% 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. \magnification=\magstep1 \vsize=227 true mm \hsize=170 true mm \voffset=-0.4 true mm \hoffset=-5.4 true mm \def\folio{\ifnum\pageno>0 \number\pageno \else\fi} \font\Largebf=cmbx10 scaled \magstep2 \font\largerm=cmr12 \font\largeit=cmti12 \font\largesc=cmcsc10 scaled \magstep1 \font\sc=cmcsc10 \def\therefore{$.\,\raise1ex\hbox{.}\,.$} \pageno=0 \null\vskip72pt \centerline{\Largebf ON SOME QUATERNION EQUATIONS} \vskip12pt \centerline{\Largebf CONNECTED WITH FRESNEL'S WAVE} \vskip12pt \centerline{\Largebf SURFACE FOR BIAXAL CRYSTALS} \vskip24pt \centerline{\Largebf By} \vskip24pt \centerline{\Largebf William Rowan Hamilton} \vskip24pt \centerline{\largerm (Proceedings of the Royal Irish Academy, 7 (1862), pp.\ 122--124, 163.)} \vskip36pt \vfill \centerline{\largerm Edited by David R. Wilkins} \vskip 12pt \centerline{\largerm 1999} \vskip36pt\eject \null\vskip36pt \centerline{ON SOME QUATERNION EQUATIONS CONNECTED WITH} \centerline{FRESNEL'S WAVE SURFACE FOR BIAXAL CRYSTALS.} \vskip 12pt \centerline{Sir William Rowan Hamilton.} \vskip12pt \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