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C M Y CM MY CY CMY K
MST209 Mathematical
methods and models
Block 2
Block 4 continued
Unit 15 Modelling heat transfer
Unit 16 Interpretation of mathematical
models
Block 5
i Damping, forcing and resonance
Unit 18 Normal modes
Unit 19 Systems of particles
Unit 20 Circular motion
Block 6
Unit 21 Fourier series
Unit 22 Partial differential equations
Unit 23 Scalar and vector fields
Unit 24 Vector calculus
Block 7
Unit 25 Multiple integrals
i Numerical methods for differential
equations
Unit 27 Rotating bodies and angular
momentum
Unit 28 Planetary orbits
MST209 Mathematical methods and models
2
Block 1
Unit 1 Getting started
Unit 2
Unit 3 Second-order differential
equations
Unit 4 Vector algebra
Block 2
Unit 5
Unit 6 Dynamics
Unit 7 Oscillations
Unit 8 Energy and consolidation
Block 3
Unit 9 Matrices and determinants
Unit 10 Eigenvalues and eigenvectors
i Systems of differential equations
Unit 12 Functions of several variables
Block 4
Unit 13 Modelling with non-linear
differential equations
Unit 14 Modelling motion in two and
three dimensions
MST209 Mathematical methods and models
ISBN 978 0 7492 5282 3
Block
2
Un t 17
Un t 26
First-order differential equations
Statics
Un t 11
MST209
The
Open
University

MST209 Mathematical methods
and models
Block 2

Contents
UNIT 5 Statics 5
Introduction 6
1 Modelling forces 7
1.1 Particles 8
1.2 Weight 9
1.3 Normal reaction 10
1.4 Tension 11
1.5 Friction 13
2 Two or more particles 18
2.1 Newton’s third law 18
2.2 Pulleys 20
2.3 Slipping 23
3 Torques 25
3.1 Extended and rigid bodies 26
3.2 Turning eﬀect of a force 26
4 Applying the principles 32
Outcomes 37
Solutions to the exercises 38
UNIT 6 Dynamics 51
Introduction 52
1 Describing motion 53
1.1 The motion of a particle 53
1.2 One-dimensional motion 58
2 A theory of motion 63
3 Predicting motion 66
4 Some more force models 72
4.1 Friction 72
4.2 Air resistance 75
4.3 Water resistance 82
Outcomes 85

UNIT 7 Oscillations 95
Introduction 96
1 A home-made oscillating system 97
1.1 Description of the experiment 97
1.2 Results for a static bag 97
1.3 Results for an oscillating bag 99
2 The model spring 102
2.1 Force law for a model spring 102
2.2 Model springs at rest 105
3 The simplest oscillating system 109
3.1 Setting up an equation of motion 109
3.2 Solving the equation of motion 111
4 Other oscillating systems 116
4.1 Oscillations of a vertical spring 116
4.2 Systems with two springs 119
Outcomes 122
UNIT 8 Energy and consolidation 131
Introduction 132
1 Energy: deﬁnitions 133
2 Applying energy conservation 141
3 Unit 5 consolidation 150
Outcomes 161
Appendix: Potential energy for a step function 161
Index 178

c
1.1
The Open University, Walton Hall, Milton Keynes, MK7 6AA.
First published 2005. Second edition 2008.
Copyright 2005, 2008 The Open University
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Edited, designed and typeset by The Open University, using the Open University TEX System.
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ISBN 978 0 7492 5282 3

UNIT 5 Statics
Study guide for Unit 5
This unit and Unit 6 lay the foundations of the subject of mechanics, which
has a large part to play in this course. Mechanics is concerned with how
and why objects stay put, and how and why they move. This unit considers
how and why objects stay put, while Unit 6 and the later mechanics units
consider how and why they move. The material in this unit and in Unit 6
lays the foundations of mechanics.
This unit assumes a good working knowledge of vectors, which you should
have obtained from Unit 4.
The recommended study pattern is to study one section per study session, in
the order in which they appear. There are only four sections in this unit, so
for the ﬁfth study session you might like to look at the multimedia package
designed to support this unit, which is described in the media guide.
1
2
3
4
1
2
3
4
5

Unit 5 Statics
Introduction
This unit and the next lay the foundations of mechanics that will be used
throughout the course. Mechanics can itself be divided into three distinct
areas.
Quantum mechanics deals with the motion of very small objects (such
as atoms, which have diameters of about 10−10 metres).
Relativistic mechanics deals with the motion of very fast objects (such
as the electrons in a television tube, which travel at speeds of about
108 metres per second).
Newtonian mechanics is concerned with the more familiar everyday
world of objects which are larger than atoms, and which move at speeds
less than a few million metres per second.
In this course only Newtonian mechanics is considered, but this still leaves
a vast range of phenomena to discuss. Isaac Newton (1642–1727) was the
great English mathematician whose name is given to this subject. His
Philosophiae Naturalis Principia Mathematica of 1687 (Mathematical Prin-
ciples of Natural Philosophy, known as Principia for short) incorporates
one of the most celebrated examples of mathematical modelling. It was in
Principia that Newton laid down the foundations of Newtonian mechanics.
This great book, which showed for the first time how earthly and heavenly
movements obey the same laws, is cast in the form of a set of propositions
all deriving from three axioms, or laws of motion. It is these, here translated
into modern English from the original Latin, that still provide the basis for
Newtonian mechanics.
Law I Every body continues in a state of rest, or moves with constant
velocity in a straight line, unless a force is applied to it.
Law II The rate of change of velocity of a body is proportional to the
resultant force applied to the body, and is made in the direction of the
resultant force.
Law III To every action (i.e. force) by one body on another there is always
opposed an equal reaction (i.e. force) — i.e. the actions of two bodies
upon each other are always equal in magnitude and opposite in direction.
These laws did not spring fully formed from Newton’s imagination. Earlier
investigators, notably the Italian mathematician and scientist Galileo Galilei
(1564–1642) and the French polymath René Descartes (1596–1650), had
formulated some similar results. But it was Newton who perceived that
these three laws were sufficient for the foundations of mechanics.
It is perhaps surprising that these seminal laws of mathematical modelling
were written entirely in prose, with no hint of a mathematical symbol. Sym-
bolic forms were developed during the eighteenth century. Newton himself
followed through the complicated chains of reasoning arising from the laws
with far less recourse to symbolism, or indeed to calculus, than later math-
ematicians have found necessary. In this course, the laws themselves will be
explained and the arguments based on them simplified by using the vector
notation developed in Unit 4.
One of the central concepts in Newtonian mechanics is that of a force. The
word ‘force’ is used in everyday conversation in a variety of ways: he forced
his way in; the force of destiny; to put into force; the labour force. In
mathematics and science, the word ‘force’ has a precise definition. However,
this definition relies on the movement of objects and so is deferred until
Sir Isaac Newton (courtesy of
Mansell/Time Inc./Katz)
6

Section 1 Modelling forces
the next unit. Essentially, though, this definition says that a force either
changes the shape of the object on which it acts, or causes movement of
the object. When we experience a force, in the mathematical sense of the
word, we feel it through contact: pulling on a rope, lifting a shopping bag,
pushing against a car, holding a child aloft. In each case, the force that we
experience has a magnitude and a direction, so we model a force as a vector
quantity.
Mathematical representation of a force
A force is represented mathematically by a vector. The magnitude of
the vector represents the magnitude of the force, and the direction of
the vector specifies the direction in which the force is applied.
Sometimes we can see the effect of a force, such as when a mattress depresses
under the weight of someone sitting on it, a washing line sags under the
weight of the washing, a rubber band is stretched over some packages, a
door is pulled open, or a bag of shopping is lifted. In each case there is
an obvious deformation or movement that indicates that a force is present.
Sometimes, however, the presence of a force is not so obvious, in situations
such as a ladder leaning against a wall (though you could appreciate the
presence of a force in this situation were you to replace the wall and hold
the ladder steady yourself), a box resting on a shelf, or a cable holding up
a ceiling lamp (consider holding up the lamp by the cable yourself).
This unit considers the conditions under which objects remain stationary
when subjected to forces, which is a topic known as statics. For example, q
?
what is the minimum angle θ that a ladder leaning against a wall can make
with the ground before the ladder slides to the ground (see Figure 0.1)?
Cases where forces cause motion are discussed in Unit 6 and elsewhere in
the course. The study of motion is called dynamics. Figure 0.1
Before forces and their effects can be analysed mathematically, they and the
objects on which they act have to be modelled mathematically. In Section 1
objects are modelled as particles, and the section shows how various forces
such as the forces of gravity, tension and friction can be modelled. It also
shows how to analyse one-particle systems in equilibrium. Section 2 extends
the ideas to systems involving two or more particles. Section 3 goes on to
consider situations where an object needs to be modelled as a solid body
rather than as a particle. Section 3 also discusses the turning effect of a force
(known as torque), which happens only if a force acts on a solid body rather
than a particle. Section 4 describes the application to statics problems of
the concepts and principles described in the earlier sections.
1 Modelling forces
This section shows how four common types of force can be modelled: the
force of gravity, the force exerted by a surface on an object in contact with
it, the tension force due to a string, and the friction force between two
surfaces. These forces and the situations in which they occur are modelled
and analysed in Subsections 1.2–1.5. First, however, we look at one way of
modelling the objects on which forces act.
7

Unit 5 Statics
1.1 Particles
When we create a mathematical model, the aim is to simplify the real situ-
ation being modelled so that only the essential features are included. This
enables us to analyse the situation mathematically. In mechanics, the most
important things to model are the forces acting on objects, and throughout
this unit and the other mechanics units you will see how to do this. However,
we also need to model the objects on which the forces act.
Sometimes, such as when the length, breadth, depth, orientation or internal
structure of an object is important, the object needs to be modelled as a
solid body, which possesses both mass and size. An example is provided by The modelling of solid bodies
a ladder leaning against a wall, where the length of the ladder is important. is discussed in Section 3.
At other times, when the size and structure of the object are not important,
the object may be modelled as a particle, which possesses mass but no size.
An example is provided by a pallet of bricks hanging on the end of a cable
from a crane, where the pallet of bricks can be modelled as a particle if we
are interested only in the forces acting on the cable.
Definition
A particle is a material object whose size and internal structure may
be neglected. It has mass but no size, and so occupies a single point in
space. A particle is often represented in diagrams by a black dot •.
Observation has shown that each force acting on an object can be modelled
as acting at a particular point on the object, this point being referred to
as the point of action of the force. In situations where a particle model
is appropriate, all the forces acting on the object are modelled as acting
through the point in space occupied by the particle. It is conventional to
show these forces in diagrams, known as force diagrams, by vector arrows
whose tails coincide with the particle and whose directions correspond to
the directions in which the forces act, as Figure 1.1 illustrates.
F4
F1
F2
F3
Figure 1.1
When several forces are acting on a particle, observation has shown that
the overall effect of these forces can be represented by a single vector given
by the sum of the vectors representing the individual forces. In this unit,
we deal with objects that do not move, i.e. objects in equilibrium. For a
particle in equilibrium, the forces acting on it must balance each other (or
else it would move), so we have the following important condition.
Equilibrium condition for particles
A particle subjected to forces F1, F2, . . . , Fn is in equilibrium if the
forces sum to the zero vector, i.e.
n
Fi = 0.
i=1
Note that in force diagrams
arrows are usually drawn with
arbitrary lengths. This
contrasts with the usual
convention for vectors where
length indicates magnitude.
This condition was first
stated by Isaac Newton as
part of his first law of motion.
We often say that the sum of
the forces is zero, with the
implication that this means
the zero vector.
8

