Sensors and Actuators Engineering System Instrumentation 2nd Silva Solution Manual

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impedance should be the complex conjugate of the other. Different matching criteria are
used depending on the applications.

14 SENSORS AND ACTUATORS
COMPONENT INTERCONNECTION AND SIGNAL CONDITIONING 14
Z + Z
Solution 2.2
1. Maximum power transfer
2. Power transfer at maximum efficiency
3. Reflection prevention in signal transmission
4. Loading reduction
Solution 2.3
When a measuring device is connected to a system, the conditions in the system itself will
change, as the measured signal flows through the measuring device. For example, in
electrical measurements, a current may pass through the measuring device, thereby altering the
voltages and currents in the original system. This is called “electrical loading,” and will
introduce an error, as the measurand itself is distorted. Similarly, in mechanical
measurements, due to the mass of the measuring device, the mechanical condition (forces,
motions) of the original system will change, thereby affecting the measurand and causing an
error. This is called mechanical loading.
Now consider the system shown in Figure P2.3. We have: v
 Z 
= K i
v
o   i
 s i 
 Zi
 vo (Zi / Zs )
For a voltage follower, K = 1 and Zo  Zi . Hence, vo = 
Z + Z
 vi , or
v
=
1+ (Z / Z )
.
This relationship is sketched in Figure S2.3.
 s i  i i s
1
0.8
0.6
0.4
0.2
00 1 2 3 4 5 6 7 8 9 10

Figure S2.3: Non-dimensional curve of loading performance.

2

i i
Some representative values of the curve are tabulated below.
Zi / Zs vo / vi
0.1 0.091
0.5 0.55
1 0.5
2 0.667
5 0.855
7 0.875
10 0.909
Note: Performance improves with the impedance ratio Zi / Zs .
Solution 2.4
Open-circuit voltage at the output port is (in the frequency domain)
 1 
R2 + 
 jC
voc =

R1 + R2 + jL +
= veq
1 

(i)
 jC
Note: Equivalent source veq is expressed here as a function of frequency. Its corresponding
time function veq (t) is obtained by using inverse Fourier transform. Alternatively, first
R +
1 
replace j by the Laplace variable s: v (s) = 
 sC

v(s)
eq
R + R + sL +
1 
. Then obtain the
1 2
sC
inverse Laplace transform, for a given v(s), using Laplace transform tables.
Now, in order to determine Zeq, note from Figure P2.4(b) that if the output port is
veq
shorted, the resulting short circuit current iSC is given by: iSC =
Zeq
. Hence,
veq v
Z = = oc
(ii)
eq
sc sc
Since we know voc (or veq ) from equation (i) we only have to determine isc . Using the
actual circuit with shorted output, we see that there is no current through the parallel
impedance R2 +
1
jC
because the potential difference across it is zero. Thus,

1
l s
s
isc =
(R
v
+ jL)
(iii)
Now substituting Equations (i) and (iii) in (ii) we have:
 
R +
1 
R + jL

  2  1 
 jC 
Zeq =
 1 
R1 + R2 + jL +
 
 j C
Solution 2.5
(a) Load power efficiency  =
Rl
=
Rl / Rs
v 2
R
(Rl + Rs ) (Rl / Rs +1)
v 2
(b) Load power p = s l
; Maximum load power (occurs at R = R ) p = s
l
[R + R ]2 l s max
4R
4Rl / Rs
➔ pl / pmax =
[Rl / Rs +1]2
We use the following MATLAB script (.m file) to generate the two curves:
% Efficiency and load power curves
lr=[]; eff=[]; pw=[]; % declare vectors
lr=0; eff=0; pw=0; %initialize variables
for i=1:100
a=0.1*i; %load resistance ratio
lr(end+1)=a; % store load resistance
eff(end+1)=a/(a+1); % efficiency
pw(end+1)=4*a/(a+1)^2; % load power
end
plot(lr,eff,'-',lr,pw,'-',lr,pw,'x')
The two curves are plotted in Figure S2.5.
It is seen that maximum efficiency does not correspond to maximum power. In
particular, the efficiency increases monotonically with the load resistance while the
maximum power occurs when Rl = Rs . Hence, a reasonable trade-off in matching the
resistances would be needed when both considerations are important.

Efficiency and Load Power Curves
1
0.9
0.8
Efficiency (fractional)
0.7 Load power/Max power
0.6
0.5
0.4
0.3
0.2
0.1
0
0 1 2 3 4 5 6 7 8 9 10
Load Resistance/Source Resistance Ratio
Figure S2.5: Variation of efficiency and maximum power with load resistance.
Solution 2.6
Voltage is an across variable. In order to reduce loading effects, the resistance of a voltmeter
should be much larger than the output impedance of system or load impedance. Then, the
voltmeter will not draw a significant part of the signal current (and will not distort the
signal). Current is a through variable. The resistance of an ammeter should be much smaller
than the output impedance of system or load impedance. Then, the ammeter will not provide
a significant voltage drop (and will not distort the signal).
Voltmeter should be able to operate with a low current (due to its high resistance) and
associated low torque, in conventional electromagnetic deflection type meters. Low torque
means, a torsional spring having low stiffness has to be used to get an adequate meter
reading. This makes the meter slow, less robust, and more nonlinear, even though high
sensitivity is realized.
Ammeter should be able to carry a large current because of its low resistance. Hence
meter torque would be high in conventional designs. This can create thermal problems,
magnetic hysteresis, and other nonlinearities. The device can be made fast, robust, and
mechanically linear, however, while obtaining sufficient sensitivity.
Note: The torque is not a factor in modern digital multi-meters.
Solution 2.7
(a) The input impedance of the amplifier = 500 MΩ.
Estimated error =
10
100% = 2%
(500 +10)
(b) Impedance of the speaker = 4 Ω.

o
o i
o i
i
Estimated error =
0.1
100% = 2.4%
(4 + 0.1)
Solution 2.8
v = F (f , f )  v =
F
1
f +
F
1
f (i)
o 1 o i o
fo fi
v = F (f , f )   v =
F2
f +
F2
f (ii)
i 2 o i i
fo fi
In terms of incremental variables about an operating point, we can define the input
impedance Zi and the output impedance Zo as
vi
Zi =
fi
with fo = 0 (iii)
v with f = 0
Z = o o
(iv)
o
fo with vo = 0
Note: fo = 0 corresponds to incremental open-circuit condition and vo = 0
incremental short-circuit condition.
corresponds to
From (ii) with f = 0 (i.e., open circuit at output) we get Z =
F2
.
fi
Now using the open-circuit by subscript “oc” and the short-circuit by subscript “sc” we have:
From (i):
v =
F
1
f (v)
From (ii):
o oc
v
fi
=
F2
f
i oc
(vi)
i oc fi
i oc
Note: vi is an independent increment, which does not depend on whether oc or sc condition
exists at the output. But fi will change depending on the output condition.
From (v) and (vi):
vo oc
=
F
1
f
F2
f
vi (vii)

From (i):
0 =
F
1
f
i i
+
F
1
f (viii)
From (ii):
fo
o sc
fi
i sc


vi =
F2
fo
fo sc
+
F2
fi
fi sc
(ix)
Eliminating fi sc
from Equations(viii) and (ix) we get,
1
fo sc
=
F2
−
F
1

F2 F
1 

vi (x)
Substitute (vii) and (x) in (iv):
f o fo f i fi 
F
Z = 1
F2
 F2
−
F1

F2 F1

=
F1
.
F2 F1
−
F1
o 
f f
 
f f f f

f f f f
 i i   o o i i  i o i o
One way to experimentally determine Zi and Zo (under static conditions) is to first
experimentally determine the two sets of operating curves given by vo = F
1(fo, fi ) and
vi = F2(fo, fi ) under steady-state conditions. For example fo is kept constant and fi is
changed in increments to measure vo and vi once the steady state is reached. This will give
two curves fi versus vo and vi versus vi for a particular value of fo. Next fo is incremented and
another pair of curves is obtained. Once these two sets of curves are obtained for the
required range for fi and fo, the particular derivatives are determined from using the general
method shown in Figure S2.8, for the case z = F(x,y) with:
z
  and
z

z
.
x y y
z y+Δy
y
Δz
α

x
Figure S2.8: Computation of local slopes.

