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Preface to the Third Edition
At the time when the second edition of this book was published the study of the liquid state
was a rapidly expanding ﬁeld of research. In the twenty years since then, the subject has
matured both theoretically and experimentally to a point where a real understanding exists
of the behaviour of “simple” liquids at the microscopic level. Although there has been a
shift in emphasis towards more complex systems, there remains in our view a place for
a book that deals with the principles of liquid-state theory, covering both statics and dy-
namics. Thus, in preparing a third edition, we have resisted the temptation to broaden too
far the scope of the book, and the focus remains ﬁrmly on simple systems, though many
of the methods we describe continue to ﬁnd a wider range of application. Nonetheless,
some reorganisation of the book has been required in order to give proper weight to more
recent developments. The most obvious change is in the space devoted to the theory of
inhomogeneous ﬂuids, an area in which considerable progress has been made since 1986.
Other major additions are sections on the properties of supercooled liquids, which include
a discussion of the mode-coupling theory of the kinetic glass transition, on theories of con-
densation and freezing and on the electric double layer. To make way for this and other
new material, some sections from the second edition have either been reduced in length or
omitted altogether. In particular, we no longer see a need to include a complete chapter on
molecular simulation, the publication of several excellent texts on the subject having ﬁlled
what was previously a serious gap in the literature. Our aim has been to emphasise what
seems to us to be work of lasting interest. Such judgements are inevitably somewhat sub-
jective and, as before, the choice of topics is coloured by our own experience and tastes. We
make no attempt to provide an exhaustive list of references, limiting ourselves to what we
consider to be the fundamental papers in different areas, along with selected applications.
We are grateful to a number of colleagues who have helped us in different ways: Dor
Ben-Amotz, Teresa Head-Gordon, David Heyes, David Grier, Bill Jorgensen, Gerhard
Kahl, Peter Monson, Anna Oleksy, Albert Reiner, Phil Salmon, Ilja Siepmann, Alan Soper,
George Stell and Jens-Boie Suck. Bob Evans made many helpful suggestions concerning
the much revised chapter on ionic liquids, George Jackson acted as our guide to the litera-
ture on the theory of associating liquids, Alberto Parola provided a valuable set of notes on
hierarchical reference theory, and Jean-Jacques Weis undertook on our behalf new Monte
Carlo calculations of the dielectric constant of dipolar hard spheres. Our task could not
have been completed without the support, encouragement and advice of these and other
colleagues, to all of whom we give our thanks. Finally, we thank the respective publishers
for permission to reproduce ﬁgures from Journal of Chemical Physics, Journal of Non-
Crystalline Solids, Physical Review and Physical Review Letters.
November 2005 J.P. H
ANSEN
I.R. MCDONALD
v

Preface to the Second Edition
The ﬁrst edition of this book was written in the wake of an unprecedented advance in
our understanding of the microscopic structure and dynamics of simple liquids. The rapid
progress made in a number of different experimental and theoretical areas had led to a
rather clear and complete picture of the properties of simple atomic liquids. In the ten
years that have passed since then, interest in the liquid state has remained very active, and
the methods described in our book have been successfully generalised and applied to a
variety of more complicated systems. Important developments have therefore been seen in
the theory of ionic, molecular and polar liquids, of liquid metals, and of the liquid surface,
while the quantitative reliability of theories of atomic ﬂuids has also improved.
In an attempt to give a balanced account both of the basic theory and of the advances of
the past decade, this new edition has been rearranged and considerably expanded relative to
the earlier one. Every chapter has been completely rewritten, and three new chapters have
been added, devoted to ionic, metallic and molecular liquids, together with substantial new
sections on the theory of inhomogeneous ﬂuids. The material contained in Chapter 10 of
the ﬁrst edition, which dealt with phase transitions, has been omitted, since it proved im-
possible to do justice to such a large ﬁeld in the limited space available. Although many
excellent review articles and monographs have appeared in recent years, a comprehensive
and up-to-date treatment of the theory of “simple” liquids appears to be lacking, and we
hope that the new edition of our book will ﬁll this gap. The choice of material again re-
ﬂects our own tastes, but we have aimed at presenting the main ideas of modern liquid-state
theory in a way that is both pedagogical and, so far as possible, self-contained. The book
should be accessible to graduate students and research workers, both experimentalists and
theorists, who have a good background in elementary statistical mechanics. We are well
aware, however, that certain sections, notably in Chapters 4, 6, 9 and 12 require more con-
centration from the reader than others. Although the book is not intended to be exhaustive,
we give many references to material that is not covered in depth in the text. Even at this
level, it is impossible to include all the relevant work. Omissions may reﬂect our ignorance
or a lack of good judgement, but we consider that our goal will have been achieved if the
book serves as an introduction and guide to a continuously growing ﬁeld.
While preparing the new edition, we have beneﬁted from the advice, criticism and help
of many colleagues. We give our sincere thanks to all. There are, alas, too many names
to list individually, but we wish to acknowledge our particular debt to Marc Baus, David
Chandler, Giovanni Ciccotti, Bob Evans, Paul Madden and Dominic Tildesley, who have
read large parts of the manuscript; to Susan O’Gorman, for her help with Chapter 4; and
to Eduardo Waisman, who wrote the ﬁrst (and almost ﬁnal) version of Appendix B. We
vi

PREFACE TO THE SECOND EDITION vii
are also grateful to those colleagues who have supplied references, preprints, and material
for ﬁgures and tables, and to authors and publishers for permission to reproduce diagrams
from published papers. The last stages of the work were carried out at the Institut Laue-
Langevin in Grenoble, and we thank Philippe Nozières for the invitations that made our
visits possible. Finally, we are greatly indebted to Martine Hansen, Christiane Lanceron,
Rehda Mazighi and Susan O’Gorman for their help and patience in the preparation of the
manuscript and ﬁgures.
May 1986 J.P. H
ANSEN
I.R. MCDONALD

Preface to the First Edition
The past ten years or so have seen a remarkable growth in our understanding of the statisti-
cal mechanics of simple liquids. Many of these advances have not yet been treated fully in
any book and the present work is aimed at ﬁlling this gap at a level similar to that of Egel-
staff’s “The Liquid State”, though with a greater emphasis on theoretical developments.
We discuss both static and dynamic properties, but no attempt is made at completeness
and the choice of topics naturally reﬂects our own interests. The emphasis throughout is
placed on theories which have been brought to a stage at which numerical comparison
with experiment can be made. We have attempted to make the book as self-contained as
possible, assuming only a knowledge of statistical mechanics at a ﬁnal-year undergraduate
level. We have also included a large number of references to work which lack of space has
prevented us from discussing in detail. Our hope is that the book will prove useful to all
those interested in the physics and chemistry of liquids.
Our thanks go to many friends for their help and encouragement. We wish, in particular,
to express our gratitude to Loup Verlet for allowing us to make unlimited use of his un-
published lecture notes on the theory of liquids. He, together with Dominique Levesque,
Konrad Singer and George Stell, have read several parts of the manuscript and made sug-
gestions for its improvement. We are also greatly indebted to Jean-Jacques Weis for his
help with the section on molecular liquids. The work was completed during a summer
spent as visitors to the Chemistry Division of the National Research Council of Canada; it
is a pleasure to have this opportunity to thank Mike Klein for his hospitality at that time
and for making the visit possible. Thanks go ﬁnally to Susan O’Gorman for her help with
mathematical problems and for checking the references; to John Copley, Jan Sengers and
Sidney Yip for sending us useful material; and to Mrs K.L. Hales for so patiently typing
the many drafts.
A number of ﬁgures and tables have been reproduced, with permission, from The Phys-
ical Review, Journal of Chemical Physics, Molecular Physics and Physica; detailed ac-
knowledgements are made at appropriate points in the text.
June 1976 J.P. H
ANSEN
I.R. MCDONALD
viii

CHAPTER1
Introduction
1.1 THE LIQUID STATE
The liquid state of matter is intuitively perceived as being intermediate in nature between
a gas and a solid. Thus a natural starting point for discussion of the properties of any given
substance is the relationship between pressureP, number densityρand temperatureT
in the different phases, summarised in the equation of statef(P,ρ,T)=0. The phase
diagram in theρ–Tplane typical of a simple, one-component system is sketched in Fig-
ure 1.1. The region of existence of the liquid phase is bounded above by the critical point
(subscript c) and below by the triple point (subscript t). Above the critical point there is
only a single ﬂuid phase, so a continuous path exists from liquid to ﬂuid to vapour; this is
not true of the transition from liquid to solid, because the solid–ﬂuid coexistence line, or
melting curve, does not terminate at a critical point. In many respects the properties of the
dense, supercritical ﬂuid are not very different from those of the liquid, and much of the
theory we develop in later chapters applies equally well to the two cases.
We shall be concerned in this book almost exclusively with classical liquids. For atomic
systems a simple test of the classical hypothesis is provided by the value of the de Broglie
thermal wavelengthΛ, deﬁned as
Λ=
λ
2πβ¯h
2
m
ρ
1/2
(1.1.1)
wheremis the mass of an atom andβ=1/k
BT. To justify a classical treatment of static
properties it is necessary thatΛbe much less thana, wherea≈ρ
−1/3
is the mean nearest-
neighbour separation. In the case of molecules we require, in addition, thatΘ
rotπT,
whereΘ
rot=¯h
2
/2IkBis a characteristic rotational temperature (Iis the molecular mo-
ment of inertia). Some typical results are shown in Table 1.1, from which we see that quantum effects should be small for all the systems listed, with the exceptions of hydrogen and neon.
Use of the classical approximation leads to an important simpliﬁcation, namely that the
contributions to thermodynamic properties which arise from thermal motion can be sepa- rated from those due to interactions between particles. The separation of kinetic and po- tential terms suggests a simple means of characterising the liquid state. LetV
Nbe the total
potential energy of a system, whereNis the number of particles, and letK
Nbe the total
1