1.2 Weight
When you hold a shoe, your fingers experience a force. The shoe, like all
objects, has a force associated with it, and if you do not provide opposition
to this force in holding the shoe, the shoe will fall to the ground. But what
is the source of the force exerted by the shoe?
This force is due to the attraction of the shoe to the Earth. The force of
attraction of objects to the Earth is called the force of gravity or the
gravitational force. The gravitational force acting on a particular object
is not constant, but depends on the position of the object relative to the
Earth: there is a small variation of this force with height above ground (or
depth below ground), and there is an even smaller variation with latitude
and longitude. When applied to a particular object, this force is called the
weight of the object. In this course, we shall assume that the weight of a
particular object is constant near the Earth’s surface.
In everyday speech, the words mass and weight are interchangeable. Mathe-
matically, however, they are different. The mass of an object is the amount
of matter in the object and is independent of the object’s position in the
universe; it is a scalar quantity, measured in kilograms (kg) in the SI sys-
tem. The weight of an object is the gravitational force on the object, and
is dependent on where the object is situated; it is a vector quantity, whose
magnitude is measured in newtons (N) in the SI system and whose direction The newton is defined
is downwards towards the centre of the Earth. formally in Unit 6.
Mass and weight are, however, related in that an object of mass m has The relationship between
weight of magnitude mg, where g is a constant known as the acceleration mass and weight is based on
due to gravity. Near the Earth’s surface, g has the value of approximately Newton’s second law of
motion, which is discussed in
9.81 m s−2, and we shall assume this value for g throughout this course. If Unit 6.
the Cartesian unit vector k points vertically upwards from the surface of the
Earth, then the weight W of an object of mass m is −mgk (where we need
the negative sign because the force of gravity acts vertically downwards, i.e.
the weight acts vertically downwards).
Weight
m
An object of mass m has weight W of magnitude |W| = mg, where g
is the acceleration due to gravity, with direction towards the centre of W
the Earth. If the object is modelled as a particle, the force of gravity
on the object can be illustrated by the force diagram in Figure 1.2. Figure 1.2
*Exercise 1.1
What is the weight of a particle of mass 3 kg in a coordinate system where
the k-direction is vertically downwards?
When modelling forces acting on objects, it is often convenient to define
Cartesian unit vectors and to express the force vectors in component form,
i.e. to resolve the vectors into their components. These Cartesian unit vec-
tors define the directions of the axes in a Cartesian coordinate system, so
we often refer to the process of defining Cartesian unit vectors as choosing
axes.
9

6
Unit 5 Statics
*Exercise 1.2
i π
Later in this unit we shall find it convenient to use axes which are not
horizontal and vertical. Express the weight W of a particle of mass 15 kg
15kg
j
in terms of the Cartesian unit vectors i and j, where i and j both lie in a
vertical plane and are oriented as shown in Figure 1.3. W
Figure 1.3
In the previous exercise, nice angles (i.e. multiples of 6
π
(30◦)) were chosen
in order to help you evaluate the cosines involved without having to use a
calculator. Another convenient angle is 4
π
(45◦). The cosines of all these
angles are given in the Handbook. Sometimes, as in the above exercise, ob-
tuse angles are used; cosines of such angles can be derived from the addition
formulae, which are also given in the Handbook. For example,
cos 2π 1
3 = cos(π
2 + π
6 ) = cos π
2 cos π
6 − sin π
2 sin π
6 = −2 .
1.3 Normal reaction
Consider an empty coffee mug resting on a table. Let us model the mug as
a particle. We know that one force, the mug’s weight, is acting on the mug.
But, since the mug is at rest (i.e. not moving), the equilibrium condition
for particles tells us that some other force(s) must be acting on the mug (so
that all the forces acting on the mug sum to zero). The only possible source
for another force on the mug is the table. So, in order for the mug to remain
at rest, the equilibrium condition tells us that the table must exert a force
on the mug, which must be equal in magnitude to the weight of the mug
and opposite in direction. The force exerted by the table on the mug, and
indeed exerted by any surface on an object in contact with it, is called the
normal reaction force or simply the normal reaction.
The situation is illustrated in Figure 1.4, which shows not only the mug and
table, but the corresponding force diagram (plus the Cartesian unit vector k
pointing vertically upwards). The normal reaction force is denoted by N and
the weight of the mug by W. Using the equilibrium condition for particles,
we have
W + N = 0.
If the mug has mass m, then W = −mgk, and hence
N = −W = mgk
N
k m
W
is a force acting vertically upwards with the same magnitude as the weight
of the mug.
The normal reaction force is remarkable in that it adjusts itself to the mag-
nitude required. For example, if the coffee mug is replaced by a full pot of
coffee, then the normal reaction increases (unless the weight of the coffee
pot is too much for the table, in which case the table collapses and the pot
is no longer at rest). Contrast this with the weight of an object, which is
fixed and constant, regardless of what is happening to the object. Our basic
modelling assumption is that the magnitude of the normal reaction force is
potentially unlimited.
There is a normal reaction force whenever one object (e.g. a mug) presses on
another (e.g. a table). Observation has shown that this force acts normally
(i.e. at a right angle) to the common tangent at the point of contact between
the objects. It therefore need not act vertically upwards.
Figure 1.4
This explains the name
normal reaction force.
10

For example, if the table on which the mug is resting is on an uneven floor,
so that the table top makes an angle θ with the horizontal, then the normal N
reaction force makes an angle θ with the vertical, as shown in Figure 1.5. (In
such a case there must be other forces acting on the mug if it is to remain
in equilibrium. These other forces are discussed later.)
q
q
1.4 Tension
Consider a lamp hanging from a ceiling on an electric cable. Let us model Figure 1.5
the lamp as a particle. As in the case of the mug and the table in the
previous subsection, we know that there is a weight associated with the
lamp, and that since the lamp is at rest, by the equilibrium condition some
other force(s) must be acting on it. The only possible source for another
force is the cable, so the cable must exert a force on the lamp. The force
exerted by the cable on the lamp is called the tension force.
Tension forces occur whenever objects are tautly joined, e.g. by cables, ropes,
strings or threads. These cables and ropes can be modelled in different ways.
For example, if we want to model the ceiling lamp and are interested only
in the force in the cable, then we can model the cable as a model string,
defined as an object possessing length, but no area, volume or mass, and
which does not stretch (i.e. it is inextensible). On the other hand, if we are
interested in how much the cable stretches under the weight of the lamp,
then we can model the cable as a model spring, which has properties similar Springs are discussed in
to those of a model string (i.e. it has no area, volume or mass), but allows Unit 7.
extension. In this unit we consider only strings.
The ceiling lamp example is illustrated in Figure 1.6. The tension force due
to the model string is denoted by T, and the weight of the lamp by W.
In a manner similar to the case of normal reaction forces, the equilibrium
condition for particles gives
W + T = 0.
If the lamp has mass m, then W = −mgk, and hence
T = −W = mgk
T
k m
W
is a force acting vertically upwards (along the length of the model string) Figure 1.6
with the same magnitude as the weight of the lamp.
We assume that the tension force due to a model string acts along the length
of the string and away from the point of its attachment to an object. As in
the case of a normal reaction, the magnitude of this force (often referred to as
the tension in the string — a scalar quantity) depends on the requirements
necessary to maintain equilibrium, so it is potentially unlimited. (In reality,
a string can exert only a certain tension force before it breaks, but a model
string supports an unlimited tension force.)
Definitions
(a) A model string is an object with a fixed finite length, and no area,
volume or mass, that exerts a force at the point of attachment.
(b) The tension force due to a string is directed along the length
of the string away from the point of attachment.
As in the case of normal reaction forces, the tension force due to a string
need not be vertically upwards, as the following example illustrates.
11

3
i
Unit 5 Statics
Example 1.1
A hanging ﬂower basket of mass 4 kg is suspended by one cord from a porch
and tied by another cord to the wall, as shown in Figure 1.7. Model the
basket as a particle and the cords as model strings. What are the magnitudes
of the tension forces due to the cords?
Solution
We choose axes as shown in Figure 1.7. Note that we need choose only
two axes because all the forces act in the same vertical plane. Denoting the
tension forces by T1 and T2, and the weight of the basket by W, we have
the force diagram shown in Figure 1.8. Figure 1.7
π
j
4 kg
π
3
π
6
π
3
T1
T2
W
Figure 1.8
In the diagram, the angle between the vector T1 and the unit vector j is
calculated by imagining the right-angled triangle shown, and using the fact
that the angles of a triangle sum to π radians. The angle between the vectors
T1 and i is calculated using the fact that the angle between the unit vectors
i and j is a right angle (so the required angle is π
2 − π
6 = π
).
3
The equilibrium condition for particles tells us that
T1 + T2 + W = 0. (1.1)
To progress further, we need to express the three forces in terms of the unit
vectors i and j. To do this, we apply the technique of resolving vectors
described in Unit 4. Starting with the weight W, we have In later examples and
W = (W . i) i + (W . j) j
solutions, this process of
resolving vectors will not be
π
2 i + |W| cos π j done explicitly for forces
aligned with the axes. The
|W
= −|W
|
= cos
| j. components will be written
down by inspection of the
Similarly, the tension force T1 can be expressed as
diagram.
T1 = (T1 . i) i + (T1 . j) j
1| cos π
3
π
j
6
T i + |T1
= | | cos
√
3
= 1
|T1| i + |T1| j.
2 2
Finally, the tension force T2 can be written as
T2 = (T2 . i) i + (T2 . j) j
= |T2| cos π i + |T2| cos
= −|T2| i.
π
j
2
Using the components of the three forces in Equation (1.1) gives two scalar
equations for the i- and j-components, respectively:
1
|T1| − |T2| + 0 = 0, (1.2)
2
√
3
|T1| + 0 − |W| = 0. (1.3)
2
But |W| = 4g, so Equation (1.3) gives
√ √
|T1| = 2|W|/ 3 = 8g/ 3 45.31.
12

