### Translator’s Preface

This is an English translation of a paper by the French physicist Alfred
Potier (1840-1905) that originally appeared in 1872aaaA. Potier, “Recherches
sur l’intégration d’un système d’équations aux différentielles partielles à
coefficients périodiques,” _Comptes Rendus de l’Association Française pour
l’Avancement des Sciences (Bordeaux)_ , vol. 1, pp. 255-272 (1872):
https://gallica.bnf.fr/ark:/12148/bpt6k201148c/f258.item . and was reprinted
in a collection of his works forty years later.bbbA. Potier, _Mémoires sur
l’Électricité et l’Optique_. Paris: Gauthier-Villars, 1912, pp. 239-256. The
paper presents an analysis of wave propagation through a periodic medium by a
method that in many ways anticipated the technique of multiple-scale
homogenization by more than a century.cccA. Bensoussan, J.-L. Lions and G.
Papanicolaou, _Asymptotic Analysis for Periodic Structures_. Amsterdam: North-
Holland, 1978.dddE. Sanchez-Palencia, _Non-Homogeneous Media and Vibration
Theory_. Berlin: Springer-Verlag, 1980. Indeed, this paper appeared nearly a
decade before Floquet’s work on ordinary differential equations with periodic
coefficients,eeeG. Floquet, “Sur les équations différentielles linéaires à
coefficients périodiques,” _Comptes Rendus Acad. Sci. (Paris)_ , vol. 91, pp.
880-882 (1880).fffG. Floquet, “Sur les équations différentielles linéaires à
coefficients périodiques,” _Ann. Sci. École Norm. Sup._ , ser. 2, vol. 12, pp.
47-88 (1883).gggG. Floquet, “Sur une classe d’équations différentielles
linéaires non homogènes,” _Ann. Sci. École Norm. Sup._ , ser. 3, vol. 4, pp.
111-128 (1887). and it would be more than 50 years before the appearance of
Bloch’s paper treating the problem of Schrödinger’s equation in a three-
dimensionally periodic potential.hhhF. Bloch, “Über die Quantenmechanik der
Elektronen in Kristallgittern,” _Zeits. Physik_ , vol. 52, pp. 555-600 (1928).

Neither of these authors seems to have been aware of Potier’s work, and his
paper remained essentially unknown until almost the end of the twentieth
century. Although Poincaré made passing reference to it in his lectures on the
mathematical theory of light,iiiH. Poincaré, _Leçons sur la Théorie
Mathématique de la Lumière_. Paris: Georges Carré, 1889, pp. 258 and 318. the
only other citation to it before 1994 that I have found is in a paper by
Chipart in 1924.jjjH. Chipart, “Sur la propagation de la lumière dans les
milieux à structure périodique,” _Comptes Rendus Acad. Sci. (Paris)_ , vol.
178, pp. 319-321 (1924). This lack of citations is perhaps not too surprising,
since Potier’s paper was published in a fairly obscure journal. The paper also
suffers from a significant number of typographical errors and confusing
notations that can hamper understanding of it, but I believe that it merits a
wider readership, and so present this translation.

The connection of Potier’s work to the modern theory of multiple-scale
homogenization is perhaps not quite a direct one. He assumes a quasi-plane
wave to be propagating in the medium that takes the form of what we now call a
Floquet-Bloch wave—an exponential function multiplied by a function periodic
with the same periodicity as the constitutive properties of the medium. In
modern homogenization technique, that exponential function would be replaced
by a more arbitrary function that varies slowly with respect to the period of
the medium. But what ties Potier’s analysis to the modern theory (and what
distinguishes his construction of the solution from that of Floquet or Bloch)
is his expansion of the periodic function in a series of ascending powers of
frequency, whose coefficients he proceeds to solve for in a recursive manner.
At each stage of the process, the solution of a static boundary value problem
is required to determine these coefficients, in the same way as is done
today.c,d Although the periodicity of the medium does not appear explicitly in
his expansion, it does so in an implicit way that he did not explore in
detail.

In 1872, the publication of Maxwell’s theory of electromagnetic fields had
taken place only seven years earlier, and the theory had yet to achieve wide
acceptance or receive full experimental confirmation. Thus, Potier regarded
light as an elastic wave phenomenon propagating in the ether. The modern
reader must allow for that, and can think of the vector whose cartesian
components are $(u,v,w)$ as the electric field $\mathbf{E}$. Of course, the
equations of elasticity are not quite the same as those describing the
electromagnetic field in an inhomogeneous medium, but the method laid out in
this paper could easily be adapted to treat Maxwell’s equations as well.

For clarity and easier readability, I have made the mathematical notation
somewhat more consistent with that normally used today. The ordinary
derivative notations $d/dx$, $d/dy$ and $d/dz$ have been replaced by the
corresponding partial derivatives $\partial/\partial x$, $\partial/\partial y$
and $\partial/\partial z$. There are also, in the French original, some
infelicities of notation wherein the same symbol is used to denote different
things in different parts of the paper. I have introduced new notations where
needed to avoid confusion arising from that, or simply to conform to common
usage today. I have endeavored to be as faithful to the original French as I
could, but in a few places I have taken liberties in the translation in the
interest of clarity. Translator’s notes were inserted when it seemed to me
that more extensive explanation was needed. I have inserted equation numbers
(T1), (T2), etc. when I needed to refer to an equation that Potier left
unnumbered. All equations in the paper have been reviewed; corrections of the
typographical errors I found in them were made where necessary, and such
equations marked with a $(*)$.

I will be happy to be informed of any incorrect translations or further errors
that I have missed in the original paper, and will include them in updated
versions of this document.

Edward F. Kuester

University of Colorado Boulder

<EMAIL_ADDRESS>
Investigation of the solution of a system of partial differential equations
with periodic coefficients111Session of 12 September 1872.

A. Potier

(Mining Engineer)

The various hypotheses that can be advanced about the nature of the medium
that transmits light ultimately rest on the differential equations governing
the movement of the molecules of this medium, and whose solutions can be
subjected to experimental verification.

This in turn will allow us to accept or reject the proposed differential
equations, with the proviso, however, that the method used to solve them is
valid without question. Now among these equations, those whose coefficients
are periodic have recently prompted a discussion concerning the method of
solution; I will briefly indicate the crux of this debate, and then I will
propose a different method that appears to me to be beyond dispute.

In a memoir published in the _Journal de Mathématiques Pures et Appliquées_
,222 E. Sarrau, “Sur la propagation et la polarisation de la lumière dans les
cristaux,” J. Math. Pures Appl., ser. 2, vol. 12, pp. 1-46 (1867) and vol. 13,
pp. 59-110 (1868). M. Sarrau, having given reasons to believe that the
coefficients in the differential equations for the movement of the ether in
crystalline media must be periodic, sought to obtain (using a method given by
Cauchy) the differential equations with constant coefficients that must be
satisfied by the average values of the displacements. This method consists in
expanding the given periodic coefficients, as well as the components of the
molecular displacement of the ether, in series of imaginary exponentials. The
equations thereby obtained are then separated into as many equations as the
number of terms it is necessary to keep in these series, that is, into an
infinite number of equations (admittedly linear in the unknowns); it is easy
to see that even though the series corresponding to the given data (that is,
the series into which the periodic coefficients characterizing the medium are
expanded) have only a finite number of terms, we nevertheless have to take the
series for the displacements to be infinite, so that the number of equations
of first order to be solved can never be finite; it is therefore not certain
that we can extend to this infinite set of equations the conclusions that can
be drawn from a study of a finite number of equations.

When we limit ourselves to the practical case when the periodic coefficients
reduce to a single one, all these difficulties remain, and M. de Saint-Venant
in his memoir “On various ways of presenting the theory of light waves”333A.
J. C. Barré de Saint-Venant, “Sur les divers manières de présenter la théorie
des ondes lumineuses,” _Ann. Chim. Phys._ , ser. 4, vol. 25, pp. 335-381
(1872). even seems to think that the method is not exact. Indeed, here is how
the eminent mathematician puts it:

> “…the only periodic coefficient that enters into his [M. Sarrau’s] equations
> is the inverse $\frac{1}{\rho}$ of the density of the ether or more
> generally, as he assumes, a function of $\rho$ called $e$. It is this
> function of the density appearing in the right side of these equations that
> he replaces by its constant average value plus periodic terms, making
> similar replacements for the displacements $u$, $v$, $w$, from which he
> obtains by Cauchy’s method (and for various crystalline forms) the equations
> for the average displacements. They yield laws of proportionality for double
> refraction, rotatory polarization, etc. that agree with experiment.
>
> “But the factor $\rho$ could just as well have stayed on the left side of
> the equation. When, according to the hypothesis, we equate this variable
> density to a periodic function, instead of its inverse $\frac{1}{\rho}$,
> since $\rho$ multiplies the second derivative with respect to time while on
> the other hand $\frac{1}{\rho}$ multiplies the second derivative with
> respect to the spatial coordinates, we might imagine that the result would
> not be the same and that it would yield a law proportional to something
> other than what was sought. This is in fact what happens. The second method
> (the more natural of the two) leads to the wrong behavior, contrary to
> experiment; namely, that the coefficient of birefringence, instead of being
> essentially independent of wavelength, would vary as its inverse square,
> whereas the rotatory power that varies as the inverse square of wavelength
> would vary as its inverse fourth power.”

In short, Cauchy’s method of solution applied to these same equations would
lead to two different results depending on whether a factor is placed on one
side of the equation or the other; if this was really the case, we would be
justified in saying that equations with periodic coefficients cannot be
solved, and as a consequence cannot be tested experimentally.

Therefore, even though this question is already an old one, I hope the
following work in which I will propose another method will be of some
interest, not least because I have placed the factor $\rho$ precisely with the
derivatives with respect to time, but nevertheless the resulting solution is
in agreement with known laws.

