% 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, 1st June 1999.
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\centerline{\Largebf ON SOME QUATERNION EQUATIONS}
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\centerline{\Largebf CONNECTED WITH FRESNEL'S WAVE}
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\centerline{\Largebf SURFACE FOR BIAXAL CRYSTALS}
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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,
7 (1862), pp.\ 122--124, 163.)}
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\centerline{\largerm Edited by David R. Wilkins}
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\centerline{\largerm 1999}
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\centerline{ON SOME QUATERNION EQUATIONS CONNECTED WITH}
\centerline{FRESNEL'S WAVE SURFACE FOR BIAXAL CRYSTALS.}
\vskip 12pt
\centerline{Sir William Rowan Hamilton.}
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\centerline{Read February 28th, 1859 and May 9th, 1859.}
\vskip12pt
\centerline{[{\it Proceedings of the Royal Irish Academy},
vol.~vii (1862), pp.\ 122--124, 163.]}
\bigskip
\centerline{[{\sc Monday, February 28, 1859.}]}
\nobreak\bigskip
1.
The ellipsoid of which the three semi-axes are usually denoted as
$a$, $b$, $c$, in statements of the Fresnelian theory of the
wave-surface in a biaxal crystal, being here represented by the
equation,
$$S \rho \phi \rho = 1,$$
where the vector function~$\phi$ has the distributive and other
properties described by Sir W.~R.~H., in his Seventh Lecture on
Quaternions, it follows from the physical principles, or
hypotheses, of Fresnel, that a small displacement, $\delta \rho$,
of a molecule of the ether in a crystal, gives rise to an elastic
force, which may be denoted by $\phi^{-1} \, \delta \rho$. But
if this displacement, $\delta \rho$, be (as is assumed)
tangential to a wave-front in the medium, to which the
vector~$\mu$ is normal, and of which the tensor $T \mu$ denotes
the slowness of propagation, so that $\mu$ may be called the
{\sc Index-Vector}, then the tangential component of the elastic
force must admit of being represented by $\mu^{-2} \, \delta
\rho$. Hence the normal component of the same force (supposed by
Fresnel to be destroyed by the incompressibility of the ether)
must admit of being denoted by the symbol,
$$(\phi^{-1} - \mu^{-2}) \, \delta \rho;$$
which symbol must, therefore, admit of being equated to a vector
of the form $\mu^{-1} \, \delta m$, $\delta m$ being a small
scalar. We are, therefore, at liberty to write the following
symbolical expression for the displacement supposed by Fresnel to
exist:
$$\delta \rho = (\phi^{-1} - \mu^{-2})^{-1} \mu^{-1} \, \delta m.$$
But it has been supposed that the displacement $\delta \rho$ is
tangential to the wave, or perpendicular to $\mu$; if therefore
we write
$$\tau \, \delta m = \mu^{-1} \, \delta \rho,
\quad\hbox{or}\quad
\tau = \mu^{-1} (\phi^{-1} - \mu^{-2})^{-1} \mu^{-1},$$
then $\tau$ is at least a {\it vector}, {\it even\/} on the
principles of Fresnel: while, on those of Mac Cullagh and of
Neumann, it would have the direction of the {\it true\/}
displacement, or vibration, within the crystal. And thus,
{\it without any labour of calculation}, but simply by the
{\it expressing\/} of the fundamental {\it conceptions\/} of
Fresnel's theory in the {\sc Language} of Quaternions,
Sir W.~R.~H.\ obtains an {\it Equation of the Index-surface},
under the following {\sc Symbolical Form}:---
$$0 = S \mu^{-1} (\phi^{-1} - \mu^{-2})^{-1} \mu^{-1};
\eqno {\rm (a)}$$
which is easily transformed into the following:---
$$1 = S \mu (\mu^2 - \phi)^{-1} \mu.
\eqno {\rm (a')}$$
He has also verified, that when he writes,
$$\phi = \alpha^{-1} S \mathbin{.} \alpha^{-1}
+ \beta^{-1} S \mathbin{.} \beta^{-1}
+ \gamma^{-1} S \mathbin{.} \gamma^{-1},$$
$\alpha$,~$\beta$,~$\gamma$, being three rectangular vectors,
whereof the lengths are $a$, $b$, $c$, an easy quaternion
{\it translation\/} enables him to pass from these last forms to
certain others, although less concise ones, for the equation of
the index surface, expressed in rectangular co-ordinates; one, at
least, of which latter forms (he believes) was assigned by
Fresnel himself.
\bigbreak
2.
To pass next to the {\it Equation of the Wave-surface}, let
$\rho$ be the vector of that surface; or the vector of
Ray-velocity; or simply, the {\sc Ray-Vector}. It is connected
with the index vector~$\mu$ (if this last vector be supposed to
be measured in the direction of wave-propagation {\it itself},
and {\it not\/} in the {\it opposite\/} direction,) by the
relations,
$$S \mu \rho = -1,\quad
S \rho \, \delta \mu = 0;$$
with which may be combined their easy consequence,
$$S \mu \, \delta \rho = 0,$$
which assists to express the {\it reciprocity\/} of the two
surfaces. Hence, by some {\it very unlaborious\/} (although,
perhaps, {\it not obvious\/}) processes, depending on the
published principles of the Quaternions, and especially on those
of the Seventh Lecture, but in which it is found convenient to
introduce an {\it auxiliary vector},
$$\nu = (\mu^2 - \phi)^{-1} \mu,$$
(which may be considered to have both geometrical and physical
significations,) Sir W.~R.~H.\ infers that $\nu$ is perpendicular
to $\rho$; and also that it may be thus expressed as a function
thereof:---
$$\nu = (\phi - \rho^{-2})^{-1} \rho^{-1}.$$
An immediate result is, that the ``Equation of the Wave'' may be
{\it symbolically expressed\/} as follows:---
$$0 = S \rho^{-1} (\phi - \rho^{-2}) \rho^{-1};
\eqno {\rm (b)}$$
or, by an easy transformation,---
$$1 = S \rho (\rho^2 - \phi^{-1})^{-1} \rho.