Z l c
n n
Zc c
Solution 2.9
(a)
We have: vi = Zcii , vt = Zlit , vr = −Zcir , vt = vi + vr , and it = ii + ir , where “i” denotes current
and the subscript “r” denotes “reflected.”
Substitute: v = Z i = Z (i + i ) = Z (
vi
−
vr
) =
Zl
(v − v ) =
Zl
(v − (v − v )) =
Zl
(2v − v )
t l t l i r l i r i t i i t
Zc Zc Zc Zc Zc
➔ (1+
Zl
)v = 2
Zl
v ➔ v =
2Zl
v
t i t
(Z + Z )
i
(b) We need vt = vi ➔
2Zl
(Zl + Zc )
=1 ➔ Zl = Zc
(c) Use a transformer with the required impedance ratio = (turns ratio)2
Solution 2.10
For the given system, n =
the frequency ratio r  2.0.
1106
100
rad/s = 100 rad/s and   200 rad/s. Hence, we have
For r = 2.0 and Tf = 0.5 we have 0.5 =
1 + 16 2
9 + 16 2
or,  =
5
. Hence,
48
b = 2nm = 2
5
100100 N.s/m
48
→ b = 6.455103
N.s/m .
With this damping constant, for r  2, we will have Tf  0.5. Decreasing b will decrease
Tf in this frequency range.
To plot the Bode diagram using MATLAB, first note that:
2 = b /m = 6.455103
/100 = 64.55 rad/s and  2
=104
(rad/s)2
The corresponding transmissibility function is Tf =
64.55s +104
s2
+ 64.55s +104 with s = j
The following MATLAB script will plot the required Bode diagram:
% Plotting of transmissibility function
clear;
m=100.0;
k=1.0e6;
b=6.455e3;
sys=tf([b/m k/m],[1 b/m k/m]);

bode(sys);

Phase
(deg)
Magnitude
(dB)
The resulting Bode diagram is shown in Figure S2.10. A transmissibility magnitude of 0.5
corresponds to 20log10 0.5 dB = -6.02 dB.
Note from the Bode magnitude curve in Figure S2.10.4 that at the frequency 200 rad/s the
transmissibility magnitude is less than -6 dB and it decreases continuously for higher
frequencies. This confirms that the designed system meets the design specification.
Bode Diagram
10
0
-10
-20
-30
-40
-50
0
-45
-90
-135
1
10
2 3 4
10 10 10
Frequency (rad/sec)
Figure S2.10: Transmissibility magnitude and phase curves of the designed system.
Solution 2.11
(a) Mechanical Loading
A motion variable that is being measured is modified due to forces (inertia, friction, etc.) of
the measuring device.
(b) Electrical Loading
The output voltage signal of the sensor is modified from the open circuit value due to the
current flowing through external circuitry (load).
Mechanical loading can be reduced by using noncontact sensors, reducing inertia and
friction, etc.

Electrical loading can be reduced by using a low-output-impedance sensor, high-
impedance load, impedance transformer, etc. Some typical values of the listed parameters are
given in the following table:
Parameter Ideal Value Typical Value
Input Impedance Infinity 1 M Ω
Output Impedance Zero 50 Ω
Gain Infinity 10
6
Bandwidth Infinity 10 kHz
Solution 2.12
The differential signal from the secondary windings is amplified by the ac amplifier and is
supplied to the demodulator. A carrier signal is used by the demodulator to demodulate the
differential ac signal. The modulating signal that is extracted in this manner is proportional
to the machine displacement. This signal is filtered to remove high-frequency noise (and
perhaps the carrier component left by the demodulator), and then amplified and digitized (using
an ADC) to be fed into the machine control computer.
The compensating resistor Rc may be connected between the points A and B or A and
C, as shown in Figure S2.12.
AC
Excitation
A
Primary
Coil
Secondary
Coils
C B
RC
Differential
Output
Figure S2.12: Null compensation for an LVDT.

Process
Action
Drive
Amplifier
DAC
Drive
Solution 2.13
(a)
Scanning
Logic
Control
Image Sensor
Monitored Light
Object
CCD or
CMOS
Matrix
Capacitive-
Coupled
Amplifier
and Filter
ADC
Computer
and Image
Acquisition
Hardware/
Software
Signal
Figure S2.13: Monitoring of an industrial process using image processing.
(b) Data rate = 488  380  8  30 bits/s = 44.5 Megabits/s
(c) Since hardware processors are faster, we prefer them for this level of high data rates
for real-time action. Also, they are cheaper when mass produced. Disadvantages include
limitations on algorithm complexity in image processing and memory size.
Solution 2.14
Since the open-loop gain K of an op-amp is very high (105
to 109
) and the output voltage
cannot exceed the saturation voltage (which is of the order of 10 V) the input voltage
vi = vip − vin is of the order of a few V, which can be assumed zero (when compared with
the operating voltages) for most practical purposes. Hence, vip = vin . Next since the input
impedance Zi is very high (M), the current through the input leads has to be very small for
this very small vi under unsaturated conditions.
(a) The saturated output of the op-amp must be 14 V in this example. The ac noise (line
noise, ground loops, etc.) in the circuit can easily exceed the saturation input (on the
order of 10 V) of the op-amp, under open-loop conditions. Hence, vi = vip − vin can

o
oscillate between + and - values of the saturation input. This provides an output, which
switches between the +ve saturated output +vsat and the -ve saturated output -vsat of the
op-amp.
(b)
Case 1: vip = −1 V , vin = +0.5 V ➔ vi = vip − vin = −1− 0.5 V = −1.5 V
➔ vo = −1.55106
V = −1.55 V = −7.5 V
This is valid since the output is not saturated.
Case 2: vip = 0 , vin = 5 V ➔ vi = 0−5 V = −5 V
➔ v = −55106
V = −25 V
➔ Op-amp is saturated ➔ The actual output would be vo = −14 V
Solution 2.15
(a) Offset Current (Typically in nA)
Bias currents are needed to operate the transistor elements in an op-amp IC. These currents
i+ and i- flow through the input leads of an op-amp. The offset current is the difference
i+ − i− . Ideally, the offset current is zero.
(b) Offset Voltage (Typically in mV or less)
Due to internal circuitry (IC) in an op-amp, the output voltage might not be zero even when
the two inputs are maintained at the same potential (say, ground). This is known as the offset
voltage at output. Furthermore, due to unbalances in the internal circuitry, the potentials at
the two input leads of an op-amp will not be equal even when the output is zero. This
potential difference at the input leads is known as the input offset voltage. This is usually
modeled as a small voltage source connected to one of the input leads
(c) Unequal Gains (Can range over 105
to 109
)
The open-loop gain of an op-amp with respect to the “+” input lead may be different from
that with respect to the “-” input lead. This is known as unequal gains.
(d) Slew Rate (Typically about 0.5 V/s)
When the input voltage is instantaneously changed, the op-amp output will not change
instantaneously. The maximum rate at which the output voltage can change (usually
expressed in V/s) is known as the slew rate of an op-amp.
Even though K and Zi are not precisely known and can vary with time and frequency,
their magnitudes are large. Hence, we can make the basic assumptions: equal potential at the
two input leads and zero current through the input leads, under unsaturated conditions. Then,
these parameters do not enter the output equations of an op-amp circuit.

v
Solution 2.16
(a)
A voltage follower is an amplifier having a unity voltage gain, a very high input impedance,
and a very low output impedance. A simple model for a voltage follower is obtained by
connecting the “-” lead of an op-amp to the output (feedback path) and using the “+” lead as
the input lead. Under unsaturated conditions we have vo = vi . It is known that the input
impedance of a voltage follower is much larger than that of the original op-amp (which itself
is quite large—megohm range) and the output impedance of a voltage follower is much
smaller than that of the original op-amp (which is also small). Hence, a voltage follower
functions primarily as an impedance transformer that provides the ability to acquire a voltage
from a high-impedance device, where the current is rather low (e.g., a high-impedance
sensor) and transmitting that voltage signal into a low-impedance device, without distorting
the acquired voltage.
(b)
Consider circuit in Figure P2.16. Since vB = 0, we have vA = 0 .
Hence, current summation at node A gives:
vi
+
vo
= 0
R Rf
Note: The current through an input lead of an op-amp has to be zero.
Hence,
vo
= −
Rf
and
R
K = −
f
➔ This is an inverting amplifier.
vi R R
Solution 2.17
Slew rate: s = 2fba (i)
where, a = output amplitude, fb = bandwidth (Hz).
The rise time Tr is inversely proportional to fb. Hence, fb =
k
where, k = constant.
Tr
Substitution gives: s =
2ka
Tr
(ii)
From (i): For constant s, bandwidth decreases as a is increased.
For a sine signal, substitute the given values in (i): f =
0.5
MHz = 31.8 kHz
b
2  2.5
Next, for a step input, use s =
y
t
where, Δy = step size, = Δt rise time
Substitute numerical values: t =
y
=
2.5
s = 5 s .
s 0.5

Solution 2.18
(a) Common-mode voltage vcm = voltage common to the two input leads of a differential
amplifier = average of the two inputs.
Common-mode output voltage vocm = output voltage of the amplifier due to vcm (i.e.,
in the absence of any voltage differential at the input.)
(b) Common-mode gain =
vocm
vcm
(c) CMRR = K
vcm
=
K
vocm common-mode gain
where, K = amplifier gain (i.e., differential gain or gain at the output for the inferential input).
Specifically: v = K(v − v ) + K 
1
(v + v )
o ip in cm
Typically CMRR  20,000 .
2
ip in
When A is closed and B is open, the flying capacitor C gets charged to the differential
voltage vi1 − vi2 and hence the common-mode voltage does not enter. When A is open and
B is closed, the capacitor voltage, which does not contain the common-mode signal, is
applied to the differential amplifier.
Solution 2.19
The textbook definition of stability relates to the dynamic model (linear or nonlinear) of a
system and hence to its natural response. In particular, in a linear system, if at least one pole
(eigenvalue) has a positive real part, the natural response of the system will diverge, and the
system is unstable.
Instrumentation stability is linked to the drift associated with change in parameters of
the instrument or change in the environmental conditions.
Temperature drift =
Change in output
Change in temperature
assuming that the other conditions and the input are maintained constant.
Long term drift =
Change in output
Duration

assuming that the other conditions and input are the same.
Ways to Reduce Drift: Regulate the power supply; Use feedback; Keep the environment
uniform; Use compensating elements and circuitry; Recalibrate the device before
each use.