In what follows, $u$, $v$ and $w$ denote the components of displacement of a
molecule of the ether. The coefficient of elastic density $\rho$ is assumed to
vary periodically, so it is consequently sufficient to know its value in the
interior of a certain parallelepiped in order to deduce its value over all
space. This parallelepiped will be called the elementary volume. Moreover,
since we are only concerned with periodic motion in the theory of light, we
can always put

$\frac{d^{2}u}{dt^{2}}=-\omega^{2}u,\qquad\frac{d^{2}v}{dt^{2}}=-\omega^{2}v,\qquad\frac{d^{2}w}{dt^{2}}=-\omega^{2}w$

where we have denoted the quotient $\displaystyle\frac{2\pi}{\tau}$ by
$\omega$, $\tau$ being the period of the oscillations.444$\omega$ was denoted
$\alpha$ in the original (_Translator’s note_).

We want to find out whether the partial differential equations

$\nabla^{2}u-\frac{\partial\theta}{\partial x}=-\rho\omega^{2}u,$
$\nabla^{2}v-\frac{\partial\theta}{\partial y}=-\rho\omega^{2}v,$
$\nabla^{2}w-\frac{\partial\theta}{\partial z}=-\rho\omega^{2}w,$ $None$

in which the symbol $\nabla^{2}$ stands for

$\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial
y^{2}}+\frac{\partial^{2}}{\partial z^{2}}$

and

$\theta=\frac{\partial u}{\partial x}+\frac{\partial v}{\partial
y}+\frac{\partial w}{\partial z},$

possess a system of solutions of the form

$u={\rm A}e^{mx+ny+pz},\qquad v={\rm B}e^{mx+ny+pz},\qquad w={\rm
C}e^{mx+ny+pz}$ $None$

with $m$, $n$ and $p$ being constants, while ${\rm A}$, ${\rm B}$ and ${\rm
C}$ are periodic functions with the same periodicities as $\rho$.

We see immediately that if $m$, $n$, $p$ differ from 0, they must depend on
$\omega$, and the same is true for ${\rm A}$, ${\rm B}$ and ${\rm C}$. In
addition, the equations above keep the same form when the coordinate axes are
changed; things can always be arranged so that $m$ and $n$ are zero.555It
seems to me that this cannot be true for a general anisotropic periodic
medium. Potier appears to assume that the medium is periodic along each of the
cartesian coordinates $x$, $y$ and $z$. If the variation of material
properties is different in these three directions, and nonseparable, then the
case $m=n=0$ can only give behavior of the solutions that is typical, but not
universal (_Translator’s note_). Thus, we only need to calculate ${\rm A}$,
${\rm B}$, ${\rm C}$ and $p$; substituting ${\rm A}e^{pz}$, ${\rm B}e^{pz}$
and ${\rm C}e^{pz}$ for $u$, $v$ and $w$ in the equations under consideration,
they become

$\displaystyle\nabla^{2}{\rm A}-\frac{\partial\Theta}{\partial
x}+p\left(2\frac{\partial{\rm A}}{\partial z}-\frac{\partial{\rm C}}{\partial
x}\right)+\left(p^{2}+\rho\omega^{2}\right){\rm A}$ $\displaystyle=$
$\displaystyle 0$ $\displaystyle\nabla^{2}{\rm
B}-\frac{\partial\Theta}{\partial y}+p\left(2\frac{\partial{\rm B}}{\partial
z}-\frac{\partial{\rm C}}{\partial
y}\right)+\left(p^{2}+\rho\omega^{2}\right){\rm B}$ $\displaystyle=$
$\displaystyle 0$ $\displaystyle\nabla^{2}{\rm
C}-\frac{\partial\Theta}{\partial z}-p\left(\frac{\partial{\rm A}}{\partial
x}+\frac{\partial{\rm B}}{\partial y}\right)+\rho\omega^{2}{\rm C}$
$\displaystyle=$ $\displaystyle 0$

with

$\Theta=\frac{\partial{\rm A}}{\partial x}+\frac{\partial{\rm B}}{\partial
y}+\frac{\partial{\rm C}}{\partial z}$

We expand ${\rm A}$, ${\rm B}$, ${\rm C}$, $p$ and $p^{2}$ in powers of
$\omega$ by putting666Evidently, ${\rm P}_{2}=p_{1}^{2}$, ${\rm
P}_{3}=2p_{1}p_{2}$, ${\rm P}_{4}=p_{2}^{2}+2p_{1}p_{3}$, etc. (_Translator’s
note_).

$\displaystyle{\rm A}$ $\displaystyle=$ $\displaystyle{\rm A}_{0}+{\rm
A}_{1}\omega+{\rm A}_{2}\omega^{2}+\ldots$ $\displaystyle{\rm B}$
$\displaystyle=$ $\displaystyle{\rm B}_{0}+B_{1}\omega+{\rm
B}_{2}\omega^{2}+\ldots$ $\displaystyle{\rm C}$ $\displaystyle=$
$\displaystyle{\rm C}_{0}+{\rm C}_{1}\omega+{\rm C}_{2}\omega^{2}+\ldots$
$\displaystyle p$ $\displaystyle=$ $\displaystyle
p_{1}\omega+p_{2}\omega^{2}+\ldots$ $\displaystyle p^{2}$ $\displaystyle=$
$\displaystyle{\rm P}_{2}\omega^{2}+{\rm P}_{3}\omega^{3}+\ldots$

If the proposed form of solution is acceptable, we should be able to determine
${\rm A}$, ${\rm B}$ and ${\rm C}$ as functions periodic like $\rho$.

Substitution of these expressions into the differential equations777 and
equating the coefficients of each power of $\omega$ separately (_Translator’s
note_) gives

$\displaystyle\nabla^{2}{\rm A}_{0}-\frac{\partial\Theta_{0}}{\partial x}=0,$
$\displaystyle\nabla^{2}{\rm A}_{1}-\frac{\partial\Theta_{1}}{\partial
x}+p_{1}\left(2\frac{\partial{\rm A}_{0}}{\partial z}-\frac{\partial{\rm
C}_{0}}{\partial x}\right)=0$ $\displaystyle\nabla^{2}{\rm
B}_{0}-\frac{\partial\Theta_{0}}{\partial y}=0,$ $\displaystyle\nabla^{2}{\rm
B}_{1}-\frac{\partial\Theta_{1}}{\partial y}+p_{1}\left(2\frac{\partial{\rm
B}_{0}}{\partial z}-\frac{\partial{\rm C}_{0}}{\partial y}\right)=0$
$\displaystyle\nabla^{2}{\rm C}_{0}-\frac{\partial\Theta_{0}}{\partial z}=0,$
$\displaystyle\nabla^{2}{\rm C}_{1}-\frac{\partial\Theta_{1}}{\partial
z}-p_{1}\left(\frac{\partial{\rm A}_{0}}{\partial x}+\frac{\partial{\rm
B}_{0}}{\partial y}\right)=0$

and in general888I. e., for $n\geq 2$. Potier evidently intends that
$\displaystyle\Theta_{n}=\frac{\partial{\rm A}_{n}}{\partial
x}+\frac{\partial{\rm B}_{n}}{\partial y}+\frac{\partial{\rm C}_{n}}{\partial
z}$. Note that the subscript $n$ used here is not the same as the quantity $n$
that appears in equation (T2) (_Translator’s note_).

$\left.\begin{array}[]{ccc}\nabla^{2}{\rm
A}_{n}-\displaystyle\frac{\partial\Theta_{n}}{\partial
x}+\displaystyle\sum_{i=0}^{n-1}\left(2\frac{\partial{\rm A}_{i}}{\partial
z}-\frac{\partial{\rm C}_{i}}{\partial
x}\right)p_{n-i}+\displaystyle\sum_{j=0}^{n-2}{\rm P}_{n-j}{\rm
A}_{j}+\rho{\rm A}_{n-2}&=&0\\\ \nabla^{2}{\rm
B}_{n}-\displaystyle\frac{\partial\Theta_{n}}{\partial
y}+\displaystyle\sum_{i=0}^{n-1}\left(2\frac{\partial{\rm B}_{i}}{\partial
z}-\frac{\partial{\rm C}_{i}}{\partial
y}\right)p_{n-i}+\displaystyle\sum_{j=0}^{n-2}{\rm P}_{n-j}{\rm
B}_{j}+\rho{\rm B}_{n-2}&=&0\\\ \nabla^{2}{\rm
C}_{n}-\displaystyle\frac{\partial\Theta_{n}}{\partial
z}-\displaystyle\sum_{i=0}^{n-1}\left(\frac{\partial{\rm A}_{i}}{\partial
x}+\frac{\partial{\rm B}_{i}}{\partial y}\right)p_{n-i}+\rho{\rm
C}_{n-2}&=&0\end{array}\right\\}$ (1)

We can thus iteratively determine ${\rm A}$, ${\rm B}$ and ${\rm C}$ to a
better and better approximation if the equations above are compatible. The
compatibility condition will be obtained by differentiating the first with
respect to $x$, the second with respect to $y$, the third with respect to $z$
and adding. The resulting sum will contain the various groups of terms
$\left(\nabla^{2}{\rm A}_{n}-\displaystyle\frac{\partial\Theta_{n}}{\partial
x}\right)\ldots$ that should be replaced by expressions for them that follow
from (1); what results is the equation

$\displaystyle\frac{\partial}{\partial x}\left(\rho{\rm
A}_{n-2}\right)+\frac{\partial}{\partial y}\left(\rho{\rm
B}_{n-2}\right)+\frac{\partial}{\partial z}\left(\rho{\rm C}_{n-2}\right)$ (2)
$\displaystyle+\rho\left(p_{1}{\rm C}_{n-3}+p_{2}{\rm
C}_{n-4}+\ldots+p_{n-2}{\rm C}_{0}\right)=0$

By the way, this equation can be deduced directly from the equation999This
equation follows from equation (T1) above, taking the $x$, $y$ and $z$
derivative of each line respectively, and adding them up (_Translator’s
note_).

$\frac{\partial(\rho u)}{\partial x}+\frac{\partial(\rho v)}{\partial
y}+\frac{\partial(\rho w)}{\partial z}=0$ $None$

Next, we impose the condition that the functions ${\rm A}$, ${\rm B}$ and
${\rm C}$ be periodic. Because of the fact that any integral taken over the
elementary volume of a derivative of a periodic function will be zero, the
integral over this volume of the non-differentiated parts of equations (1) and
(2) must be zero. We will adopt the convention that the notation
$\left[\Phi\right]$ represents the average value of a function $\Phi$ taken
over the elementary volume:

$\left[\Phi\right]=\frac{1}{V_{p}}\int\int\int\Phi\,dx\,dy\,dz\equiv\frac{1}{V_{p}}\int\Phi\,d\varpi$
$None$

where $V_{p}=\int\,d\varpi$ is the volume of the elementary parallelepiped. We
then have101010For $n\geq 2$; if $n=0$ or 1, the summations should be omitted
(_Translator’s note_).

