, we shall have
$$A_i
   =  {\textstyle {1 \over 2}}
      {(2\pi)^{2i} m
         \over 1 \mathbin{.} 2 \mathbin{.} 3 \mathbin{.} 4
         \, \ldots \, (2i + 2)}
      {\rm S}
         \left\{ - r^{-3} + 3 r^{-5} \Delta x^2 \right\}
         \Delta x^{2i+2};
   \eqno (13.)$$
and by extending the summation to particles, distant by several
times the length of an undulation from the particle which they
are supposed to attract, these sums (13.) become extremely large
and the terms of the series (11.) diverge very rapidly at first,
though they always finish by converging.  In fact, if we conceive
a sphere, whose radius $= n \lambda = n$ times the length of an
undulation ($n$ being a very large multiplier), and whose centre
is the attracted particle; and if we consider only the combined
effect of the actions of all the particles within this sphere, we
may, as a good approximation, convert each sum (13.) into a
triple definite integral, and thus obtain, for the general term
of the series (11.), the expression
$$(-1)^i A_i \lambda^{-2i}
   =  {(-1)^i \, 4 \pi m n^2 \lambda^2 \over (2i + 5) \epsilon^3}
         \mathbin{.}
      {(2\pi n)^{2i}
         \over 1 \mathbin{.} 2 \mathbin{.} 3 \, \ldots \, (2i + 3)},
   \eqno (14.)$$
$\epsilon$ being the mean interval between any two adjacent
particles of the ether, so that the number of such particles
contained in any sphere of radius~$r$, is nearly
$\displaystyle = {4 \pi r^3  \over 3 \epsilon^3}$,
if $r$ be a large multiple of $\epsilon$.  And hence we find, by
taking the sum of all these terms (14.), the expression
$$v^2
   =  {\lambda^2 m \over \pi \epsilon^3}
      \left\{
         {\textstyle {1 \over 3}}
       + {\cos 2 \pi n \over (2\pi n)^2}
       - {\sin 2 \pi n \over (2\pi n)^3};
      \right\};
   \eqno (15.)$$
so that, by taking the limit to which $v^2$ tends, when $n$ is
taken greater and greater, we get at last as a near approximation
$$v^2 = {\lambda^2 m \over 3 \pi \epsilon^3},
   \eqno (16.)$$
and
$${\lambda \over v}
   =  \sqrt{ {3 \pi \epsilon^3 \over m } }.
   \eqno (17.)$$
But
$\displaystyle {\lambda \over v}$
expresses the time of oscillation of any one vibrating particle;
this time would therefore be nearly constant, if the particles
attracted each other according to the law of the inverse square
of the distance; and consequently this law is inadmissible, as
being incompatible with the law of dispersion.  It had appeared
to Sir William Hamilton important to reproduce these results,
though he remarked that they agree substantially with those of
Cauchy, because the law of the inverse square was one which
naturally offered itself to the mind, and had, in fact, been
proposed by at least one mathematician of high talent. There was,
however, another law which had great claims on the attention of
mathematicians, as having been proposed by Cauchy to represent
the phenomena of the propagation of light {\it in vacuo}, namely
the law of a repulsive action, proportional inversely to the
fourth power, or to the square of the square of the distance.
M.~Cauchy had, indeed, supposed that this law might hold good
only for small distances, but in examining into its
admissibility, it appeared fair to treat it as extending to all
the neighbouring particles which act on any one.  But against
this law also, Sir William Hamilton brought forward objections,
which were founded partly on algebraical, and partly on numerical
calculations, and which appeared to him decisive.

