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Giancoli Physics: Principles with Applications, 6
th
Edition

© 2005 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they
currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
200
CHAPTER 24: The Wave Nature of Light

Answers to Questions

1. Huygens’ principle applies both to sound waves and water waves. Huygens’ principle applies to all
waves that form a wave crest. Both sounds and water waves can be represented in this way.

2. A piece of evidence that light is energy is that you can focus the light from the Sun with a magnifying
glass on to a sheet of paper and burn a hole in it. You have added so much energy to the paper that its
temperature rises to the point where it ignites.

3. There are certain situations where describing light as rays works well (for example, lenses) and there are
other situations where describing light as waves works well (for example, diffraction). Actually, the ray
model doesn’t work at all when describing diffraction. Thus, there are always limitations to the
“models” we use to describe nature and we need to realize what these are.

4. The main reason that we can hear sounds around corners, but not see around corners, is diffraction.
Sound waves have very long wavelengths when compared to light waves, which makes diffraction
effects much more obvious. Diffraction effects are very noticeable once the size of the object that the
wave is diffracting around is about the same size as the wavelength of the wave. The wavelength of
sound is on the order of 1 m, while the wavelength of light is on the order of 0.1 µm. A secondary
reason is reflection. Sounds waves reflect off of walls very well in a specular manner, and so can bounce
around corners, but light reflects off of the walls in a very diffuse manner.

5. The wavelength of light in a medium such as water is decreased when compared to the wavelength in air.
Thus, sindmθλ= says that θ is decreased for a particular m and d. This means that the bright spots on
the screen are more closely packed together in water than in air.

6. As red light is switched to blue light, the wavelength of the light is decreased. Thus,
sindm
θλ= says
that θ is decreased for a particular m and d. This means that the bright spots on the screen are more
closely packed together with blue light than with red light.

7. Destructive interference occurs when the path lengths of two rays of light from the same source differ by
odd half-integers of the wavelength ( ()
1
2
2, 3 2, 5 2, m
λλλ λ + , etc.). Under these conditions, the
wave crests from one ray match up with the wave troughs from the other ray and cancellation occurs
(destructive interference).

8. One reason was that the double-slit experiment allowed scientists to measure an actual wavelength,
which was something that could not be done at the time with diffraction observations. Another reason
was that using a particle model, you could explain diffraction in a qualitative way by talking about the
particles bouncing off the edges of the diffracting object or the two edges of a single slit. When a second
slit is added, the particle model had a much more difficult time explaining why there are now dark spots
where the particles used to be able to strike with only one slit open.

9. Similarities between doing a double-slit experiment with sound and light: the sources must be coherent
for the interference pattern to be observed; they both produce a pattern of high and low intensity at some
distance away from the double slit (bright and dark for the light and loud and quiet for the sound); they
both work best with a single-frequency source. Differences between doing a double-slit experiment with

Giancoli Physics: Principles with Applications, 6
th
Edition
© 2005 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they
currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
201

sound and light: The slits for light must be extremely close together when compared to sound; you don’t
actually need slits for sound (just use two speakers).

10. The reason you do not get an interference pattern from the two headlights of a distant car is that they are
not coherent light sources (they have random phases). Thus, you cannot produce zones of destructive
and constructive interference where the crests and troughs match up or the crests and crests match up.
Also, the headlights are far enough apart that even if they were coherent, the interference pattern would
be so tightly packed that it would not be observable with the unaided eye.

11. Basically there would be two overlapping single-slit diffraction patterns, with one being blue and the
other being red. Wherever the two different patterns overlapped, there would be a purplish (red + blue)
bright spot. There would not be a central bright spot, due to the different wavelengths of the two colors.

12. The two faces of the pane of glass are parallel to each other, so the white light rays that were separated
into colors as they entered the front face all come out of the second face at the same angle as each other.
When these rays reach your eyes, they all combine back together into white light. The separated colors
in the prism reach the other face of the prism at all different angles (since the second face of the prism is
not parallel to the first face), which means that all of these different colors leave the prism at all different
angles. These light rays are still separated into colors when they reach your eyes, which allows you to
see a rainbow of colors at different angles.

13. Since red light is bent less than violet light in glass (the index of refraction for red light in glass is less
than the index of refraction for violet light in glass), the focal length of both a converging lens and a
diverging lens is longer for red light and shorter for violet light.

14. By looking at the direction and the relative amount that the light rays bend at each interface, we can infer
the relative sizes of the indices of refraction in the different materials (bends toward normal = faster
material to slower material or smaller n material to larger n material; bends away from normal = slower
material to faster material or larger n material to smaller n material). From the first material to the
second material the ray bends toward the normal, thus it slows down and n
1 < n2. From the second
material to the third material the ray bends away from the normal, thus it speeds up and n
2 > n3. Careful
inspection shows that the ray in the third material does not bend back away from the normal as far as the
ray was in the first material, thus the speed in the first material is the faster than in the third material and
n
1 < n3. Thus, the overall ranking of indices of refraction is: n 1 < n3 < n2.

15. As you squeeze your fingers together, you start to see vertical bright and dark bands that are aligned
parallel to your fingers. The dark bands get wider as you continue to bring your fingers closer together,
until at one point, when your fingers are still not actually touching, the dark bands seem to quickly jump
in and darken the entire gap.

16. (a) When you immerse a single-slit diffraction apparatus in water, the wavelength of light gets
smaller, due to the increase in the index of refraction, and the diffraction pattern gets more closely
spaced. The equation
sin
D
λ
θ= for the half-width of the central maximum says that θ is
decreased for a particular D when the wavelength decreases. And the equation for the locations of
the minima, sin
m
D
λ
θ= , also indicates that θ is decreased for a particular m and D when the
wavelength decreases. This means that the bright spots on the screen are more closely spaced in
water than in air.