$\left[\sum_{j=0}^{n-2}{\rm P}_{n-j}{\rm A}_{j}+\rho{\rm
A}_{n-2}\right]=0,\quad\left[\sum_{j=0}^{n-2}{\rm P}_{n-j}{\rm B}_{j}+\rho{\rm
B}_{n-2}\right]=0,\quad\left[\rho{\rm C}_{n-2}\right]=0.\ (*)$ (3)

When conditions (2) and (3) are satisfied, the system (1) will have periodic
solutions.111111It appears to me that while conditions (2) and (3) are
necessary, Potier has not proved them to be sufficient for periodic solutions
to exist (_Translator’s note_).

The first equations to be solved are for ${\rm A}_{0}$, ${\rm B}_{0}$ and
${\rm C}_{0}$, which must satisfy the conditions

$\nabla^{2}{\rm A}_{0}-\frac{\partial\Theta_{0}}{\partial x}=\nabla^{2}{\rm
B}_{0}-\frac{\partial\Theta_{0}}{\partial y}=\nabla^{2}{\rm
C}_{0}-\frac{\partial\Theta_{0}}{\partial z}=0$ $\frac{\partial}{\partial
x}\left(\rho{\rm A}_{0}\right)+\frac{\partial}{\partial y}\left(\rho{\rm
B}_{0}\right)+\frac{\partial}{\partial z}\left(\rho{\rm C}_{0}\right)=0$
$None$ $\left[({\rm P}_{2}+\rho){\rm A}_{0}\right]=\left[({\rm
P}_{2}+\rho){\rm B}_{0}\right]=\left[\rho{\rm C}_{0}\right]=0$

From the first three equations we find that

${\rm A}_{0}=a_{0}+\frac{\partial\varphi}{\partial x},\quad{\rm
B}_{0}=b_{0}+\frac{\partial\varphi}{\partial y},\quad{\rm
C}_{0}=c_{0}+\frac{\partial\varphi}{\partial z}$

where $\varphi$ is a periodic function121212The functions $\varphi$, $\psi$
and $\chi$ called periodic in this paper are also assumed to have an average
values of 0 over the elementary volume; see Appendix 1. Since the integral of
the derivative of a periodic function over the elementary volume is zero, the
quantities $a_{0}$, $b_{0}$ and $c_{0}$ represent the average values of ${\rm
A}_{0}$, ${\rm B}_{0}$ and ${\rm C}_{0}$ respectively (_Translator’s note_).
and $a_{0}$, $b_{0}$ and $c_{0}$ are constants, satisfying the equations

$\frac{\partial}{\partial x}\left(\rho\frac{\partial\varphi}{\partial
x}\right)+\frac{\partial}{\partial y}\left(\rho\frac{\partial\varphi}{\partial
y}\right)+\frac{\partial}{\partial z}\left(\rho\frac{\partial\varphi}{\partial
z}\right)+a_{0}\frac{\partial\rho}{\partial
x}+b_{0}\frac{\partial\rho}{\partial y}+c_{0}\frac{\partial\rho}{\partial
z}=0$ (4) ${\rm
P}_{2}a_{0}+\left[\rho\left(a_{0}+\frac{\partial\varphi}{\partial
x}\right)\right]=0,\ {\rm
P}_{2}b_{0}+\left[\rho\left(b_{0}+\frac{\partial\varphi}{\partial
y}\right)\right]=0,\ \left[\rho\left(c_{0}+\frac{\partial\varphi}{\partial
z}\right)\right]=0$ (5)

These equations completely determine the function $\varphi$, the value of
${\rm P}_{2}$ and the ratios $a_{0}:b_{0}:c_{0}$ (see Appendix 1).131313In
fact, the function $\varphi$ as well as the constants $a_{0}$, $b_{0}$ and
$c_{0}$ are determined only to within a common multiplying factor, since the
equations they satisfy are homogeneous (_Translator’s note_).

Substituting the expressions thus obtained into the equations that determine
${\rm A}_{1}$, ${\rm B}_{1}$ and ${\rm C}_{1}$, we have

$\nabla^{2}{\rm A}_{1}-\frac{\partial\Theta_{1}}{\partial
x}+p_{1}\frac{\partial^{2}\varphi}{\partial x\partial
z}=0,\qquad\nabla^{2}{\rm B}_{1}-\frac{\partial\Theta_{1}}{\partial
y}+p_{1}\frac{\partial^{2}\varphi}{\partial y\partial z}=0,$ $\nabla^{2}{\rm
C}_{1}-\frac{\partial\Theta_{1}}{\partial
z}-p_{1}\left(\frac{\partial^{2}\varphi}{\partial
x^{2}}+\frac{\partial^{2}\varphi}{\partial y^{2}}\right)=0,$ $\rho p_{1}{\rm
C}_{0}+\frac{\partial}{\partial x}\left(\rho{\rm
A}_{1}\right)+\frac{\partial}{\partial y}\left(\rho{\rm
B}_{1}\right)+\frac{\partial}{\partial z}\left(\rho{\rm C}_{1}\right)=0,$
${\rm P}_{3}a_{0}+\left[\left({\rm P}_{2}+\rho\right){\rm
A}_{1}\right]=0,\quad{\rm P}_{3}b_{0}+\left[\left({\rm P}_{2}+\rho\right){\rm
B}_{1}\right]=0,\quad\left[\rho{\rm C}_{1}\right]=0.$

The solutions of the equations in the first two lines are

${\rm A}_{1}=a_{1}+\frac{\partial\psi}{\partial x},\quad{\rm
B}_{1}=b_{1}+\frac{\partial\psi}{\partial y},\quad{\rm
C}_{1}=c_{1}+p_{1}\varphi+\frac{\partial\psi}{\partial z}$ (6)

in which the periodic function $\psi$ and the constants $a_{1}$, $b_{1}$ and
$c_{1}$ are determined by the conditions

$a_{1}\frac{\partial\rho}{\partial x}+b_{1}\frac{\partial\rho}{\partial
y}+c_{1}\frac{\partial\rho}{\partial z}+\frac{\partial}{\partial
x}\left(\rho\frac{\partial\psi}{\partial x}\right)+\frac{\partial}{\partial
y}\left(\rho\frac{\partial\psi}{\partial y}\right)+\frac{\partial}{\partial
z}\left(\rho\frac{\partial\psi}{\partial
z}\right)+p_{1}\frac{\partial(\rho\varphi)}{\partial
z}+p_{1}\rho\left(c_{0}+\frac{\partial\varphi}{\partial z}\right)=0\quad(*)$
$None$ $\left.\begin{array}[]{ccc}a_{1}[\rho]+{\rm P}_{3}a_{0}+{\rm
P}_{2}a_{1}+\left[\rho\displaystyle\frac{\partial\psi}{\partial
x}\right]&=&0\quad(*)\\\ \\\ b_{1}[\rho]+{\rm P}_{3}b_{0}+{\rm
P}_{2}b_{1}+\left[\rho\displaystyle\frac{\partial\psi}{\partial
y}\right]&=&0\quad(*)\\\ \\\ \left[\left(\rho
c_{1}+p_{1}\rho\varphi\right)+\rho\displaystyle\frac{\partial\psi}{\partial
z}\right]&=&0\end{array}\right\\}$ $None$

(see Appendix 2).

Substituting the expressions for ${\rm A}_{0}$, ${\rm B}_{0}$, ${\rm C}_{0}$
and ${\rm A}_{1}$, ${\rm B}_{1}$, ${\rm C}_{1}$ into the equations that
determine ${\rm A}_{2}$, ${\rm B}_{2}$, ${\rm C}_{2}$, ${\rm P}_{4}$, …(the
first of the most general form of these conditions), we arrive at:

$\left.\begin{array}[]{ccc}\nabla^{2}{\rm
A}_{2}-\displaystyle\frac{\partial\Theta_{2}}{\partial
x}+\frac{\partial^{2}}{\partial x\partial
z}\left(p_{1}\psi+p_{2}\varphi\right)+{\rm P}_{2}a_{0}+\rho{\rm A}_{0}&=&0\\\
\\\ \nabla^{2}{\rm B}_{2}-\displaystyle\frac{\partial\Theta_{2}}{\partial
y}+\frac{\partial^{2}}{\partial y\partial
z}\left(p_{1}\psi+p_{2}\varphi\right)+{\rm P}_{2}b_{0}+\rho{\rm B}_{0}&=&0\\\
\\\ \nabla^{2}{\rm C}_{2}-\displaystyle\frac{\partial\Theta_{2}}{\partial
z}-\left(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial
y^{2}}\right)\left(p_{1}\psi+p_{2}\varphi\right)+\rho{\rm
C}_{0}&=&0.\quad(*)\end{array}\right\\}$ $None$

These equations admit periodic solutions because of how $\varphi$ and $\psi$
were determined. We will begin by finding three periodic
functions141414Observe the difference here and afterwards between the
italicized $A_{2}$, $B_{2}$, $C_{2}$ and their non-italicized versions ${\rm
A}_{2}$, ${\rm B}_{2}$, ${\rm C}_{2}$. The constants $a_{2}$, $b_{2}$ and
$c_{2}$ introduced below allow us to require that $[A_{2}]=[B_{2}]=[C_{2}]=0$
(_Translator’s note_). $A_{2}$, $B_{2}$ and $C_{2}$ that satisfy the equations

$\nabla^{2}A_{2}+\left({\rm P}_{2}a_{0}+\rho{\rm
A}_{0}\right)=0,\quad\nabla^{2}B_{2}+\left({\rm P}_{2}b_{0}+\rho{\rm
B}_{0}\right)=0,\quad\nabla^{2}C_{2}+\rho{\rm C}_{0}=0$ $None$

which, because of equations (5), is always uniquely possible. We then
put151515There is one step needed in the demonstration that the following
functions satisfy equations (T6) which is perhaps not obvious. Define
$T_{2}=\frac{\partial A_{2}}{\partial x}+\frac{\partial B_{2}}{\partial
y}+\frac{\partial C_{2}}{\partial z}$ Note that this is not the same as
$\Theta_{2}$. Now take $\partial/\partial x$ of the first part of (T7),
$\partial/\partial y$ of the second, $\partial/\partial z$ of the third and
add the results. By (T3), this gives $\nabla^{2}T_{2}=0$. Since $T_{2}$ is
periodic, from property 1 deduced in Appendix 1, we have that $T_{2}=0$
(_Translator’s note_).