2 INTRODUCTION
erutarepmet
density
F
VLS
T
T
critical point
triple point
c
t
tc
FIG. 1.1. Schematic phase diagram of a typical monatomic substance, showing the boundaries between solid (S
liquid (L) and vapour (V
TABLE1.1.Test of the classical hypothesis
Liquid T t/K Λ/Å Λ/a Θ rot/Tt
H2 14.13 .30 .97 6 .1
Ne 24 .50 .78 0 .26
CH
4 91 0 .46 0 .12 0 .083
N
2 63 0 .42 0 .11 0 .046
Li 454 0 .31 0 .11
Ar 84 0 .30 0 .083
HCl 159 0 .23 0 .063 0 .094
Na 371 0 .19 0 .054
Kr 116 0 .18 0 .046
CCl
4 250 0 .09 0 .017 0 .001
Λis the de Broglie thermal wavelength atT=T tanda=(V /N )
1/3
.
kinetic energy. Then in the liquid state we ﬁnd thatK N/|VN|≈1, whereasK N/|VN|β1
corresponds to the dilute gas andK
N/|VN|π1 to the low-temperature solid. Alternatively,
if we characterise a given system by a lengthσand an energyε, corresponding roughly
to the range and strength of the intermolecular forces, we ﬁnd that in the liquid region
of the phase diagram the reduced number densityρ
∗
=Nσ
3
/Vand reduced temperature
T
∗
=kBT/εare both of order unity. Liquids and dense ﬂuids are also distinguished from
dilute gases by the greater importance of collisional processes and short-range, positional
correlations, and from solids by the lack of long-range order; their structure is in many

INTERMOLECULAR FORCES AND MODEL POTENTIALS 3
TABLE1.2.Selected properties of typical simple liquids
Property Ar Na N 2
Tt/K 84 371 63
T
b/K(P=1 atm) 87 1155 77
T
c/K 151 2600 126
T
c/Tt 1.87 .02 .0
ρ
t/nm
−3
21 24 19
C
P/CV 2.21 .11 .6
L
vap/kJ mol
−1
6.599 5 .6
χ
T/10
−12
cm
2
dyn
−1
200 19 180
c/ms
−1
863 2250 995
γ/dyn cm
−1
13 191 12
D/10
−5
cm
2
s
−1
1.64 .31 .0
η/mg cm
−1
s
−1
2.87 .03 .8
λ/mW cm
−1
K
−1
1.3 8800 1 .6
(k
BT/2πDη)/Å4 .12 .73 .6
χT=isothermal compressibility,c=speed of sound,γ=surface tension,D=self-diffusion
coefﬁcient,η=shear viscosity andλ=thermal conductivity, all atT=T
t;Lvap=heat of vapor-
isation atT=T
b.
cases dominated by the “excluded-volume” effect associated with the packing together of
particles with hard cores.
Selected properties of a simple monatomic liquid (argon), a simple molecular liquid
(nitrogen
the properties of the liquid metal are in certain respects very different from those of the
other systems, notably in the values of the thermal conductivity, isothermal compressibility,
surface tension, heat of vaporisation and the ratio of critical to triple-point temperatures; the
source of these differences should become clear in Chapter 10. The quantityk
BT/2πDη
in the table provides a Stokes-law estimate of the particle diameter.
1.2 INTERMOLECULAR FORCES AND MODEL POTENTIALS
The most important feature of the pair potential between atoms or molecules is the harsh
repulsion that appears at short range and has its origin in the overlap of the outer electron
shells. The effect of these strongly repulsive forces is to create the short-range order that is
characteristic of the liquid state. The attractive forces, which act at long range, vary much
more smoothly with the distance between particles and play only a minor role in deter-
mining the structure of the liquid. They provide, instead, an essentially uniform, attractive
background and give rise to the cohesive energy that is required to stabilise the liquid. This
separation of the effects of repulsive and attractive forces is a very old-established concept.
It lies at the heart of the ideas of van der Waals, which in turn form the basis of the very
successful perturbation theories of the liquid state that we discuss in Chapter 5.

4 INTRODUCTION
The simplest model of a ﬂuid is a system of hard spheres, for which the pair potential
v(r)at a separationris
v(r)=∞,r<d,
=0,r>d
(1.2.1)
wheredis the hard-sphere diameter. This simple potential is ideally suited to the study of
phenomena in which the hard core of the potential is the dominant factor. Much of our un-
derstanding of the properties of the hard-sphere model come from computer simulations.
Such calculations have revealed very clearly that the structure of a hard-sphere ﬂuid does
not differ in any signiﬁcant way from that corresponding to more complicated interatomic
potentials, at least under conditions close to crystallisation. The model also has some rele-
vance to real, physical systems. For example, the osmotic equation of state of a suspension
of micron-sized silica spheres in an organic solvent matches almost exactly that of a hard-
sphere ﬂuid.
1
However, although simulations show that the hard-sphere ﬂuid undergoes
a freezing transition atρ
∗
(=ρd
3
)≈0.945, the absence of attractive forces means that
there is only one ﬂuid phase. A simple model that can describe a true liquid is obtained by
supplementing the hard-sphere potential with a square-well attraction, as illustrated in Fig-
ure 1.2(a ε, the well depth, and(γ−1),the
width of the well in units ofd, whereγtypically has a value of about 1.5. An alternative
to the square-well potential with features that are of particular interest theoretically is the
hard-core Yukawa potential, given by
v(r)=∞,r
∗
<1,
=−
ε
r
∗
exp
≈
−λ(r
∗
−1)
π
,r
∗
>1
(1.2.2)
wherer
∗
=r/dand the parameterλmeasures the inverse range of the attractive tail in the
potential. The two examples plotted in Figure 1.2(b λappropriate
either to the interaction between rare-gas atoms (λ=2) or to the short-range, attractive
forces
2
characteristic of certain colloidal systems(λ=8).
A more realistic potential for neutral atoms can be constructed by a detailed quantum-
mechanical calculation. At large separations the dominant contribution to the potential
comes from the multipolar dispersion interactions between the instantaneous electric mo-
ments on one atom, created by spontaneous ﬂuctuations in the electronic charge distribu-
tion, and moments induced in the other. All terms in the multipole series represent attractive
contributions to the potential. The leading term, varying asr
−6
, describes the dipole–
dipole interaction. Higher-order terms represent dipole–quadrupole(r
−8
), quadrupole–
quadrupole(r
−10
)interactions, and so on, but these are generally small in comparison
with the term inr
−6
.
A rigorous calculation of the short-range interaction presents greater difﬁculty, but over
relatively small ranges ofrit can be adequately represented by an exponential function of
the form exp(−r/r
0), wherer 0is a range parameter. This approximation must be supple-
mented by requiring thatv(r)→∞forrless than some arbitrarily chosen, small value.
In practice, largely for reasons of mathematical convenience, it is more usual to represent
the short-range repulsion by an inverse-power law, i.e.r
−n
, withnlying generally in the

INTERMOLECULAR FORCES AND MODEL POTENTIALS 5
0.5 1.0 1.5 2.0 2.5
/ )r(v
( - 1)d
(a) square-well potential
1
0
-1
0.5 1.0 1.5 2.0 2.5
= 8
= 2
1
0
-1
r / d
/ )r(v
(b) Yukawa potential
FIG. 1.2. Simple pair potentials for monatomic systems. See text for details.
range 9 to 15. The behaviour ofv(r)in the limiting casesr→∞andr→0 may therefore
be incorporated in a simple potential function of the form
v(r)=4ε
≈
(σ/r)
12
−(σ/r)
6
π
(1.2.3)
which is the famous 12-6 potential of Lennard-Jones. Equation (1.2.3) involves two para-
meters: the collision diameterσ, which is the separation of the particles wherev(r)=0;
andε, the depth of the potential well at the minimum inv(r). The Lennard-Jones potential
provides a fair description of the interaction between pairs of rare-gas atoms and also of
quasi-spherical molecules such as methane. Computer simulations
3
have shown that the
triple point of the Lennard-Jones ﬂuid is atρ
∗
≈0.85,T
∗
≈0.68.
Experimental information on the pair interaction can be extracted from a study of any
process that involves collisions between particles.
4
The most direct method involves the
measurement of atom–atom scattering cross-sections as a function of incident energy and
scattering angle; inversion of the data allows, in principle, a determination of the pair po-

6 INTRODUCTION
tential at all separations. A simpler procedure is to assume a speciﬁc form for the potential
and determine the parameters by ﬁtting to the results of gas-phase measurements of quan-
tities such as the second virial coefﬁcient (see Chapter 3) or the shear viscosity. In this way,
for example, the parametersεandσin the Lennard-Jones potential have been determined
for a large number of gases.
The theoretical and experimental methods we have mentioned all relate to the properties
of an isolated pair of molecules. The use of the resulting pair potentials in calculations
for the liquid state involves the neglect of many-body forces, an approximation that is
difﬁcult to justify. In the rare-gas liquids, the three-body, triple-dipole dispersion term is
the most important many-body interaction; the net effect of triple-dipole forces is repulsive,
amounting in the case of liquid argon to a few percent of the total potential energy due
to pair interactions. Moreover, careful measurements, particularly those of second virial
coefﬁcients at low temperatures, have shown that the true pair potential for rare-gas atoms
is not of the Lennard-Jones form, but has a deeper bowl and a weaker tail, as illustrated by
the curves plotted in Figure 1.3. Apparently the success of the Lennard-Jones potential in
accounting for many of the macroscopic properties of argon-like liquids is the consequence
of a fortuitous cancellation of errors. A number of more accurate pair potentials have been
developed for the rare gases, but their use in the calculation of condensed-phase properties
requires the explicit incorporation of three-body interactions.
Although the true pair potential for rare-gas atoms is not the same as the effective pair
potential used in liquid-state work, the difference is a relatively minor, quantitative one.
The situation in the case of liquid metals is different, because the form of the effective
ion–ion interaction is strongly inﬂuenced by the presence of a degenerate gas of con-
duction electrons that does not exist before the liquid is formed. The calculation of the
ion–ion interaction is a complicated problem, as we shall see in Chapter 10. The ion–
electron interaction is ﬁrst described in terms of a “pseudopotential” that incorporates both
the coulombic attraction and the repulsion due to the Pauli exclusion principle. Account
-200
-100
0
100
200
34567
K / )r(v
r / Å
Ar-Ar potentials
FIG. 1.3. Pair potentials for argon in temperature units. Full curve: the Lennard-Jones potential with parameter
valuesε/k
B=120 K,σ=3.4 Å, which is a good effective potential for the liquid; dashes: a potential based on
gas-phase data.
5