Chapter 24 The Wave Nature of Light

© 2005 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they
currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
202
(b) When you perform a single-slit diffraction experiment in vacuum, the wavelength of light gets
slightly larger, due to the decrease in the index of refraction, and the diffraction pattern spreads
farther apart. The equation
sin
D
λ
θ= for the half-width of the central maximum says that θ is
increased for a particular D when the wavelength increases. And the equation for the locations of
the minima, sin
m
D
λ
θ= , also indicates that θ is increased for a particular m and D when the
wavelength increases. This means that the bright spots on the screen are spread farther apart in
vacuum than in air.

17. (a) When you increase the slit width in a single-slit diffraction experiment, the spacing of the
fringes decreases. The equation for the location of the minima,
sin
m
D
λ
θ= , indicates that θ is
decreased for a particular m and
λ when the width D increases. This means that the bright spots on
the screen are more closely packed together for a wider slit.
(b) When you increase the wavelength of light used in a single-slit diffraction experiment, the
spacing of the fringes increases. The equation for the location of the minima,
sin
m
D
λ
θ= , indicates
that θ is increased for a particular m and D when the wavelength increases. This means that the
bright spots on the screen are spread further apart for a longer wavelength.

18. The interference pattern created by the diffraction grating with 10
4
lines/cm has bright maxima that are
more sharply defined and narrower than the interference pattern created by the two slits 10
-4
cm apart.

19. (a) The advantage of having many slits in a diffraction grating is that this makes the bright maxima
in the interference pattern more sharply defined, brighter, and narrower.
(b) The advantage of having closely spaced slits in a diffraction grating is that this spreads out the
bright maxima in the interference pattern and makes them easier to measure.

20. (a) The color at the top of the rainbow for the diffraction grating is violet. The equation
sindm
θλ= says that θ is smallest (thus, the deviation from horizontal is smallest) for the shortest
wavelength, for a given d and m. The wavelength of violet light (450 nm) is shorter than that of red
light (700 nm).
(b) The color at the top of a rainbow for the prism is red. The index of refraction for transparent
materials (like the glass that makes up the prism) is smaller for long (red) wavelengths and larger for
short (violet) wavelengths. Since the red light encounters a smaller index of refraction as it goes
through the prism, it doesn’t slow as much as the violet light, which also means that it doesn’t bend
as much as the violet. If the red light is bent away from the horizontal direction least, it will appear
at the top of the rainbow.

21. For the red and violet colors from different orders to overlap, the angles in the equation
sindm
θλ=would need to be equal. Mathematically, this means:
()( )( )m 1 400 nm m 700 nm
dd
+
< .
In other words, the
m
th
order 700 nm line overlaps the (m+1)
th
order 400 nm line. Reducing this
equation, we get
m > 4/3. So, starting with the 2
nd
order 700 nm line and the 3
rd
order 400 nm line, the
observed spectra will always overlap at this and higher orders. This answer does not depend on
d. As
you can see from the above equation, the slit spacing cancels out.

Giancoli Physics: Principles with Applications, 6
th
Edition
© 2005 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they
currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
203

22. Once the thickness of the film becomes more than a few wavelengths thick, several interference patterns
become mixed together, and it is hard to see any individual effects. When the thickness of the film is
only about 1
λ thick, then the reflections from the top and bottom surfaces of the film for each color have
path differences of just one constructive interference (path difference =
λ/2) and one destructive
interference (path difference =
λ) patterns. It is easy for our eyes to pick out these widely spaced bright
colors that are separated by dark areas on the film. Once the film gets very thick, though, there are many
constructive (
λ/2, 3λ/2, 5λ/2, etc.) and many destructive (λ, 2λ, 3λ, etc.) path differences allowed. The
resulting interference patterns are all closely spaced and overlapping, making it difficult for our eyes to
distinguish between the bright and dark areas. As the film gets even thicker, the larger amount of
overlap causes all the colors to run together, making it impossible to see the individual interference
patterns.

23. There are many, many circular tracks on a CD and each track is made up of a series of pits and high
spots. Light reflects very well off of the high spots and not the low spots. Thus, when you shine white
light on a CD, each track is a slightly different distance from you, and as the light reflects off of each
track to you, they each have a different path length. Thus, you’ll basically see a colorful diffraction
grating pattern. If a monochromatic light is used, you will see a single-color interference pattern. In
other words, instead of seeing the full rainbow of colors spreading out from the center of the CD (as
shown in Figure 24-56), there would just be several “spokes” of the same color as your source spreading
out from the center of the CD. The spacings of these “spokes” can be used just like a diffraction grating
to determine that the track spacing on the CD is approximately 1600 nm.

24. As you move farther away from the center of the curved piece of glass on top, the path differences
change more rapidly due to the curvature. Thus, you get higher order interference patterns more closely
spaced together. An air wedge has equally spaced interference patterns because as you move farther
from the contact point of the flat piece of glass on top, the path differences change linearly.

25. If yellow/green light is getting reflected back to us, then the coating must be designed to transmit violet,
blue, orange, and red light completely.

26. At the edge of the oil drop, the film is so thin that the path difference between the light reflecting off of
the top surface and the light going through the oil and reflecting off of the bottom surface is so small that
we can consider it to be zero. Thus, the two different rays of light must be in phase when they reach our
eyes. We know that the phase of the light being reflected off of the top surface of the oil must have been
flipped 180°, since the index of refraction of oil is greater than that of air. Thus, the reflection off of the
bottom surface of the film (where it touches the water), must also have flipped the phase of the light
180°. This tells us that the index of refraction of the water is higher than that of the oil. Thus, we know
that the index of refraction of the oil is greater than that of air and less than that of water: 1.00 <
n <
1.33.