${\rm A}_{2}=A_{2}+a_{2}+\frac{\partial\chi}{\partial x},$ ${\rm
B}_{2}=B_{2}+b_{2}+\frac{\partial\chi}{\partial y},$ ${\rm
C}_{2}=C_{2}+c_{2}+\frac{\partial\chi}{\partial z}+p_{1}\psi+p_{2}\varphi,$

and we have the following equations, deduced from the groups (2) and (3), to
determine $\chi$, $a_{2}$, $b_{2}$ and $c_{2}$:

$\frac{\partial}{\partial x}\left(\rho A_{2}\right)+\frac{\partial}{\partial
y}\left(\rho B_{2}\right)+\frac{\partial}{\partial z}\left(\rho
C_{2}\right)+a_{2}\frac{\partial\rho}{\partial
x}+b_{2}\frac{\partial\rho}{\partial y}+c_{2}\frac{\partial\rho}{\partial z}$
$+\frac{\partial}{\partial x}\left(\rho\frac{\partial\chi}{\partial
x}\right)+\frac{\partial}{\partial y}\left(\rho\frac{\partial\chi}{\partial
y}\right)+\frac{\partial}{\partial z}\left(\rho\frac{\partial\chi}{\partial
z}\right)+\frac{\partial}{\partial
z}\left(p_{1}\rho\psi+p_{2}\rho\varphi\right)+\rho p_{1}{\rm C}_{1}+\rho
p_{2}{\rm C}_{0}=0$ ${\rm P}_{4}a_{0}+{\rm P}_{3}a_{1}+{\rm
P}_{2}a_{2}+\left[\rho{\rm A}_{2}\right]=0$ ${\rm P}_{4}b_{0}+{\rm
P}_{3}b_{1}+{\rm P}_{2}b_{2}+\left[\rho{\rm B}_{2}\right]=0$ $\left[\rho{\rm
C}_{2}\right]=0$

These equations, in the same way as the preceding ones, will determine the
constants ${\rm P}_{4}$, $a_{2}$, $b_{2}$, $c_{2}$ and the function $\chi$.
This process can be continued indefinitely.

Since the values of ${\rm P}_{2}$ are both negative,161616See Appendix 1
(_Translator’s note_). we deduce from each of them an imaginary value171717at
least for sufficiently small $\omega$ (_Translator’s note_). for $p$ of the
form $i{\rm S}$, where $i=\sqrt{-1}$, and carrying out all the operations
indicated above, we find that the quantities

$\left.\begin{array}[]{c}a_{0},b_{0},c_{0},\quad a_{2},b_{2},c_{2},\quad{\rm
P}_{2},\varphi,{\rm P}_{4},\\\ {\rm A}_{0},{\rm B}_{0},{\rm C}_{0},\quad{\rm
A}_{2},{\rm B}_{2},{\rm C}_{2}\\\ \mbox{\rm are
real.}\end{array}\right|\begin{array}[]{c}a_{1},b_{1},c_{1},\quad\psi,p_{2},{\rm
P}_{3},\\\ {\rm A}_{1},{\rm B}_{1},{\rm C}_{1}\\\ \mbox{\rm are
imaginary.}\end{array}$

The ${\rm A}_{n}$, ${\rm B}_{n}$, ${\rm C}_{n}$ being real or imaginary
according to whether $n$ is even or odd, we can write181818The notations were
$p=\varpi+i\varpi^{\prime}$, ${\rm A}={\rm M}+i{\rm M}^{\prime}$, ${\rm
B}={\rm N}+i{\rm N}^{\prime}$ and ${\rm C}={\rm P}+i{\rm P}^{\prime}$ in the
original (_Translator’s note_).

${\rm A}={\cal M}+i{\cal M}^{\prime},\quad{\rm B}={\cal N}+i{\cal
N}^{\prime},\quad{\rm C}={\cal P}+i{\cal P}^{\prime}\quad\mbox{\rm and}\quad
p=\alpha+i\beta.$

If we only consider average values of the displacements (that is, the constant
parts of ${\rm A}$, etc.), we can put them in the form

$u_{0}+iu^{\prime}_{0}\quad,\qquad v_{0}+iv^{\prime}_{0}\quad,\qquad
w_{0}+iw^{\prime}_{0}$

where the portions ($u_{0},v_{0},w_{0}$) are even polynomials in $\omega$ and
($u^{\prime}_{0},v^{\prime}_{0},w^{\prime}_{0}$) are odd ones. As far as the
quantity $\frac{p}{\omega}$ is concerned, it must be an even function of
$\omega$ and purely imaginary, since the governing equations contain only
$\omega^{2}$. Indeed, I show directly in Appendix 2 that ${\rm P}_{3}$ and
thus $p_{2}$ are zero in general.191919Except for special cases such as the
one to be studied below (_Translator’s note_).

The values $u$, $v$ and $w$ thus take the form

$\displaystyle u$ $\displaystyle=$
$\displaystyle\left(u_{0}+iu^{\prime}_{0}\right)e^{i\beta z-i\omega t}\quad,$
$\displaystyle v$ $\displaystyle=$
$\displaystyle\left(v_{0}+iv^{\prime}_{0}\right)e^{i\beta z-i\omega t}\quad,$
$\displaystyle w$ $\displaystyle=$
$\displaystyle\left(w_{0}+iw^{\prime}_{0}\right)e^{i\beta z-i\omega t}\quad,$

where $\beta$ can take either of two values (Appendix 1), each of which
corresponds to a set of values for $u$, $v$, etc.

The real parts of $u$, $v$ and $w$ must also satisfy the proposed equations
because their coefficients are real; thus

$\displaystyle u$ $\displaystyle=$ $\displaystyle u_{0}\cos\left(\beta
z-\omega t\right)-u^{\prime}_{0}\sin\left(\beta z-\omega
t\right)\quad,\qquad(*)$ $\displaystyle v$ $\displaystyle=$ $\displaystyle
v_{0}\cos\left(\beta z-\omega t\right)-v^{\prime}_{0}\sin\left(\beta z-\omega
t\right)\quad,\qquad(*)$ $\displaystyle w$ $\displaystyle=$ $\displaystyle
w_{0}\cos\left(\beta z-\omega t\right)-w^{\prime}_{0}\sin\left(\beta z-\omega
t\right)\quad;\qquad(*)$

which represents an elliptical vibration propagating with a velocity
$\frac{\omega}{\beta}$. And because
$\frac{\beta}{\omega}=i\left(p_{1}+p_{3}\omega^{2}+\cdots\right)$, where
$p_{1}$ can take one of two distinct values,202020assumed here to be negative
imaginary so that $\beta/\omega>0$ (_Translator’s note_) we see that two
vibrations whose ellipticity depends on the wavelength can propagate as plane
waves transverse to $z$ with two different velocities, and that the first term
of the difference between these velocities is independent of the wavelength,
in agreement with experiment. If we keep only the first terms in all the
series, and thus set $u^{\prime}_{0}$, $v^{\prime}_{0}$, $w^{\prime}_{0}$,
$p_{3}$, etc. equal to zero, the double refraction remains but dispersion
disappears.

From the results of Appendix 1, we could by changes of coordinates deduce the
velocities corresponding to arbitrary directions of wave propagation. But if
we want to limit ourselves to the first terms, it would be better to
substitute the general values of $u$, $v$ and $w$ from (T2) directly into the
differential equations expanded in powers of $\omega$, and by following a
method identical to that followed above, the amplitude coefficients $a_{0}$,
$b_{0}$ and $c_{0}$ are found to be determined by equations of the form

$\displaystyle\lambda^{2}a_{0}-m\left(ma_{0}+nb_{0}+pc_{0}\right)$
$\displaystyle=$ $\displaystyle Da_{0}+Kb_{0}+Hc_{0}$
$\displaystyle\lambda^{2}b_{0}-n\left(ma_{0}+nb_{0}+pc_{0}\right)$
$\displaystyle=$ $\displaystyle Ka_{0}+Eb_{0}+Gc_{0}$
$\displaystyle\lambda^{2}c_{0}-p\left(ma_{0}+nb_{0}+pc_{0}\right)$
$\displaystyle=$ $\displaystyle Ha_{0}+Gb_{0}+Fc_{0}$ $\mbox{\rm
where}\quad\lambda^{2}=m^{2}+n^{2}+p^{2}$

which simultaneously determine the two velocities with which plane wave
vibrations can be propagated. It is easily shown that these equations lead to
a Fresnel wave surface, and produce a vibration transverse to the ray in the
plane that projects it onto the plane of the wave. This form being less
convenient than the one originally obtained for the special case of
propagation only in the $z$-direction, I will revert to the latter in what
follows.