INTERMOLECULAR FORCES AND MODEL POTENTIALS 7
-400
-200
0
200
400
345678
K / )r(v
r / Å
liquid K
1000
2000
3000
2.8 3.2 3.6
Ar K
FIG. 1.4. Main ﬁgure: effective ion–ion potential (in temperature units) for liquid potassium.
6
Inset: comparison
on a logarithmic scale of potentials for argon and potassium in the core region.
must then be taken of the way in which the pseudopotential is modiﬁed by interaction be-
tween the conduction electrons. The end result is a potential that represents the interaction
between screened, electrically neutral “pseudoatoms”. Irrespective of the detailed assump-
tions made, the main features of the potential are always the same: a soft repulsion, a deep
attractive well and a long-range oscillatory tail. The potential, and in particular the depth of
the well, are strongly density dependent but only weakly dependent on temperature. Fig-
ure 1.4 shows an effective potential for liquid potassium. The differences compared with
the potentials for argon are clear, both at long range and in the core region.
For molten salts and other ionic liquids in which there is no shielding of the electro-
static forces similar to that found in liquid metals, the coulombic interaction provides the
dominant contribution to the interionic potential. There must, in addition, be a short-range
repulsion between ions of opposite charge, without which the system would collapse, but
the detailed way in which the repulsive forces are treated is of minor importance. Polarisa-
tion of the ions by the internal electric ﬁeld also plays a role, but such effects are essentially
many-body in nature and cannot be adequately represented by an additional term in the pair
potential.
Description of the interaction between two molecules poses greater problems than for
spherical particles because the pair potential is a function both of the separation of the
molecules and of their mutual orientation. The model potentials discussed in this book di-
vide into two classes. The ﬁrst consists of highly idealised models of polar liquids in which
a point dipole–dipole interaction is superimposed on a spherically symmetric potential. In
this case the pair potential for particles labelled 1 and 2 has the general form
v(1,2)=v
0(R)−μ
1·T(R)·μ
2 (1.2.4)

8 INTRODUCTION
whereRis the vector separation of the molecular centres,v 0(R)is the spherically sym-
metric term,μ
iis the dipole-moment vector of particleiandT(R)is the dipole–dipole
interaction tensor:
T(R)=3RR/R
5
−I/R
3
(1.2.5)
whereIis the unit tensor. Two examples of (1.2.4
of dipolar hard spheres, wherev
0(R)is the hard-sphere potential, and the Stockmayer po-
tential, wherev
0(R)takes the Lennard-Jones form. Both these models, together with ex-
tensions that include, for example, dipole–quadrupole and quadrupole–quadrupole terms,
have received much attention from theoreticians. Their main limitation as models of real
molecules is the fact that they ignore the angle dependence of the short-range forces. A sim-
ple way to take account of such effects is through the use of potentials of the second main
type with which we shall be concerned. These are models in which the molecule is repre-
sented by a set of discreteinteraction sitesthat are commonly, but not invariably, located at
the sites of the atomic nuclei. The total potential energy of two interaction-site molecules
is then obtained as the sum of spherically symmetric, interaction-site potentials. Letr
iαbe
the coordinates of siteαin moleculeiand letr
jβbe the coordinates of siteβin moleculej.
Then the total intermolecular potential energy is
v(1,2)=
1
2
β
α
β
β
vαβ
Θ
|r
2β−r1α|
σ
(1.2.6)
wherev
αβ(r)is a site–site potential and the sums onαandβrun over all interaction
sites in the respective molecules. Electrostatic interactions are easily allowed for by inclu-
sion of coulombic terms in the site–site potentials. Let us take as an example the particu-
larly simple case of a homonuclear diatomic, such as that pictured in Figure 1.5. A crude
interaction-site model would be that of a “hard dumb-bell”, consisting of two overlapping
hard spheres of diameterdwith their centres separated by a distanceL<2d. This should
be adequate to describe the main structural features of a liquid such as nitrogen. An obvious
improvement would be to replace the hard spheres by two Lennard-Jones interaction sites,
with parameters chosen to ﬁt, say, the experimentally determined equation of state. Some
homonuclear diatomics also have a large quadrupole moment, which plays a signiﬁcant
role in determining the short-range angular correlations in the liquid. The model could in
that case be further reﬁned by placing point chargesqat the Lennard-Jones sites, together
L
q q
-2q
FIG. 1.5. An interaction-site model of a homonuclear diatomic.

EXPERIMENTAL METHODS 9
with a compensating charge−2qat the mid-point of the internuclear bond; such a charge
distribution has zero dipole moment but a non-vanishing quadrupole moment proportional
toqL
2
. Models of this general type have proved remarkably successful in describing the
properties of a wide variety of molecular liquids, both simple and complicated.
1.3 EXPERIMENTAL METHODS
The experimental methods available for studying the properties of simple liquids may be
placed in one of two broad categories, depending on whether they are concerned with
measurements on a macroscopic or microscopic scale. In general, the calculated micro-
scopic properties are more sensitive to the approximations used in a theory and to the
assumptions made about the pair potentials, but the macroscopic properties can usually be
measured with considerably greater accuracy. The two types of measurement are there-
fore complementary, each providing information that is useful in the development of a
statistical-mechanical theory of the liquid state.
The classic macroscopic measurements are those of thermodynamic properties, partic-
ularly of the equation of state. Integration of accurateP–ρ–Tdata yields information
on other thermodynamic quantities, which can be supplemented by calorimetric measure-
ments. For most liquids the pressure is known as a function of temperature and density only
in the vicinity of the liquid–vapour equilibrium line, but for certain systems of particular
theoretical interest experiments have been carried out at much higher pressures; the low
compressibility of a liquid near its triple point means that highly specialised techniques
are required. The second main class of macroscopic measurements are those relating to
transport coefﬁcients. A variety of experimental methods are used. The shear viscosity,
for example, can be determined from the observed damping of torsional oscillations or
from capillary-ﬂow experiments, while the thermal conductivity can be obtained from a
steady-state measurement of the transfer of heat between a central ﬁlament and a surround-
ing cylinder or between parallel plates. A direct method of determining the coefﬁcient of
self-diffusion involves the use of radioactive tracers, which places it in the category of mi-
croscopic measurements; in favourable cases the diffusion coefﬁcient can be measured by
nuclear magnetic resonance (NMR
Raman) are also useful in the study of reorientational motion in molecular liquids, while
dielectric-response measurements provide information on the slow, structural relaxation in
supercooled liquids near the glass transition.
Much the most important class of microscopic measurements, at least from the theoreti-
cal point of view, are the radiation-scattering experiments. Elastic scattering of neutrons or
x-rays, in which the scattering cross-section is measured as a function of momentum trans-
fer between the radiation and the sample, is the source of our experimental knowledge of
the static structure of a ﬂuid. In the case of inelastic scattering the cross-section is measured
as a function of both momentum and energy transfer. It is thereby possible to extract in-
formation on wavenumber and frequency-dependent ﬂuctuations in liquids at wavelengths
comparable with the spacing between particles. This provides a very powerful method of
studying microscopic time-dependent processes in liquids. Inelastic light-scattering exper-
iments give similar information, but the accessible range of momentum transfer limits the

10 INTRODUCTION
method to the study of ﬂuctuations of wavelength of order 10
−5
cm, corresponding to the
hydrodynamic regime. Such experiments are, however, of considerable value in the study
of colloidal dispersions and of critical phenomena.
Finally, there are a range of techniques of a quasi-experimental character, referred to
collectively as computer simulation, the importance of which in the development of liquid-
state theory can hardly be overstated. Simulation provides what are essentially exact results
for a given potential model; its usefulness rests ultimately on the fact that a sample con-
taining a few hundred or few thousand particles is in many cases sufﬁciently large to sim-
ulate the behaviour of a macroscopic system. There are two classic approaches: theMonte
Carlomethod and the method ofmolecular dynamics. There are many variants of each,
but in broad terms a Monte Carlo calculation is designed to generate static conﬁgurations
of the system of interest, while molecular dynamics involves the solution of the classical
equations of motion of the particles. Molecular dynamics therefore has the advantage of
allowing the study of time-dependent processes, but for the calculation of static properties
a Monte Carlo method is often more efﬁcient. Chapter 2 contains a brief discussion of the
principles underlying the two types of calculation.
NOTES AND REFERENCES
1. Vrij, A., Jansen, J.W., Dhont, J.K.G., Pathmamanoharan, C., Kops-Werkhoven, M.M. and Fijnaut, H.M.,Fara-
day Disc.76, 19 (1983).
2. See, e.g., Meijer, E.J. and Frenkel, D.,Phys. Rev. Lett.67, 1110 (1991
colloidal suspension can be modelled by a Yukawa potential with a positive tail.
3. Hansen, J.P. and Verlet, L.,Phys. Rev.184, 151 (1969
4. Maitland, G.C., Rigby, M., Smith, E.B. and Wakeham, W.A., “Intermolecular Forces”. Clarendon Press, Ox-
ford, 1981.
5. Model BBMS of ref. 4, p. 497.
6. Dagens, L., Rasolt, M. and Taylor, R.,Phys.Rev.B11, 2726 (1975

CHAPTER2
Statistical Mechanics
This chapter is devoted to a summary of the principles of classical statistical mechanics,
a discussion of the link between statistical mechanics and thermodynamics, and the de-
ﬁnition of certain equilibrium and time-dependent distribution functions of fundamental
importance in the theory of liquids. It also establishes much of the notation used in later
parts of the book. The focus throughout is on atomic systems; some of the complications
that arise in the study of molecular liquids are discussed in Chapter 11.
2.1 TIME EVOLUTION AND KINETIC EQUATIONS
Consider an isolated, macroscopic system consisting ofNidentical, spherical particles
of massmenclosed in a volumeV. An example would be a one-component, monatomic
gas or liquid. In classical mechanics the dynamical state of the system at any instant is
completely speciﬁed by the 3Ncoordinatesr
N
≡r1,...,r Nand 3Nmomentap
N
≡
p
1,...,p Nof the particles. The values of these 6Nvariables deﬁne aphase pointin a
6N-dimensionalphase space.LetHbe the hamiltonian of the system, which we write in
general form as
H
Θ
r
N
,p
N
σ
=K
N
Θ
p
N
σ
+V
N
Θ
r
N
σ
+Φ
N
Θ
r
N
σ
(2.1.1)
where
K
N=
N
β
i=1
|pi|
2
2m
(2.1.2)
is the kinetic energy,V
Nis the interatomic potential energy andΦ Nis the potential energy
arising from the interaction of the particles with some spatially varying, external ﬁeld.
If there is no external ﬁeld, the system will be both spatially uniform and isotropic. The
motion of the phase point along itsphase trajectoryis determined by Hamilton’s equations:
˙r
i=
∂H
∂pi
, ˙p i=−
∂H
∂ri
(2.1.3)
These equations are to be solved subject to 6Ninitial conditions on the coordinates and mo-
menta. Since the trajectory of a phase point is wholly determined by the values ofr
N
,p
N
11

