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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 A MODE OF DEDUCING THE EQUATION OF}
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\centerline{\Largebf FRESNEL'S WAVE}
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\centerline{\Largebf By}
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\centerline{\Largebf William Rowan Hamilton}
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\centerline{\largerm (Philosophical Magazine, 19 (1841), pp.\ 381--383.)}
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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 a Mode of deducing the Equation of\/ {\largerm Fresnel's} Wave. By\/}
\hskip 0pt plus10pt minus0pt
{\largerm Sir} {\largesc William Rowan Hamilton},
{\largeit LL.D., P.R.I.A., Member of several Scientific Societies at
Home and Abroad, Professor of Astronomy in the University of
Dublin, and Royal Astronomer of Ireland\/}\footnote*{Communicated
by the Author.}.
\bigbreak
\centerline{[{\it The London, Edinburgh and Dublin Philosophical Magazine
and Journal of Science},}
\centerline{vol.~xix (1841), pp. 381--383.]}
\bigbreak
The following does not pretend to be the {\it best}, but merely
to be {\it one\/} way, of deducing the known equation of
Fresnel's wave, from a known geometrical construction. It
requires only the first principles of the application of algebra
to the geometry of three dimensions, and does not introduce any
of the geometrical properties of the auxiliary ellipsoid
employed, except those which are immediately suggested by the
equation of that ellipsoid. It has, therefore, in the
algebraical point of view, a certain degree of directness,
although it might be rendered easier and shorter by borrowing
more largely from geometry.
\bigbreak
1.
The known construction referred to is thus enunciated by Sir John
Herschel, in his Treatise on Light, {\it Encyclop{\ae}dia
Metropolitana}, article 1017. ``M.~Fresnel gives the following
simple construction for the curve surface bounding the wave in
the case of unequal axes, which establishes an immediate relation
between the length and direction of its radii. Conceive an
ellipsoid having the same semiaxes $a$,~$b$,~$c$; and having cut
it by any diametral plane, draw perpendicular to this plane from
the centre two lines, one equal to the greatest, and the other to
the least radius vector of the section. The loci of the
extremities of these perpendiculars will be the surfaces of the
ordinary and extraordinary waves.''
\bigbreak
2.
The coordinates of the wave being $x$~$y$~$z$, and those of the
ellipsoid $X$~$Y$~$Z$, we have the six equations,
$${X^2 \over a^2} + {Y^2 \over b^2} + {Z^2 \over c^2}
= 1,
\eqno {\rm (1.)}$$
$${X \, dX \over a^2} + {Y \, dY \over b^2} + {Z \, dZ \over c^2}
= 0,
\eqno {\rm (2.)}$$
$$x X + y Y + z Z = 0,
\eqno {\rm (3.)}$$
$$x \, dX + y \, dY + z \, dZ = 0,
\eqno {\rm (4.)}$$
$$X \, dX + Y \, dY + + Z \, dZ = 0,
\eqno {\rm (5.)}$$
$$x^2 + y^2 + z^2 = X^2 + Y^2 + Z^2;
\eqno {\rm (6.)}$$
between which we are to eliminate $X$, $Y$, $Z$, and the ratios
of their differentials.
\bigbreak
3.
The equations (1.) and (2.) are satisfied by assuming
$$X = a \sin \theta \cos \phi,\quad
Y = b \sin \theta \sin \phi,\quad
Z = c \cos \theta;
\eqno {\rm (7.)}$$
and then the equation (3.) gives
$$\tan \theta = {-cz \over ax \cos \phi + by \sin \phi};
\eqno {\rm (8.)}$$
while the comparison of the two values of
$\displaystyle \tan \theta {d\phi \over d\theta}$,
deduced from (4.) and (5.), gives
$${ax \cos \phi + by \sin \phi - cz \tan \theta
\over ax \sin \phi - by \cos \phi}
= {a^2 \cos^2 \phi^2 + b^2 \sin \phi^2 - c^2
\over (a^2 - b^2) \sin \phi \cos \phi};
\eqno {\rm (9.)}$$
and the equation (6.) becomes
$$(x^2 + y^2 + z^2) (1 + \tan \theta^2)
= (a^2 \cos \phi^2 + b^2 \sin \phi^2) \tan \theta^2 + c^2.
\eqno {\rm (10.)}$$
It remains therefore to eliminate $\theta$ and $\phi$ between the
three equations (8.) (9.) (10.).