The results above can be regarded as established only insofar as the ratios
$a_{0}:b_{0}:c_{0}$ are really determined by equations (A3) of Appendix 1.
However, the orientation of the plane of the wave relative to the medium might
be such that these three equations reduce to two, one determining ${\rm
P}_{2}$, the other only fixing the polarization of the wave (this is the
general case for cubic media, the case of a wave bitangential to the wave
surface by the bi-axes, or of a wave perpendicular to the axis). If we
designate by ${\rm M}$, ${\rm M}^{\prime}$, ${\rm M}^{\prime\prime}$, ${\rm
N}$, ${\rm N}^{\prime}$ and ${\rm N}^{\prime\prime}$ the six
integrals212121Explicitly,  ${\rm
M}=\left[\rho\left(\displaystyle\frac{\partial\varphi^{\prime}}{\partial
x}+1\right)\right],\quad{\rm
M}^{\prime}=\left[\rho\left(\displaystyle\frac{\partial\varphi^{\prime\prime}}{\partial
y}+1\right)\right],\quad{\rm
M}^{\prime\prime}=\left[\rho\left(\displaystyle\frac{\partial\varphi^{\prime\prime\prime}}{\partial
z}+1\right)\right]$ ${\rm
N}=\left[\rho\displaystyle\frac{\partial\varphi^{\prime\prime\prime}}{\partial
y}\right]=\left[\rho\displaystyle\frac{\partial\varphi^{\prime\prime}}{\partial
z}\right],\quad{\rm
N}^{\prime}=\left[\rho\displaystyle\frac{\partial\varphi^{\prime\prime\prime}}{\partial
x}\right]=\left[\rho\displaystyle\frac{\partial\varphi^{\prime}}{\partial
z}\right],\quad{\rm
N}^{\prime\prime}=\left[\rho\displaystyle\frac{\partial\varphi^{\prime\prime}}{\partial
x}\right]=\left[\rho\displaystyle\frac{\partial\varphi^{\prime}}{\partial
y}\right]$ (see Appendix 1 )_(Translator’s note)_. that appear in these
equations, they take the form

$\displaystyle\left({\rm P}_{2}+{\rm M}\right)a_{0}+{\rm
N}^{\prime\prime}b_{0}+{\rm N}^{\prime}c_{0}$ $\displaystyle=$ $\displaystyle
0,$ $\displaystyle{\rm N}^{\prime\prime}a_{0}+\left({\rm P}_{2}+{\rm
M}^{\prime}\right)b_{0}+{\rm N}c_{0}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle{\rm N}^{\prime}a_{0}+{\rm N}b_{0}+{\rm M}^{\prime\prime}c_{0}$
$\displaystyle=$ $\displaystyle 0.\ (*)$

If we have ${\rm M}^{\prime\prime}{\rm N}^{\prime\prime}={\rm N}{\rm
N}^{\prime}$ and ${\rm M}{\rm M}^{\prime\prime}-{\rm N}^{\prime 2}={\rm
M}^{\prime}{\rm M}^{\prime\prime}-{\rm N}^{2}=-{\rm M}^{\prime\prime}{\rm
P}_{2}$, the values of ${\rm P}_{2}$ will be equal, and there will remain only
the one equation

${\rm N}^{\prime}a_{0}+{\rm N}b_{0}+{\rm M}^{\prime\prime}c_{0}=0$

to determine the ratios $a_{0}:b_{0}:c_{0}$; they are thus indeterminate. So
if we want to determine $a_{1},b_{1},c_{1}$ using equations (A4) from Appendix
2, under these conditions (A4) become

${\rm M}^{\prime\prime}{\rm P}_{3}a_{0}+{\rm N}^{\prime}({\rm
N}^{\prime}a_{1}+{\rm N}b_{1}+{\rm M}^{\prime\prime}c_{1})+p_{1}{\rm
M}^{\prime\prime}\left[\rho\frac{\partial\psi}{\partial x}\right]=0,$ ${\rm
M}^{\prime\prime}{\rm P}_{3}b_{0}+{\rm N}({\rm N}^{\prime}a_{1}+{\rm
N}b_{1}+{\rm M}^{\prime\prime}c_{1})+p_{1}{\rm
M}^{\prime\prime}\left[\rho\frac{\partial\psi}{\partial y}\right]=0,$ ${\rm
N}^{\prime}a_{1}+{\rm N}b_{1}+{\rm
M}^{\prime\prime}c_{1}+\left[\rho\frac{\partial\psi}{\partial
z}\right]p_{1}+p_{1}\left[\rho\varphi\right]=0;$

Multiplying these equations by $a_{0}$, $b_{0}$ and ${\rm
M}^{\prime\prime}c_{0}$ respectively, then adding the results gives

${\rm M}^{\prime\prime}{\rm
P}_{3}(a_{0}^{2}+b_{0}^{2})+p_{1}\left[a_{0}\rho\frac{\partial\psi}{\partial
x}+b_{0}\rho\frac{\partial\psi}{\partial
y}+c_{0}\rho\frac{\partial\psi}{\partial z}+\rho\varphi\right]{\rm
M}^{\prime\prime}=0.$

The term inside the brackets is always zero (Appendix 2), but since the ratio
$a_{0}:b_{0}$ is not necessarily real, we cannot further put ${\rm P}_{3}=0$.
On the contrary, solving the equations (Appendix 4)

$\frac{{\rm M}^{\prime\prime}\left({\rm
P}_{3}a_{0}+p_{1}\left[\rho\displaystyle\frac{\partial\psi}{\partial
x}\right]\right)}{{\rm N}^{\prime}}=\frac{{\rm M}^{\prime\prime}\left({\rm
P}_{3}b_{0}+p_{1}\left[\rho\displaystyle\frac{\partial\psi}{\partial
y}\right]\right)}{{\rm
N}^{\prime}}=p_{1}\left[\rho\frac{\partial\psi}{\partial
z}\right]+p_{1}\left[\rho\varphi\right]$ $None$

we find that $\displaystyle\frac{a_{0}}{b_{0}}=\pm i$, and that ${\rm P}_{3}$
is real; consequently $p_{2}$ is an imaginary quantity whose sign changes with
that of $\displaystyle\frac{a_{0}}{b_{0}}$. For this particular orientation,
the wave can therefore propagate as two vibrations whose projection onto the
plane of the wave is circular, because $(a_{0}^{2}+b_{0}^{2})=0$, and with
different velocities, the plane of vibration being ${\rm N}^{\prime}x+{\rm
N}y+{\rm M}^{\prime}z=0$ since we are not looking for an approximation of any
higher order than this.

For $p$ we obtain the two values $\omega p_{1}\pm\omega^{2}p_{2}$ $(*)$, both
imaginary; if $\beta_{1}$ and $\beta_{2}$ are the imaginary parts of $p_{1}$
and $p_{2}$,222222$\beta_{1,2}$ are $\varpi_{1,2}$ in the original
(_Translator’s note_). the values of the displacements being

$u=a_{0}e^{i[(\beta_{1}\omega\pm\beta_{2}\omega^{2})z-\omega t]}$ $None$
$v=\pm ia_{0}e^{i[(\beta_{1}\omega\pm\beta_{2}\omega^{2})z-\omega t]}$ $None$

and their real parts being

$u=a_{0}\cos[(\beta_{1}\omega\pm\beta_{2}\omega^{2})z-\omega t]$ $None$ $v=\mp
a_{0}\sin[(\beta_{1}\omega\pm\beta_{2}\omega^{2})z-\omega t],$ $None$

these are the equations of two circular vibrations propagating with the
velocities $\displaystyle\frac{1}{\beta_{1}\pm\beta_{2}\omega}$ and
wavelengths $\displaystyle\frac{2\pi}{\beta_{1}\omega\pm\beta_{2}\omega^{2}}$
$(*)$. Since the rotatory power is given by the difference between the
inverses of the wavelengths of the two rays, it will be
$\beta_{2}\omega^{2}/\pi$ $(*)$; thus it will vary as the inverse square of
the wavelength since $\omega=2\pi/\tau$.

When the medium possesses a certain symmetry (if it is not merohedral), the
values of $p_{2}$ and ${\rm P}_{3}$ are zero, and the ratio $a_{0}:b_{0}$ is
really indeterminate, which is most often the case.

There is no interest in pursuing this discussion by the examination of the
terms that depend on higher powers of $\omega$ or of the inverse of the
wavelength; the main goal I had set myself seems to have been achieved, since
it follows from the calculations above that, under the sole condition that the
expansions in ascending powers of the wavelength232323Potier undoubtedly meant
increasingly negative powers of the wavelength here (_Translator’s note_). are
admissible (and all analysts must make this assumption), equations with
periodic coefficients account for the phenomena exhibited by transparent
crystalline media.

## APPENDICES

### Appendix 1

Suppose a periodic function242424In what follows, periodic functions except
for $\rho$ are supposed to have had their constant parts (which do not
contribute to their derivatives) removed. _Translator’s note:_ In other words,
all periodic functions except for $\rho$ are assumed to have an average value
of $0$ over the elementary volume. satisfies the differential equation

$\frac{\partial}{\partial x}\rho\frac{\partial F}{\partial
x}+\frac{\partial}{\partial y}\rho\frac{\partial F}{\partial
y}+\frac{\partial}{\partial z}\rho\frac{\partial F}{\partial z}+U=0$ $None$

($U$ being a function periodic like $\rho$); if $V$ is another periodic
function, $d\varpi$ is a volume element, and the integral $\int$ is carried
out over the elementary volume determined by the periodicity, we will have

$\int UV\,d\varpi=\int\rho\left(\frac{\partial F}{\partial x}\frac{\partial
V}{\partial x}+\frac{\partial F}{\partial y}\frac{\partial V}{\partial
y}+\frac{\partial F}{\partial z}\frac{\partial V}{\partial z}\right)\,d\varpi$
$None$

Thus:

  * 1.

If $U=0$ and $V=F$, then

$\int\rho\left[\left(\frac{\partial F}{\partial
x}\right)^{2}+\left(\frac{\partial F}{\partial
y}\right)^{2}+\left(\frac{\partial F}{\partial
z}\right)^{2}\right]\,d\varpi=0$

and as a consequence the periodic function that satisfies the equation

$\frac{\partial}{\partial x}\rho\frac{\partial F}{\partial
x}+\frac{\partial}{\partial y}\rho\frac{\partial F}{\partial
y}+\frac{\partial}{\partial z}\rho\frac{\partial F}{\partial x}=0$

must be identically zero, and thus equation (A1) admits only one periodic
solution.252525Evidently, Potier assumes here that $\rho>0$ everywhere
(_Translator’s note_). If there were two such solutions, their difference
would satisfy the last equation above and would have to be zero.

  * 2.