12 STATISTICAL MECHANICS
at any given time, it follows that two different trajectories cannot pass through the same
point in phase space.
The aim of equilibrium statistical mechanics is to calculate observable properties of a
system of interest either as averages over a phase trajectory (the method of Boltzmann),
or as averages over an ensemble of systems, each of which is a replica of the system of
interest (the method of Gibbs). The main features of the two methods are reviewed in later
sections of this chapter. Here it is sufﬁcient to recall that in Gibbs’s formulation of statisti-
cal mechanics the distribution of phase points of systems of the ensemble is described by
aphase-space probability densityf
[N]
(r
N
,p
N
;t). The quantityf
[N]
dr
N
dp
N
is the prob-
ability that at timetthe physical system is in a microscopic state represented by a phase
point lying in the inﬁnitesimal, 6N-dimensional phase-space element dr
N
dp
N
. This deﬁ-
nition implies that the integral off
[N]
over all phase space is

f
[N]
Θ
r
N
,p
N
;t
σ
dr
N
dp
N
=1 (2.1.4)
for allt. Given a complete knowledge of the probability density it would be possible to
calculate the average value of any function of the coordinates and momenta.
The time evolution of the probability density at a ﬁxed point in phase space is governed
by the Liouville equation, which is a 6N-dimensional analogue of the equation of conti-
nuity of an incompressible fluid; it describes the fact that phase points of the ensemble are
neither created nor destroyed as time evolves. The Liouville equation may be written either
as
∂f
[N]
∂t
+
N
β
i=1
λ
∂f
[N]
∂ri
·˙ri+
∂f
[N]
∂pi
·˙pi
ρ
=0 (2.1.5)
or, more compactly, as
∂f
[N]
∂t
=

H,f
[N]

(2.1.6)
where{A,B}denotes the Poisson bracket:
{A,B}≡
N
β
i=1
λ
∂A
∂ri
·
∂B
∂pi
−
∂A
∂pi
·
∂B
∂ri
ρ
(2.1.7)
Alternatively, by introducing the Liouville operatorL, deﬁned as
L≡i{H,} (2.1.8)
the Liouville equation becomes
∂f
[N]
∂t
=−iLf
[N]
(2.1.9)

TIME EVOLUTION AND KINETIC EQUATIONS 13
the formal solution to which is
f
[N]
(t)=exp(−iLt)f
[N]
(0) (2.1.10)
The Liouville equation can be expressed even more concisely in the form
df
[N]
dt
=0 (2.1.11)
where d/dtdenotes the total derivative with respect to time. This result is called the
Liouville theorem. The meaning of the Liouville theorem is that the probability density,
as seen by an observer moving with a phase point along its phase trajectory, is indepen-
dent of time. Consider the phase points that at timet=0 are contained within a phase-
space element dr
N
(0)dp
N
(0). As time increases, the element will change in shape but no
phase points will enter or leave, otherwise phase trajectories would cross each other. The
Liouville theorem therefore implies that the volume of the element must remain the same:
volume in phase space is said to be “conserved”. In mathematical terms, conservation of
volume in phase space is equivalent to the statement that the jacobian corresponding to
the transformationr
N
(0),p
N
(0)→r
N
(t),p
N
(t)is equal to unity; this is easily proved
explicitly.
1
The time dependence of any function of the phase-space variables,B(r
N
,p
N
)say, may
be represented in a manner similar to (2.1.9). AlthoughBis not an explicit function oft,it
will in general change with time as the system moves along its phase trajectory. The time
derivative ofBis therefore given by
dB
dt
=
N
β
i=1
λ
∂B
∂ri
·˙ri+
∂B
∂pi
·˙pi
ρ
(2.1.12)
or, from Hamilton’s equations:
dB
dt
=
N
β
i=1
λ
∂B
∂ri
·
∂H
∂pi
−
∂B
∂pi
·
∂H
∂ri
ρ
=iLB (2.1.13)
which has as its solution
B(t)=exp(iLt)B(0) (2.1.14)
Note the change of sign in the propagator compared with (2.1.10
The description of the system that the full phase-space probability density provides is for
many purposes unnecessarily detailed. Normally we are interested only in the behaviour
of a subset of particles of sizen, say, and the redundant information can be eliminated
by integratingf
[N]
over the coordinates and momenta of the other(N−n)particles. We
therefore deﬁne areduced phase-space distribution functionf
(n)
(r
n
,p
n
;t)by
f
(n)
Θ
r
n
,p
n
;t
σ
=
N!
(N−n)!

f
[N]
Θ
r
N
,p
N
;t
σ
dr
(N−n)
dp
(N−n)
(2.1.15)

14 STATISTICAL MECHANICS
wherer
n
≡r1,...,r nandr
(N−n)
≡rn+1,...,r N, etc. The quantityf
(n)
dr
n
dp
n
yields
the probability of ﬁnding a subset ofnparticles in the reduced phase-space element
dr
n
dp
n
at timet, irrespective of the coordinates and momenta of the remaining particles;
the combinatorial factorN!/(N−n)!is the number of ways of choosing a subset of sizen.
To ﬁnd an equation of motion forf
(n)
we consider the special case when the total
force acting on particleiis the sum of an external forceX
i, arising from an external
potentialφ(r
i), and of pair forcesF ijdue to other particlesj, withF ii=0. The second of
Hamilton’s equations (2.1.3
∂H
∂ri
=−X i−
N
β
j=1
Fij (2.1.16)
and the Liouville equation becomes

∂
∂t
+
N
β
i=1
pi
m
·
∂
∂ri
+
N
β
i=1
Xi·
∂
∂pi

f
[N]
=−
N
β
i=1
N
β
j=1
Fij·
∂f
[N]
∂pi
(2.1.17)
We now multiply through byN!/(N−n)!and integrate over the 3(N−n)coordinates
r
n+1,...,r Nand 3(N−n)momentap n+1,...,p N. The probability densityf
[N]
is zero
whenr
ilies outside the volume occupied by the system and must vanish asp i→∞to
ensure convergence of the integrals over momenta in (2.1.4). Thusf
[N]
vanishes at the
limits of integration and the derivative off
[N]
with respect to any component of position
or momentum will contribute nothing to the result when integrated with respect to that
component. On integration, therefore, all terms disappear for whichi>nin (2.1.17
remains, given the deﬁnition off
(n)
in (2.1.15

∂
∂t
+
n
β
i=1
pi
m
·
∂
∂ri
+
n
β
i=1
Xi·
∂
∂pi

f
(n)
=−
n
β
i=1
n
β
j=1
Fij·
∂f
(n)
∂pi
−
N!
(N−n)!
n
β
i=1
N
β
j=n+1

F
ij·
∂f
[N]
∂pi
dr
(N−n)
dp
(N−n)
(2.1.18)
Because the particles are identical,f
[N]
is symmetric with respect to interchange of parti-
cle labels and the sum of terms forj=n+1toNon the right-hand side of (2.1.18
replaced by(N−n)times the value of any one term. This simpliﬁcation makes it possible
to rewrite (2.1.18 f
(n)
to that off
(n+1)
:

∂
∂t
+
n
β
i=1
pi
m
·
∂
∂ri
+
n
β
i=1

X
i+
n
β
j=1
Fij

·
∂
∂pi

f
(n)
=−
n
β
i=1

F
i,n+1·
∂f
(n+1)
∂pi
drn+1dpn+1 (2.1.19)

TIME EVOLUTION AND KINETIC EQUATIONS 15
The system of coupled equations represented by (2.1.19
subsequently rederived by others. It is known as the Bogolyubov–Born–Green–Kirkwood–
Yvon or BBGKY hierarchy. The equations are exact, though limited in their applicability to
systems for which the particle interactions are pairwise additive. They are not immediately
useful, however, because they merely express one unknown function,f
(n)
, in terms of
another,f
(n+1)
. Some approximateclosure relationis therefore needed.
In practice the most important member of the BBGKY hierarchy is that corresponding
ton=1:
λ
∂∂t
+
p
1
m
·
∂
∂r1
+X1·
∂
∂p1
∞
f
(1)
(r1,p1;t)
=−

F
12·
∂
∂p1
f
(2)
(r1,p1,r2,p2;t)dr 2dp2 (2.1.20)
Much effort has been devoted to ﬁnding approximate solutions to (2.1.20
of expressions that relate the two-particle distribution functionf
(2)
to the single-particle
functionf
(1)
. From the resultingkinetic equationsit is possible to calculate the hydrody-
namic transport coefﬁcients, but the approximations made are rarely appropriate to liquids
because correlations between particles are mostly treated in a very crude way.
2
The sim-
plest possible approximation is to ignore pair correlations altogether by writing
f
(2)
(r,p,r

,p

;t)≈f
(1)
(r,p;t)f
(1)
(r

,p

;t) (2.1.21)
This leads to the Vlasov equation:
λ
∂
∂t
+
p
m
·
∂
∂r
+
≈
X(r,t)+F(r,t)
π
·
∂
∂p
∞
f
(1)
(r,p;t)=0 (2.1.22)
where the quantity
F(r,t)=

F(r,r

;t)f
(1)
(r

,p

;t)dr

dp

(2.1.23)
is the average force exerted by other particles, situated at pointsr

, on a particle that at
timetis at a pointr; this is an approximation of classic mean-ﬁeld type. Though obvi-
ously not suitable for liquids, the Vlasov equation is widely used in plasma physics, where
the long-range character of the Coulomb potential justiﬁes a mean-ﬁeld treatment of the
interactions.
Equation (2.1.20) may be rewritten schematically in the form
λ
∂
∂t
+
p
1
m
·
∂
∂r1
+X1·
∂
∂p1
∞
f
(1)
=
λ
∂f
(1)
∂t
∞
coll
(2.1.24)
where the term(∂f
(1)
/∂t)collis the rate of change off
(1)
due to collisions between par-
ticles. The collision term is given rigorously by the right-hand side of (2.1.20