Solution 2.20
DC
Power
Supply
High -
Frequency
Oscillator
Input
Filter Voltage
Follower
Modulator Demodulator
Transformer
Coupling
Low-
Pass
Filter
DC
Amplifier
Output
Input
Section
Output
Section
Figure S2.20: An isolation amplifier.
Solution 2.21
Possible causes:
1. Faulty cellphone charger and it not having a ground lead and pin
2. Faulty laptop charger and it not having a ground lead and pin
Faulty Cellphone Charger: Due to a short-circuit, the high voltage (110-240 VAC) will
leak into its cable and reach the cellphone. If the cellphone is not properly
grounded/isolated, the voltage will form a path through the user’s body. According to the
burns, this path has to include the chest and the ears (possibly through the
headphone cable).
Faulty laptop Charger: Due to a short, the high voltage (110-240 VAC) will leak from the
charger into the DC cable segment that is connected to the laptop. If the laptop is not
properly grounded/isolated, the voltage will form a path through the user’s body.
According to the burns, this path has to include the chest and the ears (possibly through
the headphone cable).
On the one hand, the newspaper report indicated that there were inexpensive and non-
compliant cellphone chargers in the market. However, since the power consumption of
the cellphone charger is relatively low and since the electricity path through the body
included the ears (Note: The headphones were connected to the laptop, not to the
cellphone) the other possibilities of fault need to be investigated as well. Typically,
however, the laptop chargers (particularly those provided by reputed laptop

manufacturers) are subjected to rigorous standards, inspection, and quality control (so
are cellphone chargers from reputed manufacturers).
Solution 2.22
Passive filters are circuits made of passive elements, which do not require an external power
supply to operate. These circuits allow through those signal components in a certain frequency
range and block off the remaining frequency components.
Advantages and disadvantages of passive filters: See disadvantages and advantages of active
filters.
The voltage follower is an impedance transformer. It reduces loading problems by providing
a very high input impedance and very low output impedance. Furthermore, it does not change
the voltage gain.
Solution 2.23
Applications:
(a) Anti-aliasing filters in digital signal processing
(b) To remove dc components in ac signals
(c) As tracking filters
(d) To remove line noise in signals.
Each single-pole stage will have a transfer function of the form: G (s) =
ki (zis +1)
Hence, the cascaded filter will have the transfer function:
i
( pis +1)
G( s ) = ΠG ( s ) = Π k
( zis +1)
, where ““ denotes the product operation.
i i
( pi s +1)
Note that the poles are at −
1
 pi
there cannot be resonant peaks.
and these are all real; there are no complex poles. Hence,
Solution 2.24

It provides the flattest magnitude over the pass band among all filters of the same order
(same pole count).
Also, we prefer a very sharp cutoff (i.e., steep roll-up and roll-down).

A o
Solution 2.25
(a)
Op-amp properties: 1. Voltages at input leads are equal; 2. Currents through input leads = 0
Op-amp property: vB = vP = vo (i)
Current Balance at Node A:
(vi −vA )
=
(vA −vB )
+
(vA −vP )
(ii)
Zc Zc R
Current Balance at Node B:
(vA −vB )
=
vB
(iii)
Note:
Zc R
Z =
1
= impedance of capacitor
c
Cs
Substitute (i) and (iii) in (ii):
(vi
−vA
)
=
vo
+
(vA
−vo
)
=
vA
→ v = (1+
1
)v (iv)
Zc
R R R i
τs
A
Substitute (i) in (iii):
(vA −vo )
=
vo
→ v = (1+
1
)v (v)
Zc R τs
Note: τ = RC = time constant
v (s)2
Substitute (iv) in (v): G(s) = o
=
vi (s +1)2
This is a 2nd order transfer function ➔ 2-pole filter
(b)
With s = j in G(s), we have G( j) =
− 2
2
(1+ j )
Filter magnitude G( j)
=
2
= 2
=
(1+  2
 2
)
The magnitude of the filter transfer function is sketched in Figure S2.25. This represents a
high-pass filter.
(c)
When,  
1
: G( j)

When,  
1
: G( j)

  2
 2
=
2
=
2
 = 1
 2
 2
Hence, we may use  c =
1
as the cutoff frequency.

Note: G( j) → 1 as  →  For small : Roll-up slope of G( j)

curve is 40 dB/decade.

Magnitude
G( j)
(Log)
0 dB
40 dB/decade
Frequency 
(Log)
Figure S2.25: Filter transfer function magnitude.
Solution 2.26
Strain Gauge for force Sensing: Low-frequency noise due to ambient temperature
fluctuations. These may be compensated for (using abridge circuit) and also through
high-pass filtering
Wearable Ambulatory Monitoring (WAM): In human mobility monitoring (e.g., in
telehealth applications) a popular WAM sensor is a combined accelerometer and
gyroscope. Both sensors will be affected by bias, removal of which would need high-
pass filtering). High-frequency artifacts may be generated in the sensed signal due to
muscle tremor and low-frequency artifacts may be formed due to respiration. These
may be removed using band-pass filtering.
Microphone (Robotic Voice Commands): A band-pass filter for the human vocal range
(80Hz to 1100Hz).
AC-powered Tachometer for Speed Sensing: Line noise (60 Hz) may be removed using a
notch filter.
Solution 2.27
(a)
The main signal component appears sinusoidal with frequency ~ 1 rad/s (period ~ 6.3 s).
From the figure it is not clear whether there is a superimposed sinusoidal signal of high
frequency and/or high-frequency noise, even though some oscillations are observed in the
noise.

5
Signal
(V)
Signal
(V)
(b)
We use the following MATLAB command to obtain the four-pole Butterworth low-pass
filter with cut-off frequency at 2.0 rad/s:
>> [b,a]=butter(4,2.0,'s')
b =
0 0 0 0 16.0000
a =
1.0000 5.2263 13.6569 20.9050 16.0000
Then, we use the following MATLAB commands to filter the data signal using this filter, and
plot the result shown in Figure S2.27(a):
>>y1=lsim(b,a,u,t);
>> plot(t,y1,'-')
(a) Low-pass Filtered Signal with Cut-off 2.0 rad/s
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0 1 2 3 4 5 6 7 8 9 10
Time (s)
-3
x 10 (b) Filtered Signal with Pass-band [9.9,10.1] rad/s
4
3
2
1
0
-1
-2
-3
-4
-5
0 1 2 3 4 5 6 7 8 9 10
Time (s)

Signal
(V)
Signal
(V)
0.25
(c) Filtered Signal with Pass-band [9.0,11.0] rad/s
0.2
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
-0.25
0 1 2 3 4 5 6 7 8 9 10
Time (s)
0.25
(d) Filtered Signal with Pass-band [8.0,12.0] rad/s
0.2
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
-0.25
0 1 2 3 4 5 6 7 8 9 10
Time (s)
Figure S2.27: Filtered signals. (a) Low-pass at 2.0 rad/s; (b) Band-pass over [9.9,
10.1]; (c) Band-pass over [9.0, 11.0]; (d) Band-pass over [8.0, 12.0].
It is seen that the filtered signal has a frequency of 1.0 rad/s with the correct amplitude (1.0)
and negligible phase shift. Initially some signal distortion is seen due to the transient nature
of the output. However, the steady-state value is reached in half a period of the signal.
(c)
Band-pass filtering for the three cases are obtained using the following MATLAB
commands:
>> Wn=[9.9,10.1];
>> [b2,a2] = butter(4,Wn,'bandpass','s');
>> y2=lsim(b2,a2,u,t);
>> plot(t,y2,'-')

>> Wn=[9.0,11.0];
>> plot(t,y2,'-')
>> Wn=[8.0,12.0];
>> plot(t,y2,'-')
The results are shown in Figures S2.27 (b)-(d). The very narrow pass-band produced a
filtered result that took a rather long time to reach the steady state of amplitude 0.2 (i.e., the
filter had a larger time constant). When the pass-band was increased, the steady state was
reached quicker (i.e., smaller filter time constant). However, the amplitude distortion of the
filtered signal was noticeable as a result.
Solution 2.28
If a characteristic of a signal “B” is changed with respect to time, depending on some
characteristic parameter of another signal “A,” this process is termed modulation. The
modulating signal (or data signal) is the signal A. The carrier signal is the signal B. The output
signal of the modulation process is the modulated signal. The process of recovering the data
signal (A) from the modulated signal is known as demodulation.
(a) Amplitude Modulation (AM)
The carrier is a periodic signal (typically a sine wave). The amplitude of the carrier signal is
varied in proportion to the magnitude of the data signal. Specifically, the carrier signal is
multiplied by the data signal. In one form of AM, the carried signal is added again to the
resulting product signal. The AM technique is used in radio transmission and in sensing (e.g.,
differential transformer). The sign of the data signal is represented by a 180 phase change
in the carrier signal.
(b) Frequency Modulation (FM)
The carrier is typically a sine wave signal. The frequency of the carrier signal is varied in
proportion to the magnitude of the data signal. This process is commonly used in radio
transmission and data storage. Sign of the data signal is represented by changing the carrier
phase angle by 180.
(c) Phase Modulation (PM)
The carrier signal is typically a sine wave. The phase angle of the carrier signal is varied in
proportion to the magnitude of the data signal. Used in signal transmission. Sign of the data
signal is represented by positive or negative phase change in the carrier.
(d) Pulse-width Modulation (PWM)