Equation (4) therefore completely determines262626The constants $a_{0}$,
$b_{0}$ and $c_{0}$ being given (_Translator’s note_). the function $\varphi$,
and it is evident that if we put

$\frac{\partial}{\partial x}\rho\frac{\partial\varphi^{\prime}}{\partial
x}+\frac{\partial}{\partial y}\rho\frac{\partial\varphi^{\prime}}{\partial
y}+\frac{\partial}{\partial z}\rho\frac{\partial\varphi^{\prime}}{\partial
x}=-\frac{\partial\rho}{\partial x}$ $\frac{\partial}{\partial
x}\rho\frac{\partial\varphi^{\prime\prime}}{\partial
x}+\frac{\partial}{\partial
y}\rho\frac{\partial\varphi^{\prime\prime}}{\partial
y}+\frac{\partial}{\partial
z}\rho\frac{\partial\varphi^{\prime\prime}}{\partial
x}=-\frac{\partial\rho}{\partial y}$ $\frac{\partial}{\partial
x}\rho\frac{\partial\varphi^{\prime\prime\prime}}{\partial
x}+\frac{\partial}{\partial
y}\rho\frac{\partial\varphi^{\prime\prime\prime}}{\partial
y}+\frac{\partial}{\partial
z}\rho\frac{\partial\varphi^{\prime\prime\prime}}{\partial
x}=-\frac{\partial\rho}{\partial z}$

then we will have
$\varphi=a_{0}\varphi^{\prime}+b_{0}\varphi^{\prime\prime}+c_{0}\varphi^{\prime\prime\prime}$.

Equations (5) can then be written

$\left.\begin{array}[]{c}a_{0}{\rm
P}_{2}+\left[\rho\left(\displaystyle\frac{\partial\varphi^{\prime}}{\partial
x}+1\right)\right]a_{0}+\left[\rho\displaystyle\frac{\partial\varphi^{\prime\prime}}{\partial
x}\right]b_{0}+\left[\rho\displaystyle\frac{\partial\varphi^{\prime\prime\prime}}{\partial
x}\right]c_{0}=0,\\\ \\\
\left[\rho\displaystyle\frac{\partial\varphi^{\prime}}{\partial
y}\right]a_{0}+b_{0}{\rm
P}_{2}+\left[\rho\left(\displaystyle\frac{\partial\varphi^{\prime\prime}}{\partial
y}+1\right)\right]b_{0}+\left[\rho\displaystyle\frac{\partial\varphi^{\prime\prime\prime}}{\partial
y}\right]c_{0}=0,\ (*)\\\ \\\
\left[\rho\displaystyle\frac{\partial\varphi^{\prime}}{\partial
z}\right]a_{0}+\left[\rho\displaystyle\frac{\partial\varphi^{\prime\prime}}{\partial
z}\right]b_{0}+\left[\rho\left(\displaystyle\frac{\partial\varphi^{\prime\prime\prime}}{\partial
z}+1\right)\right]c_{0}=0,\ (*)\end{array}\right\\}$ $None$

which, after eliminating $a_{0}:b_{0}:c_{0}$, give a quadratic equation whose
roots are the values of ${\rm P}_{2}$.272727That is, setting the determinant
of the system of equations (A3) equal to zero (_Translator’s note_).

Of the nine integrals that appear in these equations, only six are
independent. In fact, if in equation (A2) we first put $F=\varphi^{\prime}$,
$U=\frac{\partial\rho}{\partial x}$, $V=\varphi^{\prime\prime}$, then
$F=\varphi^{\prime\prime}$, $U=\frac{\partial\rho}{\partial y}$,
$V=\varphi^{\prime}$, we get282828Here and below I have added the notations
${\rm M},\ldots,{\rm N}^{\prime\prime}$ used in the main body of the paper to
designate the various integrals (_Translator’s note_).

$\int\,d\varpi\,\rho\left(\frac{\partial\varphi^{\prime}}{\partial
x}\frac{\partial\varphi^{\prime\prime}}{\partial
x}+\frac{\partial\varphi^{\prime}}{\partial
y}\frac{\partial\varphi^{\prime\prime}}{\partial
y}+\frac{\partial\varphi^{\prime}}{\partial
z}\frac{\partial\varphi^{\prime\prime}}{\partial
z}\right)=\int\varphi^{\prime\prime}\frac{\partial\rho}{\partial
x}\,d\varpi=\int\varphi^{\prime}\frac{\partial\rho}{\partial
y}\,d\varpi\equiv\frac{1}{V_{p}}{\rm N}^{\prime\prime}.$

where $V_{p}$ is the volume of the elementary parallelepiped. But the
integrals

$\int\,d\varpi\frac{\partial}{\partial
x}\left(\rho\varphi^{\prime\prime}\right)\quad\mbox{\rm
and}\quad\int\,d\varpi\frac{\partial}{\partial
y}\left(\rho\varphi^{\prime}\right)$

evidently being zero, the two integrals appearing in the previous equation
will be (apart from the signs)

$\int\rho\frac{\partial\varphi^{\prime\prime}}{\partial
x}\,d\varpi\quad\mbox{\rm
and}\quad\int\rho\frac{\partial\varphi^{\prime}}{\partial y}\,d\varpi$

Likewise we can demonstrate the equalities

$\left[\rho\frac{\partial\varphi^{\prime}}{\partial
z}\right]=\left[\rho\frac{\partial\varphi^{\prime\prime\prime}}{\partial
x}\right]\equiv{\rm
N}^{\prime},\qquad\left[\rho\frac{\partial\varphi^{\prime\prime}}{\partial
z}\right]=\left[\rho\frac{\partial\varphi^{\prime\prime\prime}}{\partial
y}\right]\equiv{\rm N}.$

The left sides of equations (A3) are therefore the derivatives with respect to
$a_{0}$, $b_{0}$ and $c_{0}$ of the same quadratic form; the ratios
$a_{0}:b_{0}:c_{0}$ are thus real.292929The quadratic form in question is
$Q=\frac{{\rm P}_{2}+{\rm M}}{2}a_{0}^{2}+\frac{{\rm P}_{2}+{\rm
M}^{\prime}}{2}b_{0}^{2}+\frac{{\rm M}^{\prime\prime}}{2}c_{0}^{2}+{\rm
N}^{\prime\prime}a_{0}b_{0}+{\rm N}^{\prime}a_{0}c_{0}+{\rm N}b_{0}c_{0}$
(_Translator’s note_).

Moreover, if we put $F=V=\varphi^{\prime}$, $U=\frac{\partial\rho}{\partial
x}$, equation (A2) becomes

$\int\,d\varpi\,\rho\left[\left(\frac{\partial\varphi^{\prime}}{\partial
x}\right)^{2}+\left(\frac{\partial\varphi^{\prime}}{\partial
y}\right)^{2}+\left(\frac{\partial\varphi^{\prime}}{\partial
z}\right)^{2}\right]=-\int\varphi^{\prime}\frac{\partial\rho}{\partial
x}\,d\varpi=+\int\rho\frac{\partial\varphi^{\prime}}{\partial x}\,d\varpi$

and so the three integrals

${\rm M}\equiv\left[\rho\frac{\partial\varphi^{\prime}}{\partial
x}\right],\quad{\rm
M}^{\prime}\equiv\left[\rho\frac{\partial\varphi^{\prime\prime}}{\partial
y}\right],\quad{\rm
M}^{\prime\prime}\equiv\left[\rho\frac{\partial\varphi^{\prime\prime\prime}}{\partial
z}\right]$

are positive, and the values of ${\rm P}_{2}$ will both be real and negative.

### Appendix 2

Equation (T4) allows us to present an expression for $\psi$ composed of four
parts.303030The citation to this equation is corrected from the original
(_Translator’s note_). In fact, if we denote the unique periodic solution of
the equation

$\frac{\partial}{\partial x}\rho\displaystyle\frac{\partial\Psi}{\partial
x}+\displaystyle\frac{\partial}{\partial
y}\rho\displaystyle\frac{\partial\Psi}{\partial y}+\frac{\partial}{\partial
z}\rho\displaystyle\frac{\partial\Psi}{\partial
z}+\rho\left(c_{0}+\frac{\partial\varphi}{\partial
z}\right)+\frac{\partial(\rho\varphi)}{\partial z}=0$

by $\Psi$, we must have

$\psi=a_{1}\varphi^{\prime}+b_{1}\varphi^{\prime\prime}+c_{1}\varphi^{\prime\prime\prime}+p_{1}\Psi$
$None$

so that equations (T5) become313131The citation to this equation is corrected
from the original (_Translator’s note_).

$\left.\begin{array}[]{c}{\rm P}_{3}a_{0}+\left[{\rm
P}_{2}+\rho\left(\displaystyle\frac{\partial\varphi^{\prime}}{\partial
x}+1\right)\right]a_{1}+\left[\rho\displaystyle\frac{\partial\varphi^{\prime\prime}}{\partial
x}\right]b_{1}+\left[\rho\displaystyle\frac{\partial\varphi^{\prime\prime\prime}}{\partial
x}\right]c_{1}+p_{1}\left[\rho\displaystyle\frac{\partial\Psi}{\partial
x}\right]=0,\\\ \\\ {\rm
P}_{3}b_{0}+\left[\rho\displaystyle\frac{\partial\varphi^{\prime}}{\partial
y}\right]a_{1}+\left[{\rm
P}_{2}+\rho\left(\displaystyle\frac{\partial\varphi^{\prime\prime}}{\partial
y}+1\right)\right]b_{1}+\left[\rho\displaystyle\frac{\partial\varphi^{\prime\prime\prime}}{\partial
y}\right]c_{1}+p_{1}\left[\rho\displaystyle\frac{\partial\Psi}{\partial
y}\right]=0,\\\ \\\
\left[\rho\displaystyle\frac{\partial\varphi^{\prime}}{\partial
z}\right]a_{1}+\left[\rho\displaystyle\frac{\partial\varphi^{\prime\prime}}{\partial
z}\right]b_{1}+\left[\rho\left(\displaystyle\frac{\partial\varphi^{\prime\prime\prime}}{\partial
z}+1\right)\right]c_{1}+p_{1}\left[\rho\varphi+\rho\displaystyle\frac{\partial\Psi}{\partial
z}\right]=0,\end{array}\right\\}$ $None$

If we wish to use these equations to determine $a_{1}$, $b_{1}$, $c_{1}$, we
observe that the matrix multiplying these unknowns is the same as that in
equations (A3) that determine $a_{0}$, $b_{0}$, $c_{0}$; its determinant is
thus zero. What is more, it is evident by comparing these two sets of
equations that if $a_{1}$, $b_{1}$, $c_{1}$ satisfy equations (A4), then
$a_{1}+\lambda a_{0}$, $b_{1}+\lambda b_{0}$, $c_{1}+\lambda c_{0}$ also
satisfy them; a term $\lambda\varphi$ will be added to the function $\psi$,
and $\lambda{\rm A}_{0}$, $\lambda{\rm B}_{0}$, $\lambda{\rm C}_{0}$ to ${\rm
A}_{1}$, ${\rm B}_{1}$, ${\rm C}_{1}$ respectively. Comparison of the
equations that determine an arbitrary ${\rm A}_{n}$, ${\rm B}_{n}$, ${\rm
C}_{n}$ with those that determine ${\rm A}_{0}$, ${\rm B}_{0}$, ${\rm C}_{0}$
shows that it must always turn out this way: the addition of $\lambda{\rm
A}_{0}$, $\lambda{\rm B}_{0}$, $\lambda{\rm C}_{0}$ to ${\rm A}_{n}$, ${\rm
B}_{n}$, ${\rm C}_{n}$ respectively amounts to multiplying $u$, $v$, $w$ by
the factor $(1+\lambda\omega^{n})$.