Substituting this into Equation (1.2) gives
√
|T2| = 4g/ 3 22.66.
So the model predicts that the tension force due to the cord from the porch
has magnitude about 45.3 N and that the tension force due to the cord from
the wall has magnitude about 22.7 N.
The procedure that was used in Example 1.1 can be used to solve many
problems in statics, and may be summarized as follows.
Procedure 1.1 Solving statics problems
Given a statics problem, perform some or all of the following steps.
(a) Draw a sketch of the physical situation, and annotate it with any
relevant information.
(b) Choose axes, and mark them on your sketch.
(c) Draw a force diagram or diagrams.
(d) Use the equilibrium condition and any other appropriate law(s) to
obtain equation(s).
(e) Solve the equation(s).
(f) Interpret the solution in terms of the original problem.
�Draw picture�
�Choose axes�
�Draw force diagram�
�Apply law(s)�
�Solve equation(s)�
�Interpret solution�
In this unit, the steps in this procedure will often be identiﬁed (using the
marginal abbreviations above) in the solutions to examples and exercises.
The procedure is intended to be a guide rather than a rigid set of rules. For
example, if it is not obvious which set of axes to choose, then draw the force
diagram ﬁrst, and the best choice may become more apparent. Try using
the procedure in the following exercise.
*Exercise 1.3
During December, a large plastic Christmas tree of mass 10 kg is suspended
by its apex using two ropes attached to buildings either side of the high
street of Trappendorf. The ropes make angles of π
6 and π
4 with the horizontal.
Model the Christmas tree as a particle and the ropes as model strings. What
are the magnitudes of the tension forces due to the two ropes?
1.5 Friction
Consider a book resting on a horizontal surface. There are two forces acting
on the book: the weight downwards and the normal reaction upwards. Sup-
pose that you push the book gently sideways (see Figure 1.9). If you do not
push hard enough, the book will not move; it will remain in equilibrium. We
know that the vertical forces, i.e. the weight and the normal reaction, bal-
ance each other (i.e. they are equal in magnitude and opposite in direction),
so they cannot be preventing the book from moving sideways. Therefore
there must be another force present. This force is known as the friction
force. It is considered to act parallel to the surface, i.e. at right angles to
the normal reaction, and in a direction that opposes any (possible) motion
along that surface. Modelling the book as a particle, and denoting the push-
ing force by P, the friction force by F, the weight by W and the normal
reaction by N, the force diagram for this example is shown in Figure 1.10.
sideways
push
Figure 1.9
N
F P
W
Figure 1.10
13

Unit 5 Statics
Friction forces are caused by the roughness of even seemingly very smooth
surfaces — a roughness that serves to inhibit the smooth movement of one
surface over another. So friction forces are present only where there is
movement or the possibility of movement. There is no friction force present
when an object is resting on a horizontal surface, where the only two forces
acting on the object are its weight and the normal reaction. But when an
object is being pushed or pulled, or is resting on a sloping surface, then a
friction force is present (see Figure 1.11).
F
N
W
N
W
no friction friction force
force present
Figure 1.11
Unlike the normal reaction, which is potentially unlimited in magnitude,
there is a limit to the magnitude of the friction force; if this limit is reached,
then slipping occurs. The limiting value of the magnitude of the friction
force depends almost entirely on the materials of the two surfaces and on the
magnitude of the normal reaction force between them. It does not usually
depend on the area of contact between the two surfaces, or on the angle
at which the two surfaces are inclined to the horizontal. Experiments show
that the limiting value of the magnitude of the friction force F (which just
prevents slipping for two given surfaces) is approximately proportional to
the magnitude of the normal reaction force N between the two surfaces. So,
on the verge of slipping, we have |F| = µ|N|, where µ is the coefficient of
static friction, which depends on the materials of the two surfaces. Some
approximate values of µ for different materials are given in Table 1.1.
Example 1.2
A steel fork of mass 0.05 kg rests on a horizontal wooden table. Model the
fork as a particle. What is the maximum sideways force that can be applied
before the fork starts to move?
Solution
The situation is illustrated in Figure 1.12. Since all the forces act in a vertical
plane, we can choose axes as shown. The force diagram is also shown in the
figure, where F is the friction force, P is the sideways force, W is the weight,
and N is the normal reaction.
In this unit we consider only
cases where objects remain at
rest, so that there is only the
possibility of movement.
Friction in cases where there
is movement is considered in
Unit 6.
Table 1.1 Approximate
coefficients of static friction
Surface µ
Steel on steel (dry) 0.58
Steel on steel (oiled) 0.1
Plastic on plastic 0.8
Rubber on tarmac 1.3
Steel on wood 0.4
Wood on wood 0.35
N
�Draw picture�
sideways j
force 0.05kg F P �Choose axes�
i
m = 0.4 �Draw force diagram�
W
Figure 1.12
14

The equilibrium condition for particles gives �Apply law(s)�
F + N + P + W = 0. (1.4)
When the fork is on the point of moving (slipping), we have
|F| = µ|N|,
where µ = 0.4 is the coeﬃcient of static friction of steel on wood.
To be able to use Equation (1.4), we need to express the forces in terms of �Solve equation(s)�
the unit vectors i and j. Looking at Figure 1.12, the forces can be written
in component form as
F = −|F| i, N = |N| j, P = |P| i, W = −|W| j.
Resolving Equation (1.4) in the i-direction gives
−|F| + 0 + |P| + 0 = 0,
so (as expected)
|F| = |P|.
Resolving (1.4) in the j-direction gives
|N| = |W|.
Therefore, when the fork is on the point of moving,
|P| = |F| = µ|N| = µ|W| = 0.4 × 0.05g = 0.02g 0.196.
So the model predicts that a sideways force of magnitude about 0.196 N can �Interpret solution�
be applied without moving the fork.
Here is a summary of how we go about modelling problems that involve
static friction, i.e. problems involving friction but no motion.
Modelling static friction
Consider two surfaces in contact.
(a) The friction force F acts in a direction perpendicular to the nor-
mal reaction N between the surfaces and opposite to any possible
motion along the common tangent to the surfaces.
(b) |F| ≤ µ|N|, where µ is a constant called the coeﬃcient of static |F| cannot exceed its limiting
friction for the two surfaces involved. value µ|N|. Slipping occurs if
a friction force of magnitude
(c) |F| = µ|N| when the object is on the verge of slipping. This equal- greater than µ|N| would be
ity is sometimes referred to as describing a situation of limiting needed to prevent it.
friction.
(d) If one of the surfaces is designated as being smooth, it may be
assumed that there is no friction present when this surface is in
contact with another, regardless of the roughness of the other sur-
face.
Let us now apply these ideas to some examples, in which we shall also apply
the steps of Procedure 1.1. In most of the situations that we investigate, we
shall be concerned with limiting friction.
*Exercise 1.4
A wood block of mass 5 kg rests on a horizontal plank of wood. Model the
block as a particle. What horizontal force is required to start it moving?
15

Unit 5 Statics
Exercise 1.5
A steel block of mass 0.5 kg rests on a horizontal dry steel surface and is
pulled by a horizontal force of 2 N. Model the block as a particle. Does the
block move? What is the magnitude of the friction force?
*Exercise 1.6
A shallow box made of a uniform material and without a lid can be placed
on a horizontal table in two possible ways (as shown in Figure 1.13):
(a) with its base in contact with the table surface;
(b) with its open top in contact with the table surface.
(a) (b)
Which of these two positions requires the smaller sideways force to start the
box slipping? Figure 1.13
Inclined planes
Consider now an object resting on a sloping plane surface, often referred to
as an inclined plane, such as the one shown in Figure 1.14. Provided that
N
the angle of inclination is not large, the object can remain at rest and does F
not slide down the slope. The forces acting on the object are its weight,
the normal reaction and friction. The weight W acts vertically downwards.
The normal reaction N acts normally to the surface between the object and
W
the slope. The friction force F is perpendicular to the normal reaction and
hence parallel to the slope, and it acts up the slope to counteract the natural
tendency of the object to move down the slope. Figure 1.14
Example 1.3
A crate of empty bottles of total mass 30 kg is to be hauled by a rope up
a ramp from the cellar of a pub. The rope is parallel to the ramp, and the
π
ramp makes an angle of radians with the horizontal. The coeﬃcient of
6
static friction between the plastic crate and the wooden ramp is 0.2.
What is the tension force due to the rope when the crate is on the point of
moving upwards?
Solution
The situation is illustrated in Figure 1.15. �Draw picture�
π
6
m = 0.2 30 kg
i
j
Figure 1.15
All the forces act in a vertical plane, so we need only two axes. We could �Choose axes�
choose i to be horizontal and j vertical as before, but it makes calculations
easier if we choose i to be parallel to the slope and j perpendicular to it, as
shown in Figure 1.15. This is because, when we come to resolve the forces
in the i- and j-directions, three of the four forces (all except W) will then
act along one or other of the axes, making resolving them much simpler.
Modelling the crate as a particle and the rope as a model string, the force �Draw force diagram�
diagram is as shown in Figure 1.16, where W is the weight, N the normal
reaction, F the friction force, and T the tension force.
16

π
6
π
6
π
3
T N
W
F
i
j
Figure 1.16
The equilibrium condition for particles gives �Apply law(s)�
T + N + F + W = 0. (1.5)
When the crate is on the point of moving, we have
|F| = µ|N|,
where µ = 0.2 is the coefficient of static friction.
As before, the first step in solving the equations involves resolving the force �Solve equation(s)�
vectors into components. In this case, three of the force vectors are aligned
with the axes and can be written down immediately:
F = |F| i, N = |N| j, T = −|T| i.
To find the weight of the crate in component form, we use the formula
derived in Unit 4:
W = (W . i) i + (W . j) j Note that
π π π
| cos( ) j
W| cos i + W π π
cos( )
+
= | | +
3 2 3 2 3
√ π π π π
− sin sin
= cos cos
3
Now Equation (1.5) can easily be resolved in the i-direction, giving
1
W i − W| j
| | | 2 3 2 3
= . √
2 2
= − 3
2 .
1
2
| + 0 + W| = 0,
−|T F| +
| |
so
1
2 | (1.6)
T = F W
| +
| | | |.
Similarly, resolving Equation (1.5) in the j-direction gives
√
3
2 |
0 + N + 0 − | = 0,
W
| |
so
√
| = 3
2 (1.7)
N |W|
| .
At the point of moving, |F| = 0.2|N| and Equations (1.6) and (1.7) give
√
3
2 |
| + 1
2 |W | + 1
2 |
T| = 0.2 | = 0.2 ×
|N W W
| |.
Thus, since |W| = 30g,
√
T| = ( 3
10 + 1
2 ) × 30g 198.
|
Therefore, when the crate is on the point of moving, the model predicts that �Interpret solution�
the tension force due to the rope is about 198 N up the ramp.
Mathematically, different choices of axes make no difference to the final
solution obtained to a mechanics problem. However, a sensible choice of
axes, as in Example 1.3, can reduce the amount of calculation. You will find Choice of axes is discussed
that, with experience, you will be able to choose axes that reduce the work again in Unit 6.
involved.
17