16 STATISTICAL MECHANICS
Vlasov equation it is eliminated by replacing the true external forceX(r,t)by an effec-
tive force – the quantity inside square brackets in (2.1.22) – which depends in part on
f
(1)
itself. For this reason the Vlasov equation is called a “collisionless” approximation. In
the most famous of all kinetic equations, derived by Boltzmann more than a century ago,
(∂f
(1)
/∂t)collis evaluated with the help of two assumptions, which in general are justi-
ﬁed only at low densities: that two-body collisions alone are involved and that successive
collisions are uncorrelated.
2
The second of these assumptions, that of “molecular chaos”,
corresponds formally to supposing that the factorisation represented by (2.1.21
prior to any collision, though not subsequently. In simple terms it means that when two
particles collide, no memory is retained of any previous encounters between them, an as-
sumption that clearly breaks down when recollisions are frequent events. A binary collision
at a pointris characterised by the momentap
1,p2of the two particles before collision and
their momentap

1
,p

2
afterwards; the post-collisional momenta are related to their pre-
collisional values by the laws of classical mechanics. With Boltzmann’s approximations
the collision term in (2.1.24) becomes
λ
∂f
(1)
∂t
∞
coll
=
1
m

σ(Ω,Δp)
≈
f
(1)
(r,p

1
;t)f
(1)
(r,p

2
;t)
−f
(1)
(r,p1;t)f
(1)
(r,p2;t)
π
dΩdp 2 (2.1.25)
whereΔp≡|p
2−p1|andσ(Ω,Δp)is the differential cross-section for scattering into a
solid angle dΩ. As Boltzmann showed, this form of the collision term is able to account for
the fact that many-particle systems evolve irreversibly towards an equilibrium state. This
irreversibility is described by Boltzmann’s H-theorem; the source of the irreversibility is
the assumption of molecular chaos.
Solution of the Boltzmann equation leads to explicit expressions for the hydrodynamic
transport coefﬁcients in terms of certain “collision” integrals.
3
The differential scattering
cross-section and hence the collision integrals themselves can be evaluated numerically for
a given choice of two-body interaction, though for hard spheres they have a simple, ana-
lytical form. The results, however, are applicable only to dilute gases. In the case of hard
spheres the Boltzmann equation was later modiﬁed semi-empirically by Enskog in a man-
ner that extends its range of applicability to considerably higher densities. Enskog’s theory
retains the two key assumptions involved in the derivation of the Boltzmann equation, but
it also corrects in two ways for the ﬁnite size of the colliding particles. First, allowance is
made for the modiﬁcation of the collision rate by the hard-sphere interaction. Because the
same interaction is also responsible for the increase in pressure over its ideal-gas value,
the enhancement of the collision rate relative to its low-density limit can be calculated if
the hard-sphere equation of state is known. Secondly, “collisional transfer” is incorporated
into the theory by rewriting (2.1.25) in a form in which the distribution functions for the
two colliding particles are evaluated not the same point,r, but at points separated by a
distance equal to the hard-sphere diameter. This is an important modiﬁcation of the the-
ory, because at high densities interactions rather than particle displacements provide the
dominant mechanism for the transport of energy and momentum.
The phase-space probability density of a system in thermodynamic equilibrium is a func-
tion of the time-varying coordinates and momenta, but is independent oftat each point in

TIME AVERAGES AND ENSEMBLE AVERAGES 17
phase space. We shall use the symbolf
[N]
0
(r
N
,p
N
)to denote the equilibrium probability
density; it follows from (2.1.6) that a sufﬁcient condition for a probability density to be de-
scriptive of a system in equilibrium is that it should be some function of the hamiltonian.
Integration off
[N]
0
over a subset of coordinates and momenta in the manner of (2.1.15
yields a set of equilibrium phase-space distribution functionsf
(n)
0
(r
n
,p
n
). The casen=1
corresponds to the equilibrium single-particle distribution function; if there is no external
ﬁeld the distribution is independent ofrand has the familiar maxwellian form, i.e.
f
(1)
0
(r,p)=
ρexp(−β|p|
2
/2m)
(2πmkBT)
3/2
≡ρfM(p) (2.1.26)
wheref
M(p)is the Maxwell distribution of momenta, normalised such that

f
M(p)dp=1 (2.1.27)
The corresponding distribution of velocitiesuis
φ
M(u)=
λ
m
2πkBT
∞
3/2
exp
∗
−mβ|u|
2
/2
→
(2.1.28)
2.2 TIME AVERAGES AND ENSEMBLE AVERAGES
Certain thermodynamic properties of a physical system may be written as averages of func-
tions of the coordinates and momenta of the constituent particles. These are the so-called
“mechanical” properties, which include internal energy and pressure; “thermal” properties
such as entropy are not expressible in this way. In a state of thermal equilibrium these av-
erages must be independent of time. To avoid undue complications we again suppose that
the system of interest consists ofNidentical, spherical particles. If the system is isolated
from its surroundings, its total energy is constant, i.e. the hamiltonian is a constant of the
motion.
As before, letB(r
N
,p
N
)be some function of the 6Nphase-space variables and letB
be its average value, where the angular brackets represent an averaging process of a nature
as yet unspeciﬁed. Given the coordinates and momenta of the particles at some instant, their
values at any later (or earlier) time can in principle be obtained as the solution to Newton’s
equations of motion, i.e. to a set of 3Ncoupled, second-order, differential equations which,
in the absence of an external ﬁeld, have the form
m¨r
i=Fi=−∇ iVN
∗
r
N
→
(2.2.1)
whereF
iis the total force on particlei. It is therefore natural to viewBas a time average
over the dynamical history of the system, i.e.
B
t=lim
τ→∞
1
τ

τ
0
B
≈
r
N
(t),p
N
(t)
π
dt (2.2.2)

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and melancholy glance at his comrades, who witnessed his

departure with aching hearts and tears in their eyes. Henceforth the
divorce between the count and the adventurers was accomplished.
General Guerrero, on the count's entrance to Guaymas, ordered the
honours due to a commander-in-chief to be paid him. Don Louis
smiled with disdain. What did he care for these empty ceremonies?
The count and the general had a lengthened conversation together.
The general had not yet given up his projects of seduction; but this
time, like the first, the count answered with a positive refusal.
The company was henceforth surrendered defencelessly to the
machinations of Señor Pavo. This man lost no time, and by his
advice the adventurers sent as a deputation to the count two
ignorant sailors, with orders to come to a settlement with him at any
price. These two emissaries were selected by Señor Pavo, for the
worthy man knew perfectly well what he was about. The two sailors
presented themselves at the count's house, who sent out a message
to them that he was engaged at the moment, and begged them to
wait a little while. The ambassadors, ruffled in their self-esteem, and
puffed up with the importance of the mission intrusted to them, left
the count's house immediately, swearing at his insolence, and went
straight to the palace of General Guerrero.
The latter, advised beforehand, knew what would happen, and was
impatiently awaiting them. He ordered them to be admitted at once
so soon as they sent in their names, and received them most
graciously: then, when he had sufficiently intoxicated them with
flattery, he made them sign—that is to say, make a cross at the foot
of—a treaty, in which they recognised that, having been deceived
and abandoned in a cowardly manner by their chief, they pledged
themselves to lay down their arms and quit the country for a sum of
eleven thousand piastres.
[1]
We must confess that General Guerrero
made a capital bargain, for the arms came into his possession. Oh!
the Mexicans are famous negotiators, and, above all, most crafty
diplomatists.

Unable to vanquish the company, the Mexicans bought it of two
scoundrels, by the intervention of a third, whose duty it was to
defend it.
Thus the Atrevida Company had committed suicide. It effected its
own dissolution without even attempting to see once again that chief
who had been its idol, and whom it abandoned writhing on a bed of
suffering.
We must mention, to the honour of the French plenipotentiaries
that, in the treaty they signed, the liberty of the count was formally
guaranteed.
Now let us see by what extraordinary concourse of circumstances
the count, when in such a critical position, was thus abandoned by
all his friends. How was it that General Guerrero, his obstinate foe,
had shown himself so kind and almost generous toward Don Louis
during the last events we have narrated?
We will proceed to explain this; but, in order to do so, we must take
up events further back, and return to Valentine and his comrades,
whom we left galloping at full speed along the road to the hacienda.
[1] A little over £2000.
CHAPTER XXIII.
THE HACIENDA DEL MILAGRO.
The road from Hermosillo to the Hacienda del Milagro is perfectly
well traced, straight and wide along the entire distance. Although
the night was gloomy and unlit by the moon, as the five horsemen
galloped on side by side, it would have been impossible for them to
pass Don Cornelio without seeing him, had they caught him up; but
they reached the hacienda without receiving any tidings of him.

The road had been so trampled in every direction during the last few
days, both by French and Mexicans, that it was impossible for these
experienced hunters to distinguish or take up any imprint which
could serve to guide them in their researches. The traces of horses,
wagons, and men were so interlaced in each other, that they were
completely illegible, even to the most practised eye. Several times
Valentine tried, though in vain, to read this book of the desert.
Hence, the nearer the hunters drew to their destination, the more
alarmed and anxious they became.
It was about eight in the morning when they reached the hacienda:
they had travelled the whole night through without stopping, save to
search for traces of the man they were pursuing. The hacienda was
tranquil; the peons were engaged in their ordinary labours; the
ganado was grazing at liberty on the prairies. When the hunters
entered, Don Rafaël was preparing to mount on horse, seemingly to
take a ride round his farm. A peon was holding a magnificent
mustang, which champed its bit and snorted impatiently at being
held so long. When the hacendero perceived the newcomers he ran
toward them, playfully menacing them with his chicote.
"Ah!" he said with a laugh, "here are my deserters returned. Good
morning, gentlemen."
The latter, astonished at this merry reception, which they did not at
all comprehend, remained dumb. Don Rafaël then noticed their
gloomy and embarrassed air.
"Hilloh! what is the matter with you?" he asked seriously. "Are you
the bearers of ill news?"
"Perhaps so," Valentine answered sadly. "May Heaven grant that I
am mistaken!"
"Speak—explain yourself. I was mounting to go and obtain news
about you; but as you are here, it is unnecessary."
The hunters exchanged an intelligent glance.
"Of course we will furnish you with all the details you may wish for."