The spirit of these objections consisted in showing that the law
in question would give too great a preponderance to the effect of
the immediately adjacent particles, and would thereby produce
irregularities which are not observed to exist.  In particular,
if it be supposed that
$$\eqalign{
{\rm S} \mathbin{.} r^i \, \Delta x^2
   &= {\rm S} \mathbin{.} r^i \, \Delta y^2
    = {\rm S} \mathbin{.} r^i \, \Delta z^2,\cr
{\rm S} \mathbin{.} r^i \, \Delta x^4
   &= {\rm S} \mathbin{.} r^i \, \Delta y^4
    = {\rm S} \mathbin{.} r^i \, \Delta z^4,\cr
{\rm S} \mathbin{.} r^i \, \Delta x^2 \, \Delta y^2
   &= {\rm S} \mathbin{.} r^i \, \Delta y^2 \, \Delta z^2
    = {\rm S} \mathbin{.} r^i \, \Delta z^2 \, \Delta x^2,\cr}$$
and also, in (5.), that $c = 0$, $a = b$, and that $\lambda$ is
much greater than $\epsilon$, it is found that the two values
$v^2$ and $v_\prime^2$ of the square of the velocity~$v$,
corresponding to vertical and to horizontal but transversal
vibrations, are connected by the relation
$$v_\prime^2 = - {2 \over 3} v^2,$$
being expressed as follows:
$$\eqalign{
v_\prime^2
   &= {m \over 4} {\rm S}
      \left( 5 r^{-7} \, \Delta x^4 - r^{-3} \right),\cr
v^2
   &= {3m \over 8} {\rm S}
      \left( r^{-3} - 5 r^{-7} \, \Delta x^4 \right);\cr}$$

In conclusion, he offered reasons for believing that the law of
action of the particles of the ether on each other resembles more
the law which Poisson has in one of his memoirs proposed as
likely to express the mutual action of the particles of ordinary
and solid bodies, being perhaps of some such form as the
following:---
$$r f(r)
   =  - a \mathbin{.}
            b^{- \left( {r \over g \epsilon} \right)^h}
      + a_\prime \mathbin{.}
            b_\prime^{- \left( {r \over g_\prime \epsilon} \right)^h_\prime};
   \eqno (18.)$$
$b$ and $b_\prime$ being each greater than unity, and
$g$, $g_\prime$, $h$, $h_\prime$ being some large positive
numbers, while $a$ and $a_\prime$ are constant and positive
multipliers, and $\epsilon$ is, as before, the mean or average
interval between two adjacent particles.  With such a law there
would be a nearly constant repulsion, if $a$ be greater than
$a_\prime$, and if $g$ be less than $g_\prime$, as long as
$\displaystyle {r \over g \epsilon}$
is sensibly less than unity; but the force would rapidly change,
as the distance~$r$ approached to $g \epsilon$, and would then
become a nearly constant attraction, until $r$ became nearly $=
g_\prime \epsilon$; it would then diminish rapidly, and soon
become insensible.  Sir William Hamilton did not, however, intend
to exclude the hypothesis, that the function $r f(r)$ may contain
several alternations of such repulsive and attractive
terms,---much less did he deny that at great distances it may
reduce itself to the law of the inverse square.

\vfill\eject

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{\largeit\noindent
On the Propagation of Light in Crystals.  By Professor\/}
{\largerm Sir} {\largesc W.~R. Hamilton}.

\bigbreak

\centerline{[{\it Report of the Eighth Meeting of the British
Association for the Advancement of}}
\centerline{{\it Science; held at Newcastle in August 1838}.}
\centerline{(John Murray, London, 1839), Part~II, p.~6.]}

\bigbreak

By continuing to modify the analysis of M.~Cauchy in the manner
already explained, he had succeeded in deducing, more
satisfactorily than had in his opinion been done before, from
dynamical principles, a large and important class of the
phenomena of light in crystals; though much still remained to be
done before it could be said that a perfect theory of light was
obtained. He had employed, for the purposes of calculation, the
supposition that the arrangement of the particles of the ether in
a crystal differs from an exactly cubical arrangement only by
very small displacements, caused by the action of the particles
of the crystalline body; and had attended only to those indirect
or reflex effects of the latter particles which are owing to the
disturbances which they produce in the arrangement of the former
particles: but he did not mean to assert that he had established
any strong physical probability for this being the true
{\it modus operandi\/} in crystals, though he thought the
hypothesis had explained so much already that it deserved to be
still further developed.

\bye