Unit 5 Statics
*Exercise 1.7
A full crate of bottles of mass 60 kg is at the top of the ramp described in
Example 1.3, ready to be lowered into the cellar. What force needs to be
applied to the rope to keep the crate from sliding down the ramp?
End-of-section Exercises
Exercise 1.8
On a building site, a pallet of bricks of mass 1800 kg is suspended from the
cable of a crane. The length of the cable is 10 m. One of the site workers is
pulling with a force of magnitude 800 N horizontally on a rope attached to
the pallet, in order to position the pallet over the lorry into which it is to
be lowered. How far can the pallet be moved horizontally by the worker?
Exercise 1.9
(a) A box of mass m is resting on a surface inclined at an angle α to the
horizontal. If the box is on the point of slipping, what is the coefficient
of static friction?
(b) Two identical mugs are placed on a tray. One mug is half full of coffee,
the other is empty. The tray is tilted slowly. Use your answer to part (a)
to determine which mug will start to move first.
2 Two or more particles
In the previous section we considered the action of forces on one particle
and introduced the equilibrium condition for particles. In this section we
extend these ideas to situations involving two or more particles. Subsec-
tion 2.1 shows how Newton’s third law can be applied to such situations.
Subsection 2.2 introduces a new modelling device — the model pulley —
and Subsection 2.3 considers friction in the two-particle case.
2.1 Newton’s third law
The equilibrium condition for particles extends to systems involving two or
more particles, in that, if the system is in equilibrium, then each particle
must be in equilibrium, so the sum of the forces acting on each particle must
be zero.
Consider, for example, the simple situation of one book lying on top of
another on a desk. Both books are in equilibrium. Let us model each book
as a particle. We can apply the equilibrium condition for particles to each
book in turn. We therefore need to determine the forces on each book.
For the upper book, the situation is similar to the case of an object resting
on a surface, discussed in Section 1. There is the weight W1 of the book
acting vertically downwards, and the normal reaction N1 of the surface (of
the lower book) acting vertically upwards. For the lower book, we have its
weight W2 acting vertically downwards, and the normal reaction N2 of the
surface (of the desk) acting vertically upwards, but this time there is also
another force — the normal reaction N3 from the upper book. The situation
and the force diagrams are shown in Figure 2.1.
18
Figure 2.1
N1
W1
upper
book
N2
W2
N3
lower
book

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What do you want? demanded Peaches huskily. What are you
doing here?
She was retreating toward the bell as she spoke, the man's gaze
following her action without protest. Coming well into the room he
removed his hat, shaking a few drops from it as he did so. The
shoulders of the coat were also wet. Evidently it was raining heavily
outside. His face as revealed in the stronger light was less alarming,
and he spoke in an even tone.
Ring by all means! said he. Bring help as soon as possible! As for
who I am, he went on, throwing back his wet coat and revealing a
silver badge, I am Pedro, the missing night watchman, and I have a
warrant of extradition for the arrest of Sandro di Monteventi, alias
The Eel—wanted by the International Secret Service for the theft of
the Scarpia panels and sundry charges.
Go on, ring, miss, said a second man, following in on the heels of
the first; a man whom Peaches instantly recognized as the face at
the window. Ring, please—we know he is in the house—and
incidentally don't you try to get away. We want to talk to you—you
seemed to know him rather well.

XV
With a violent movement Peaches rang the bell. And almost at once
the house was again in confusion. The two newcomers, backed by
the cursing Markheim and aided by Mr. Pegg, made straight for the
room occupied by Sandro. Peaches followed in their wake, and saw
them batter down the door—to find an empty room and a gaping
window.
Of course! The idiots! Now if they had only had sense enough to
wake me up I could have told them better! But no, they let me sleep
—sleep, mind you, when all this, as it were, human motion picture
was proceeding right under my very nose! I feel outraged, indignant,
as I consider the lack of forethought and consideration which this
lack of attention evidenced. Of course the duke escaped—the
ninnies should have left some one outside in the garden—and their
excuse that they did not believe that he could escape so rapidly from
the third story of the house would have been made quite
unnecessary if I had been there to inform them of his nocturnal
wanderings as known to me.
Really, as I listened to Peaches' recital I became quite distinctly
vexed. The fate by which I seemed doomed to remain a bystander
looking on at life from a safe distance or merely to be told about it
at secondhand or to read of it in printed form was really too
annoying. Despite my utmost endeavor I was apparently to be
cheated of active participation in the great drama of existence.
But no one could look at Peaches' pale and suffering beauty for long
and remain unindulgent. And as I lay in the great bed enjoying the
tea and toast which she had so thoughtfully brought me I restrained
the comments which sprang to my lips and merely asked, What
happened then?

We came downstairs, said Peaches slowly, twisting the amber
beads about her throat, Mark, pa and myself along with these two
cowbird detectives. I tell you, Free, I just could hardly believe the
story they told. But I had to, in the end. You see, for one thing, as I
sat there I began to realize I had seen the Pedro once before.
Where?
In a London movie house—and in a hotel bedroom at Monte Carlo,
said she significantly.
There! I cried. I foiled him twice, you see! Now it's a lucky thing I
wasn't there last night, isn't it? Humph! I'd probably have defeated
justice again! But what did he say?
He's been after Sandro for years, she narrated. I am afraid there
isn't the shadow of a doubt, Free, but that Sandy is the cleverest
picture thief in the world. They have almost got him half a dozen
times, but never with conclusive evidence. And thank God, they
didn't get him this time, either—not yet at least! Why, do you know,
they are certain that he took the Scarpia panels? It seems, if you
remember, that they thought that they had been found in the cellar.
But it wasn't the originals that they found. They were reproductions
—synthetic pictures, like a near-ruby—do you get me?
But the recovery was reported in the papers, I objected.
The French Government hushed the matter up in order to try and
catch him off his guard, she went on. And, Free, that's just what
he has done in this very house.
How do you mean—explain yourself grammatically if possible, said
I.
I mean that the Madonna of the Lamp which is hanging in the
library at this moment is the bunk, replied Peaches earnestly. It's a
fake—painted on new canvas and nicely antiqued. The cops took it
down and showed it to us.
And what did he want to steal a fake for? I demanded.

He didn't want to steal a fake, you dear old prune! said Peaches,
half laughing. He wanted to steal the original, and that's exactly
what he did.
And got away with it! I gasped, astonished into a colloquialism.
But when and how on earth?
Very simple, but clever, she told me, quite as if it were to the
young man's credit. He had this fake all ready on a stretcher in his
room. He took the original, stretcher and all, out of the frame and
upstairs, where he unmounted it and hid it—it isn't large, you know.
And then, before he could slip the substitute into place, you and I
came in from the garden—from the garden where we had been
waiting for him to—to——
Here she broke off and began to laugh hysterically.
Come, come, my dear! I cried. Don't do that—just remember
what a lucky escape you have had. So we interrupted him before he
could put the substitute in place! Well, land of goodness! I do recall
that he was all dressed when he came down stairs at Mr. Markheim's
command! Go on, do, my dear!
Well, said Peaches, complying with renewed composure, this
Pedro-bird claims that Sandy slipped it in while we were all out in
the hall with the servants and he was in and out apparently taking
care of Markheim's orders. If the secret-service men hadn't been on
the job Sandy would in all probability have simply stayed his two
weeks out as a quiet well-behaved servant, and then gone away
with a first-class reference and the original Madonna, and the
substitution might never have been found out, or it might have been
years—until some feast was held by a lot of experts at Mark's
invitation—who knows! And he's been doing this sort of thing for
years and years!
Extraordinary! Most extraordinary! I exclaimed, pulling off my
nightcap and starting to rise. I must really dress and descend to
take a look at that picture and the scene of the crime!

You can't! said Peaches, suddenly listless. You can't—we are both
locked in!
I could scarcely believe my ears. But Peaches was in earnest, there
was no doubt about that.
Locked in! I repeated incredulously. What on earth are you
saying, Alicia Pegg?
I was saying a mouthful! she responded. Pa has locked us in.
But what for? I demanded with proper indignation.
I told him I was going to follow Sandro, said Peaches, as if the
explanation was the most obvious thing possible and she were just a
trifle impatient of my stupidity.
Are you crazy? I cried. Follow him—follow that thief—that—that
scoundrel? Aren't the police following him? Isn't that following
enough?
That's just why, she announced. Wherever he is—wherever he
goes, I am going too. After last night I can't do anything else. And if
it's to jail—all right, I'll go to jail. But I won't stay away from him,
and I will find him if the secret-service can't, and I hope most
heartily they will make a flivver of it. And I'll never leave him again—
believe me!
I was obliged to believe her. I had, indeed, only to look at her in
order to do so. And as I looked, a gleam of human intelligence broke
into my brain.
Peaches, I said solemnly, did you tell on Markheim?
Of course not! she said, flushing hotly. He—wasn't himself; I
realize that now.
So you just told your father that you are through with Markheim
and are in love with the duke?
She nodded dumbly.
No wonder he locked you up! I gasped, falling back on the pillows.