\bigbreak
4.
Substituting for $\tan \theta$, in (9.) and (10.), its value
given by (8.), we can easily obtain
$$A \tan \phi + B \mathop{\rm cotan} \phi = C;
\eqno {\rm (I.)}$$
$$A' \tan \phi + B' \mathop{\rm cotan} \phi = C';
\eqno {\rm (II.)}$$
if we put for abridgement
$$\eqalign{
A &= (c^2 - b^2) abxy;\cr
B &= (a^2 - c^2) abxy;\cr
C &= (b^2 - c^2) a^2 x^2 + (c^2 - a^2) b^2 y^2 + (b^2 - a^2) c^2 z^2;\cr
r^2 &= x^2 + y^2 + z^2;\cr
A' &= r^2 (b^2 y^2 + c^2 z^2) - c^2 b^2 (y^2 + z^2);\cr
B' &= r^2 (a^2 x^2 + c^2 z^2) - c^2 a^2 (x^2 + z^2);\cr
C' &= - 2 (r^2 - c^2) abxy.\cr}$$
And eliminating $\phi$ between the equations (I.) and (II.), we
find
$$(AB' - A'B)^2 + (AC' - A'C) (BC' - B'C) = 0;
\eqno {\rm (III.)}$$
a form for the equation of the wave, which we have now only to
develope and depress.
\bigbreak
5.
Expanding it first under the form
$$W_8 + W_{10} + W_{12} = 0,$$
in which $W_8$, $W_{10}$, $W_{12}$ are, respectively, homogeneous
functions of $x$,~$y$,~$z$, of the 8th, 10th, and 12th
dimensions, we soon discover that these three functions have a
common factor, of the 8th dimension, namely, $c^2 z^2 r^2 R$, in
which
$$R^2 = C^2 + 4 a^2 b^2 (c^2 - a^2) (c^2 - b^2) x^2 y^2,$$
$C$ having the same meaning as in (I.), so that
$$R^2 > 0,\quad \hbox{if } c^2 > b^2 > a^2,\quad \hbox{or if }
c^2 < b^2 < a^2,$$
conditions which we may suppose to be satisfied. And rejecting,
as evidently foreign to the question, this common factor
$c^2 z^2 r^2 R$, the known equation of the wave results, under
the form
$$u_0 + u_2 + u_4 = 0,
\eqno {\rm (IV.)}$$
in which
$$\eqalign{
u_0 &= a^2 b^2 c^2,\cr
u_2 &= - \{ a^2 (b^2 + c^2) x^2 + b^2 (c^2 + a^2) y^2
+ c^2 (a^2 + b^2) z^2 \},\cr
u_4 &= (x^2 + y^2 + z^2) (a^2 x^2 + b^2 y^2 + c^2 z^2).\cr}$$
\bigbreak
6.
The foregoing investigation is taken from a manuscript Report
which I had the honour of drawing up in July~1830, when, in
conjunction with the late and present Provosts of Trinity
College, Dublin, I was appointed to examine the first
communication of Professor MacCullagh to the Royal Irish Academy,
since published in the second part of the sixteenth volume of the
Transactions of that body. A far more concise and
elegant deduction of the same known equation of the wave from the
same geometrical construction, depending, however, a little more
on the geometrical properties of the ellipsoid, has since been
communicated by Professor MacCullagh himself, and is published in
the second part of the seventeenth volume of the
{\it Transactions\/} of the same Academy. Others have published
other demonstrations.
My own mode of deducing the equation of the wave from the
principles of Fresnel, without any reference to the ellipsoid
above referred to, may be seen in the `Third Supplement' to my
Theory of Systems of Rays contained in the first part of the
last-mentioned volume.
\nobreak\bigskip
Observatory of Trinity College, Dublin,
\qquad October~13, 1841.
\bigbreak
P.S. Since writing out and sending off the foregoing paper, I
have had opportunity to refer to Fresnel's own deduction of the
same equation of his wave from the same geometrical construction,
entitled ``Calcul tr\`{e}s simple qui conduit de l'\'{e}quation
d'un ellipsoide \`{a} celle de la surface des ondes.''
({\it M\'{e}m.\ de l'Acad.\ des Sci.\ de l'Inst.\ Royale de
France}, tom.~vii., page~137.) It is much simpler than
mine, and nearly coincident with that of Professor MacCullagh,
but seems to have been overlooked by both of us.
\bye