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% David R. Wilkins
% School of Mathematics, Trinity College, Dublin 2, Ireland
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% Trinity College, 1st June 1999.
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\centerline{\Largebf ON THE UNDULATORY TIME OF PASSAGE}
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\centerline{\Largebf OF LIGHT THROUGH A PRISM}
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\centerline{\Largebf By}
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\centerline{\Largebf William Rowan Hamilton}
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\centerline{\largerm (Philosophical Magazine, 2 (1833), pp.\ 284--287.)}
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\centerline{\largerm Edited by David R. Wilkins}
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\centerline{\largerm 1999}
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{\largeit\noindent
On the undulatory Time of Passage of Light through a Prism. By\/}
\hskip 0pt plus10pt minus0pt
{\largesc William R. Hamilton,} {\largeit Esq.\
Andrews' Professor of Astronomy in the University of Dublin, and
Royal Astronomer of Ireland\/}\footnote*{Communicated by the
Author.}.
\bigbreak
\centerline{[{\it The London and Edinburgh Philosophical Magazine
and Journal of Science},}
\centerline{vol.~ii (1833), pp.\ 284--287.]}
\bigbreak
Since I communicated my little paper, On the Effect of
Aberration in prismatic Interference, I have seen Professor
Airy's remarks on Mr.~Potter's experiment; in which it is
suggested, that the observed central points, which tended towards
the thickness of the prism, were not the points of simultaneous
arrival of two homogenenous streams. From the well-known
experience and skill of Professor Airy as an observer, I think it
likely that he has assigned the true physical explanation of
Mr.~Potter's instructive experiment; though I wish that this
experiment were repeated, with careful micrometrical measures.
But I continue to think the mathematical correction just, which I
proposed in my recent paper. In that paper, I took Mr.~Potter's
own account of his experiment; namely, that he had found, in the
plane perpendicular to the edge, a tendency {\it towards\/} the
thickness, and {\it from\/} a certain intermedial line, in the
locus of points of simultaneous arrival to two near homogeneous
streams: and I endeavoured to show, that according to the
undulatory theory, this locus {\it ought}, during a considerable
range, to tend in this direction and not in the opposite;---a
mathematical result, which was contrary to Mr.~Potter's
conclusion. It is, I hope, unnecessary to repeat the expression
of my sincere respect for the gentleman from whom I have found
myself obliged to differ on this mathematical question. But as I
only stated, in my former paper, a correction of Mr.~Potter's
formula for the difference of times of arrival of two homogeneous
streams, arising from the prismatic aberration of figure, and
showed the influence of this aberrational correction on the
course of the sought locus, without showing how I obtained the
correction itself,---it may be useful to give here an outline of
the method which I employ, for the treatment of this, and of
other similar questions; referring, for more full details, to the
recent and forthcoming volumes of the Transactions of the
Royal Irish Academy.
Let light be supposed to go, in a bent path $ABCD$, from an
initial point~$A$ to a final point~$D$, through any prism
ordinary or extraordinary, undergoing a first refraction at the
point of entrance~$B$, and a second refraction at the point of
emergence~$C$, the prism being placed {\it in vacuo}, and its
angle being small or large; and let the position of the final
point~$D$ be marked by three rectangular coordinates
$x$,~$y$,~$z$, of which the origin is taken on the edge of the
prism; and let the position of the initial point~$A$ be marked by
three other rectangular coordinates $x'$,~$y'$,~$z'$, having the
same origin, but not necessarily the same axes: let
$\alpha$,~$\beta$,~$\gamma$, be the cosines of the angles which
the emergent or final direction $CD$ makes with the rectangular
axes of $x$,~$y$,~$z$; and let $\alpha'$,~$\beta'$,~$\gamma'$, be
the cosines of the angles which the incident or initial direction
$AB$ makes with the rectangular axes of $x'$,~$y'$,~$z'$;
finally, let $V$ be the undulatory time of propagation from the
initial to the final point, measured by the equivalent path
{\it in vacuo\/}; and let it be considered as a function of the
initial and final coordinates, which, by the position that we
have assigned to the origin, is homogeneous of the first
dimension. We shall then have the two following equations,
deduced from my general methods,
$$V = \alpha x + \beta y + \gamma z
- \alpha' x' - \beta' y' - \gamma' z',
\eqno {\rm (1)}$$
$$0 = x \, \delta \alpha + y \, \delta \beta + z \, \delta \gamma
- x' \, \delta \alpha' - y' \, \delta \beta' - z' \, \delta \gamma';
\eqno {\rm (2)}$$
that is, $V$ is to be determined as a function of the extreme
coordinates $x$~$y$~$z$ $x'$~$y'$~$z'$, which I have called
in my Theory of Systems of Rays the {\it Characteristic
Function}, by the condition that it shall be a maximum or minimum,
with respect to the quantities
$\alpha$,~$\beta$,~$\gamma$, $\alpha'$,~$\beta'$,~$\gamma'$,
of the expression (1): attending to the two general relations,
$$\alpha^2 + \beta^2 + \gamma^2 = 1,\quad
\alpha'^2 + \beta'^2 + \gamma'^2 = 1,
\eqno {\rm (3)}$$
and to two other relations between the final and initial cosines
of direction
$\alpha$~$\beta$~$\gamma$ $\alpha'$~$\beta'$~$\gamma'$,
which result, in each particular case, from the prismatic
connexion between the incident and emergent directions. And when
the form of the {\it characteristic function\/}~$V$ is known, the
six extreme cosines of direction may be deduced from it, by
differentiation, as follows:
$$\left.