The carrier is a pulse signal. The pulse width of the carrier is changed in proportion to the
magnitude of the data signal. Both the spacing between the pulses (pulse period) and the
pulse amplitude are kept constant. Used in dc motor speed control, other control applications,
and digital-to-analog conversion (DAC). Sign of data is accounted for by using both +ve and
-ve pulses.
(e) Pulse-Frequency Modulation (PFM)
The carrier is a pulse signal. The frequency of the pulses is changed in proportion to the
magnitude of the data signal. Pulse width and pulse amplitude are maintained constant (and
the pulse period is varied). Used in dc motor speed control. Sign of data is accounted for by
using both +ve and -ve pulses.
(f) Pulse-Code Modulation (PCM)
Carrier signal is a pulse sequence. The value of the data signal at a given time instant is
represented in the binary form and this value is represented in the carrier (of by equally
spaced pulses) using the fact that the presence of a pulse can be used to represent binary 0.
Then for a given word size, say n bits, a maximum of n pulses have to be transmitted. The
sign of the data word may be represented by an additional bit, known as the sign bit (using,
say 1 to represent “+” and 0 to represent “-”). Separation between one data word and the
next may be detected through “framing” a data word using “start bits” and “end bits.” Used
in digital communication.
Solution 2.29
Intentional AM
• Radio broadcast
AM will improve signal communication with reduced distortion by noise and
transmission loss. It will also facilitate making several broadcasts
simultaneously in the same geographic area (due to the frequency-shifting
property of AM)
• Signal conditioning
AM enables us to exploit advantages of ac signal conditioning hardware
(improved stability, reduced drift, etc.). Also, the AM process will improve
the signal level and noise immunity as a result of the use of the original signal
(to be conditioned) to modulate a high-frequency, high-power carrier signal.
Natural AM
• Any device that uses the transformer action (primary winding and secondary
winding with the primary coil being excited by an AC; e.g., linear variable
differential transducer or LVDT, ac tachometer).
• A rotating machine with a fault; e.g., a gearbox with a fault on a tooth, a turbine
rotor with eccentricity or damaged blade.
Yes. In the first device AM provides the advantages of ac signal conditioning, and
improves noise immunity and signal level.

In the second device, the AM principle is useful for fault detection and diagnosis (through
identification of the associated frequencies).
Solution 2.30
(a) Ball passing frequency = carrier frequency fc =
840 + 960
= 900 Hz
2
Hence, Estimated shaft speed = fb = 900 − 840 rev/s
Number of balls =
ball passing frequency
=
900
=15
shaft speed 60
= 60rev/s = 3600 r.p.m.
Peak frequencies = 900  60 Hz = [840 and 960]Hz, (see Figure P2.30).
(b) Yes.
Solution 2.31
(a) Phase-Sensitive Demodulation
If the sign of the data signal (modulating signal) is preserved at all times during modulation
and is properly recovered in the demodulation process, we have a phase-sensitive
demodulation. Note: A sign change corresponds to a 180 phase change of the modulated
signal; hence, the name.
(b) Half-Wave Demodulation
In half-wave demodulation, an output is generated during every other half-period of the
carrier signal.
(c) Full-Wave Demodulation
Here, an output (demodulated value) is generated continuously for every point of the carrier
signal.
The rotating frequency (rev/s) fo modulates the responses produced by the forcing
functions such as gear meshing, ball or roller hammer, and blade passing in rotating machinery.
Hence, if the forcing frequency is fc, according to the modulation theorem, peaks occur at fc
 fo instead of at fc.
Solution 2.32
Resolution corresponds to 1 LSB ➔ 10/ 28
= 10/256 = 0.04 V
Quantization error 9rounding off) corresponds to half this value → 0.02 V

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"You, at least," he says, reflectively, "will miss her greatly. You have
so long honored her by your preference for her society and
companionship. How will you fill up the long months of her
absence?"
She sighs softly.
"She has left me a precious charge—all her poor to look after, her
heathen fund, her sewing society—much that has been her sole
charge heretofore, and which I fear may be but imperfectly fulfilled
by me. Still I will do my best."
"You always do your best, I think, in all that you undertake," says
this loyal heart.
"Whatever is worth doing at all is worth doing well, I think," she
answers, with a faint flush evoked by his quiet meed of praise.
Then people begin to flock in to look at the wedding gifts and at
Grace Winans, who is the loveliest thing of all. She has on a wedding
garment in the shape of pale violet silk, with overdress of cool
muslin, trimmed with Valenciennes, white kid gloves and turquois
ornaments set in pearls. The wedding guests wore their bonnets,
and she had a flimsy affair of white lace studded with pansies on the
top of her graceful head. Her dress was somewhat after the style of
fashionable half-mourning. She had selected it purposely because
not knowing if she were wife or widow a more showy attire was
repugnant to her feelings.
"This," she said, touching a costly little prayer-book with golden
cross, monogram, and clasps. "This, I fancy, is from you."
"You are right," he answered. "This set of the poets so handsomely
bound is from mother. But are you not weary of looking at all these
things? Shall we not go and find Lulu?"
"By the way," she says, idly, as they slowly pass through the politely
staring throng, exchanging frequent nods and smiles with
acquaintances, and occasional compliments with more intimate

friends, "there is a report—have you heard it?—from Memphis,
Tennessee—of the yellow fever."
"Yes," he answers, slowly. "I have heard the faintest rumor of it,"
looking down with a cloud in his clear eyes at the fair inscrutable
face. "Are you worried about it? I remember to have heard you say
your nearest relatives were there."
"Only distant relatives," she answers, composedly. "I am no more
worried about them than about the other inhabitants of that city. My
relatives had little sympathy for me in the days of my bereavement
and destitution, and though one may overlook and forgive such
things one does not easily forget."
He was looking at her all the time she was speaking, though her
eyes had not lifted to his. On the sweet, outwardly serene face he
saw the impress of a growing purpose. What it was he dared not
whisper to his own heart.
The cloud only leaves his brow when they reach his radiant sister.
She stands beneath a bridal arch of fragrant white blossoms, roses,
and lilies, and orange blossoms dropping their pendant leaves down
over her head as she receives the congratulations and adieus of her
friends before she goes to change her bridal robe for the traveling-
dress in which she is to start for the other shores of the Atlantic.
Conway is beside her, nonchalant, smiling, handsome, very well
satisfied with himself and the world. As his glance falls on the fair,
pensive face of the Senator's deserted wife, the smile forsakes his
lip, one sigh is given to the memory of "what might have been," and
turning again to his young bride, the past is put away from him
forever, and he is content.
And presently the new-made Mrs. Conway flits up stairs with Gracie,
to array herself in the sober gray traveling-silk.
Grace parts the misty folds of the bridal vail and kisses the pearl-fair
forehead.
"Oh, darling!" she whispers, "may God be very good to you—may he
bless you in your union with the man of your choice."

Lulu's tears, always lying near the surface, begin to flow.
"Oh, Gracie," she says, suddenly, "if all should not be as we fear—if I
should chance to see your husband on the shores of Europe, may I
tell him—remember he has suffered so much—may I tell him that
you take back the words you said in the first agony of your baby's
loss?"
"What was it I said?" asked Gracie, with soft surprise.
"Do you not remember the night you were taken ill, when you were
half delirious, and he came to see you——"
"Did he come to see me?" interrupts Grace.
"Certainly—don't you remember? You were half delirious, and you
fancied your husband had hidden away the child to worry you, and
you said——"
"I said—oh, what did I say, Lulu?" breathed the listener, impatiently.
Lulu stopped short, looking, in surprise, at the other.
"Gracie, is it possible that you were entirely delirious, and that you
recollect nothing of your husband's visit and your refusal to see
him?"
"This is the first I ever knew of it," said Grace, sadly; "but go on,
Lulu, and tell me, please, what I did say."
"You refused to see him, though entreated to do so by Mrs. Conway;
you said you would never see him—never, never—unless he came
with the missing child in his arms."
"Did I say all that, Lulu?" asked Grace, in repentant surprise.
"All that, and more. You said that if he attempted to enter your room
you would spring from the window—and he was in the parlor; he
heard every word from your own lips."
"Oh, Lulu, I must have been delirious; I remember nothing of all
that, and it has, perhaps, kept him from me all the time," came in a

moan from the unhappy young creature, as she leaned against the
toilet-table, with one hand clasping her heart.
Lulu caught up a bottle of eau-de-cologne and showered the fine,
fragrant spray over the white face, just as Mrs. Clendenon hurried in.
"My darling, do you know you should have been down stairs before
this time—hurry, do."
And too much absorbed in her own grief to observe the ill-concealed
agitation of Mrs. Winans, or attributing it to her sorrow at losing
Lulu, the mother assisted the young bride to change her white silk
for her traveling one.
Then for one moment Lulu flung herself in passionate tears on her
friend's breast, with a hundred incoherent injunctions and promises,
from which she was disturbed by the entrance of Mrs. Conway,
radiantly announcing that the carriage waited and they had no time
to spare. And Lulu, lingering only for a blessing from her mother's
lips, a prayerful "God bless you" from her brother's, went forth with
hope on her path, love in her heart, and the sunshine on her head,
to the new life she had chosen.
When the last guest had departed, the "banquet fled, the garland
dead," Mrs. Winans removed her bonnet, and spent the remainder of
the day in diverting the sad mother whose heart was aching at the
loss of her youngest darling.
"It seems as if all the sunshine had gone out of the house with her,"
Willard said, sadly, to Grace, as they stood looking together at the
deserted bridal arch that seemed drooping and fading, as if in grief
for the absent head over which it had lately blossomed. "I fancied
we should keep our baby with us always in the dear home nest; but
she is gone, so soon—a wife before I had realized she had passed
the boundaries of childhood."
"The months of absence will pass away very quickly," she said,
gently, trying to comfort him as best she could, "and you will have
her back with you."