27. Polarization tells us that light is a transverse wave. Longitudinal waves cannot be polarized.

28. Polarized sunglasses completely (100%) block horizontally polarized glare and block all other
polarizations of light 50%. Regular sunglasses just block 50-75% of all light coming in. The advantage
of polarized sunglasses is the total elimination of glare. Even if regular sunglasses block a glare at 75%,
the glare is so intense that it still makes it difficult for our eyes.

29. To determine if a pair of sunglasses is polarizing or not, first find the glare (strong reflection) from a
light source on a horizontal non-metallic surface (water, tiled floor, polished tabletop, top of car). Then
look at the reflection through the sunglasses and rotate them. If they are polarizing sunglasses, then there

Chapter 24 The Wave Nature of Light

© 2005 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they
currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
204
should be orientations of the glasses that almost completely block out the glare and other orientations
that let 50% of the glare through. If they are not polarizing sunglasses, the glare will be slightly
diminished by the same amount at all orientations of the glasses, but it will never completely go away.

30. The first sheet of polarizer will diminish the intensity of the incoming non-polarized light by 50%:
1
102
II= . The next polarizer will diminish the light again, but this time only by a factor of
2
cos
θ:
2o 33
11 1 0 48
cos 30II I I=== . The next polarizer will diminish the light by the same factor:
2o 39
32 2 0 432
cos 30II I I=== . The last polarizer will again diminish the light by the same factor:
2o 327
43 3 0 0 4 128
cos 30 0.211II I I I==== . Thus, about 21% of the light gets through these four polarizers
when each one is rotated by 30°.

31. If Earth had no atmosphere, the “color” of the sky would be black (and dotted with stars and planets) at
all times. This is the condition of the sky that the astronauts found on the Moon, which has no
atmosphere. The reason the sky is blue for Earth, is that the air molecules scatter light from the Sun in
all directions, and preferentially scatter blue light down to the surface. If there were no air molecules to
scatter the light from the Sun, the only light we would see would be from the stars/planets and directly
from the Sun and the rest would be black.

32. If the atmosphere were 50% more dense, sunlight would be much redder than it is now. As the
atmosphere increased in density, more and more of the blue light would be scattered away in all
directions, making the light that reaches the ground very red. Think of the color of a deep red sunset, but
this would be the color even at noon.

Solutions to Problems

1. For constructive interference, the path difference is a multiple of the wavelength:
sin , 0, 1, 2, 3, ...
dmm
θλ== .
For the fifth order, we have

(
) ()
5
1.6 10 m sin8.8 5 ,λ
−
×°= which gives
7
4.9 10 m.λ
−
=×

2. For constructive interference, the path difference is a multiple of the wavelength:
sin , 0, 1, 2, 3,...
dmm
θλ== .
For the third order, we have

()(
)
9
sin18 3 610 10 m ,d
−
°= × which gives
6
5.9 10 m 5.9 m.d µ
−
=× =

3. For constructive interference, the path difference is a multiple of the wavelength:
sin , 0, 1, 2, 3, ...dmmθλ== .
We find the location on the screen from
tan .yLθ=
For small angles, we have

sin tan ,
θθ≈ which gives
.
mmL
yL
dd
λ λ⎛⎞
==
⎜⎟
⎝⎠

Giancoli Physics: Principles with Applications, 6
th
Edition
© 2005 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they
currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
205

For adjacent fringes, 1,m∆= so we have

;
Lm
y
d
λ∆
∆=

(
)()
()
3
5.00m 1
0.065m ,
0.048 10 mλ
−
=
×
which gives
7
6.2 10 m.λ
−
=×
The frequency is

( )
()
8
7
3.00 10 m s
6.24 10 m
c
f
λ
−
×
== =
×
14
4.8 10 Hz.×

4. For constructive interference, the path difference is a multiple of the wavelength:
sin , 0, 1, 2, 3, ...dmmθλ== .
We find the location on the screen from
tan .yLθ=
For small angles, we have

sin tan ,
θθ≈ which gives
.
mmL
yL
dd
λ λ⎛⎞
==
⎜⎟
⎝⎠

For adjacent fringes, 1,m∆= so we have
;
Lm
y
d
λ∆
∆=

(
)() ()
()
9
2
3
3.6m 656 10 m 1
3.9 10 m
0.060 10 m
−
−
−
×
==×=
×
3.9 cm.

5. For constructive interference, the path difference is a multiple of the wavelength:
sin , 0, 1, 2, 3, ...dmmθλ== .
We find the location on the screen from
tan .
yL
θ=
For small angles, we have
sin tan ,θθ≈ which gives
.
mmL
yL
dd
λ λ⎛⎞
==
⎜⎟
⎝⎠

For the fourth order we have

()(
)()
9
3
1.5m 680 10 m 4
48 10 m ,
d
−
−
×
×= which gives
5
8.5 10 md
−
=×= 0.085 mm.