Locked me up and said the marriage would go ahead as per
schedule, she announced grimly. Which is bunk of course. The
point is—what shall we do about it?
Have they caught the duke? I inquired.
I don't believe so, said she. There is nothing to that effect in the
early afternoon newspapers from New York, though there's plenty
about the robbery. Take a look!
Let me see! I exclaimed, stretching out my hand for the paper.
And forthwith she spread the lurid sheets before my distressed eyes.
The headlines were of the variety known as scare. Not the German
ex-Kaiser himself, or even a Bolshevist labor leader was ever
presented in larger type than was the lurid announcement of the
attempted robbery. And all our names were mentioned—even that of
Talbot—the sacred family name, which we had kept inviolate for
generations against all newspaper publicity excepting only mention
in the society and political columns. For, of course, the difference
between one's appearing as a social or political item and as a piece
of mere vulgar news must at once be apparent to any reader of
refined upbringing. And never before had the Talbots been news. I
dreaded to think how my sister Euphemia would take it should the
article chance to meet her eye. She might eventually forgive me
much; but I seriously doubted whether her charity would ever
extend over newspaper headlines. Alas! This was but a foretaste of
what was to come!
But much as the reporters had to say of the splendor of Sebastian
Markheim's mansion and the beauty of Sebastian Markheim's
fiancée, whose coming marriage would be of the greatest social
consequence, uniting the greatest fortune of the East with the
greatest fortune of the Western Coast, and so on, and though it was
further replete with details of the method by which the robbery had
been committed, together with a florid account of the robber's high
station in life, his heroic action in battle, where he was supposed to
have been killed while defending a position single-handed in a rocky

pass during the Austrian invasion, thereby enabling the rest of his
brigade to escape—nothing indicated that his capture was at this
time considered very likely. The authorities were full of assurances
but rather short on facts, to all appearances.
Well, now, Alicia, my dear, I remarked when I had satisfied myself
that no detail of importance had escaped me in my perusal of the
printed account of our affair—now, Alicia, my dear, said I. I feel it
incumbent to be quite sure that you know what you are saying when
you announce your intention of linking your life with that of this wild
young Italian—always provided that the gallows does not get him
before you do. Can't you reconcile yourself to the idea that he is a
thief, no matter how titled, and that therefore he is no match for an
honest American girl?
Oh, cut the moralizing, Free! interrupted Peaches. I am in love
with him, I tell you. And I have sufficient faith in my own integrity to
believe that this wouldn't be true if he really was the yellow dog
everybody seems bent on trying to make him out. Now I've got a
hunch—a mighty straight hunch that he is O. K. There's more to this
than we know. Maybe the old picture belonged to his great-
grandmother or something, and he's only taking it back. How do you
know he isn't doing just that very thing?
But the Scarpia panels didn't belong to his grandmother, I
answered smartly.
But they haven't got the goods on him for those other deals, she
retorted. And if they had, I'd still be crazy about him. Freedom, this
is a question of the rest of my life. You've got to take my side.
But what are you—we going to do? I pleaded, bewildered by her
intensity. And what is all this nonsense about our being locked in
these rooms?
You just try to get out and see if it's nonsense, replied Peaches.
You were asleep when they locked me in, and as there is no lock on
the doors between our rooms they locked you too. I wouldn't let
them disturb you, not only because you were so tired but because I

knew damn well that if I let you out I wouldn't get this chance to
talk to you.
Well, this is outrageous! I exclaimed, rising in good earnest this
time. We shall see whether your father can imprison two adult
women in a free country to suit his whim! I shall make my toilet at
once and then we shall see what we shall see!
Better hurry up then! replied Peaches. Because they—he and
Mark—are going to the city on the twelve-o'clock train. Don't you
remember why we came home early last night?
Last night seemed a thousand years ago. But she was quite right; I
did recall the fact, and accordingly made all possible haste, Peaches
assisting me.
Now look here, you flighty young thing! she warned. Don't do
anything rash! Remember, you are the only person I have to depend
on for help. Don't go get yourself kept away from me now!
I must and shall interview your father, I protested. But perhaps if
you would be kind enough to give me an idea of what you intend
doing I shall be in a better position to be of assistance.
I'm going to leave this house before another twenty-four hours are
over, she declared firmly. If you can persuade pa to let me go like
a human, and come along with me, so much the better. If not, I'll
have to go some other way that may not be as agreeable to him in
the long run.
Why not let me tell him about that terrible performance of Mr.
Markheim's? I suggested. That will be sufficient, or I mistake your
father greatly.
Sure it would be sufficient, said Peaches. But then I'd have to give
myself away pretty badly, wouldn't I? And there might be a
roughhouse. Pa is a dead shot and I'd rather get him out of shooting
distance before I break the information to him. At present he just
about thinks I'm crazy in the head.

Well, I'll do what I can to persuade him that this is the twentieth
century and not the middle ages! I responded. This indignity
certainly cannot be allowed to continue. But suppose you—we do
get away from here to-day, what then? How do you propose to find
a thief that the police will have a hard time discovering?
I don't propose, said Peaches. I intend. That's a whole lot
stronger. How, I haven't the remotest idea. But it's plain enough I
can't do anything while they've got me cooped up like a marketable
yearling, can I? Let's get out of this, that's the first thing to
accomplish.
Very well, I agreed, gathering up my reticule and taking up the
house-telephone receiver.
I asked to speak with Mr. Pegg. The request was at once attended to
by the footman who responded, and in a tone which brooked no
delay I commanded the Citrus King to come upstairs and release
me. My tone must have foreshadowed the mood I was in, for he
responded as if by magic. In less than five minutes I was face to
face with him in the hall.
Come on over and sit down in the conservatory, Miss Free, he
entreated as soon as he saw my face. We want to keep the
servants out of this much as we can, you know!
All right, Mr. Pegg, I agreed, for this was my own thought. All
right. But if you allow the situation to continue you will have a hard
time in doing that!
Accordingly we repaired down the corridor to a little glass room full
of plants, where we could talk in seclusion. Mr. Pegg, as usual,
chewed upon an unlighted cigar and looked at me thoughtfully over
the top of it, his shrewd eyes half closed.
You've got awfully pretty hair, Miss Free, said he unexpectedly. I'm
glad you've took back to them curls again.
Now see here, Mr. Pegg, I said severely, not to be diverted by any
frivolous remarks. Now see here, Mr. Pegg, what is the meaning of

this outrageous performance?
When I was a cattleman, said Mr. Pegg, looking at the ornate
ceiling, we used to lock 'em in a corral until they cooled off a little.
What—who? I demanded.
The ones we was breaking, he informed me. Then his manner
changed and he brought his big fist down on his knee with a thump.
Now, my dear lady, he said firmly, I know what I'm doing. Why, I
had to keep her on the ranch, watched like a hawk—and simply
because she kept thinking she was in love with some undesirable or
other. I've seen her do this before. So I'm just going to detain her
where she'll be safe until she comes to her senses.
Mr. Pegg, you are taking the wrong track with Peaches this time! I
warned him. You can't play the Roman father with your child and
marry her out of hand—you cannot! You engaged me as a social
mentor and I would be doing less than my duty if I didn't inform you
that this sort of thing is no longer being done in the best families!
Say! remarked Mr. Pegg, removing the cigar and staring at me.
Are you trying to be humorous, or what?
I assure you I am far from any such idea! I replied with hauteur. I
merely affirm that you cannot, even legally, keep an adult female
child imprisoned against her will and then marry her off to—to a
swindler!
A swindler! exclaimed Mr. Pegg. Oh, come now, Miss Free—
smuggling in that picture wasn't Mark's fault. You can't say he did it
—because you don't know it. Why, you and he have always been
good friends; you're not going back on him now? Peaches is just a
kid. By the end of the week she will have changed her mind again.
Good heavens, look at the fix it would put us in if she insisted on
breaking her engagement now! The invitations out, the presents
coming in—trousseau bought! We'd be the laughingstock of the
country. Not that I'd give a—cuss—if it wasn't that I know Alicia.
She'd up and go back to him when it was all thoroughly broken off.

You see that what she needs is the high hand. I've had to use it
before.
Mr. Pegg, said I, you are mistaken. What is worse, you are a cave
man! I am convinced Peaches really is in love with Sandro di
Monteventi and that you will break her heart if you persist in your
heroic attitude. I beg you will desist.
Nothing doing! said Mr. Pegg, rising and lighting the cigar—a sign
that the interview was closed. I'm not in a desisting mood. I may as
well add that I am wise to the fact that she's been mooning round
after that fellow ever since she came into this house. Kimball's
Commercial Arithmetic, indeed!
I don't know to what you refer, I assure you! I said stiffly. And I
insist upon at least having a key to our rooms.
Will you give me your word of honor not to use that key to let her
out with? asked my employer doubtfully.
Certainly, if you wish, I replied promptly. You may have my word
for that!
Well, here you are, then, he answered, taking a key from a great
cluster on his ring. You'll keep the letter of your word, I know, no
matter how uneasy the spirit gets. And now I must mosey along.
Mark and I have to run up to town on business, and he wants to see
the family-doctor about his eye—he ran into his bedpost in the dark
last night, and maybe it's just as well to keep Peaches from seeing
him wearing that beauty spot.
With which intelligent and discerning remark Mr. Pegg left me to my
own devices, and of course I promptly returned to my apartment
and the waiting Peaches, who greeted my entrance the more eagerly
when she observed I let myself in with a key.
You wonder! cried she, embracing me with a look of rapture. So
he gave in to you—you enchantress!
He did not! I said dryly. He put me on my honor not to let you
have this key, and my honor is sacred, and I'm going to keep it that

way!
Free—you beast! cried Peaches. Give it to me. Don't be absurd!
Keeping one's freely given word is never absurd, I observed.
Besides, if I were to break it and let you walk out, do you think for
one minute that the servants would let you get away without
protest? Or without notifying your father by telephone? It is you who
are absurd!
That's so! said Peaches, suddenly weary. Oh, Free—you think it
out! Help me, I am so tired.
Lack of sleep, I pronounced. And I'll wager you have eaten
nothing. The first thing to do is to have a nice hot luncheon sent
upstairs—I presume your father's instructions permit the service of
food. And then you must get a few hours of complete rest while I
take a stroll in the fresh air and perfect some course of action.
Then you will help me? said Peaches eagerly.
It was really pathetic to see her so comparatively tired and helpless.
She was never more than comparatively so, I may state. However,
my compassion for her was not lessened by this fact.
Of course I am going to help you, I declared. That any mere man
should attempt a performance of this kind outside of Bolshevik
Russia is too outrageous to be endured. But first take some hot soup
and a nap. I will have a plan when you wake up, I feel sure.
Meekly as a little girl she submitted to my ministrations, hot broth
and all. And when at length she lay sleeping amidst the golden glory
of her loosened hair, her face like a pale sage lily in its midst, I stole
downstairs, first faithfully locking the door behind me and pocketing
the key.
The garden between walls was filled with the roseate glow of sunset
as I stepped forth into it, and the night promised fair. The earth was
damp and fragrant from the April storm of the night before, and the
new buds seemed to have doubled their endeavor to make the world
green overnight. On the edges of the paths the frail hothouse-born