"I don't know," he said, with a half-sob in his manly voice, lifting a
long, trailing spray of white blossoms that an hour before he had
seen resting against the dear brown head of his sister, touching it
tenderly to his lips—"I don't know, Mrs. Winans. I don't believe in
presentiments—I am not at all superstitious—but to-day, when I
kissed my sister's lips in farewell, a chill crept through my frame, a
voice, that seemed as clear and distinct as any human voice,
seemed to whisper in my ear, 'Never again on this side of eternity!'
What did it mean?"
Ah! Willard Clendenon—that the fleshly vail that separates your pure
spirit from the angels is so clear that a gleam of your near
immortality glimmered through!

CHAPTER XXI.
"RUE."
"Hope, cheated too often when life's in its spring,
From the bosom that nursed it forever takes wing;
And memory comes as its promises fade
To brood o'er the havoc that passion has made."
—C. F. Hoffman.
The gossips of Norfolk are weary of wondering at the vagaries of the
Hon. Mrs. Winans. They admired and envied her very much in the
role of queen of beauty and fashion; they are simply amazed when
she glides before the foot-lights in the garb of a "ministering angel."
When she first began to aid and assist Miss Clendenon in her
charitable undertakings they thought it only natural, in view of the
sudden intimacy that had sprung up between the two, that the one
should be found wherever the other was. But it was quite a different
thing when the Senator's lovely and exclusive wife assumed those
duties alone. Society, wounded by her quiet and almost complete
withdrawal from its fascinations, set it down to a lack of a new
sensation, and predicted that as soon as the novelty wore out Mrs.
Winans would seek some newer and fresher hobby.
But quietly oblivious to it all, the young lady went her way,
smoothing with gentle advice and over thoughtful bounty many a
thorny path where poverty walked falteringly on, lending a patient
and sympathetic ear to the grievous complaints that rose from the
homes of want and distress, strangely gentle to all little children,
careful of their needs, thoughtful of their future, dropping the gentle
promises of Christ along darkened paths barren of such precious
seeds, and often society was scandalized by the not unfrequent sight

of the young lady taking out for an airing on the cool, breezy
suburbs or sea-shore some puny child or ailing adult from the haunts
of poverty and making them comfortable by her side in that darling
little phaeton that all Norfolk ran to their windows to gaze at when it
passed.
Miss Lavinia Story—dear old spinster!—undertook to interview the
lady on the subject of her going so far in alleviating the "fancied
wants and grievances of those wretched poor trash," and was fairly
driven from the field when Mrs. Winans, with a glimmer of mischief
under her black lashes and a very serious voice asked her if her
leisure would admit of her joining the sewing society, of which she
was manager.
"For indeed," said Grace, half playfully, half in earnest, "we are in
want of workers very badly. A lady from 'our set' volunteered very
kindly last week as operator on the sewing-machine I donated the
society, and they are so dreadfully in want of basters. Surely, Miss
Lavinia, you will enlist as baster—that, if not more. Think of the poor
people who need clothing so badly, and say 'yes.'"
"I? I would not spoil my eyesight with everlasting stitching for poor
people, who are always lazy and shiftless, and smell of onions," said
Miss Story, loftily.
"I beg your pardon, I am sure," smoothly returned her merciless
tormentor. "I forgot that your eyesight cannot be as strong as it
once was. Perhaps you would not object to becoming a visitor of the
sick, or something of that sort."
"My eyesight not as strong as it once was?" returned the lady, in
perceptible anger. "You mistake very much, Mrs. Winans; my eyes
are as young as they ever were" (she was fifty at the least), "but I
can use them to better advantage than by wearing them out in the
service of your sewing circle."
"It is rather tedious—this endless stitching," confessed the zealous
advocate of the sewing society, "but perhaps you would not object
to taking a little sewing at home occasionally—little dresses or

aprons, and such trifling things for the little folks—even that would
be a help to us in the present limited number of workers—won't you
try to help us out that much?"
Miss Lavinia adjusted her spectacles on her high Roman nose, the
better to annihilate with a flashing glance the persistent young lady
whom she felt dimly persuaded in her own mind was "laughing in
her sleeve at her," and Mrs. Winans, with the pearly edge of one
little tooth repressing the smile that wanted to dimple on her lip, sat
demurely expectant.
"I did not call on you, Mrs. Winans, I assure you, to solicit a situation
as seamstress. I never allow myself to be brought into personal
contact with the filthy and odious poor. I do my share in taking care
of them by contributing to the regular poor fund of the church."
"Oh, indeed?" said the listener, still unmoved and demure. "I am
sure it is very considerate of you and very comforting to the poor
people besides."
"I think, my dear," answered Miss Lavinia, pacified by the rather
equivocal compliment, "that it would be better for you to confine
yourself to the same plan. Let those who have not our refined and
delicate instincts minister to those of the poor class who are really
deserving of pity and of assistance, while we can do our part just as
well by placing our contributions in the hands of some worthy
person who can undertake its proper distribution. It hardly looks well
for a lady of your standing to be brought into such frequent and
familiar intercourse with the vulgar and low people to be met in the
homes of poverty, if you will pardon such plain speaking from an old
friend and well-wisher."
"And so you will not undertake to help us sew," persisted the placid
little tormentor, as the rustle of Miss Story's brown silk flounces
announced impending departure.
"No, indeed—quite out of the question," answered the irate spinster,
as she hurried indignantly away to report to her gossips, and only
sorry that it was out of her power or that of any of her peers to

socially ostracize the self-possessed young advocate of the sewing
society.
"The most persistent little woman you ever saw," she said. "I fairly
thought she'd have coaxed me into that low sewing-circle, or sent
me away with a bundle of poor children's rags to mend. I won't
undertake to advise her again in a hurry; and my advice to all of her
friends is to let her alone. She is 'joined to her idols.'"
And the "persistent little woman" ran up stairs and jotted down a
spirited account of her pleasant sparring with the spinster in her
friendly, even sympathetic journal—the dear little book to which was
confided the gentle thoughts of her pure young heart.
"Dear little book," she murmured, softly fluttering the scented leaves
and glancing here and there at little detatched jottings in her pretty
Italian text, "how many of my thoughts, nay hopes and griefs are
recorded here."
Now and then a smile dawns in her blue eyes, and anon her sweet
lip quivers as the written record of a joy or grief meets her gaze.
Looking back over earlier years, the pleasures of the fleeting hours,
the dawning hopes of maidenhood, the deep, wild sorrow of her
slighted love, she suddenly pauses, her finger between the pages,
and says to herself with a half-sad smile:
"And this was about the time when I fancied myself a poet. Why
have I not torn this out long ago? I wonder why I have kept this
foolish rhyme all these years?"
In soft, murmuring tones she read it aloud, a faint inflection of scorn
running through her low, musical voice:
rue!
"Violets in the spring
You gave me with the dew-tears in their eyes,
I said, in faint surprise:
Love do not tearful omens round them cling?

You answered: Pure as dew
Our new-born love, no omens sad have we
From morning violets, save that love shall be
Forever fresh and new.
"Roses, through summer's scope,
You brought me when the violets were flown—
Flushed, like the dawn—full-blown;
No folded leaves where hope could 'live in hope,'
I moaned; the perfume soon departs;
The scented leaves fall from the thorny stem.
You said: But they were sun-kissed, child, what then?
The fragrance lingers yet within our hearts.
"November's 'flying gold'
Drives through the 'ruined woodlands,' drift on drift,
Nor violet nor rose, your later gift,
Love's foolish, sun-kissed story has been told.
Dear, were you false or true?
I know not—only this: Love had its blight;
Nor dews nor fragrance fill my heart to-night—
But only—Rue!
"Ocean View, November, 1866."
"Rue!" she repeats, with a low, bitter laugh; "ah, me, I have been
gathering a harvest of rue all my life."
The leaves fall together over the sorrowful, girlish rhyme, the book
drops from her hand, and, sighing, she throws herself down on a
low divan of cushioned pale blue silk, looking idly out of the open
window at the evening sky glowing with the opalescent hues of a
summer's sunset.
"I daresay it's quite natural to make a dunce of one's self once in a
life-time," she muses, "and I presume there is a practical era in
every one's life. All the same I wish it had never come to me; the
consequences have followed me through life."

Her small hand goes up to her throat, touching the spring of the
pearl-studded locket she wears there. The lid flying open shows the
dusk glory of Paul Winans' pictured face smiling on her through a
mist of her own tears.
"And I drove you from me. Lulu says I did it; spoke my own doom
with fever-parched, delirious lips! Why did they believe me? Why did
they not tell me of it long ago? They should have known I could not
have been so cruel! All this time you have thought I hated you, all
this time I have thought you hated me! You did come; you did want
to make peace with your wronged though willful wife. It is joy to
know that though too late for hope even. Why did I go to
Washington? Why did I go in defiance of his will? All might have
been well with us ere this. Both of them—the darling baby and the
darling father—might have been mine now. Instead—oh, Heaven,
Paul dearest, you will never know now—unless, perchance, you are
in heaven—how deeply, how devotedly I loved you! Who is to
blame? Ah, me! It is all rue!"
A moment her lips trembled against the pictured face, then she
shuts it with a snap, and lies with closed eyes and compressed lips,
thinking deeply and intensely, as "hearts too much alone" always
think. But with the passing moments her sudden heart-ache softens
a little. Rousing herself she walks over to the window, saying, with a
faint, fluttering sigh:
"Ah, well! 'Fate is above us all.'"
How sweet the air is! The salt breeze catches the odor of the
mignonette in her window, and wafts it to her, lifting the soft tresses
from her aching temples with its scented breath, and with the
sublime association that there is in some faint flower perfumes and
grief, the bitter leaven at her heart swells again with all the painful
luxury of sorrow.
"I am so weary of it all—life's daily treadmill round! What is it worth?
How is it endurable when love is lost to us?"