6. For constructive interference, the path difference is a multiple of the wavelength:
sin , 0, 1, 2, 3, ...dmmθλ== .
We find the location on the screen from
tan .yLθ=
For small angles, we have

sin tan ,
θθ≈ which gives

Chapter 24 The Wave Nature of Light

© 2005 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they
currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
206

.
mmL
yL
dd
λ λ⎛⎞
==
⎜⎟
⎝⎠

For the second order of the two wavelengths, we have

()( )()
()
9
3
3
1.6m 480 10 m 2
2.84 10 m 2.84mm;
0.54 10 m
a
a
Lm
y
dλ
−
−
−
×
== =×=
×

( )() ()
()
9
3
3
1.6m 620 10 m 2
3.67 10 m 3.67mm.
0.54 10 m
b
b
Lm
y
dλ
−
−
−
×
== =×=
×

Thus the two fringes are separated by 3.67mm 2.84mm
− =0.8 mm.

7. For constructive interference of the second order for the blue light, we have

(
)( )sin 2 460nm 920nm.
b
dmθλ== =
For destructive interference of the other light, we have
( )
1
2
sin , 0, 1, 2, 3, ...dmmθλ=′+ ′= .
When the two angles are equal, we get

( )
1
2
920nm , 0, 1, 2, 3, ...mm λ=′+ ′= .
For the first three values of ,m′ we get

( )
1
2
920nm 0 ,
λ=+ which gives
3
1.84 10 nm;λ=×
()
1
2
920nm 1 ,
λ=+ which gives 613nm;λ=
( )
1
2
920nm 2 ,
λ=+ which gives 368nm.λ=
The only one of these that is visible light is 613 nm.

8. For destructive interference, the path difference is
( )
1
2
sin , 0, 1, 2, 3, ... ;dmmθλ=+ = or

( )( )
()
() ()
1
2 1
2
2.5cm
sin 0.50 , 0, 1, 2, 3, ...
5.0cm
m
mm
θ
+
==+= .
The angles for the first three regions of complete destructive interference are

()
() ()
1
2 1
00 2
sin 0 0.50 0.25, 15 ;
m
d
λ
θθ+
==+==°

( )
()()
1
2 1
11 2
sin 1 0.50 0.75, 49 ;
m
d
λ
θθ+
==+==°

( )
() ()
1
2 1
2 2
sin 2 0.50 1.25,
m
d
λ
θ+
==+= therefore, no third region.

We find the locations at the end of the tank from
tan ;yL
θ=

(
)
0
2.0m tan15 0.52m;y=°=

(
)
1
2.0m tan 49 2.3m.y=°=
Thus you could stand
0.52 m, or 2.3 m away from the line perpendicular to the board midway between the openings.

Giancoli Physics: Principles with Applications, 6
th
Edition
© 2005 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they
currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
207

9. The 180° phase shift produced by the glass is equivalent to a path length of
1
2
.
λ For constructive
interference on the screen, the total path difference is a multiple of the wavelength:

1
max2
sin , 0, 1, 2, 3, ... ;dmm
λ θλ+==±±± or ( )
1
2
sin , 0, 1, 2, 3, ...dmmθλ=− =±±± .
For destructive interference on the screen, the total path difference is

( )
11
max22
sin , 0, 1, 2, 3, ... ;dmmλθ λ+=+=±±± or sin , 0, 1, 2, 3, ...dmm
θλ= =±±± .
Thus the pattern is just the reverse of the usual double-slit pattern.

10. For constructive interference, the path difference is a multiple of the wavelength:
sin , 0, 1, 2, 3, ...dmmθλ== .
We find the location on the screen from
tan .yLθ=
For small angles, we have
sin tan ,θθ≈ which gives
.
mmL
yL
dd
λ λ⎛⎞
==
⎜⎟
⎝⎠

For the third order we have

()( )(
)
9
3
3 1.6m 500 10 m
12 10 m ,
d
−
−
×
×= which gives
4
2.0 10 m.d
−
=×
With the new wavelength, then, the second-order maximum is located a distance of

mL
y
d
λ
=

(
)( )()
9
4
2 1.6m 650 10 m
0.010m
2.0 10 m
−
−
×
==
×

10mm= from the central maximum.

11. For constructive interference, the path difference is a multiple of the wavelength:
sin , 0, 1, 2, 3, ...dmmθλ== .
We find the location on the screen from
tan .yLθ=
For small angles, we have

sin tan ,
θθ≈ which gives
.
mmL
yL
dd
λ λ⎛⎞
==
⎜⎟
⎝⎠

For adjacent fringes,
1,m∆= so we have

Lm
y
dλ∆
∆=

() (
)()
9
3
3
5.0m 544 10 m 1
2.7 10 m 2.7mm.
1.0 10 m
−
−
−
×
==×=
×

12. The presence of the water changes the wavelength:
water
water
480nm
361nm.
1.33
n
λ
λ
== =
For constructive interference, the path difference is a multiple of the wavelength in the water:

water
sin , 0,1,2,3,... .dmm
θλ==

Chapter 24 The Wave Nature of Light

© 2005 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they
currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
208
We find the location on the screen from
tan .
yL
θ=
For small angles, we have

sin tan ,
θθ≈ which gives

water water
.
mmL
yL
ddλλ⎛⎞
==
⎜⎟
⎝⎠

For adjacent fringes, 1,
m∆= so we have

water
Lm
y
d
λ∆
∆=

(
)() ()
()
9
3
5
0.400m 361 10 m 1
2.41 10 m 2.41mm.
6.00 10 m
−
−
−
×
==×=
×

13. To change the center point from constructive interference to destructive interference, the phase shift
produced by the introduction of the plastic must be an odd multiple of half a wavelength, corresponding
to the change in the number of wavelengths in the distance equal to the thickness of the plastic. The
minimum thickness will be for a shift of a half wavelength:

()
plastic
1
plastic 2
plastic
1;
tntt tt
Nn
λλλλλ
⎛⎞ ⎛⎞⎛⎞ ⎛⎞⎛⎞
=−= −= −=⎜⎟ ⎜⎟⎜⎟ ⎜⎟⎜⎟
⎜⎟
⎝⎠ ⎝⎠⎝⎠⎝⎠⎝⎠

()
()
1
2
1.60 1 ,
640nm
t⎡⎤
−=⎢⎥
⎢⎥⎣⎦
which gives
533nm.t=

14. We find the speed of light from the index of refraction,.
c
v
n
= For the change, we have

()
red violetred violet
violet
violet
cc
nnvv
v c
n⎡⎤⎛⎞⎛ ⎞
−⎢⎥⎜⎟⎜ ⎟
− ⎝⎠⎝ ⎠⎣⎦
=
⎛⎞
⎜⎟
⎝⎠

(
)( )
()
violet red
red
1.665 1.617
0.030 3.0%.
1.617
nn
n
−−
== ==

15. We find the angles of refraction in the glass from

112 2
sin sin ;nnθ θ=

() ( )
2,450
1.00 sin 60.00 1.4820 sin ,θ°= which gives
2,450
35.76 ;
θ= °

() ( )
2,700
1.00 sin 60.00 1.4742 sin ,θ°= which gives
2,700
35.98 .
θ= °
Thus the angle between the refracted beams is

2,700 2,450
35.98 35.76 0.22 .θθ−=°−°=°

Giancoli Physics: Principles with Applications, 6
th
Edition
© 2005 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they
currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
209

16. For the refraction at the first surface, we have

air
sin sin ;
ab
nn
θ θ=

() ( )
1
1.00 sin 45 1.642 sin ,
b
θ°= which gives
1
25.51 ;
b
θ= °

() ( )
2
1.00 sin 45 1.619 sin ,
b
θ°= which gives
2
25.90 .
b
θ= °
We find the angle of incidence at the second surface from

(
)( )90 90 180 ,
bc
Aθθ°− + °− + = ° which gives

11
60.00 25.51 34.49 ;
cb
A
θ θ=− = °− °= °

22
60.00 25.90 34.10 .
cb
A
θ θ=− = °− °= °
For the refraction at the second surface, we have

air
sin sin ;
cd
nn
θ θ=

(
) ()
1
1.642 sin34.49 1.00 sin ,
d
θ°= which gives
1
68.4 from the normal;
d
θ=°

() ()
2
1.619 sin34.10 1.00 sin ,
d
θ°= which gives
2
65.2 from the normal.
d
θ=°

17. We find the angle to the first minimum from

()( )
()
9
1min 1min
3
1580 10m
sin 0.0132, so 0.755 .
0.0440 10 m
m
Dλ
θθ
−
−
×
== = = °
×

Thus the angular width of the central diffraction peak is

( )
11min
2 2 0.755 1.51 .θθ∆= = °= °

18. The angle from the central maximum to the first minimum is 17.5 .°
We find the wavelength from

1min
sin ;Dm
θ λ=

(
)()()
6
2.60 10 m sin 17.5 1 ,λ
−
×°= which gives
7
7.82 10 m 782nm.λ
−
=× =

19. For constructive interference from the single slit, the path difference is
( )
1
2
sin , 1, 2, 3, ...Dmmθλ=+ = .
For the first fringe away from the central maximum, we have

(
) ()( )
69 3
1 2
3.20 10 m sin 520 10 m ,θ
−−
×=× which gives
1
14.1 .
θ=°
We find the distance on the screen from

()
11
tan 10.0m tan14.1 2.51m.yL θ== °=

20. We find the angle to the first minimum from

()()
9
1min 3
1 450 10 m
sin 0.00045.
1.0 10 m
m
Dλ
θ
−
−
×
== =
×

We find the distance on the screen from
tan .yL
θ=
For small angles, we have
sin tan ,θθ≈ which gives

()( )sin 5.0m 0.00045 0.00225m.yLθ== =

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Chapter 24 The Wave Nature of Light

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210
Thus the width of the central maximum is

2 0.0045m 0.45cm.y==

21. The angle from the central maximum to the first bright fringe is
16 .°
For constructive interference from the single slit, the path difference is

( )
1
2
sin , 1, 2, 3, ...Dmmθλ=+ = .
For the first fringe away from the central maximum, we have

(
)()( )
93
2
sin 16 653 10 m ,D
−
°= × which gives
6
3.6 10 m 3.6 m.D µ
−
=× =

22. We find the angle to the first minimum from

()( )
()
9
1min 1min
3
1 589 10 m
sin 0.0169, so 0.970 .
0.0348 10 m
m
Dλ
θθ
−
−
×
== = = °
×

We find the distance on the screen from

()
2
11
tan 2.30m tan 0.970 3.89 10 m 3.89cm.yLθ
−
== °=×=

Thus the width of the peak is

(
)
11
2 2 3.89cm 7.79cm.yy∆= = =

23. We find the angular half-width θ of the central maximum from
sin ;
D
λ
θ=

9
55.0 440 10 m
sin ,
2 D
−
°×⎛⎞
=
⎜⎟
⎝⎠
which gives
7
9.53 10 m.D
−
=×

24. We find the angle to the first minimum from the distances:

( )
()
1
1min 1min2
9.20cm
tan 0.0180 sin ,
255cm
θθ=== because the angle is small.
We find the slit width from

1min
sin ;Dm
θ λ=

(
)()( )
9
0.0180 1 415 10 m ,D
−
=× which gives
5
2.30 10 m 0.0230mm.D
−
=× =

25. Because the angles are small, we have

()
11
1min 1min2
tan sin .
y
Lθθ
∆
==
The condition for the first minimum is

11
1min 2
sin .
y
DD
L
θ λ
∆
==
If we form the ratio of the expressions for the two wavelengths, we get