"All the better. In the first place, then, dismount and come into the
house, where we shall converse more at our ease."
The hunters obeyed, and followed Don Rafaël into a vast apartment
which served as the hacendero's business room. When they entered
Valentine opposed the closing of the door.
"In that way," he said, "we shall not have to fear listeners."
"Why such precautions?"
"I will tell you. Where are Doña Angela and Doña Luz at this
moment?"
"They are probably still asleep."
"Very good. Tell me, Loyal Heart, have you received any visitor
during the last twenty-four hours?"
"I have not seen a living soul since the departure of the Count de
Prébois Crancé."
"Ah!" the hunter said, "then a courier did not arrive last night?"
"None."
"So that you are ignorant of the deeds accomplished yesterday?"
"Utterly."
"You are not aware that the count fought a battle yesterday?"
"No."
"That he took Hermosillo by assault?"
"No."
"And that General Guerrero's army is utterly routed?"
"No. Is what you tell me really the truth?"
"The most perfect truth."
"In that case the count is victor?"
"Yes, and is now installed at Hermosillo."

"It is almost incredible. And now, my friend, as I have answered all
your questions frankly and without comment, will you do me the
kindness to tell me why you asked them?"
"Yesterday, so soon as the count was master of Hermosillo, he
thought of you, perhaps of somebody else, and he sent off a courier
ordered to give you a letter."
"Me! That is strange. The courier was doubtlessly a native, an
Indian?"
"No, he was Don Cornelio Mendoza, a Spanish gentleman, whom
you probably remember."
"Certainly—a jolly, excellent companion, who was continually
strumming the vihuela."
"The same man," Valentine said ironically. "Well, this jolly, excellent
companion, who was continually strumming the vihuela, my dear
Loyal Heart, is simply a traitor who sold all our secrets to the
enemy."
"Oh, Valentine! you must be very sure ere you bring such an
accusation against a caballero."
"Unfortunately," the hunter said sadly, "the slightest doubt on the
subject is impossible; the count holds in his hands all the fellow's
correspondence with General Guerrero."
"Cuerpo de Cristo!" Don Rafaël exclaimed, "do you know, my friend,
this is very serious?"
"I am so fully of your opinion that, in spite of the fatigue that
overpowered me, I begged these gentlemen to accompany me, and
started at full gallop, hoping to surprise him on the road and seize
him; for, beside the letter he had to deliver to you, he had others of
a most compromising nature, addressed to several influential
persons in the province."
"That is an awkward affair," Loyal Heart said with a pensive air: "it is
evident that the scoundrel, instead of coming here, has gone

straight to hand the papers to the general."
"There is not, unfortunately, the least doubt of that."
"What is to be done?" Don Rafaël muttered mechanically.
There was a moment's silence: each reflected on the means to be
employed in order to neutralise the effects of this treachery.
Curumilla and Eagle-head rose, and prepared to leave the room.
"Where are you going?" Valentine asked them.
"While our brothers are consulting," the Araucano replied, "the
Indian chiefs will go on the discovery."
"You are right, chief: go, go," the hunter said. "I do not know why,"
he added mournfully, "but I have a foreboding of misfortune."
The two Indians went out.
"Do you know the contents of the letter the count wrote me?" Don
Rafaël asked presently.
"On my faith, no; but it is probable that he told you of the victory,
and begged you to conduct Doña Angela to Hermosillo. In any case
the letter was most compromising."
"As for that, I am very slightly alarmed, for General Guerrero will
think twice before he attacks me.
"What is the use of this long deliberation, and such a loss of
precious time? We have only one thing to do, and that is to go to
Hermosillo as escort to Doña Angela," Belhumeur said.
"In truth, that is the most simple," Valentine said in confirmation.
"Yes," Don Rafaël remarked; "the count can only be pleased with
that course."
"Come, let us carry out the plan without further delay," Belhumeur
continued. "While Black Elk and myself make all the preparations for
the journey, do you, Loyal Heart, go and inform Doña Angela of the
determination we have come to."

"Do so, and, above all, make haste," Valentine said. "I do not know
why, but I should have liked to be off already."
Without further words they separated, and the hunter remained
alone. In spite of himself Valentine was a prey to the most poignant
uneasiness. He walked in agitation up and down the room, stopping
at times to listen or look out of the windows, as if he expected to
see an enemy rise. At length, no longer able to endure the
uncertainty, he went out.
The two hunters were busily engaged in lassoing horses and
saddling them, while the peons were bringing in mules to carry the
baggage. Valentine felt his disquietude augmented with every
moment. He helped his comrades with feverish impatience, and
urged each to make haste. An hour passed away. All was then ready,
and they only awaited Doña Angela, who arrived, accompanied by
Doña Luz and Don Rafaël.
"At last!" Valentine exclaimed. "To horse, to horse! Let us start at
once!"
"Let us go," his friends repeated.
Each mounted; but suddenly a great noise was heard outside, and
Curumilla appeared with agitated features, and panting violently.
"Fly, fly!" he shouted; "they are coming."
"Forward!" Valentine exclaimed.
But an insurmountable obstacle rose before them. At the moment
they were passing through the gate of the hacienda it was suddenly
blocked up by the cattle the peons were driving back from the fields,
probably to prevent them being carried off by marauders. The poor
beasts pressed into the gateway, each anxious to be first, while
uttering lamentable moans, and goaded behind by the peons. It was
useless to hope getting out before the ganado had entered, and
there was no chance of clearing the gateway by driving it back.
Hence the fugitives were compelled to wait, whether they would or
not. Valentine was half mad with anger.

"I knew it, I knew it," he muttered in a hoarse voice, and clenching
his fists in rage.
At length, after nearly an hour (for Don Rafaël possessed numerous
herds), the gate was free.
"Let us be off in Heaven's name!" Valentine shouted.
"It is too late," Eagle-head said, appearing suddenly in the gateway.
"Maldición!" the hunter yelled as he rushed forward.
Valentine looked around him, and uttered a cry of alarm. The
hacienda was completely surrounded by nearly five hundred Mexican
cavalry, in the midst of whom General Guerrero could be
distinguished.
"Ah, the wretched traitor!" the hunter exclaimed.
"Come, let us not be discouraged," Loyal Heart said. "Cuerpo de
Cristo! it is not so long since I gave up desert life that I should have
forgotten all its stratagems. We will not give these troops time to
look about them. Let us charge, and make a hole through them."
"No," Valentine said authoritatively; "close and bar the gate,
Belhumeur."
The Canadian hastened to obey.
"Stay," Don Rafaël said.
"Loyal Heart," Valentine continued, "you are no longer the master to
act as you please, and throw yourself headlong into desperate
enterprises. You must live for your wife and your children; besides,
can we expose Doña Angela to the risk of being killed among us?"
"That is true," he answered. "Pardon me; I was mad."
"Oh!" Doña Angela exclaimed, "what do I care about death if I am
not to see again the man I love?"
"Señorita," the hunter said sententiously, "allow events to follow
their course. Who knows if things are not better so? For the present
return to the house, and leave us to manage this affair."

"Come, my child, come," Doña Luz said to her affectionately; "your
presence is useless here, and perhaps it may soon become
injurious."
"I obey you, señora," the maiden said sadly; and she retired slowly,
leaning on the arm of Doña Luz, who lavished on her all the
consolations her heart dictated. Don Rafaël had given all his servants
orders to arm, and hold themselves in readiness to offer a vigorous
resistance in case the hacienda was attacked, an event which, from
the orders given by the general to his troops, might be expected at
any moment. The peons of the hacienda were numerous, and
devoted to their master; hence the struggle threatened to be
serious.
Suddenly repeated blows were struck on the gate. Valentine, who
had been thinking deeply for several moments, bent down to Don
Rafaël's ear, and whispered a few words.
"Oh!" the latter replied, "that is almost cowardice, Don Valentine."
"You must," the hunter said obstinately.
And while Loyal Heart proceeded very unwillingly to the gate, he
quickly entered the house. Don Rafaël opened a trap door in the
gate, and asked who was there, and what was wanted; then, to the
great surprise of all, after negotiating for a few moments with the
men who demanded entrance in so peremptory a manner, he
ordered the gate to be unbarred. In an instant it was thrown open,
and the general appeared, accompanied by several officers, with
whom he rode boldly in.
"I beg your pardon for keeping you waiting, general, but I did not
know it was you," Don Rafaël said to him.
"Caramba! amigo," the general remarked with a smile as he looked
round, "you have a numerous garrison here, as far as I can judge."
"After the late events that have taken place in Sonora the roads are
infested with marauders," Don Rafaël replied: "it is wise to take
precautions."

The general shrugged his shoulders.
"Very good, caballero," he replied dryly; "but it does not please me
to see so many men armed without any legal motive. Lay down your
arms, gentlemen."
The peons looked at their master; the latter bit his lips, but made
them a sign to obey. All the weapons were then thrown on the
ground.
"I am very vexed, Don Rafaël, but I am about to leave a garrison in
your hacienda. You and all the persons present are my prisoners.
Get ready to follow me to Guaymas."
"Is that the reward for allowing you to enter my house?" Don Rafaël
said bitterly.
"I should have entered in any case," the general replied sternly. "And
now send my daughter here at once."
"Here I am, my father," the young lady said as she appeared at the
head of the steps.
Doña Angela came down slowly into the courtyard, walked toward
her father, and stopped two paces from him.
"What would you of me?" she said to him.
"Give you the order to follow me," he answered dryly.
"I can do no other than obey you. Still you know me, father: my
resolution is inflexible. I have in my hands the means to liberate
myself from your tyranny when it appears to me too heavy for
endurance. Your conduct will regulate mine. Now let us start."
The only affection that remained warm and pure in the heart of the
ambitious man was his love for his daughter; but that love was
immense and unbounded. This man, who recoiled before no deed,
however cruel it might be, to attain the object he proposed to
himself, trembled at a frown from this child of sixteen, who, knowing
the tyrannical power she exercised over her father, abused it
unscrupulously. On his side, Don Sebastian knew the iron will and

untamable character of his daughter. Hence he trembled in his heart
on listening to her cold declaration, although he allowed nothing to
be seen. He turned away with an air of disdain, and gave orders for
departure.
A quarter of an hour later all the prisoners were en route for
Guaymas, and no one was left at the hacienda but General Don
Ramon and Doña Luz, who were watched by a garrison of fifty men,
commanded by an officer, who had orders not to let them
communicate with anybody.
Valentine, on seeing the general so speedily recovered from his
defeat, judged the position of affairs at a glance. With his usual
perspicuity he understood that, owing to Don Cornelio's treachery,
the pueblos would not rise, that the hacenderos who had pledged
their word would keep aloof, that the revolt would prove abortive,
and that the count, ill and abandoned by everybody, would probably
soon be reduced to treat with the man he had conquered. This was
the reason why he urged Don Rafaël not to attempt a useless
resistance, which could only have compromised him; and, at the
same time, he persuaded Doña Angela to feign acceptance of her
father's conditions, and return with him.
We see that the hunter had reasoned well, and that his previsions
were correct. Still he was mistaken in supposing that he would
manage to advise his foster-brother of all that had occurred. The
orders given by the general in reference to the prisoners were
executed with such extreme precision, that it was impossible even to
tell the count of his whereabouts. And now that we have recounted
the events that took place at the hacienda, we will approach the
conclusion of this long drama.
CHAPTER XXIV.