tulips lay beaten into the earth. But in the meadow toward the river
the wild crocuses marched bravely. Robins were warbling their
mellow sunset note, and the world seemed sweetly peaceful and
greatly at variance with my mood.
With my mind continually revolving the problem at hand I walked
about the bordered barren beds with a step that was listless enough
in good sooth, pausing now and again to glance up at the walls of
the fine dwelling, which was now to all intents and purposes a
prison. And after a few turns I began to realize that my attention
was turning more and more frequently to the window that had been
Sandro's and to the problem of his escape.
That he had come out by the window upon the first occasion of my
discovering him in the library, and simply let himself in at the
casement door, was plain enough, leaving his door locked from the
inside to avoid invasion by the other servants; indeed it had
developed that it had been his habit to keep his door locked during
the entire period of his employment in the house. But how had he
got there? That was the question. So far as one could see there was
absolutely no means of reaching the ground from that third story,
unless one excepted a frail and narrow wooden lattice intended for
the encouragement of vines, which extended upward to the level of
the higher windows.
Obeying an impulse I went over and made examination of this
lattice, and the riddle was a riddle no longer.
I wonder, I wonder! I said aloud.
I often have, myself! agreed a cheerful voice behind me.
With a guilty start I turned about, and there, of all people on earth,
was Richard, the chauffeur, big nose and all, smiling at me in his
familiar, friendly manner.
Richard! I cried warmly. What brought you here?
I—say, Aunt Mary, I had to come, that was all, he said with
troubled eyes. It's Peaches. You know how I feel about her—how I

have felt all along. I had to see her. It was as if she needed me. Just
a fool hunch. But I came. I couldn't help it—you understand?
Understand? I cried. Bless the boy, I do! Then a way out of our
situation began to make itself clear in my brain and I seized him by
the arm, dragging him to a bench out of general sight from the
house and making him sit beside me, greatly to his bewilderment.
Richard, I said solemnly, have you been at the house yet?
Why, no! said he. I came right into the garden when I saw you
from the drive.
Does anybody know you are coming?
Not a soul! declared Dicky. Why all this mystery?
Listen! I said rapidly. Something awful has happened. Peaches is a
prisoner. Your intuition was right. She—we need your help, and need
it badly.
Is she hurt? he asked. A prisoner? What in the name——
I want you to get a big powerful automobile and have it at the
entrance of the park at twelve o'clock to-night. As soon as you
arrive, park your car, and come to the foot of that trellis over there.
When you get there give the whistle you used to call Peaches with.
If you get an answer, wait for us. If after half an hour you don't hear
anything, call me on the telephone first thing in the morning. Is that
clear?
Yes—but Great Scott! What's wrong?
Never you mind, except that something is very wrong here.
Markheim is an unspeakable beast, and Mr. Pegg is trying to force
Peaches into going through with the marriage in spite of what she
has found out. He has locked her in her room, which opens into
mine.
Well, why not unlock her, then? he asked with stupid masculine
simplicity. Haven't you got a key?

I have, I said. But I have given him my word not to unlock it to
let her out!
But you'll break your word! he said with a satisfied grin.

Not at all! I disclaimed the suggestion. Not at all. However, I
made no promise in regard to the window. And with your assistance
——
I get you! cried Dicky, springing to his feet. Twelve sharp to-night
it is. And I'd better be off now before the old boys get back from
town and spot me—eh, what?
Yes, I agreed.
Then I hesitated. Should I tell him of the duke? Was it possible that
he had not seen the afternoon papers? Evidently so, since he had
not commented upon the robbery. Assuredly they had escaped his
notice. And why tell the poor lovesick boy about Alicia's part in it? I
had a feeling that he would be even more effective in assisting us if
he did not know until we were well on our way that night. So I
merely repeated my instructions and hurried from him to impart the
glad tidings to my charge and then to secure my knitting, in order
that I might be flaunting that badge of womanly innocence in the
drawing-room when those wretched cave men, Markheim and Mr.
Pegg, came down dressed for dinner.

XVI
My dear father used to say that the test of good breeding lay in the
ability to maintain the social amenities toward some one who had
wronged you. Kipling, I think it is, cites the instance of an
Englishman who continued to dress for dinner alone in the jungle, as
a perfect example of breeding. But then, Kipling had only the
Englishman's word for it, because if he were alone when he dressed,
which seems probable—indeed is so stated—how could any one
have seen him? Whereas I have watched my dear father turn the
other cheek to the barber who used to visit our establishment
weekly, when one cheek had been badly scraped, and not utter
anything stronger than an inquiry about the man's health!
And the art of behaving naturally, yet not too naturally, if you
understand me, through the routine of living under trying domestic
conditions, certainly appears to come more easily to persons whose
traditional training has been in the line of self-restraint rather than
that of self-expression; in other words, to those of aristocratic
forbears. Perhaps that is why the purest aristocracy so seldom
attains anything except good manners. But I digress. My intent was
merely to make a passing philosophic comment upon the dinner
party of three—Mr. Markheim, Mr. Pegg and myself—which was held
that evening at the villa.
For though no one could deny Mr. Pegg's sterling worth there were
times when his, as it were, silver needed repolishing. And this was
such a time. As for Sebastian Markheim, for all his wealth, the
veneer of culture, which had never been much more than tailor-
deep, now showed the common clay beneath all too plainly; and the
bandage which his New York physician had arranged over one eye
did nothing to make his behavior more becoming. Whereas on the
other hand I was my own cheery, chatty self, only more so, if

possible, entertaining both gentlemen with a pleasant account of a
railroad accident of which I had read that day, and an explanation of
the main differences between knitting and crochet work.
However, they were not very responsive, proving conclusively my
dear father's theory. In point of fact they were both so
uncommunicative that it was necessary for me to exercise
considerable tact and ingenuity before I could get out of them the
fact that Sandro di Monteventi was still at large, though he had been
traced as far as New York City.
Indeed I cannot imagine why these two gentlemen should have
been suspicious of my trustworthiness, yet their reticence could have
no other implication. However, when I made quite sure that no
further information was to be had out of them I continued to be
quite as delightful as before, even insisting upon serving their after-
dinner coffee with my own hands as soon as the footman had
carried it into the library for us.
I confess that my solicitation about the serving of this was not
wholly disinterested, inasmuch as I administered a small dose of
veronal in each cup—a mere five grains to insure their sleeping—and
sleeping early. And in truth my dear father never approved the
taking of coffee in the evening, and I knew that neither of these
men had had sufficient sleep during the past forty-eight hours. Also,
I did not wish my project to fail through any oversight on my part.
Moreover, neither being a good judge of coffee, they made no
comment on the flavor.
Thus it was that when, shortly after nine o'clock, first one and then
the other excused himself and went off to bed, I did not seek to
detain either, but remained myself in the library for half an hour,
ostensibly engaged in the perusal of a volume of Carlyle's French
Revolution but in reality with one eye fixed upon the clock, and my
attention absorbed with waiting for the moment when I might retire
to my chamber without apparent undue haste.

At length the clock struck ten, having been considerably longer than
its usual time in getting round to it, or so I fancied, and I rose in a
leisurely fashion, putting away my book and ringing for the footman.
When he appeared I bade him a cheerful good night and told him to
put out the lights. Then I made my way upstairs to Peaches, my
heart beating with excitement but my head quite cool and collected
as I admitted myself to our, as it were, joint prison.
I found the dear girl already dressed in a dark suit and small hat, her
face still pale, though her sleep had greatly refreshed her and her
eyes were once more the great fiery cat eyes of amber that I loved
to watch.
Free, she began at once, is there any news of him? Have they
caught him?
Not yet, I replied, but he's in New York somewhere—at least
that's what they think. Don't forget to take your toothbrush.
And you are sure that Dicky understands what to do?
Of course! I replied, going to my top bureau drawer and regarding
the contents critically. Now let me see what I shall take.
I guess father will never forgive us, remarked Peaches dolefully.
But it seems a person never can do what they think right without
getting in wrong with some one.
I shall take my father's chronometer, I mused half aloud, smelling
salts and a pack of cards, for solitaire. Also my small folding check
book. These, together with my toothbrush and clean handkerchief,
will just about fill my reticule.
I was putting these articles into their receptacle as I talked, but my
attention was fixed upon Alicia's face. She looked as if she were
seeing a vision; never have I beheld such an expression of anxious
beatitude, if one may say so, on any human countenance either
before or since. It was hardly wholesome.
Did you put on low-heeled shoes? I asked practically. Peaches
came to with a start.

Yes, she replied. Free, do they let you get married in jail?
They send you there for getting married too often, I replied. Now
keep your mind on the excitement of the moment and hook up my
shirt waist for me, there's a good girl.
A shirt waist that hooks up the back is a blouse, Free, she replied,
smiling wanly. How am I ever going to make your sense of luxury
as strong as your pocket-book?
This blouse by any other name was just as dear, I replied.
And so with light chaffing we made the interval of our preparation
and waiting durable to each other; and at length I sat down by the
opened, darkened window for the third night in succession, to listen
for Richard, the chauffeur, to signal. One by one the other lights in
the house were extinguished and gradually complete silence reigned
over the massive pile of what had but a brief three days ago been
Peaches' future home, and which we were about to forswear forever
in the cause of love and spiritual freedom, not to mention actual
physical freedom. At five minutes of the hour Peaches broke the
silence with an impatient whisper.
All this stage stuff is the greatest bunk! she exclaimed under her
breath. I wish to goodness you'd open the door and let us walk
downstairs like rational human beings!
And break a Talbot's word? I retorted. Never! What I promise
your dear father I keep my word about.
Freedom Talbot, I sometimes think you are stuck on pa,
commented Peaches reflectively.
And then, before I was obliged to reply to this most inconsiderate
comment and indefensible charge, a low whistle sounded from the
garden, the old familiar whistle with which I had heard Peaches
signal to Richard, the chauffeur, a thousand times. At once she was
upon her feet, her body tense, her foolish remark mercifully
forgotten as she responded. Three liquid notes, soft yet clear. Then
silence.