Ah! poor child! Love is not all of life. When love is lost life's cares
and duties still remain. We must endure it. Well for us that God's
love is over all.
Some thought like this calms the seething waves of passion in her
heart. She picks up her journal from the floor where it had fallen,
and listlessly tears out the page that holds the simple rhyme of her
girlhood's folly. Leaning on the window she takes it daintily between
her fingers and tears it into tiny bits that scatter like snow-flakes
down on the graveled path of the garden below.
"Loved by two," she says, musingly. "What was Bruce Conway's love
worth, I wonder? Or Paul Winans' either, for that matter? The one
fickle, unstable, the other jealous, proud, unbending as Lucifer! Not
quite my ideal of perfect love, either one of them! After all, what is
any man's love worth, I wonder, that it should blight a woman's
life?"
Loved by three she might have said, but she did not know. How
much the fleshly vail between our spirits hides from our finite eyes.
How often and often a purer, better, stronger love than we have ever
known is laid in silence at our feet, over which we walk blinded and
never know the truth.
And yet by some odd chance, nay, rather unconscious prescience,
she thinks of Willard Clendenon, recalling his words on the day of his
sister's marriage:
"Never again on this side of eternity."
"What did it mean?" she mused aloud. "It was strange at the least. I
trust no harm will come to Lulu, little darling. She is still well and
happy, or at least her letters say so." And drawing from her pocket a
letter lately received from Lulu, she ran over its contents again with
all a woman's innocent pleasure in re-reading letters.
"How happy she seems," a faint smile curving the perfect lips; "and
how devoted is Mr. Conway; how her innocent, joyous, loving heart
mirrors itself in her letters! Sunshine, roses, honeymoon, bliss. Ah,
me," with a light sigh chasing the smile away, "how evanescent are

all things new and sweet; like that sky late aglow with the radiance
of day, now darkening with the shades of twilight."
Norah comes in to light the gas, and is gently motioned away.
"Not yet, Norah. I have a fancy to sit in the twilight. You can come in
later."
And Norah goes obediently.
Then she incloses the perfumed pink epistle in the dainty envelope
bearing the monogram of the newly made wife, and laying it aside
rests her head upon her hands, watching with dusk pained eyes the
shadows that darken over the sky and over her golden head as she
sits alone, her heart on fire with that keenest refinement of human
suffering—"remembering happier things." All her brightness, all her
love lies behind her in the past, in the green land of memory. The
present holds no joy, the future no promise. The dimness of
uncertainty, of doubt, of suspense, lies darkly on the present hour,
the hopelessness of hope clouds the future. Heaven seems so far
away as she lifts her mournful gaze to the purple, mysterious
twilight sky, life seems so long as she remembers how young she is,
and what possibilities for length of days lie before her. What wonder
that her brave, long-tried strength fails her a little, that her sensitive
spirit quails momentarily, and the angel of the human breast, hope,
"Comes back with worn and wounded wing,
To die upon the heart she could not cheer."

CHAPTER XXII.
ON TIPTOE FOR A FLIGHT.
"If it be a sin to love thee,
Then my soul is deeply dyed
With a stain more dark than crimson,
That hath all the world defied;
For it holds thine image nearer
Than all else this earth hath given,
And regarded thee as dearer
Almost than its hopes of heaven!"
A period of three months goes by after Lulu's marriage, swiftly to
those who are gone, slowly to those who remain. Mrs. Clendenon, in
quiet household employments, in prayerful study of her Bible, fills up
the aching void of her daughter's absence. Grace, in pursuance of
the charge Lulu has left her, finds much of her leisure employed in
scenes and undertakings that gently divert her mind from her own
troubles to those of others. Under it all, the wound that time has
only seared lies hidden, as near as she can hide it, from the probing
of careless fingers.
Captain Clendenon shuts himself up in his dusty law office with his
red-tape documents and law books. Of late he has covered himself
with glory in the winning of a difficult suit at law, and Norfolk is loud
in praise of the one-armed soldier, the maimed hero who has grown
into such an erudite lawyer. He takes the adulation very quietly. "The
time has passed when he sighed for praise." A shadow lies darkly on
his life—the shadow of Grace Winans' unhappiness. In that strong,
pure heart of his, no thought of himself, no selfish wish for his own
happiness ever intrudes. Had peace folded his white wings over her
fair head she would long ago have become to his high, honorable

heart, a thing apart from his life, as something fair and lovely that
was dead; and with her safe in the shelter of another man's love he
would have tutored his heart to forget her. As it was, when he
looked on the fair face that was to him but a reflex of the saintly
soul within, his whole soul yearned over her; his love, which had in it
more of heaven than earth, infolded her within the sphere of its own
idolizing influence. She became to him, not the fair, fascinating, but
sometimes faulty mortal woman the world saw in her, but rather a
goddess, a creature most like
"That ethereal flower—
No more a fabled wonder—
That builds in air its azure bower,
And floats the starlight under.
Too pure to touch our sinful earth,
Too human yet for heaven,
Half-way it has its glorious birth,
With no root to be riven."
Such worship as this has always been the attribute of the purest,
most unselfish love.
He sat alone in his office one day, his head bowed idly over
Blackstone, his thoughts far away, when the sharp grating of wheels
on the street outside startled him into rising and glancing out of the
window. She was springing from her little pony-phaeton, and in
another moment came flitting down the steps and into the room like
a ray of sunshine.
"Moping, are you?" she asked with her head on one side, and a
glimmer of her old-time jaunty grace.
"Not exactly," he answered, cheerfully bowing over the gloved hand
she extended with frank sweetness—"only thinking; our life is too
short for moping."
She might have added:

"I myself must mix with action
Lest I wither by despair."
"Are you busy?" glancing, as he offered her a seat, at the table
littered with books and papers.
"Not at all; I am at your service," he replied.
"I want to talk to you; but—excuse me—your office looks so gloomy
—makes me blue," she shivered a little. "Is your mother quite well?"
"Quite well—thanks. Will you not go up and see her?—or shall I
bring her down?"
"Thanks—neither, I believe. I saw her a day or two since, and I am
come on business now. Captain Clendenon, is it quite comme il faut
for a lady to ask you to take a drive? If so, my phaeton is at your
service. I want to ask you something; I cannot here. Some of your
tiresome clients may disturb me."
The soft appealing eyes and voice work their will with this infatuated
man. If she had asked him to lie down under the hoofs of her
cream-white ponies and be trampled on, I fear he would have done
it. A man's love for a woman sometimes rises above its ordinary
ridiculousness into the sublimity of pathos, and how little it is for him
to consent to sit by her side and hear those magical tones, perhaps
give some advice to that ever restless young spirit. He calls his office
boy, takes his hat, and goes. Presently they are rolling over one of
Norfolk's handsome drives, and censorious people, looking from their
windows, exchange opinions that Mrs. Winans is "rather fast."
"Alas! for the rarity
Of Christian charity
Under the sun."
"I have been over to Portsmouth this morning," she says, in the
midst of their small talk. "It is rather a nice little jaunt over there on
the ferry-boat over the Elizabeth River—don't you think so?"

"Yes, I do think so; had you a nice time?"
"I don't know—yes, I suppose so. I visited some friends, and we
went down and saw the beautiful grounds of the Naval Hospital—
what a handsome building it is! The pride of Portsmouth. And what
romantic grounds! I sat there a long time and looked at the sea."
To what is all this idle chatter leading, he wonders, seeing perfectly
well with what consummate art she is leading the subject whither
she wants it to go.
"We were all talking of that dreadful fever at Memphis," she
resumes, constrainedly. "What swift progress it is making! The
newspaper accounts of it are just terrible—heart-rending, indeed;
and they are so fearfully in need of nurses and money. I have sent
them a small sum—a mere 'drop in the ocean.'"
"So have I," he answers, white to the lips. He knows what is coming.
She gives him a flitting glance, fanning herself energetically the
while. A useless proceeding, for the sea-breeze, that flutters her fair
curls like golden banners, is simply delicious.
"I heard something about you over there," she ventures. "One
always has to go abroad to hear news from home, you know."
"Very likely; you can hear anything you want to over there. Little
Portsmouth is the hot bed of gossip," he answers, smiling dryly.
"Well, for that matter, all places are," she returns. "But you do not
ask what it was that I heard?"
"Is it worth the repetition?"
"I think so, but you are not interested, I see;" and she leans back
with some displeasure—a pout on the curve of her crimson lip.
He rouses himself, all penitence and forced gallantry.
"I beg your pardon, Mrs. Winans. Any remark from yourself cannot
fail to be interesting."