THE BOAR AT BAY.
We must beg the reader to follow us to Guaymas, about a year after
the events described in the last chapter.
A man dressed in a military garb, bearing considerable resemblance
to the Mexican uniform, was walking, with his arms behind his back,
up and down the sumptuously furnished room. This man appeared
to be deep in thought; his brows were drawn together; and at times
he turned an impatient glance toward a clock placed on a bracket.
This man was evidently expecting somebody who did not arrive, for
his impatience and ill-temper increased with every moment. He took
up his hat, which he had thrown on a sofa, probably with the
intention of withdrawing, when a door opened, and a servant
announced,—
"His Excellency Don Sebastian Guerrero."
"At last," the visitor growled between his teeth.
The general appeared. He was in full uniform.
"Pardon me, my dear count," he said in an affectionate tone,
"pardon me for having kept you waiting so long, I had infinite
difficulty in getting rid of the troublesome people who bored me. At
length I am quite at your service, and ready to listen with proper
attention to the communications it may please you to make to me.
"General," the count answered, "two motives bring me here today:
in the first place, the desire to obtain from you a clear and
categorical answer on the subject of the propositions I had the
honour of making to you a few days back; and next the complaints I
have to make to you on the matter of certain very grave facts which
have occurred to the prejudice of the French battalion, and of which
I have not the least doubt," he added with a certain tinge of irony in
his voice, "you were ignorant."
"This is the first I hear of them, sir. Believe me that I am resolved to
do good and ample justice to the French battalion, of which I have

had only to speak in terms of praise since its organisation, not only
through the good conduct of the men without distinction, but also
for the services it has not ceased to render."
"Those are handsome words, general. Why must they be so barren?"
"You are mistaken, count, and I hope soon to prove to you the
contrary. But let this be for the present, and come to the grievances
of which you have to complain. Explain yourself."
The two persons who were talking in this friendly manner and
lavishing smiles were General Guerrero and Count Louis de Prébois
Crancé, the two men we have seen in such bitter enmity. What had
happened, then, since the treaty of Guaymas? What reason was
sufficiently powerful to make them forget their hatred? What
community of ideas could have existed between them to produce a
change so extraordinary and inexplicable?
We will ask our readers' permission to explain this before going
further, the more so as the events we have to narrate throw a
perfect light on the Mexican character.
The general, after the success of the treaty of Guaymas, and the
way in which, thanks to the treachery of Don Cornelio, the
insurrection of the pueblos was prevented, thought he had
completely gained his cause, and believed that he had got rid of the
count for ever. The latter, sick almost unto death, and incapable of
connecting two ideas, had received orders to leave Guaymas
immediately. His friends, who were restored to liberty after the
signature of the treaty, hastened to join him. Valentine had him
carried to Mazatlan, where he gradually recovered; then both set out
for San Francisco, leaving Curumilla in Sonora, who was ordered to
keep them acquainted with the progress of events.
The general had held up before his daughter as a merit the
generosity with which he had treated the count; then he had left her
ostensibly free to act as she pleased, hoping that with time she
would forget her love, and consent to second certain projects he did
not as yet let her see, but which consisted in marrying to one of the

most influential persons in Mexico. Still months had slipped away.
The general, who built on the count's absence, and, before all, the
want of news about him, to cure his daughter of what he called her
mad passion, was greatly astonished, when he one day began
talking to her about his plans and the marriage he had projected, to
hear her answer,—
"My father, I have told you that I will marry the Count de Prébois
Crancé: no other will obtain my hand. You yourself consented to that
union: hence I consider myself bound to him, and, so long as he
lives, I will remain faithful to him."
The general was at first greatly taken aback by this answer; for,
although he was well aware of his daughter's firmness of character,
he was far from expecting such pertinacity. Still, after a moment, he
regained his presence of mind, and bending over to her, kissed her
on the forehead, saying, with pretended kindness,—
"Come, you naughty child, I see I must do what you please, though
I confess it is a heavy sacrifice. Well, I will try. It will not depend on
me whether you see the man you love again."
"Oh, father! can it be possible?" she exclaimed with a joy she could
not restrain. "Are you speaking seriously?"
"Most seriously, wicked one; so dry your tears—re-assume your
gaiety and your bright colour of former days."
"Then I shall see him again?"
"I swear it to you."
"Here?"
"Yes, here, at Guaymas."
"Oh!" she exclaimed impetuously, as she threw her arms round his
neck and embraced him tenderly, at the same time melting into
tears. "Oh, how kind you are, my father, and how I will love you if
you do that!"

"I will do it, I tell you," he said, affected, in spite of himself, by this
love so true and so passionate.
The general had already arranged his scheme in his head—the
scheme which we shall soon see unfolded in all its hideousness. Of
the reply his daughter had made him Don Sebastian only
remembered one sentence: "So long as the count lives I will remain
faithful to him."
Poor Doña Angela had, without suspecting it, germinated in her
father's brain the most horrible project that can be imagined. Two
days later Curumilla started for San Francisco, bearer of a letter from
the young lady for the count—a letter destined to have an immense
influence on Don Louis' ulterior determination.
The Mexicans had been so magnificently beaten by the French at
Hermosillo that they had kept up a most touching and respectful
recollection of them. General Guerrero, who, as the reader has been
in a position to see, was a man of imagination, had made a
reflection full of logic and good sense on this subject. He said to
himself that if the French had so thoroughly thrashed the Mexicans,
who are very terrible soldiers as we know, a fortiori, they would
defeat the Indians, and, if necessary, the Yankees, those gringos, as
the Americans of the South call them, whom they hold in mortal
terror, and expect at any moment to see invade Mexico. In
consequence of this reasoning, General Guerrero had formed at
Guaymas a battalion entirely composed of French volunteers,
commanded by their own officers, and whose services were for the
present limited to acting as police of the port, and maintaining order
in the town.
Unfortunately the commandant of the battalion, though an upright
officer and good soldier, was not exactly the man to be placed at the
head of these volunteers. His ideas, rather narrow and paltry, were
not up to the position he occupied, and grave misunderstandings
soon broke out between the Mexicans and the foreigners—
misunderstandings probably encouraged in an underhand manner by
certain influential persons, but which placed the battalion, in spite of

the conciliatory temper of its chief, and the attempts he made to
restore harmony, in a very difficult position, which naturally became
more aggravated with each day.
Two parties were formed in the battalion: one, hostile to the
commandant, spoke affectionately of the count, the memory of
whom was still maintained in Sonora, regretted his absence, and
formed vows for his return; the other, though not devoted to the
commandant, yet remained attached to the honour of the flag. But
the devotion was lukewarm, and there was no doubt, if any
unforeseen event occurred, that these men would let themselves be
led away by circumstances.
In this state of affairs General Alvarez pronounced against Santa
Anna, President of the Republic, and summoned the chiefs of all the
corps scattered through the provinces to revolt. General Guerrero
hesitated, or pretended to hesitate, ere declaring himself. Suddenly
it was heard with amazement, almost with stupor, that the Count de
Prébois Crancé had landed at Guaymas. This is what had occurred.
Immediately after that conversation with his daughter, of which we
have quoted a part, the general paid a visit to Señor Don Antonio
Mendez Pavo. This visit was a long one. The two gentlemen
conversed secretly together, after which the general returned to his
house rubbing his hands.
In the meanwhile Don Louis was at San Francisco, sorrowful and
gloomy, ashamed of the result of an expedition so well begun,
furious with the traitors who had caused its failure, and burning—
shall we confess it?—in spite of Valentine's wise exhortations, to take
his revenge. From several quarters simultaneously influential persons
invited the count to undertake a second expedition. The money
requisite for the purchase of arms and enrolment of volunteers was
offered him. Louis had also had secret interviews with two bold
adventurers, Colonel Walker and Colonel Fremont, who at a later
date was a candidate for the presidency of the United States. These
two men made him advantageous offers; but the count declined
them, owing to the omnipotent intervention of the hunter.

Still the count had fallen into a black melancholy. The man once so
gentle and benevolent had become harsh and sardonic. He doubted
himself and others. The treachery to which he had been a victim
embittered his character to such a degree that his best friends
began to be seriously apprehensive.
He never spoke of Doña Angela—her name never rose from his heart
to his lips; but his hand frequently sought on his breast the relic she
gave him on their first meeting, and when he was alone he kissed it
fondly with many a tear. The arrival of Curumilla at San Francisco
produced a complete change; the count appeared to have suddenly
recovered all his hopes and all his illusions; the smile reappeared on
his lips, and fugitive rays of gaiety illumined his brow.
Two men arrived soon after Curumilla, whose names we will not
mention, lest we should sully the pages of this book. In a few days
these men, doubtlessly following the instructions they had received,
took complete possession of the count's mind, and hurled him back
into the torrent from which his foster-brother had found such
difficulty in drawing him.
One evening the two were seated in a room of the house they
occupied in common, and smoking a pipe after dinner.
"You will come with me, my brother, I trust?" the count said, turning
to Valentine.
"Then you really mean to go?" the latter said with a sigh.
"What are we doing here?"
"Nothing, it is true. My life is a burden to me, as yours is to you; but
we have before us the boundless desert, the immense horizon of the
prairies. Why not recommence our happy life of hunting and liberty,
instead of trusting to the fallacious promises of these heartless
Mexicans, who have already made you suffer so deeply, and whose
infamous treachery brought you to your present condition?"
"I must," the count said with resolution.