The solvability condition for equations (A4) is obtained by multiplying them
by $a_{0}$, $b_{0}$, $c_{0}$ respectively, adding the results and taking
equations (A3) and the equalities demonstrated above for the six integrals
into account; we get

${\rm
P}_{3}\left(a_{0}^{2}+b_{0}^{2}\right)+p_{1}\left[a_{0}\rho\frac{\partial\Psi}{\partial
x}+b_{0}\rho\frac{\partial\Psi}{\partial
y}+c_{0}\rho\frac{\partial\Psi}{\partial z}\right]=0$ $None$

Since ${\rm P}_{3}=2p_{1}p_{2}$, we see that $p_{2}$ will be real because
$p_{1}$ is of the form $i{\rm S}$.

The first equation in this Appendix determines the function $\Psi$ $(*)$;
since it is in the form of equation (A1) we deduce from it that

$\int\rho\,d\varpi\left(\frac{\partial\Psi}{\partial
x}\frac{\partial\varphi}{\partial x}+\frac{\partial\Psi}{\partial
y}\frac{\partial\varphi}{\partial y}+\frac{\partial\Psi}{\partial
z}\frac{\partial\varphi}{\partial z}\right)=c_{0}\int\rho\varphi\,d\varpi$

On the other hand, putting

$U=a_{0}\frac{\partial\rho}{\partial x}+b_{0}\frac{\partial\rho}{\partial
y}+c_{0}\frac{\partial\rho}{\partial z},\qquad\mbox{\rm whence}\qquad
F=\varphi\quad\mbox{\rm and}\quad V=\Psi,$

equation (A2) of Appendix 1 gives

$\int\rho\,d\varpi\left(\frac{\partial\Psi}{\partial
x}\frac{\partial\varphi}{\partial x}+\frac{\partial\Psi}{\partial
y}\frac{\partial\varphi}{\partial y}+\frac{\partial\Psi}{\partial
z}\frac{\partial\varphi}{\partial
z}\right)=\int\,d\varpi\left(a_{0}\frac{\partial\rho}{\partial
x}+b_{0}\frac{\partial\rho}{\partial y}+c_{0}\frac{\partial\rho}{\partial
z}\right)\Psi$

This last integral is equal to

$-\int\,d\varpi\,\rho\left(a_{0}\frac{\partial\Psi}{\partial
x}+b_{0}\frac{\partial\Psi}{\partial y}+c_{0}\frac{\partial\Psi}{\partial
z}\right);$

and thus, finally,

$c_{0}\int\rho\varphi\,d\varpi+\int\,d\varpi\,\rho\left(a_{0}\frac{\partial\Psi}{\partial
x}+b_{0}\frac{\partial\Psi}{\partial y}+c_{0}\frac{\partial\Psi}{\partial
z}\right)=0,$

and equation (A5) reduces to

${\rm P}_{3}\left(a_{0}^{2}+b_{0}^{2}\right)=0$

When $a_{0}:b_{0}$ is real, ${\rm P}_{2}$ and $p_{2}$ are zero.

### Appendix 3

In this Appendix I obtain an expression for the quantity ${\rm
P}_{4}=p_{2}^{2}+2p_{1}p_{3}$ $(*)$ that determines the dispersion of the
medium. We define a function $\Omega$ by the condition

$\frac{\partial}{\partial x}\rho\frac{\partial\Omega}{\partial
x}+\frac{\partial}{\partial y}\rho\frac{\partial\Omega}{\partial
y}+\frac{\partial}{\partial z}\rho\frac{\partial\Omega}{\partial
z}=\frac{\partial}{\partial x}\left(\rho A_{2}\right)+\frac{\partial}{\partial
y}\left(\rho B_{2}\right)+\frac{\partial}{\partial z}\left(\rho C_{2}\right)$
$+p_{1}\frac{\partial}{\partial
z}\left(\rho\varphi_{1}\right)+p_{1}^{2}\frac{\partial}{\partial
z}\left(\rho\Psi\right)+p_{1}^{2}\rho\varphi+p_{1}\left(\rho\frac{\partial\varphi_{1}}{\partial
z}+c_{1}\right),$

where
$\varphi_{1}=a_{1}\varphi^{\prime}+b_{1}\varphi^{\prime\prime}+c_{1}\varphi^{\prime\prime\prime}$,
and we end up with

${\rm
P}_{4}\left(a_{0}^{2}+b_{0}^{2}\right)+a_{0}\left[\rho\left(A_{2}+\frac{\partial\Omega}{\partial
x}\right)\right]+b_{0}\left[\rho\left(B_{2}+\frac{\partial\Omega}{\partial
y}\right)\right]$

It can be verified that the coefficient of each ${\rm P}_{n}$ will always be
$\left(a_{0}^{2}+b_{0}^{2}\right)$.

### Appendix 4

We have shown in Appendix 2 that the solvability condition (A5) reduces to

${\rm P}_{3}\left(a_{0}^{2}+b_{0}^{2}\right)=0.$

We need to show that in the case when the equation for ${\rm P}_{2}$ has two
equal solutions, we must have

$a_{0}^{2}+b_{0}^{2}=0$

and that in general323232Presumably Potier means here that ${\rm P}_{3}$ is
real and not equal to zero; see below (_Translator’s note_).

${\rm
P}_{3}\>\mathchoice{\raisebox{-3.01385pt}{$\displaystyle\stackrel{{\scriptstyle\textstyle<}}{{\textstyle>}}$}}{\raisebox{-3.01385pt}{$\textstyle\stackrel{{\scriptstyle\textstyle<}}{{\textstyle>}}$}}{\raisebox{-3.01385pt}{$\scriptstyle\stackrel{{\scriptstyle\textstyle<}}{{\textstyle>}}$}}{\raisebox{-3.01385pt}{$\scriptscriptstyle\stackrel{{\scriptstyle\textstyle<}}{{\textstyle>}}$}}\>0.$

In fact, the equation that defines the function $\Psi$ shows that it is of the
form

$a_{0}\Psi_{1}+b_{0}\Psi_{2}+c_{0}\Psi_{3},$

if the $\Psi_{n}$ satisfy the equations

$\frac{\partial}{\partial x}\rho\frac{\partial\Psi_{1}}{\partial
x}+\frac{\partial}{\partial y}\rho\frac{\partial\Psi_{1}}{\partial
y}+\frac{\partial}{\partial z}\rho\frac{\partial\Psi_{1}}{\partial
z}+\frac{\partial}{\partial
z}\left(\rho\varphi^{\prime}\right)+\rho\frac{\partial\varphi^{\prime}}{\partial
z}=0,$ $\frac{\partial}{\partial x}\rho\frac{\partial\Psi_{2}}{\partial
x}+\frac{\partial}{\partial y}\rho\frac{\partial\Psi_{2}}{\partial
y}+\frac{\partial}{\partial z}\rho\frac{\partial\Psi_{2}}{\partial
z}+\frac{\partial}{\partial
z}\left(\rho\varphi^{\prime\prime}\right)+\rho\frac{\partial\varphi^{\prime\prime}}{\partial
z}=0,$ $\frac{\partial}{\partial x}\rho\frac{\partial\Psi_{3}}{\partial
x}+\frac{\partial}{\partial y}\rho\frac{\partial\Psi_{3}}{\partial
y}+\frac{\partial}{\partial z}\rho\frac{\partial\Psi_{3}}{\partial
z}+\frac{\partial}{\partial
z}\left(\rho\varphi^{\prime\prime\prime}\right)+\rho\frac{\partial\varphi^{\prime\prime\prime}}{\partial
z}=0.$

The functions $\Psi_{n}$ defined in this way possess the following properties:

$\left[\rho\frac{\partial\Psi_{1}}{\partial
x}\right]=\left[\rho\frac{\partial\Psi_{2}}{\partial
y}\right]=\left[\rho\frac{\partial\Psi_{1}}{\partial
y}+\rho\frac{\partial\Psi_{2}}{\partial x}\right]=0,$
$\left[\rho\frac{\partial\Psi_{1}}{\partial
z}+\rho\frac{\partial\Psi_{3}}{\partial
x}+\rho\varphi^{\prime}\right]=\left[\rho\frac{\partial\Psi_{2}}{\partial
z}+\rho\frac{\partial\Psi_{3}}{\partial
y}+\rho\varphi^{\prime\prime}\right]=\left[\rho\frac{\partial\Psi_{3}}{\partial
z}+\rho\varphi^{\prime\prime\prime}\right]=0,$

which are proved in the same way as equation (A5) of Appendix 2.