Now for it! I whispered. You follow me—I know the way! And
carrying my shoes in my hand I stepped forth across that window
sill, which must, so I believe, bear about it the odor of romance
forevermore.
I am pained to relate that the first thing Peaches did upon reaching
the ground was to embrace Dick Talbot and kiss him upon both
cheeks. But such is the distressing truth, inappropriate as the action
was in view of the fact that she was escaping from one fiancé in
order to go in search of another, and that Dick was neither of them.
But he did not seem to object in the least, though the moment she
freed him he very properly turned his attention to helping me on
with my shoes.
All set, Aunt Mary! he whispered then. This way, please, and
watch your step in case the enemy sets up a barrage!
In silence we followed him through the garden and out across the
meadow, keeping in the shadow of the trees and hedges whenever
possible, and trampling the brave little white crocuses underfoot. At
length we reached the fence which separated the grounds from the
highroad, and as it was fortunately not very high he helped us over
without difficulty, the main gates at the lodge being, as he informed
us, locked for the night.
Drawn close to the fence was a powerful car with the engine running
softly. Richard assisted me into the rear seat and Peaches sprang up
beside him in front; there was a grinding sound from the creature's
innards and we slid smoothly out into the open road.
The river road from Ossining to New York is one of surpassing
beauty, even at night, when the smooth winding ribbon of it is
practically without traffic. But I was not much concerned with its
loveliness, as the night was too dark, for one thing, to permit more
than a speculation as to what lay behind the hedges and rows of
trees with which it is lined, and the Hudson lay hidden in the black
depth of its own valley save when a moving light or two from a
nocturnal vessel betrayed its whereabouts. Overhanging clouds now

threatened rain, and a mist crept up from the broad stream,
obscuring the lamps and blurring the occasional lighted window by
our way. At any moment I expected that, as The Duchess would say,
the heaven would open to emit a torrential storm; and I wished
heartily that I had worn my other hat.
Furthermore, if I had been able to see anything of the landscape as
we passed I could not have focussed much attention upon it
because of the terrific rate of speed at which Richard, the chauffeur,
had determined to drive. At each and every curve I anticipated an
accident of some sort—a collision with some unfortunate night
traveler, a possibly fatal encounter with a train or trolley car. But
miraculously nothing of the kind happened. I made one or two futile
attempts to dissuade him from his reckless course, inasmuch as the
discovery of our flight was extremely unlikely to occur for many
hours to come. My words were merely blown back into my face, and
solicitude for my hat and feathers at length caused me to relinquish
my efforts and sit dumbly clinging to the seat with one hand and to
my headgear with the other. I assume that he was driving as much
from the stress of his emotions as by reason of Peaches' urging him
to haste, but I could not help reflecting, sorry as I was for the young
man's hopeless passion, that love is a selfish thing—a remark which
has doubtless been made by earlier writers.
I could not hear a word of what conversation was going on in the
front seat, but there seemed to be little enough of it, and all of
Dick's energies were obviously bent on driving—a fact for which I
dumbly thanked the Almighty, and it was not until almost an hour
later, when the outskirts of the city had been reached and our driver
drew up at the curb before a species of nocturnal dairy, or all-night
lunch, as I believe such places are called, that we had any real
conversation regarding further plans.
Richard insisted that we get down from the machine and enter the
humble eating establishment, whose window displayed nothing more
inviting than a few dozen oranges, which my practiced eye

recognized as inferior sweated Southern fruit, and a black cat, the
latter sound asleep.
But once entering its tiled interior, which made me oddly
uncomfortable, conveying as it did a sense of being in a most
dreadfully public bathroom, the refreshing odor of coffee and hot
cakes revived our more material senses, and over a generous supply
of both we told Dick the whole story, beginning with the moment of
our arrival in the East up to the point of the aforementioned
pancakes and coffee.
While Peaches was telling him about the duke and how she loved
him, young Talbot could not endure to look at her—a fact of which
she appeared oblivious, so wrapped was she in her recital. And it
was only when she had quite finished and was waiting for him to
speak that he mastered his emotions sufficiently to look at her with
his honest, suffering eyes.
So he is alive? he said simply. And, of course, you have to go to
him, old girl. There is something wrong with this crook idea. That
man is not a crook.
Thanks, Dicky! said Peaches, her eyes filling as she covered his
hand with hers for an instant. I know there isn't any reason to
believe in him—but I do, just the same.
But there is a reason, said Dick unexpectedly. Look here, Peaches,
I suppose I ought to have told you this when I first came back. But I
didn't first off, because I found you engaged to another man and
apparently happy. I didn't want to go raking over old wounds. So I
didn't even speak of him except to say that I'd heard he was killed in
a gallant action—and I never even said that much until you
mentioned it first—do you remember?
Yes, she nodded. Go on, Dicky!
But I'd seen him while I was over there, he said. I—well, it was
rather by accident but I happened to save his life. Oh, not the last
time! Up to to-night I thought he was dead, the same as you did.
But before that. It was the time I got the Italian medal——

So that was why you wouldn't talk about it! I ejaculated. But
neither paid any attention to me.
He asked a lot about you, Dicky went on. And I told him all I
could. About the ranch, and what you and Miss Freedom were doing.
He was just crazy to hear. But he didn't want me to tell you about
him. 'I'm not fit for her, Dick,' he says to me. We was both getting
over scalp wounds then and used to sit out in front of the hut and
talk a lot. 'I got out of her life for her own good,' he says. 'And if it
ever comes natural tell her I didn't intend to kill the chap at the
railway station—it was in self-defense.' That's what he told me. And
then he tried to give me a ring he had, because of me having the
luck to save him, see? But I wouldn't take it. So he give me his
address in case I ever needed anything.
His address? said Peaches chokingly. Why, Monteventi is his
address, surely?
Yeh—but he give me another one besides, said Dick. Though, of
course, I heard after that he had gone West, and so I kind of forgot
about it.
If he had another address it must have been where he could be
reached in an emergency! cried Peaches. Can't you remember it,
Dicky? Oh, think! Please try to remember it!
I guess maybe I got it on me, said he with a curious shyness. I—
wrote it on the back of your picture. I—I carried it along through the
war. I might have it now, at that.
From the inside of his coat he took a thin wallet, through which he
pretended to search while we watched breathlessly. And there, as I
had anticipated, was the portrait of Alicia—Alicia at sixteen with her
heavy hair in braids over either shoulder and a Mexican sombrero
shading her laughing eyes. He turned it over and she gave a little cry
as she recognized her lover's name—followed by an address in
Hoboken!
We exchanged a look of wonder.

By gosh, I'll bet a dollar that's where he is to-night! exclaimed
Talbot. Not a very tasty neighborhood, but just the kind of a place a
bird like him would fly to for cover. And see the way I was to address
him. S. M., care of Smith! He said they forwarded his mail for him.
Peaches, I'll go there for you the minute I get you two girls safe at a
hotel!
You will not! said Peaches. Because we are going with you.
Oh, come—that's not right! protested Dick. But nothing would
dissuade Peaches.
Well—we may need some money, said he, at length consenting to
the mad scheme. I've a few dollars, but eventually we'll have to get
some more. Did you bring any, Peaches?
Her face dropped in dismay.
I never thought of it! she gasped And my purse was on the
dressing table too!
Never mind! said I, plunging my hand into my reticule. I have
brought a check book and I have a lot of money in the bank.
With which I drew out—not my check book at all, but the black
leather wallet which Peaches had thrown into the pond out at the
ranch, and which I had subsequently rescued.
For a moment we all gazed at it stupidly. Then Peaches recognized it
and snatched it from the table.
Sandy's wallet! she cried. Freedom Talbot, where did you get this
thing?
I—I found it in the garden out at home, I stammered, blushing
violently, and I kept it in case—that is, I thought that perhaps
sometime——
I see! said she in a tone which led me greatly to fear that she did.
What is it? our escort now wanted, not unnaturally, to know.

It's something of his—the duke's, I said. Peaches has had it for
years.
Give us a look-see! asked Dick, stretching out his hand for it.
Rather reluctantly she allowed him to take it.
I bet there's something sewed inside that lining! he commented
after a moment's examination. Let's open her up!
No! cried Peaches, snatching it back. If there is it's none of our
business. I'll just take care of it, thanks! And now about money—our
not having any lets us out of the hotel plan, Dick; and anyhow if we
cash a check we can't do it before to-morrow. In order to get into a
decent hotel without any bags we'd have to prove who we are, and
then pa would spot us first thing in the morning.
Besides which, if Sandro is really at this Hoboken address, he will
very likely be gone by morning, I added; if indeed he has not
already left.
You said it! cried Peaches. Come on, let's go! The Lord only
knows when that ex-sheriff of a parent of mine will have a posse on
my trail!
We acted upon this, the combined wisdom of all three of us, and
paying our modest indebtedness to the midnight-luncheon
establishment, betook ourselves back to the automobile and the
pursuit of our quest.
How silent are the busy marts of Manhattan in the small hours of the
night! With her pearl-like lamps the only sentinels along our way, we
sped into Broadway and thence across the park and down Fifth
Avenue almost as rapidly as we had proceeded along the Albany
highway from Ossining, turning west at some side street evidently
familiar to Richard, the chauffeur, since the days of his debarkation,
and sped toward a westbound ferryboat.
It was a great comfort to me to realize that the city of Hoboken itself
would not be wholly unfamiliar to him either, inasmuch as he had left
for Europe from that port as a soldier, and had again visited it in the

same capacity two years later upon his return. Therefore, he could,
of course, be relied upon to know something about the place, and
just how undesirable he considered the section for which we were
headed might be. It did not, however, occur to me to question him
on this point until the lights of the opposite shore were drawing near.
We had remained seated in the auto, which was driven bodily upon
the lower section of the ferryboat.
Richard, I said, do you consider the section for which we are
bound a residential one?
I do not! he responded promptly. I'll say the inhabitants usually
make about a week-end of it before they are invited to Sing Sing. I
wish I had thought to bring a gun along!
If a revolver will do as well, said I, I have one upon my person. It
is that which I obtained from that gambling creature in Monte
Carlo.
Good girl, Aunt Mary! he exclaimed. Slip it to me, will you?
In order to do so I must retire to the ladies' cabin, I replied with
dignity, inasmuch as it is attached to my—my garter.
Well, if you aren't a caution to rattlesnakes! exclaimed he. All
right, sport, only hurry up, for we'll be landing in a few minutes
now.
I alighted from the rear of the machine with all possible celerity and
made my way upstairs to the higher deck and the retreat which I
sought. Putting the firearm into my reticule I was about to descend
when the sight of a familiar figure standing on the front deck of the
vessel, his face sharply outlined against the light, arrested my action
and my attention.
It was the detective named Pedro—he who had posed as night
watchman at the villa—and he was standing right where he could
not fail to see our car and recognize its occupants the moment we
drove out to land.

It was an emergency and I steeled myself to meet it intelligently. If I
were to go below at once all I could accomplish would be the
warning of my companions. Still, what better course offered? None
that I could see at first. Pedro had not seen me as yet, but
continued to stand looking out toward the Jersey shore. And while I
hesitated as to what I should do the Divine Providence which looks
after lovers put a means of eluding him into my very hands, as it
were.
From a door close beside me and which was marked Private in
large letters, there at this moment emerged a man in overalls. The
door swung to behind him, locking with a snap, and an instant later
he discovered that he had left something in the cabin and being in a
great hurry swore shockingly as he fumbled with his keys, for he
was obliged to unlock the door, which fastened with a spring lock,
before he could get back into the place. The dock was very close
now, and the bell was clanging loudly. In another moment we would
have touched. The mechanic's haste was frantic, which, of course,
caused him some further delay, but at length he succeeded in
opening the door again. On the instant finding myself unobserved I
slid about a quarter of my little pack of playing cards into the jamb
of the door. They were just of a sufficient thickness to allow the door
to shut without permitting it to lock. The mechanic having found
what he wanted came out, swung the door, as he supposed, closed,
and went on his way.
Hardly had he vanished down the stairs when Pedro saw me and at
once approached, raising his hat with a sarcastic politeness that
thinly veiled a sneer. And as he came I knew for certain that he was
the man whom it had twice already been my pleasure to foil.
Nevertheless, I greeted him pleasantly enough.
Ah—good evening! said I. You are looking for Mr. Markheim, I
suppose?
Well, the fellow looked a good deal surprised at that, but he wouldn't
admit it—not he.