"Listen," Valentine went on. "You no longer possess that ardent
enthusiasm which sustained you on your first expedition. You lack
faith. You do not yourself believe in success."
"You are mistaken, brother. I am more certain of, success now than I
was then; for I have as my allies the men who were formerly my
most obstinate foes."
Valentine burst into a mocking laugh.
"Do you still believe in that?" he said to him.
The count blushed.
"Well, no," he said. "I will conceal nothing from you. My destiny
drags me on. I know that I am proceeding, not to conquest, but to
death. But no matter; I must, I will see her again. Here, read!"
The count drew from his breast the letter Curumilla brought him,
and handed it to Valentine; the latter read it.
"Well," he said, "I prefer your being frank with me. I will follow you."
"Thanks! Good heavens!" he added sadly, "I do not deceive myself: I
know the old Latin proverb which says Non bis in idem: what is once
missed is so for ever. I do not allow myself to be deceived by the
hypocritical protestations of General Guerrero and his worthy
acolyte, Señor Pavo. I know perfectly well that both will betray me
on the first opportunity. Well, be it so. I shall have seen again the
woman who expects me, who summons me, who is all in all to me.
If I fall I shall have a tomb worthy of me. The road I have traced
others happier than I will follow, and bear civilisation to those
countries which you and I once dreamed of emancipating."
Valentine could not restrain a sad smile at these words, which
completely revealed the count's character—a strange composite of
the most varying elements, and in which passion, pride, and
enthusiasm waged an unceasing contest.
The next day Louis opened a recruiting office, and a week later
embarked on board a schooner with his volunteers. The voyage

commenced with an evil augury, for the adventurers were wrecked.
Had it not been for Curumilla, who saved him at the risk of his life, it
would have been all over with the count. The adventurers remained
twelve days abandoned on a rock.
"The Romans would have seen a foreboding in our shipwreck," the
count said with a sigh, "and would have given up an expedition so
inauspiciously begun."
"We should do wisely in following their example," Valentine said
sadly: "there is yet time."
The count shrugged his shoulders in reply. A few days later they
arrived at Guaymas. Señor Pavo received the count most kindly, and
proposed, himself, to present him to the general.
"I wish to make your peace," he said to him.
Don Louis allowed him to do so. His heart beat at the thought that
he was possibly about to see Doña Angela again, but nothing of the
sort took place. The general was extremely gracious to the count,
spoke to him with feigned candour, and appeared ready to accept his
propositions. Don Louis brought with him two hundred men and
arms, and placed his sword at his disposal, if he intended to join the
Governor-General Alvarez. Don Sebastian, while not replying
absolutely to these advances, still allowed it to be seen that they
were not displeasing to him; he even went further, for he almost
promised the count to give him the command of the French battalion
—a promise which, on his side, the count feigned to hear with the
greatest pleasure.
This interview was followed by several others, in which, always
excepting the numberless protestations the general lavished on the
count, the latter could obtain nothing except a species of tacit
permission to take the command of the volunteers, in concert with
the chief of the battalion. This permission was more injurious than
useful to the count, however, as it rendered a great part of the
Frenchmen indisposed toward him, for they were angry at the
general appointing them a new leader.

During the week the count had been at Guaymas the general had
not said a word to him about Doña Angela, and it had been
impossible for him to see her. On the day when we find him again at
Don Sebastian's house, matters had reached such a pitch between
the inhabitants and the French, that immediate repression was
urgent in order to prevent great calamities. Several Frenchmen had
been insulted—two had even been stabbed in the public streets; the
cívicos and inhabitants made growling threats against the
volunteers; and there was in the air that something which forebodes
a great catastrophe, which no one, however, can explain.
The general pretended to feel deeply the insults offered the French.
He promised the count that prompt and full justice should be done,
and the assassins arrested. The truth was that the general, before
striking the great blow he was meditating, wished for the arrival of
the powerful reinforcements he expected from Hermosillo in order to
crush the French, and he only sought to gain time.
The count withdrew.
The next day the insults began again, and the French saw the
assassins, whom the general had promised to punish, walking
impudently about the streets. The battalion began to grow fearfully
excited, and a fresh deputation, at the head of which the count was
placed, was sent to the general. The count peremptorily demanded
that justice should be done, two cannon given to the battalion for its
security, and that the cívicos should be at once disarmed; for these
men, drawn from the dregs of the populace, occasioned all the
disorders.
Once again the general protested his kindly feeling toward the
French, and promised to deliver to them two guns; but he would not
hear a word about disarming the cívicos, alleging as his reason that
such a step might irritate the population and produce an ill effect.
While accompanying the Frenchmen to the very door of the saloon
he told them that, in order to prove the confidence he placed in
them, he would himself come without an escort to their barracks,
and hear their complaints.

The step the general took was a bold one, and therefore sure to
succeed, especially with Frenchmen, who are good judges of
bravery, and correct appreciators of everything that is daring. The
general kept his promise; he really proceeded alone to the French
quarters, in spite of the recommendations of his officers; he even
answered them in a way which proves how thoroughly he was
acquainted with the character of Frenchmen.
A colonel, among others, demonstrated to him the imprudence of
thus placing himself defencelessly in the hands of men exasperated
by the vexations of every description from which they had suffered
so long.
"You do not know what you are saying, colonel. The Gauls in no way
resemble the Mexicans: with them the point of honour is everything.
I know very well that the question will be discussed of keeping me
prisoner; but there is one man who will never consent, and who will
defend me if necessary: that man is the Count de Prébois Crancé."
The general judged correctly: all happened as he said. It was the
count who energetically opposed his arrest, which was already
almost resolved. The general left the barracks in the same way as he
entered them. No one dared to utter a word of reproach in his
presence. On the contrary, thanks to the honeyed eloquence with
which he was gifted, he succeeded so well in turning opinions in his
favour, that every one overwhelmed him with protestations of
devotion, and an ovation was almost offered him.
The result of this audacious visit was immense for the general; for,
through the effect he had contrived to produce on the mass of
volunteers, a division commenced among them almost immediately
after his departure, and they no longer agreed. One party wished for
peace at any price; the others demanded war with loud shouts,
insisting that he was deceiving them, and that they would be once
again the dupes of the Mexicans.
The latter were right, for they saw clearly; but, as ever happens,
they were not listened to, and in conclusion they came to a

compromise, which is always bad in such circumstances; that is to
say, a committee was appointed to come to an understanding with
the government, and regulate the affairs of the battalion.
As may be seen, the mine was charged: a spark would be sufficient
to enkindle an immense fire.
CHAPTER XXV.
THE BEGINNING OF THE END.
It was night. In a small house at Guaymas, Louis and Valentine were
conversing by the light of a meagre candle, which only spread a
smoking and trembling illumination. They were discussing the
measures by which to expedite the finale of the gloomy
machinations in which General Guerrero had managed to enfold
them with diabolical cunning, while Curumilla was peacefully
sleeping in a corner of the room.
"I foresaw it," Valentine said. "Now it is too late to draw back. We
must act energetically: if not, you are lost."
"Eh, my friend? I am so in every way."
"What! will you really break down when the hour of danger has
pealed?"
"I do not fear it: it will be welcome. I should wish to die, brother."
"Come, be a man. Regain your courage, but make haste. Have you
noticed the arms and ammunition continually arriving? Believe me,
we must make an end of it, one way or other, as speedily as
possible."
"Yes, I know as well as you that the general is deceiving us; but
these volunteers are not the men I had at Hermosillo. These fellows
hesitate and are afraid. Their commandant is incapable of acting: he

is a vacillating, irresolute man. With such people we can achieve
nothing."
"I am afraid so: still it is better to know at once on what you have to
depend than to remain any longer in this state of uncertainty."
"Tomorrow the delegates will go and see the general."
"Let them go to the deuce: they will be at least certain of obtaining
a categorical answer from him," Valentine said impatiently.
At this moment two light taps were heard at the street door.
"Who can arrive so late?" the count said. "I expect nobody."
"No matter; let us see," Valentine said. "It is often the case that the
people we least expect are the most agreeable visitors."
And he went to open the door. It was scarce ajar ere a woman
rushed into the house, crying to the hunter in a voice rendered
hoarse by terror,—
"Look, look! I am pursued!"
Valentine rushed out.
Although this woman was tapada—that is to say, her features were
completely hidden by a rebozo—the count recognised her at once.
What other woman but Doña Angela could come to see him in this
way? It was, in reality, the general's daughter. The count received
her half fainting into his arms, laid her on a butaca, and began
lavishing on her all those attentions which her condition demanded.
"In Heaven's name, speak! What is the matter with you?" he
exclaimed. "What has happened?"
In a little while the young lady recovered, passed her hand over her
forehead several times, and gazed at the count with an expression
of intense happiness.
"At length I see you again, my love!" she exclaimed as she burst into
tears, and threw herself headlong into his arms.

Don Louis returned her caresses, and tried to calm her. The maiden
was suffering from a strange nervous excitement, her large black
eyes were haggard, her face pallid as that of a corpse, and her
whole body was agitated by a convulsive tremor.
"Tell me, my child, what is the matter with you? In Heaven's name,
explain! I implore you, speak. Angela, speak, if you love me."
"If I love you, poor cherished one of my heart!" she said with a sigh
as she laid her hand in his. "If I love you! Alas! I love you to death,
Don Louis; and this love will kill me."
"Speak not so, my well-beloved angel! Dispel these gloomy
thoughts: let us only think of our love."
"No, Don Louis, I have not come to you to speak of love: I have
come to save you."
"To save me!" he said with feigned gaiety. "Do you believe me, then,
to be in great peril?"
"Don Louis, you are running an immense risk. Take heed of my
words. Do not look at me so with a smile: tomorrow you will be a
lost man. All the measures are taken. I heard all: it is horrible! And
that is the way I learnt your return to Guaymas, of which I was
ignorant. Then I ran off madly, wildly to you, in order to say to you,
'Fly, fly, Don Louis!'"
"Fly!" he repeated thoughtfully. "And you, Angela, must I lose you
again this time and for ever? No, I prefer death."
"I will go with you; for am I not your affianced, your wife in the sight
of Heaven? Come, come, Don Louis, let us go—not lose a minute, a
second. Your black horse will carry us beyond pursuit in two hours.
But take your weapons, for I was followed by a man as I came here
from my father's house."
She spoke with strange volubility, like a person talking in a fever. The
count knew not what resolution to follow, when suddenly a loud
noise was heard in the street, and the door, which was only leant to,
flew wide open.

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Theory of Simple Liquids Third Edition Jean-Pierre Hansen

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