1
1
;
bb
aa
y
y
λ
λ
∆
=
∆

()
( )
()
1
420nm
,
4.0cm 650nm
b
y∆
= which gives
1
2.6cm.
b
y∆=

Giancoli Physics: Principles with Applications, 6
th
Edition
© 2005 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they
currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
211

26. There will be no diffraction minima if the angle for the first minimum is greater than 90 .°
Thus the limiting condition is

1min
sin ;Dm
θ λ=

()
max max
sin90 1 , or .DD λ λ°= =

27. We find the angle for the second order from

sin ;dm
θλ=

(
) ()( )
59
1.45 10 m sin 2 560 10 m ,θ
−−
×=× which gives sin 0.0772,θ= so 4.43 .θ=°

28. We find the wavelength from
sin ;dmθλ=

()
()
21
10 m cm sin 28.0 3 ,
3500lines cm
λ
−
⎡⎤
°=⎢⎥
⎢⎥⎣⎦
which gives
7
4.47 10 m 447nm.λ
−
=× =

29. We find the slit separation from
sin ;dmθλ=

()(
)
9
sin18.0 3 630 10 m ,d
−
°= × which gives
64
6.12 10 m 6.12 10 cm.d
−−
=× =×
The number of lines/cm is

()
3
411
1.64 10 lines cm.
6.12 10 cmd
−
==×
×

30. Because the angle increases with wavelength, to have a complete order we use the largest wavelength.
The maximum angle is 90°, so we have

sin ;dm
θλ=

()
() ( )
291
10 m cm sin90 700 10 m ,
8300lines cm
m
−−
⎡⎤
°= ×⎢⎥
⎢⎥⎣⎦
which gives 1.70.m=
Thus only one full order can be seen on each side of the central white line.

31. We find the slit separation from
sin ;dmθλ=

(
)( )
9
sin15.5 1 589 10 m ,d
−
°= × which gives
6
2.20 10 m 2.20 m.d µ
−
=× =
We find the angle for the fourth order from
sin ;dmθλ=

(
) ()( )
69
4
2.20 10 m sin 4 589 10 m ,θ
−−
×=× which gives
4
sin 1.069,
θ= so there is no fourth order.

32. We find the angles for the second order from

sindmθλ= with 2.m=
()
7
151
sin 2 7.0 10 m
6.0 10 lines m
θ
−
=×
× gives
1
sin 0.84,
θ= so
1
57.1 .θ=°
()
7
251
sin 2 4.5 10 m
6.0 10 lines m
θ
−
=×
×
gives
2
sin 0.54,
θ= so
2
32.7 .θ=°
Therefore, 57.1 32.7 24 .θ∆= °− °= °

Chapter 24 The Wave Nature of Light

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currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
212
33. We find the wavelengths from
sin ;dm
θλ=

()
() ()
2
11
10 m cm sin31.2 1 ,
9700lines cm
λ
−
⎡⎤
°=⎢⎥
⎢⎥⎣⎦
which gives
7
1
5.34 10 m 534nm;λ
−
=× =
()
() ()
2
21
10 m cm sin36.4 1 ,
9700lines cm
λ
−
⎡⎤
°=⎢⎥
⎢⎥⎣⎦
which gives
7
2
6.12 10 m 612nm;λ
−
=× =
()
() ()
2
31
10 m cm sin 47.5 1 ,
9700lines cm
λ
−
⎡⎤
°=⎢⎥
⎢⎥⎣⎦
which gives
7
3
7.60 10 m 760nm.λ
−
=× =

34. The maximum angle is 90°, so we have
sin ;dmθλ=

()
() ( )
291
10 m cm sin90 633 10 m ,
6000lines cm
m
−−
⎡⎤
°= ×⎢⎥
⎢⎥⎣⎦
which gives
2.63.m=
Thus two orders can be seen on each side of the central white line.

35. Because the angle increases with wavelength, to have a full order we use the largest wavelength.
The maximum angle is 90°, so we find the minimum separation from
sin ;
dm
θλ=

(
)( )
9
min
sin90 2 750 10 m ,d
−
°= × which gives
64
min
1.50 10 m 1.50 10 cm.d
−−
=× =×
The maximum number of lines/cm is

()
3
4
min11
6.67 10 lines cm.
1.50 10 cm
d
−
==×
×

36. We find the angles for the first order from
sin ;
dm
θλλ==

()
()( )
29
4101
10 m cm sin 410 10 m ,
8500lines cm
θ
−−
⎡⎤
=×⎢⎥
⎢⎥⎣⎦
which gives

410 410
sin 0.3485, so 20.4 ;
θ θ==°

()
()( )
29
7501
10 m cm sin 750 10 m ,
8500lines cm
θ
−−
⎡⎤
=×⎢⎥
⎢⎥⎣⎦
which gives

750 750
sin 0.6375, so 39.6 .
θ θ==°
The distances from the central white line on the screen are

()
410 410
tan 2.30m tan 20.4 0.855m;yL θ== °=

()
750 750
tan 2.30m tan39.6 1.90m.yL θ== °=
Thus the width of the spectrum is

750 410
1.90m 0.855m 1.05m.yy−= − =

37. We find the angle for the first order from

sin ;dm
θλλ==

7
sin 21.5 6.328 10 m,d
−
°= × which gives
6
1.73 10 m.d
−
=×

Giancoli Physics: Principles with Applications, 6
th
Edition
© 2005 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they
currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
213