\eqalign{
\alpha &= {\delta V \over \delta x},\cr
- \alpha' &= {\delta V \over \delta x'},\cr}
\quad \eqalign{
\beta &= {\delta V \over \delta y},\cr
- \beta' &= {\delta V \over \delta y'},\cr}
\quad \eqalign{
\gamma &= {\delta V \over \delta z},\cr
- \gamma' &= {\delta V \over \delta z'},\cr}
\right\}
\eqno {\rm (4)}$$
When the prism is ordinary, such as glass, or when being
extraordinary its edge is an axis of elasticity; and when we
take the edge for the axis of $z$ and of $z'$, and consider only
rays in a plane perpendicular to this edge, we may make,
$$\left. \eqalign{
&z = 0,\quad z' = 0,\quad \gamma = 0,\quad \gamma' = 0,\cr
&\alpha = \cos \theta,\quad
\beta = \sin \theta,\quad
\alpha' = \cos \theta',\quad
\beta' = \sin \theta',}
\right\}
\eqno {\rm (5)}$$
$\theta$ being the emergent inclination to the axis of $x$, and
$\theta'$ being the incident inclination to the axis of $x'$; and
the undulatory time~$V$, corresponding to any given coordinates
$x$~$y$~$x'$~$y'$, is the maximum or minimum, relatively to
$\theta$, of the expression
$$V = x \cos \theta + y \sin \theta
- x' \cos \theta' - y' \sin \theta',
\eqno {\rm (6)}$$
in which $\theta'$ is to be considered as a function of $\theta$,
depending on the prismatic connexion between the initial and final
directions.
For an ordinary prism {\it in vacuo}, having its angle $=
\varpi$, and its index $= \mu$, so that
$$\sin i = \mu \sin {\varpi \over 2},
\eqno {\rm (7)}$$
$i$ being the angle of external incidence corresponding to the
minimum of deviation, the relation between $\theta$, $\theta'$,
is
$$\mu^2 \sin \varpi^2
= \sin (i + \theta)^2 + \sin (i - \theta')^2
+ 2 \cos \varpi \mathbin{.} \sin (i + \theta)
\mathbin{.} \sin (i - \theta'),
\eqno {\rm (8)}$$
if the positive semiaxis of $x$ be an emergent ray of minimum
deviation, and the positive semiaxis of $x'$ the corresponding
incident ray prolonged, while the positive semiaxes of $y$,~$y'$,
lie on the same side of the axes of $x$,~$x'$, as the prism. The
relation (8) may be put under the approximate form,
$$\theta' = \theta - {m \over 4} \mathbin{.} \theta^2,
\eqno {\rm (9)}$$
when the angles $\theta$, $\theta'$, are small, that is, when we
consider rays having nearly the minimum of deviation, $m$ being
the same positive number as in my last paper, namely,
$$m = {\displaystyle
8 \sin \left( i + {\varpi \over 2} \right)
\sin \left( i - {\varpi \over 2} \right)
\over \displaystyle
\sin 2i \mathbin{.}
\left( \cos {\varpi \over 2} \right)^2};
\eqno {\rm (10)}$$
and if, besides, we consider the ordinates $y$, $y'$, as small,
that is, if we suppose the light to pass near the edge of the
prism, and neglect terms of the fourth dimension with respect to
the small quantities $y$, $y'$, $\theta$, we shall have the
undulatory time or characteristic function $V =$ the maximum or
minimum, relatively to $\theta$, of the expression,
$$V = x - x' + (y - y') \theta
- {\textstyle {1 \over 2}}
\left( x - x' - {m \over 2} y' \right) \theta^2
- {m \over 4} x' \theta^2.
\eqno {\rm (11)}$$
In this manner we find, with the same order of approximation,
$$V = x - x'
+ {\textstyle {1 \over 2}} \mathbin{.} {(y - y')^2 \over x - x'}
+ {m \over 4} \mathbin{.}
{(x y' - x' y) (y - y')^2 \over (x - x')^3},
\eqno {\rm (12)}$$
a result which may also be thus expressed:
$$V = {m \over 2} y'
+ \sqrt{ \left( x - x' - {m \over 2} y' \right)^2 + (y - y')^2 }
- {m x' \over 4} \left( {y - y' \over x - x'} \right)^3.
\eqno {\rm (13)}$$
If we neglected the last term of this last expression, it would
give, by (4) and (5), the following formula for the tangent of
the inclination of the emergent ray,
$$\tan \theta = {\beta \over \alpha}
= {y - y' \over \displaystyle x - x' - {m \over 2} y'};
\eqno {\rm (14)}$$
it would therefore imply that all the rays which diverged before
incidence from the luminous point or primary image $x'$,~$y'$,
diverge after emergence from a prismatic focus or secondary
image, having for coordinates,
$$x = x' + {m \over 2} y',\quad y = y':
\eqno {\rm (15)}$$
so that the last term,
$\displaystyle - {m x' \over 4} \left( {y - y' \over x - x'} \right)^3$,
of the expression (13) for the undulatory time~$V$, may be
considered as an aberrational term, arising from and determining
the aberration (of figure, not of colour) of the prism.
Accordingly Mr.~Potter, neglecting this aberration of figure, did
not perceive this term in the expression of the undulatory time,
and was led to the results respecting the locus of points of
simultaneous arrival of two near homogeneous streams, which I
attempted in my last paper to correct. In comparing my present
notation with my former, we are to make $x' = -l$, and
$y' = \pm a$; and we are also to observe that the present origin
of $x$ and $y$ is on the edge of the prism. It seemed useful to
give the present outline of a proof of the results stated in my
former paper; because the methods which I have introduced for
the solution of optical problems differ much from those usually
received; and because it would perhaps be difficult, by those
usual methods, to investigate the influence of the prismatic
aberration of figure, on the undulatory time of propagation of
homogeneous light.
\nobreak\bigskip
Dublin Observatory, March~12, 1833.
\bye