% 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, 1st June 1999.

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\centerline{\Largebf ON THE EFFECT OF ABERRATION IN}

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\centerline{\Largebf PRISMATIC INTERFERENCE}

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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.\ 191--194.)}

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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 Effect of Aberration in prismatic Interference.  By\/}
\hskip 0pt plus10pt minus0pt
{\largesc William R. Hamilton,} {\largeit Esq.\
\hskip 0pt plus10pt minus0pt
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.\ 191--194.]}

\bigbreak

The experiments and reasonings of Mr.~Potter respecting the
ph{\ae}nomena of prismatic interference, published in the last
Number of the London and Edinburgh Philosophical Magazine
(for February 1833), deserve attention; for, if correct, they
would furnish a formidable and, perhaps, fatal objection against
the undulatory theory of light.  I have not repeated the
experiments, but have endeavoured to examine the mathematical
part of the question, and have obtained results which differ from
the mathematical results of Mr.~Potter, and which appear to show
that the ph{\ae}nomena described by him are consistent with the
undulatory theory.  It may, therefore, be useful to state briefly
some of my results, in a form adapted for comparison with those
of which they profess to be corrections.  In stating them, it
cannot be supposed that I intend any personal attack on
Mr.~Potter, for whose talents and industry I feel a sincere
respect.

Mr.~Potter believes it to be a mathematical consequence of the
undulatory theory of light, that when rays, in a plane
perpendicular to the edge of a prism of glass, diverge from a
luminous point {\it in vacuo}, and emerge from the prism after
refraction, into a vacuum again, the locus of the points
simultaneously attained by the emergent light is a
circle;---either rigorously, or at least with an accuracy
sufficient for the investigation of the positions of the central
points of interference of two emergent streams of homogeneous
light, which had set out together from two near luminous origins,
namely, from the images of a luminous point formed by two plane
mirrors inclined at a small angle to each other:---from which he
concludes that these central points of interference, in a given
plane perpendicular to the edge, are situated on a certain
hyperbola, tending {\it towards the angle of the prism}, whereas
he found by experiment a tendency {\it from that angle}.  I find,
however, that in consequence of the {\it prismatic aberration\/}
(which is greater than the aberration of a lens), the section of
an emergent wave differs sensibly from the circular form, and the
time of arrival of the light at any proposed point of
interference requires a sensible correction; by allowing for
which I find, as the locus of the points of central interference
in the plane perpendicular to the edge, a curve not hyperbolic,
and {\it not tending\/} towards {\it but\/} from {\it the angle
of the prism\/}: so that the ph{\ae}nomenon observed by
Mr.~Potter is a consequence of the undulatory theory.

To simplify the question I shall suppose, with him, that the line
joining the two near luminous origins is perpendicularly bisected
by a line which, if considered as an incident ray, would undergo
the minimum of deviation, and would emerge in a certain
direction, which I shall take, as he does, for the axis of $x$;
supposing also, with him, that this emergent line passes through,
or very near the edge, and measuring the positive ordinates~$y$
towards the thickness of the prism, while the positive
absciss{\ae}~$x$ are measured from the incident towards the
emergent light.  The problem is then to find, at least
approximately, the equation in $x$,~$y$, of the locus of points
of central interference, or of simultaneous arrival of the light
from the two luminous origins, with the undulatory law of
velocity; and, in particular, to examine whether this locus tends
{\it to\/} or {\it from\/} the angle of the prism, by examining
whether the ordinate~$y$ {\it decreases\/} or {\it increases},
while the abscissa~$x$ increases from its value at the prism.

Denoting, as Mr.~Potter does, the coordinates of the prismatic
focus or image corresponding to one luminous origin by the values
$$x = ma,\quad y = a,$$
and those of the prismatic image of the other luminous origin by
$$x = 0,\quad y = - a,$$
in which $a$ is half the interval between the two near luminous
origins, and $m$ is a positive number depending on the angle and
index of the prism, Mr.~Potter finds for the difference of times
of arrival of the two streams of emergent light at any point
$x$,~$y$, not far from the axis of $x$, the expression
$$\sqrt{ x^2 + (y + a)^2 }
      - \sqrt{ (x - ma)^2 + (y - a)^2 } - ma,
   \eqno {\rm (1)}$$
and equating this expression to zero, he finds for the locus of
the points of central interference, the equation of a common
hyperbola, which may be put under the following approximate form,
$$y = {ma^2 \over 4x}.
   \eqno {\rm (2)}$$
If then this analysis were sufficient, it would show, as
Mr.~Potter has concluded, that $y$ {\it decreases}, and that the
locus tends {\it towards the angle\/} of the prism; whereas the
experiment showed a contrary tendency.

But I find, that on account of the prismatic aberration, the
expression (1) for the difference of times of arrival, requires
this correction, namely,
$${ml \over 4} \left( {y + a \over x} \right)^3
      - {ml \over 4} \left( {y - a \over x} \right)^3,
   \eqno {\rm (3)}$$
in which $l$ is a positive quantity, namely, the length of the
path traversed by the light in arriving at the edge of the prism;
and after allowing for this correction (3), the equation of the
sought locus, of the points of central interference, gives the
following approximate expression for the ordinate~$y$,
$$y = {ma^2 \over 4x} - {ma^2 l \over 4x^2},
   \eqno {\rm (4)}$$
the second term being introduced by aberration, but being of the
same order as the first.  And taking account of this new term in
the expression of the ordinate, we have, by differentiation,
$${dy \over dx} = - {ma^2 \over 4x} + {ma^2 l \over 2 x^3}
   = {ma^2 (2l - x) \over 4 x^3},
   \eqno {\rm (5)}$$
so that while $x$ increases from its value~$l$ at the prism to
the value $2l$, the {\it ordinate~$y$ increases\/} from $0$ to
$\displaystyle {ma^2 \over 16 \mathbin{.} l}$,
and {\it the curve tends towards the thickness of the prism}, as
it was found in the experiment to do.  Indeed, when $x$ increases
still further, that is, when the eye is withdrawn from the prism
to a distance greater than the length of the incident path, that
is, greater than the distance of the prism from the two near
luminous origins, the curve begins to tend the other way, though
much more slowly; but the experiments of Mr.~Potter do not seem
to have been made at so great a distance from the prism, and
therefore the ph{\ae}nomenon, which he observed, appears to be
explained by the undulatory theory.

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Dublin Observatory, Feb.~12, 1833.

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