The number of lines per meter is

5
611
5.79 10 lines m.
1.73 10 m
d
−
==×
×

38. Because the angles on each side of the central line are not the same, the incident light is not normal to the
grating. We use the average angles:

(
)
1
26 38 26 48
26 43 26.72 ;
2
θ
°′+ °′
==°′=°

()
2
41 08 41 19
41 14 41.23 .
2
θ
°′+ °′
==°′=°
We find the wavelengths from
sin ;
dm
θλ=

()
() ()
2
11
10 m cm sin 26.72 1 ,
9500lines cm
λ
−
⎡⎤
°=⎢⎥
⎢⎥⎣⎦
which gives
7
1
4.73 10 m 473nm;λ
−
=× =
()
() ()
2
21
10 m cm sin 41.23 1 ,
9500lines cm
λ
−
⎡⎤
°=⎢⎥
⎢⎥⎣⎦
which gives
7
2
6.94 10 m 694nm.λ
−
=× =
Note that the second wavelength is not visible.

39. We equate a path difference of one wavelength with a phase
difference of 2 .
π With respect to the incident wave, the wave
that reflects at the top surface from the higher index of the
soap bubble has a phase change of

1
.
φπ=
With respect to the incident wave, the wave that reflects
from the air at the bottom surface of the bubble has a phase
change due to the additional path-length but no phase change
on reflection:

2
film
2
20.t
φπ
λ
⎛⎞
=+⎜⎟
⎝⎠

For constructive interference, the net phase change is

()
11
film22
film
2
2 2 , 0, 1, 2, ..., or , 0, 1, 2, ...t
mm t m m
φπππ λ
λ
⎛⎞
=−== =+=⎜⎟
⎝⎠
.
The wavelengths in air that produce strong reflection are given by

()
()( )
()
( )
()
film
1
2
4 1.34 120nm 643nm2
.
21 21nt
n
mmm
λλ== = =
+++

Thus we see that, for the light to be in the visible spectrum, the only value of
m is 0:

(
)
()
643nm
643nm,
01
λ==
+
which is an orange-red.

40. Between the 25 dark lines there are 24 intervals. When we add the half-interval at the wire end, we have
24.5 intervals, so the separation is

26.5cm
1.08cm.
24.5intervals
=

Chapter 24 The Wave Nature of Light

© 2005 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they
currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
214
41. We equate a path difference of one wavelength with a phase
difference of
2.
π With respect to the incident wave, the wave
that reflects at the top surface from the higher index of the
soap bubble has a phase change of

1
.
φπ=
With respect to the incident wave, the wave that reflects
from the air at the bottom surface of the bubble has a phase
change due to the additional path-length but no phase change
on reflection:

2
film
2
20.t
φπ
λ
⎛⎞
=+⎜⎟
⎝⎠

For destructive interference, the net phase change is

()
111
film222
film
2
2 2 , 0, 1, 2, ..., or , 0, 1, 2, ...t
mm tmmm
n λ
φππ π λ
λ⎛⎞ ⎛⎞
=−=−= == =⎜⎟ ⎜⎟
⎝⎠⎝⎠
.
The minimum non-zero thickness is

( )
()
()
1
min 2
480nm
1 169nm.
1.42
t
⎡⎤
==⎢⎥
⎢⎥⎣⎦

42. With respect to the incident wave, the wave that reflects
from the top surface of the coating has a phase change of

1
.
φπ=
With respect to the incident wave, the wave that reflects
from the glass
()1.5n≈ at the bottom surface of the coating
has a phase change due to the additional path-length and
a phase change of
π on reflection:

2
film
2
2.t
φ ππ
λ
⎛⎞
=+⎜⎟
⎝⎠

For constructive interference, the net phase change is

11
film22
film film
2
2 2 , 1, 2, 3, ..., or , 1, 2, 3, ...t
mm t m mm
n λ
φππππ λ
λ⎛⎞ ⎛⎞
=+−== == =⎜⎟ ⎜⎟
⎝⎠ ⎝⎠
.
The minimum non-zero thickness occurs for 1:
m
=

( )
()
min
film
570nm
228nm.
2 2 1.25
t
n
λ
== =
570 nm is in the middle of the visible spectrum. The transmitted light will be stronger in the wavelengths
at the ends of the spectrum, so the lens would emphasize the red and violet wavelengths.

43. The phase difference for the reflected waves from the
path-length difference and the reflection at the
bottom surface is

2
2.tφ ππ
λ
⎛⎞
=+
⎜⎟
⎝⎠

For the dark rings, this phase difference must be an odd
multiple of ,
π so we have

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In the sitting room of a small forty-five pound flat, upon the fourth
floor of a tall red-brick building in West Kensington known as Queens
Mansions, Ethel Harrison, the girl who lived with Rita Wallace, sat
sewing by the window.
It was seven o'clock in the evening and though dusk was at hand
there was still enough light to sew by. The flat, moreover, was on
the west side of the building and caught the last rays of the sun as
he sank to rest behind the quivering vapours of London.
Last week in August as it was, the heat which hung over the
metropolis for so long was in no way abated. All the oxygen was
gone from the air, and for those who must stay in London—the
workers, who could only read in the papers of translucent sunlit seas
in Cornwall where one bathed from the beaches all day long; of
bright northern moors where dew fell upon the heather at dawn—life
was become stifling and hard.
In the window hung a bird-cage and the canary within it—the pet of
these two lonely maidens—drooped upon its perch. It was known as
"The Lulu Bird" and was a recurring incident in their lives.
Ethel was six and twenty, short, undistinguished of feature and with
sandy hair. She was the daughter of a very poor clergyman in
Lancashire, and she was the principal typist in the busy office of a
firm of solicitors in the city. She had ever so many certificates for
shorthand, was a quick and accurate machine-writer, understood the
routine of an office in all its details, and was invaluable to her
employers. They boasted of her, indeed, trusted her in every way,
worked her from nine to six on normal days, to any hours of the
night at times of pressure, and paid her the highest salary in the
market.
That is to say, that this girl was at the very top of her profession and
received two pounds ten shillings a week. Dozens of girls envied her,
she was more highly paid than most of the men clerks in the city.