Yes, of course, said he, to draw me out.
This is splendid! I said heartily. We were afraid our telegram
hadn't reached you. He's just inside in this cabin. Won't you go in?
The room lighted automatically as the door was pushed inward. He
entered, I pulled out the cards and slammed the door behind him
just as the clamor of our arrival at the hospitable Hoboken shores
drowned out all immediate danger of his cries being heard.
But I ran down the stairs to the car like—like the very deuce, as my
dear father used to say. And climbing into my place I leaned over
and slipped the revolver into Dick's pocket.
Drive like Sam Hill! I commanded in a fierce undertone. I've just
locked Pedro into the fireman's washroom and he's not going to like
it very much!

XVII
I made this remark with a pleasant smile to give the appearance of
passing a joke, in case Pedro's partner should prove to be on board
and watching us. Dicky smiled back, but nevertheless acted upon my
hint without delay; and as a combined result of our smiling faces the
gateman grinned as well and permitted our car to debark first.
The delay on the pier, where we were obliged to proceed at a snail's
pace, was a dreadful strain. Suppose that Pedro's cries were to be
heard, and, rescued, he bore down upon us? I shuddered at the
thought. But at length we were past officialdom and speeding up the
hill and into the city's silent and deserted ways. Dicky turned his
head to question me, almost colliding with a lamp-post by so doing,
but his usual nonchalant skill saving us by a hair—or so it appeared
to me.
Now what the devil did you say you did? he wanted to know.
Pedro—the detective, I said—I locked him up on the boat! I
repeated.
Good heavens, Freedom! How? cried Peaches.
I told them briefly. Richard, the chauffeur, gave a long whistle.
Then it's more than likely we are headed right! said he. Gosh
Almighty, Aunt Mary, I hope I never get in wrong with you!
Why? I demanded. I simply do the obvious thing as occasion
arises.
Well, give us a little advance notice when you are going to pull
something out of the usual, he replied cryptically, and turned his
attention back to the car—for which I felt profoundly grateful—and
to scanning the corner lamps for the name of the avenue for which
we were seeking.

Fortunately the streets were literally deserted and so we escaped
notice. If any one had followed us from the ferry he would have
been visible many blocks away. The only living creature we passed in
fifty squares was a maraudering cat which shot across our path like
a black arrow.
Good luck! commented Peaches.
But the remark failed to reassure me, for by now we had discovered
and turned into our avenue, and its aspect was most decidedly not
residential. In point of fact it could hardly be said to contain houses,
much less anything worthy of being dignified by the name of
residence. It was quite unlike any part of Boston with which I was
acquainted, and I did not fancy its atmosphere, which was redolent
of gas, to say the least. Moreover, it was not at all a suitable place
for a duke to live, even when in retirement from the police. I should
have felt something on upper Fifth Avenue much more fitting—say,
in a secret chamber in the neighborhood of the Plaza. Or in the half-
ruinous mansion of some aristocrat out at, let us say at Hempstead,
which I understand contains many fine old estates.
The quarter through which we were proceeding was impossible—
simply impossible! I trust that there is very little of the snob in me,
at least of that species of snob which cannot distinguish between
genteel poverty and common poverty. Mere shabbiness is no cause
for losing caste, as I myself know full well. And so I would have said
nothing to a shabby neighborhood. But this was not even, properly
speaking, a neighborhood, being as it was, chiefly composed of gas
tanks which towered heavenward in shadowy menace, of
warehouses with blank faces, and unpleasant odors.
Between these at rare intervals were sandwiched little groups of
houses—part of what might originally have been rather a fine
terrace. Three-story brick affairs, they were, that once might have
looked out upon the river before their giant neighbors had risen to
obstruct the view. They stood in little groups of three or four,
huddled together and squeezed on either hand by elbowing dirty
lofts or other commercial tramps of buildings. Most of them

appeared to be used for the storing of hides, to judge from the
refuse in the street before them; some had been ruined by fire
without being demolished, others gaped with broken windows
behind their For Sale signs—drearily awaiting purchasers who
never came.
But here and there among them were a few which gave indication
that human beings still used them as habitations—a dirty window
curtain, a set of battered shades, a stoop less cluttered than those of
the neighbors. And occasionally a dingy notice that there were
furnished rooms to be had. But nowhere any light. It was like a city
of the dead,—or like a town long abandoned. It was difficult indeed
to realize that on the morrow—nay, later on in this very morning—
the place would be a busy waterfront.
It was before one of these poor houses that Richard, the chauffeur,
at length came to a halt; and exceptionally moldy and uninviting
specimen it was, with the storage terminal of some exporting
company on the one hand of it and a string of unsavory-looking
lodgings upon the other. The number for which we were looking was
discernible, though scarcely legible above its closed storm doors—
Number 1162. There could be no mistake. It was our destination.
But it certainly did not look inviting, from cellar to attic the shutters,
though sagging precariously on their hinges, were closed, and the
areaway was obstructed by empty crates, evidently refuse from its
business neighbor.
It doesn't look as if a soul were home, I observed. How very
disappointing!
Houses that refugees are hiding in don't exactly open up like
hotels, observed Dicky dryly. The question now is, how do we get
invited in without bringing a lot of attention on ourselves?
Well, there's no use sitting here discussing such things! I snapped,
taking out my dear father's chronometer and looking at it under the
light of the nearest lamp. It is now fifteen minutes of three o'clock.
I suggest we take some action. We can't stay here, that's plain.

Listen to that thunder, will you? I wish I had worn my other hat! I
just knew it was going to rain!
We might go up and ring the bell, suggested Peaches, climbing to
the sidewalk. That hasn't failed yet, you know.
Since we have been fools enough to come without any definite
plan, agreed Dick Talbot, I suppose we may as well act as if it
were an ordinary call. But first I'm going to run the bus round the
corner and park it out of sight. They'll be more apt to open up.
He left the motor running and assisted me to alight and then drove
off to fulfil this plan, returning presently on foot, whereat we
ascended the broken steps together, and Richard gave the old-
fashioned bell knob a vigorous pull. Faintly from below came the
sound of it in due time, a harsh jangle as when a bell clangs in an
empty echoing room. Then he waited, but no other sound broke the
stillness.
Try again, said Peaches after several minutes had elapsed.
And there really being nothing else to do, Dicky obeyed, with no
better result. Once the faint echoes of its ringing had died away
within the building all was as silent as the tomb. A cat wailed
suddenly from some hidden fence, causing us to start, but that was
all.
There may be some other way in, said Richard in a low voice.
Though this is certainly the right number.
And it may be that nobody lives here too, said I dryly, and that we
have come upon a fool's errand!
You knew we were chancing that! snapped Peaches. But I won't
be satisfied to go away now—let's try the lower door!
Well, I could not see what sense there was in that, though our
escort agreed. And so the two descended from the high stoop and
vanished into the darkness of the areaway, amid the crates that
were heaped within it, while I remained at the main entrance. The
few drops of rain which had been falling when we arrived were

rapidly increasing in number and force, and the thunder drew nearer
and nearer with angry mutterings.
Bitterly regretting that I had ever risked my best hat upon an
adventure which seemed doomed to so tame an ending I withdrew
myself from the open stoop and sought what scant shelter the outer
ledge of the storm door afforded, flattening myself as much as
possible and hoping devoutly that my ostrich tips would recurl nicely.
From below came the sound of a bell, another bell this time, but
ringing in just as desolate a way as that of the front door. Again
silence except for that wretched feline. Then came the sound of
approaching footsteps. Some one was coming down the street!
The steps were not very loud to be sure, the newcomer being soft
shod, and after a moment I realized that Peaches and Dicky, being
intent upon their immediate occupation, and furthermore, cut off
from this approach by being on the far side of the solid masonry of
the high stoop, did not hear him. It flashed across my mind that
policemen did not usually wear sneakers or rubber soles to their
shoes, and that therefore this was not the roundsman of the beat. In
confirmation of this supposition was the fact that whoever was
approaching was in a hurry—not running, but coming on with a
quick light step, very unlike the heavy deliberate tread of a night
watchman wearing away the hours at his post.
Therefore I very cautiously stuck my head round the corner, only to
withdraw it instantly and remain motionless, soundless, against the
door. It was a man who was approaching, his arms filled with
bundles such as would indicate a visit to some all-night grocery or,
more likely, delicatessen store; and his enormous height made him
unmistakable. It was Sandro.
All unknowing what awaited him, he ran lightly up the steps,
glancing up and down the street as he did so. And as he reached the
top step I fell upon him from the shadow, throwing both my arms
round his neck and causing him to spill a half dozen oranges, which
bounded down into the street and areaway—one of them, I later

learned, striking Richard upon the head and thus giving him notice
that he was wanted.
Sandro! I cried. Thank goodness you came home—my hat would
have been ruined in another five minutes!
Good Lord! Miss Talbot! he stammered, making a futile effort to
free himself of me.
But I hung on like a leech. I feared that if I relaxed my embrace for
an instant he would make a dash for liberty.
Oh, but I'm glad to see you! I said. Fear not, we know all, but are
still your friends.
By that time Peaches and Dicky were with us. Seeing this I let him
go, and for a moment he stood there looking dazedly from one to
the other, a side of bacon sticking grotesquely out from under one
arm, a bottle of milk held firmly in the other hand.
Alicia! he murmured, scarcely able to believe his eyes. I don't
understand. And Dick——
Neither do we quite get it, responded Dick cheerfully. That's why
we are here. Just hand over the eats, old man, and let us into this
palace of yours, where we can chin a little less conspicuously! Hurry
now, before some unwelcome party tries to join us!
Spurred into a sort of hypnotic life the duke obeyed, finding a key
and entering first. Peaches went next, slipping her hand through his
arm as she went; and hastily picking up two of the oranges and a
loaf of bread, which fortunately was nicely wrapped in glazed paper,
I followed them, Dicky bringing up the rear and closing the door
behind us.
Then the duke turned on a light, after a brief interval which can only
be explained by—well, it was probably Peaches' fault. At any rate he
turned on a light, which disclosed a shabby, threadbare hallway, and
then opening the door at his right indicated that we should enter.

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Mst209 Mathematical Methods And Models Block 2 The Open University

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