Now multiplying both sides of the equation that determines $\Psi_{2}$ by
$\varphi^{\prime\prime\prime}$ and both sides of the one that determines
$\Psi_{3}$ by $\varphi^{\prime\prime}$, integrating over the elementary volume
gives

$\left[\rho\left(\frac{\partial\Psi_{2}}{\partial
x}\frac{\partial\varphi^{\prime\prime\prime}}{\partial
x}+\frac{\partial\Psi_{2}}{\partial
y}\frac{\partial\varphi^{\prime\prime\prime}}{\partial
y}+\frac{\partial\Psi_{2}}{\partial
z}\frac{\partial\varphi^{\prime\prime\prime}}{\partial
z}\right)\right]\qquad\qquad\qquad\qquad\qquad\,$
$=\int\,d\varpi\frac{\varphi^{\prime\prime\prime}}{\varphi^{\prime\prime}}\frac{\partial}{\partial
z}\left(\rho\varphi^{\prime\prime
2}\right)=-\int\,d\varpi\rho\varphi^{\prime\prime 2}\frac{\partial}{\partial
z}\frac{\varphi^{\prime\prime\prime}}{\varphi^{\prime\prime}}=\int\rho\,d\varpi\left(\varphi^{\prime\prime}\frac{\partial\varphi^{\prime\prime\prime}}{\partial
z}-\varphi^{\prime\prime\prime}\frac{\partial\varphi^{\prime\prime}}{\partial
z}\right)\quad(*)$

and

$\left[\rho\left(\frac{\partial\varphi^{\prime\prime}}{\partial
x}\frac{\partial\Psi_{3}}{\partial
x}+\frac{\partial\varphi^{\prime\prime}}{\partial
y}\frac{\partial\Psi_{3}}{\partial
y}+\frac{\partial\varphi^{\prime\prime}}{\partial
z}\frac{\partial\Psi_{3}}{\partial
z}\right)\right]\qquad\qquad\qquad\qquad\qquad\qquad\qquad\,$
$\qquad\qquad\quad\quad=-\int\rho\,d\varpi\left(\varphi^{\prime\prime\prime}\frac{\partial\varphi^{\prime\prime}}{\partial
z}-\varphi^{\prime\prime}\frac{\partial\varphi^{\prime\prime\prime}}{\partial
z}\right)+\int\,d\varpi\,\rho\varphi^{\prime\prime\prime}.\quad(*)$

But the two integrals on the left sides are, thanks to equation (A2) of
Appendix 1, equal to

$-\int\,d\varpi\rho\frac{\partial\Psi_{2}}{\partial z}\qquad\mbox{\rm
and}\qquad-\int\,d\varpi\rho\frac{\partial\Psi_{3}}{\partial z}$

respectively, if we put $F=\varphi^{\prime\prime\prime}$, $V=\Psi_{2}$, or
$F=\varphi^{\prime\prime}$, $V=\Psi_{3}$. We thus have

$\int\rho\frac{\partial\Psi_{2}}{\partial
z}\,d\varpi=\int\,d\varpi\rho\left(\varphi^{\prime\prime}\frac{\partial\varphi^{\prime\prime\prime}}{\partial
z}-\varphi^{\prime\prime\prime}\frac{\partial\varphi^{\prime\prime}}{\partial
z}\right),$ $\int\rho\frac{\partial\Psi_{3}}{\partial
y}\,d\varpi=\int\,d\varpi\rho\left(\varphi^{\prime\prime\prime}\frac{\partial\varphi^{\prime\prime}}{\partial
z}-\varphi^{\prime\prime}\frac{\partial\varphi^{\prime\prime\prime}}{\partial
z}\right)-\int\,d\varpi\varphi^{\prime\prime\prime},$

and upon adding we get

$\int\rho\left(\frac{\partial\Psi_{2}}{\partial
z}+\frac{\partial\Psi_{3}}{\partial
y}\right)\,d\varpi=-\int\,d\varpi\rho\varphi^{\prime\prime\prime},$

and likewise for the other relations to be proved.

Substituting the expressions deduced above for
$\left[\rho\displaystyle\frac{\partial\Psi}{\partial x}\right]$,
$\left[\rho\displaystyle\frac{\partial\Psi}{\partial y}\right]$ and
$\left[\rho\displaystyle\frac{\partial\Psi}{\partial z}\right]$ into equation
(T5), and using (T8),333333The citations to these equations are corrected from
the original (_Translator’s note_). they become

$\left[\rho\frac{\partial\Psi}{\partial
x}\right]=b_{0}\left[\rho\frac{\partial\Psi_{2}}{\partial
x}\right]+c_{0}\left[\rho\frac{\partial\Psi_{3}}{\partial x}\right],$
$\left[\rho\frac{\partial\Psi}{\partial
y}\right]=a_{0}\left[\rho\frac{\partial\Psi_{1}}{\partial
y}\right]+c_{0}\left[\rho\frac{\partial\Psi_{3}}{\partial y}\right],$
$\left[\rho\frac{\partial\Psi}{\partial
z}\right]=a_{0}\left[\rho\frac{\partial\Psi_{2}}{\partial
z}\right]+b_{0}\left[\rho\frac{\partial\Psi_{2}}{\partial
z}\right]+c_{0}\left[\rho\varphi^{\prime\prime\prime}\right].$

But $a_{0}$, $b_{0}$ and $c_{0}$ are related by

${\rm N}^{\prime}a_{0}+Nb_{0}+{\rm M}^{\prime\prime}c_{0}=0,$

by which $c_{0}$ can be eliminated to get

${\rm M}^{\prime\prime}\left[\rho\frac{\partial\Psi}{\partial x}\right]=-{\rm
N}^{\prime}a_{0}\left[\rho\frac{\partial\Psi_{3}}{\partial
x}\right]+b_{0}\left\\{{\rm
M}^{\prime\prime}\left[\rho\frac{\partial\Psi_{2}}{\partial
x}\right]-N\left[\rho\frac{\partial\Psi_{3}}{\partial x}\right]\right\\},$
${\rm M}^{\prime\prime}\left[\rho\frac{\partial\Psi}{\partial
y}\right]=a_{0}\left[{\rm
M}^{\prime\prime}\rho\frac{\partial\Psi_{1}}{\partial y}-{\rm
N}^{\prime}\rho\frac{\partial\Psi_{3}}{\partial
y}\right]-Nb_{0}\left[\rho\frac{\partial\Psi_{3}}{\partial y}\right],$ ${\rm
M}^{\prime\prime}\left[\rho\frac{\partial\Psi}{\partial
z}\right]=a_{0}\left[{\rm
M}^{\prime\prime}\rho\frac{\partial\Psi_{1}}{\partial z}+{\rm
N}^{\prime}\rho\varphi^{\prime\prime\prime}\right]+b_{0}\left[{\rm
M}^{\prime\prime}\rho\frac{\partial\Psi_{2}}{\partial
z}+N\rho\varphi^{\prime\prime\prime}\right].$

Equation $(7)$:

${\rm M}^{\prime\prime}{\rm P}_{3}a_{0}={\rm
N}^{\prime}p_{1}\left[\rho\left(\frac{\partial\Psi}{\partial
z}+\varphi\right)\right]-{\rm
M}^{\prime\prime}\left[\rho\frac{\partial\Psi}{\partial z}\right]p_{1}$

therefore reduces (taking into account the relations among the integrals that
were established above) to

$a_{0}{\rm M}^{\prime\prime}{\rm P}_{3}+b_{0}p_{1}\left[{\rm
M}^{\prime\prime}\rho\frac{\partial\Psi_{2}}{\partial
x}-N\rho\frac{\partial\Psi_{3}}{\partial x}+{\rm
N}^{\prime}\rho\frac{\partial\Psi_{3}}{\partial y}\right]=0;$

the other equation gives

$b_{0}{\rm M}^{\prime\prime}{\rm P}_{3}+a_{0}p_{1}\left[{\rm
M}^{\prime\prime}\rho\frac{\partial\Psi_{1}}{\partial
y}+N\rho\frac{\partial\Psi_{3}}{\partial x}-{\rm
N}^{\prime}\rho\frac{\partial\Psi_{3}}{\partial y}\right]=0;$

from which we deduce

$\frac{a_{0}}{b_{0}}+\frac{b_{0}}{a_{0}}=0\qquad\mbox{\rm or}\qquad
a_{0}^{2}+b_{0}^{2}=0$

and

$M_{2}^{\prime\prime}{\rm P}_{2}^{3}=-p_{1}^{2}\left[{\rm
M}^{\prime\prime}\rho\frac{\partial\Psi_{2}}{\partial
x}-N\rho\frac{\partial\Psi_{3}}{\partial x}+{\rm
N}^{\prime}\rho\frac{\partial\Psi_{3}}{\partial y}\right].$

Since $p_{1}^{2}$ is negative, we see that ${\rm P}_{3}^{2}$ will be positive,
and that ${\rm P}_{3}$ will be real and can take one of two equal but opposite
values, its sign being linked to that of $\displaystyle\frac{a_{0}}{b_{0}}$.
Moreover, ${\rm P}_{3}=2p_{1}p_{2}$ $(*)$; $p_{2}$ will thus be imaginary if
$p_{1}$ is, and will change sign with $\displaystyle\frac{a_{0}}{b_{0}}$.

The quantity

$\left[{\rm M}^{\prime\prime}\rho\frac{\partial\Psi_{2}}{\partial
x}-N\rho\frac{\partial\Psi_{3}}{\partial x}+{\rm
N}^{\prime}\rho\frac{\partial\Psi_{3}}{\partial y}\right]$

thus indicates, through its sign and magnitude, the direction and the rotation
of the plane of polarization.

If we consider a uniaxial medium, and waves polarized transverse to the axis,
these formulas simplify; in that case we have ${\rm N}={\rm N}^{\prime}=0$,
${\rm C}=0$ and there remains

${\rm P}_{3}^{2}=-p_{1}^{2}\left[\rho\frac{\partial\Psi_{2}}{\partial
x}\right]^{2},$

whence

$p_{2}=\pm i\left[\rho\frac{\partial\Psi_{2}}{\partial
x}\right]\qquad\mbox{\rm or}\qquad\mp
i\left[\rho\frac{\partial\Psi_{1}}{\partial y}\right];$

these will remain the same for a medium with cubic symmetry as well.

### Appendix 5

If it is desired merely to account for the possible effects of periodicity, it
is convenient to examine the case where $\rho$ reduces to the sum of three
functions, of $x$, of $y$ and of $z$ respectively; the solutions of the
equations then reduce to quadratures.