"I heard—I wonder you did not tell me of it yourself—that you and
your mother are going next week to Memphis to help to nurse the
fever patients."
No answer.
"Is this true?"
Her eyes are seeking his. He looks down on her, answering
constrainedly:
"It is perfectly true, Mrs. Winans."
"Why have you kept it from me?" in some wonder this.
"We intended telling you, of course, before we left; but it is such a
harrowing topic—the sufferings of those poor yellow fever patients—
that I have hesitated in mentioning it to you."
"Was that your only reason?"
No answer. He cannot bear to speak.
"I know," she resumes, "why you have not told me. You feared I
would want to go, too, and so kept it from me in your good, true,
brotherly love; but in this case," smiling willfully up into his disturbed
face, "your painstaking has been 'Love's Labor Lost.' I have been
making my mind up to go all along, and now I mean to make the
trip there under the protection of your mother and yourself—if you
will permit me."
The murder is out. She looks away from him demurely, waiting his
reply. It comes, full of a shocked horror.
"Mrs. Winans, are you mad?"
"Not at all; are you? I am quite as strong, quite as able to help those
poor sufferers, as your mother is; and yet you do not think she is
mad," she answers, half offended.
"No; for she has had the fever, and so have I. You have heard of the
fever that desolated Norfolk and Portsmouth in 1855? Mother and I
both had the yellow fever in its worst form then, so you see it is

perfectly safe and our bounden duty for us to go to the relief of
those poor sufferers. But you are frail and delicate yourself. You
have never had the fever; you are not acclimated there, and would
only fall a victim. It would be a sort of disguised suicide, for you
would be voluntarily rushing into the jaws of death."
"No, no," she answered, half bitterly. "I bear a charmed life. Nothing
seems to check the current of my doomed existence. And you forget
that Memphis is my native home. I lived there the first sixteen years
of my life, and am quite accustomed to the peculiar climate. And
what if death should come? It would only be to 'leave all
disappointment, care, and sorrow, and be at rest forever,' But no, I
shall not die. I have borne illness, suffering, sorrow—everything that
breaks the heart, and snaps the frail threads of existence—yet here I
am still, quite healthy, passably rosy, and willing to devote my
strength to those who need it. I have been 'through the fire,'
Captain Clendenon, and really," with a subdued smile, "I think I am
fireproof."
"Some are refined in the furnace of affliction," he repeats, very
gently.
Soothed by the softly spoken words, she asks, timidly:
"Tell me if I may go under your care?"
"If you will go, I shall be most happy to take all the annoyances of
travel off your hands; but, little friend, think better of it, and give up
this mad, quixotic scheme."
"Do you think it such a mad scheme?" she asks, mortified and
humiliated. "Do you think I could do no good to those poor suffering
victims who need gentle womanly tending so badly? Do you think
the sacrifice of my ease, and luxury, and comfort, would count as
nothing with Christ? If you think this, Captain Clendenon, tell me so
frankly, and I will remain in Norfolk—not otherwise."
There is nothing for him to urge against this appeal. She touches up
the ponies with her slim, little whip, lightly and impatiently. They are

off, like the wind, for home again, as he makes the last appeal he
can think of to this indomitable young spirit.
"News may come of your husband at any time, Mrs. Winans. Were
you to go, and he, returning, found you gone, he would be most
bitterly displeased. Remember, it was his express desire that you
should remain in your home here. I beg your pardon, if I seem
persistent, but it is only through friendly interest in you and yours
that I take the liberty."
"Ah! but," a gleam of triumph lightening under her black lashes,
"you forget that I have my husband's consent to visit Memphis? You
brought it to me. I'm returning to the home of my childhood. I am
not violating any command or desire of his."
"Once more," he says, desperately "let me beg you not to think, for
the sake of all those who love you, all you love, of going to that
plague-stricken city."
"It is useless." She set her lips firmly. "I am sorry to refuse your
request, but the call of duty I must obey. My arrangements are all
made. Norah is to stay and to take care of my home. My visit to
Portsmouth this morning was for the purpose of leaving Lulu's
precious charge in the hands of a dear Christian friend; so," trying to
win him to smile by an affected lightness, "you may tell your mother
she will have company she did not anticipate, though you were so
ungallant as to persuade me not to come."
"When a woman will she will."
She carried her point against the entreaties of all her friends, and in
less than two weeks, three dusty travelers—weary in body, but very
strong in prayerful resolves and hopes—were entered as assistants
in nursing in one of the crowded hospitals of the desolated, plague-
stricken city of Memphis.

CHAPTER XXIII.
IN MEMPHIS.
"To be found untired,
Watching the stars out by the bed of pain,
With a pale cheek, and yet a brow inspired,
And a true heart of hope, though hope be vain,
Meekly to bear with wrong, to cheer decay,
And, oh! to love through all things—therefore pray."
—Hemans.
One of Grace's first acts after reaching Memphis was to inquire for
her relatives, whom she had not seen, and but seldom heard from,
since leaving Memphis, in her sixteenth year, to make her own way
in the hard world. Not that she owed them much affection, or any
gratitude—only the natural respect and remembrance of kinship
induced her to seek them out. But her efforts were not crowned with
success, for she learned that they had been among the first of the
native families to flee the city at the approach of the pestilence, and
Grace was greatly disappointed thereat.
For a few weeks her voluntarily assumed duties were arduous and
embarrassing in the extreme. Mrs. Clendenon and Willard, having
had the fever themselves, and having been witnesses of its ravages
in their own city, entered at once with confidence and experience on
the task of caring for the poor victims who filled the hospitals, and
even private houses. To Grace it was all new, and strange, and
terrible, and though her will was strong, her sensitive spirit quailed
at the horrors she daily saw, so unused was she to these scenes,
and so diffident of her own powers for service, that she half doubted
her abilities, and was, for a time, overwhelmed by the feeling which

we have all experienced at times of willingness to perform duties
from which we are deterred by ignorance, or lacking self-confidence.
But this feeling was not long suffered to deter her from usefulness.
Laborers were too sadly needed in the terrible harvest of death, and
as her increasing familiarity with the details of the awful disease
rendered her more efficient, she became an invaluable nurse to the
patients, and a reliable and prized aid to the physician of the ward
where she was placed.
The Clendenons were in the same hospital, and in the performance
of their several duties the trio often met, when a sweet sentence of
praise from the lady, and a cheerful word of encouragement from
him, went far to keep up her flagging spirits, and stimulate her to
renewed exertions.
Her strong, healthy constitution upheld her well in those days; for
the fiery scourge rolled on and on like some great prairie fire, hourly
seizing fresh victims, and erecting its everlasting monuments in the
long rows of new-made graves in the cemeteries that swelled
upward, side by side, close and many, like the green billows of old
ocean, save that they gave back no solemn, tolling dirge, to tell
where youth and love, hope and beauty, old age and infancy, joy
and sorrow, went down to the stillness of the grave.
In the universal suffering, the universal grief of those around her,
the anguish of those bereaved of whole families, of friends the
young lady put away her own griefs from her heart, and threw
herself, heart, and soul, and body, into her work; and, though her
two friends were doing precisely the same thing, they pleaded,
expostulated, scolded and warned in turn.
All in vain; for a rock would have flown from "its firm base" as soon
as Grace Winans from the position she has taken. She had, as she
pathetically protested, so little to live for, that she was all the more
willing and desirous to sacrifice herself for the sake of saving others
who had more ties in life than herself.

"That is a poor policy," Mrs. Clendenon argued, stoutly. "You have no
right to commit a moral suicide, however few your ties in life may
be. Your life is God's, and He has some plan in life for you, or He
would not have placed you on the earth."
"And this may have been His plan for me, then," persisted the
candidate for self-sacrificial honors. "He may have meant for me to
take up this very cross. I have been brought to it by many subtle
windings."
"I do not know," Mrs. Clendenon answers, with sweet seriousness,
"that God gives it to us to fathom exactly what are His plans for us. I
think He means for us to take proper care of the health and strength
He has given us, and to do His will in all things as near as we can,
leaving to Him the fulfillment of the grand plan under which, by His
fixed laws, every created being is a necessary and responsible
agent."
And Grace answers only by silence and sadness. For Captain
Clendenon, he has long ceased to argue the question with her willful
spirit, having very implicit confidence in the grand old adage that
"When a woman will, she will—you may depend on't!
And when she won't, she won't—and there's an end
on't!"
"Oddly enough," he says, trying to change the conversation from its
theological turn, "I met with an old comrade to-day—one of the boys
from my company—a Virginian, and one of the bravest in the
regiment. He had drifted down here since the war."
"What was he doing to-day? Nursing in the hospital?" Mrs.
Clendenon asks, curiously.
"Dying in the hospital," the captain answers, with a break in his clear
voice. "Down with the fever—died this evening."
"Poor boy!" his mother said, pityingly, and a tear in the younger
woman's eye echoed it.