\eqno {\rm (b')}$$
Of these formul{\ae}, likewise, the agreement with known results
(including one of his own) has been verified by Sir W.~R.~H., who
has also found that it is as easy to {\it return}, in the
quaternion calculations, from the wave to the index-surface, as
it had been to {\it pass\/} from the latter to the former: the
only difference worth mentioning between the two processes being
this, that when we interchange $\mu$ and $\rho$, in any one of
the formul{\ae}, we are at the same time to change the
{\it symbol of operation\/} to the {\it inverse operational
symbol}, $\phi^{-1}$.
\bigbreak
3.
From the expression (b), by the introduction of two auxiliary and
constant vectors, $\iota$, $\kappa$, such that (as in the Lecture
above cited) the following identity holds good:---
$$S\rho \phi \rho
= \left(
{T (\iota \rho + \rho \kappa) \over \kappa^2 - \iota^2}
\right)^2,$$
Sir W.~R.~H.\ has lately succeeded in deducing, in a new way, a
less symbolical, but more developed, {\it quaternion form\/} for
the Equation of the Wave, which he communicated in 1849 to a few
scientific friends, and which he wishes to be allowed to put on
record here: namely the equation,
$$(\kappa^2 - \iota^2)^2
= \{ S (\iota - \kappa) \rho \}^2 +
( T V \iota \rho \pm T V \kappa \rho )^2;
\eqno {\rm (c)}$$
which exhibits the {\it physical property\/} of the two vectors,
$\iota$, $\kappa$, as {\it lines of single ray-velocity\/}; and
is also adapted to {\it express}, and even to {\it suggest},
certain {\it conical cusps\/} and {\it circular ridges\/} on the
Biaxal Wave, discussed many years ago.
In the course of a recent correspondence, on the subject of the
quaternions, with Peter G. Tait, Esq., Professor of Mathematics
in the Queen's College, Belfast, Sir W.~R. Hamilton has learned
that Professor Tait has independently arrived at this last form
(c) of the Equation of Fresnel's Wave; and he hopes that the
{\it method\/} employed by Mr.~Tait will soon be, through some
channel, made public. In the meantime he desires to add, for
himself, that he is not to be understood as here offering any
{\it opinion\/} of his own on the rival merits of any
{\it physical hypotheses\/} which have been proposed respecting
the {\it directions\/} of the {\it vibrations\/} in a crystal, or
other things therewith connected; but merely as {\it applying\/}
the {\sc Calculus of Quaternions}, considered as a
{\sc Mathematical Organ}, to the {\it statement\/} and
{\it combination\/} of a few of those hypotheses, especially as
bearing on the {\sc Wave}.
\nobreak\bigskip
\centerline{\vbox{\hrule width 72pt}}
\nobreak\bigskip
\centerline{ON CERTAIN EQUATIONS IN QUATERNIONS, CONNECTED WITH}
\centerline{THE THEORY OF FRESNEL'S WAVE SURFACE.}
\vskip 12pt
\vskip12pt
\centerline{[{\sc Monday, May 9, 1859.}]}
\nobreak\bigskip
If $S \rho \phi \rho = 1$ be the {\it equation of an ellipsoid\/}
(or, indeed, of any other central surface of the second order),
then the {\it identity},
$$\rho^{-1} V \rho \phi \rho
= \phi \rho - \rho^{-1}
= (\phi - \rho^{-2}) \rho,$$
proves that the vector $\sigma = \phi \rho - \rho^{-1}$, is
perpendicular at once to $\rho$ and to $V \rho \phi \rho$. But
$V \rho \phi$ has the direction of a line tangent to the surface,
which is also perpendicular to the semidiameter~$\rho$, because
$\phi \rho$ has the direction of the normal to the surface, at
the end of that semidiameter. Hence $\sigma$ is {\it normal\/}
to the {\it plane\/} of the {\it section}, whereof $\rho$ is (not
merely a {\it semidiameter}, but) a {\it semiaxis\/}; the other
semiaxis having the direction of $V \rho \phi \rho$. But
$\rho = (\phi - \rho^{-2})^{-1} \sigma$; $\rho$ and
$\perp \sigma$;
$$\hbox{\therefore}\quad
0 = S \sigma (\phi - \rho^{-2})^{-1} \sigma;
\eqno (1)$$
and this last formula, which (when developed either by the
$\alpha \, \beta \, \gamma$ or by the $\iota \, \kappa$ form of
$\phi$), is found to lead to a {\it quadratic equation},
relatively to $\rho^2$ (or to $T \rho^2$), must, therefore, give,
in general, the {\it two\/} scalar values of the {\it square\/}
of a {\it semiaxis\/} of the {\it section\/} perpendicular to
$\sigma$, when the {\it direction\/} of this normal~$\sigma$, or
of the plane itself, is {\it given}.
Suppose now that the normal~$\sigma$ is erected {\it at the
centre\/} of the ellipsoid, and that its {\it length\/} is made
equal to the length of one of the semiaxes~$\rho$ of the section,
we shall have, of course, $T \sigma = T \rho$, and we may write
$$0 = S \sigma (\phi - \sigma^{-2})^{-1} \sigma,
\eqno (2)$$
as the equation of the {\it locus\/} of the extremity
of~$\sigma$: that is, according to Fresnel, of the {\it wave
surface}. But this is just the form (b), when we write $\rho$
for $\sigma$.
\bye