"The worst of it is," the captain goes on, "he leaves a poor, timid
little wife, and two rosebuds of children—the mother as childish and
fragile as the rest."
"And what is to become of her?" query both ladies at once.
"I want to send her home to her relatives. She was a Richmond girl.
I remember meeting her there once when my company passed
through on its way to Manassas. Arthur, poor fellow! invited me to
call on her. She was then a charming little creature, very different
from the heart-broken little thing she is now. Mother, I would like it
very much if you would call on her to-morrow, and try to comfort
her a little—she seems so friendless and desolate. You, too, Mrs.
Winans, if you can conveniently do so."
Both ladies expressed a desire to visit the bereaved young widow
and her little ones.
"Then I will take you down there to-morrow," he said, gratefully,
with a smile in his honest gray eyes. "Ah! how it pains me to meet,
as one must frequently do here, old friends and old faces, only to
close the lids over eyes that have been so dear! Poor Arthur! poor
boy! but it is one of the sad inevitable experiences of life."
"Grace, my love," Mrs. Clendenon went on to say, "I have Doctor
Constant's authority to forbid your appearance at the hospital to-
night. He says you are so unremitting in attentions to his patients
that there is danger of your falling sick, and our losing your valuable
services altogether, if you persist in taking no rest at all."
In the quiet hotel at which all three are registered they are seated at
supper in the small private dining-room. The round, neatly appointed
table at which they sit is loaded with luxuries to which they are
doing but meager justice. It is late in October, and a small fire burns
on the hearth, tempering the slightly chilly air, and lending
cheerfulness to the room. Bright gas-light glimmers down on
crimson carpets, curtains, chairs, that throw into vivid relief the
faces of our three friends—pale all of them, and thin, earnest, and
full of thoughtful gravity. It is no child's play, this nursing the yellow

fever patients in houses and hospitals. These faces bear the impress
of sleepless nights and days, and the silver threads on the elder
lady's brow are more abundant, while in Captain Clendenon's curly
brown locks one or two snow-flakes from the winter of care, not
time, are distinctly visible. There are slight hollows in the smooth
cheeks of Grace, faint blue circles around her large eyes, and no
color at all in her face except the vivid line of her red lips. She looks
like a little Quakeress in the pale gray dress that clings closely about
the slight figure, relieved only by white frills at throat and wrists. All
her bright hair is drawn back in soft waves from her face, and
confined at the back with a silver arrow that lets it fall in a soft,
bright mass of natural curls below the waist—lovely still, though
pallid, sad, and worn; and in this quiet nun-like garb, with a beauty
that grows daily less earthly, and more heavenly.
The pensive shade of a smile dwells on her lip a moment as she
looks across on Mrs. Clendenon in mute rebellion at the physician's
mandate.
"You need not look defiance," the lady returns, "for I shall add my
commands to those of Doctor Constant. This is Thursday, and you
have not slept a single night this week, while I have had two nights'
rest. My dear child, listen to reason, and remain at the hotel to-night
and get some sleep."
"I am not so very tired. I can hold out to watch to-night."
"Oh, of course! and die at your post. What can you be thinking of,
Grace? Flesh and blood cannot stand such a strain. You must take
needful rest, or you will fall a victim to the fever through sheer
exhaustion."
"I cannot rest," she answered, wearily. "It is a physical impossibility
for me to take rest and sleep when I know how many are suffering
and needing attention that I could render them."
"There are others who will supply your place," interposed the
captain. "I learned this evening that you were at two death-beds to-
day. This, I think, is too much strain on your nervous system, and

did I dare I should add my commands to the rest that you remain in
your room and take needful repose to-night. As it is, I can only offer
my earnest entreaties."
The resolute look on her face relaxes a little. She looks up to this
quiet, clear-headed captain much as Lulu does; has great respect for
his judgment; wishes sometimes that he were her brother, too—that
her tired young heart might rest against his brave and grand
strength. He sees the half-relenting in her face, and desists for fear
of saying too much.
"Two death-beds!" Mrs. Clendenon echoes. "Why, Allie Winters was
only taken ill last night, and you have been nursing her ever since.
Gracie, you don't mean to tell me that Allie Winters is dead—so
soon!"
"She died this evening with her arms about my neck," Grace
answers, in low, pained tones. "She had the fever in its worst and
most rapid form."
"Ah, me, that poor child! So young, so sweet, so beautiful, and
scarcely sixteen, I think. Was it not hard to be taken away from this
bright world so young?" sighed Mrs. Clendenon.
"Well, opinions may differ as to that," Mrs. Winans answers, half
bitterly. "The most fervent prayer I breathed over her still form was
one of thankfulness that she was taken perhaps from 'evil to come.'
She was the last of the family. They have all died with the fever. She
was poor, and almost friendless—beautiful—and beauty is often the
cause of poverty. Had she lived her life must inevitably have been a
sad one. Better, perhaps, that she is at rest."
She pushes back her chair, folds her napkin, and makes a motion to
rise. Mrs. Clendenon remonstrates.
"Gracie, you have not taken a mouthful, child."
"No, but I have taken my cocoa. Andrew," sinking listlessly back into
her chair, and speaking to the white-aproned waiter, "you may give
me another cup."

"There seems to be no abatement of the fever?" she says,
interrogatively, to the captain, as she balances her spoon on the
edge of her cup.
"On the contrary," his grave face growing graver, "the number of
victims is daily augmented."
Her grieved sigh is echoed by Mrs. Clendenon's as they rise from the
table. The next moment a sharp rap sounds at the door. Andrew
opens it, admitting Doctor Constant himself—fine-looking, noble,
with the snows of sixty earnest winters on his head and on the
beautiful beard flowing over his breast—genial, cheerful, gentle, as a
physician should always be—he makes a bow to our three friends,
but declines to be seated at all.
"I have but a moment. I came out of my rounds to make sure that
Mrs. Winans does not go out to-night," and as an eager
remonstrance formed itself on her lip, he said, resolutely: "It is no
use; you must not think of going. It is imperative that you should
sleep. You are not more than half alive now."
"But, doctor, there are so many who need me," she says, with a last
endeavor to go.
"Others can take your place. We had new and fresh nurses to come
in to-day. Pardon me if I appear persistent, madam, but I was your
mother's family physician, and thoroughly understood her condition.
Your own resembles it in a high degree, and I warn you that you
have stood as much as you can without rest. You are your own
mistress, of course, and can do as you please; but if you go to-night
you are very apt to fall from exhaustion."
"Very well," she answers, wearily, as if not caring to contest the
point longer. "Since I do not wish to commit suicide, I will stay at
home and rest to-night."
"That is right. Your nervous system is disordered, and needs
recuperation. You will feel better to-morrow, and may come back to
the hospital. As for Mrs. Clendenon and the captain, they may come
back to-night."

She does not really know how tired she is until she goes up to her
room and throws herself on the lounge, face downward, like a weary
child, to rest. But, exhausted as she is, it is hours before she sleeps.
Nervous temperaments like hers are not heavy sleepers. After long
seasons of wakefulness she finds it almost impossible to regain the
habit of natural repose. Now she lies quite still, every tense nerve
quivering with weariness, but with eyelids that seem forced open by
some intangible power, and busy, active brain that repeats all the
exciting scenes of the past week. When twelve o'clock sounds
sharply on the still of the night she rises, chilled and unrefreshed,
and crouches over the dying fire that has smoldered into ashes on
the hearth long since. She looks down at it vacantly, with a passing
thought that it is like her life, from which the sunshine and
brightness have faded long since, leaving only the chill whiteness of
despair. Often in still moments like these her young heart rises in
half angry bitterness, and beats against the bars of life, longing to
be free. "Only half alive," Dr. Constant had said to her, and patient
and long-suffering as she was, I fear it had sent a half-thrill of joy to
her bosom. Life held so little for her, was so full of repressed agony
and pathos, pressed down its heavy crosses so reluctantly on her fair
young shoulders, and sometimes even the love of God failed to fill
up that empty heart that hungered, as every human soul must,
while bounded in human frame, for human, mortal, tangible love.
Resignation to her fate she tried sedulously to cultivate, and
succeeded generally. Only in hours like this, when oppressed with a
sense of her great loneliness, the past rushed over her, with all its
sweet and bitter memories, and was put away from her thoughts
with uncontrollable rebellion against—what she scarcely dared
speak, since a higher power than mortal ruled the affairs of her
destiny.
"God help me!" she murmured, as, pushing up a window-sash, she
leaned out and looked at the quiet city of Memphis lying under the
starry midnight sky, silent save for the occasional rumble of wheels
in the distance telling the watcher that the work of death still went

grimly on—the dead being hustled out of the way to make room for
the sick and dying.
The chilly night air, the cold white glimmer of the moon and stars,
cooled the feverish blood that throbbed in her temples, a soft awe
crept into her heart—the deep, all-pervading presence of God's
infinity; and as she shut down the window and went back to the
lounge, her pained, half-bitter retrospections were overflowed by
something of that "peace which passeth all understanding."
Sleep fell on her very softly—a deep, refreshing slumber—from
which only the morning sunshine aroused her. She rose with
renewed energy for her labor of love, and kept at her post for weeks
afterward, with only occasional seasons of rest and sleep. Her
superb organism kept her up through it all, aided and abetted by her
unfaltering will. Through it all there came no tidings of her husband
or child. Letters came often from the absentees in Europe, but
without mention of either, and Grace began to feel herself a widow
indeed.
The Clendenons, too, were indefatigable in their exertions for the
victims of the fever. They were always devoted and earnest in their
efforts, and kept a watchful care, too, over Grace, whose zeal and
willingness often outran her strength and power of endurance.
Mrs. Clendenon's gentle, placid old face began to look many months
older, but it was in Willard that the greatest change was perceptible.
His cheerful spirit never flagged, but gradually the two women who
loved him each in her own way, began to see that the tall, fine form
grew thinner and slighter, the face paler, and a trifle more serious,
while silver threads began to sprinkle themselves thickly in his dark
hair. He was wearing out his strong young manhood in hard,
unremitting toil, and leaving his constitution enfeebled and open to
the attacks of disease. The idolizing heart of his mother noted all
this with secret alarm, and she would fain have persuaded him to
retire from his arduous duties and return to Norfolk. His gentle but
firm refusal checked all allusion to the matter, and, as the weeks
wore away, and the fever began to lose its hold and abate its

virulence, she hoped that they would soon be released from their
wearing tasks, and allowed time for recuperation.
The contents of a letter received more than two months previous
from Lulu weighed also on Mrs. Clendenon's mind, and she could
not, as she often did in other matters, seek the sympathy of Grace,
as Lulu had desired she should not know anything of it. So Mrs.
Clendenon bore her burden of anxiety all alone, save for Him who
carries the half of all our burdens.

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