She knew herself to be a very fortunate girl. She gave high technical
ability, a good intelligence, unceasing, unwearying and most loyal
service for fifty shillings a week.
Each year she had a holiday of fourteen days, when she clubbed
with some other girls and they all went to some farmhouse in the
country, or even for a cheap excursion abroad, with everything
calculated to the last shilling. This girl did all this, dressed like a lady,
had a little home of her own with Rita, preserved her dignity and
independence, and sent many a small postal order to help the poor
curate's wife, her mother, with the hungry brood of younger ones.
Mr. and Mrs. Harrison in Lancashire spoke of their eldest daughter
with pride. She had "her flat in town." She was "doing extraordinarily
well"; "Sister Ethel" was a fairy godmother to her little brothers and
sisters.
She was a good girl, good and happy. The graces were denied her;
she had made all sweet virtues her own. No man wooed her, no man
looked twice at her. She had no religious ecstasies, and—instead of a
theatre where one had to pay—asked no thrills from sensuous
ceremonial. She simply went to the nearest church and said her
prayers.
It is the shame of most of us that when we meet such women as
these, we pass them by with a kindly laugh or a patronising word.
Men and women of the world prefer more decorative folk. They like
to watch holiness in a picturesque setting, Elizabeth of Hungary
washing the beggar's feet upon the palace steps. . . .
A little worker-bee saint, making a milk pudding for a sick
washerwoman on a gas-stove in a flat—that comes rather too close
home, does it not?
The light was really fading now, and Ethel put down her sewing,
rose from her basket-work chair, and lit the gas.

It was an incandescent burner, hanging from the centre of the
ceiling, and the girls' living room was revealed.
It was a very simple, comely, makeshift little home.
On one side of the fireplace—now filled with a brown and gasping
harts-tongue fern in an earthen pot—was Ethel's bookshelf.
Up-to-date she had a hundred and thirty-two books, of the
"Everyman" and "World's Classics" series. She generally managed a
book and a half each fortnight, and her horizon was bounded by the
two-hundredth volume. Dickens she had very much neglected of
late, the new Ruskin had kept the set at "David Copperfield" for
weeks, but she was getting on steadily with her Thackeries.
Rita had no books. She was free of that Kingdom at the Podley
Institute, but the little black piano was hers. The great luxury of the
Chesterfield was a joint extravagance. Both ends would let down to
make a couch when necessary, and though it had cost the girls three
pounds ten, it "made all the difference to the room."
All the photographs upon the mantel-shelf were Ethel's. There was
her father in his cassock—staring straight out of the frame like a
good and patient mule. . . . Her sisters and brothers also, of all ages
and sizes, and all clothed with an odd suggestion of masquerading,
of attempting the right thing. Not but what they were all perfect to
poor Ethel, whose life was far too busy and limited to understand
the tragedy of clothes.
Rita's photographs were on the piano.
There were several of her school-friends—lucky Rita had been to a
smart school!—and the enigmatic face of Muriel Amberley with its
youthful Mona Lisa smile looked out from an oval frame of red
leather stamped with an occasional fleur-de-lys in gold.

There was a portrait of Mr. Podley, cut from the Graphic and framed
cheaply, and there were two new photographs.
One of them was that of a curly-headed, good-looking young man
with rather thick lips and a painful consciousness that he was being
photographed investing the whole picture with suspense.
Ethel had heard Rita refer to the original of this portrait once or
twice as "Dicker" or "Curly."
But, then, there was another photograph. A large one this time,
done in cloudy browns, nearly a foot square and with the name of a
very famous artist of the camera stamped into the card.
This was a new arrival, also, of the past few weeks, and it was held
in a massive frame of thick plain silver.
The frame, with the portrait in it, had arrived at the flat some
fortnight ago in an elaborate wooden box.
Ethel had recognised the portrait at once. It was of Mr. Gilbert
Lothian, the great poet. Rita had met him at a dinner-party, and, if
she didn't exaggerate, the great man had almost shown a disposition
to be friendly. It was nice of him to send Rita his photograph, but
the frame was rather too much. All that massive silver!—"it must
have cost thirty shillings at least," she had thought in her innocence.
When the gas was turned up, for some reason or other her eye had
fallen at once upon the photograph upon the top of the piano.
She had read some of Lothian's poems, but she had found nothing
whatever in them that had pleased her. Even when her father had
written to her and recommended them for her to read the poems
meant less than nothing, and the face—no! she didn't like the face.
"I hardly think that it's quite a good face," she said to herself, not
recognising that—the question of morality quite apart—her hostility
rose from the fact that it was a face utterly outside her limited

experience, a face that was eloquent of a life, of things, of thoughts
that she could never even begin to understand.
In the middle of the room the small round table was spread with a
fair white cloth and set for a meal. There was a green bowl of
bananas, a loaf of brown bread, some sardines in a glass dish. But a
place was laid for one person only.
Rita was in their mutual bedroom dressing. Rita was going to dine
out.
The two girls had lived together for a year now. At the beginning of
their association one thing had been agreed between them. Their
outside lives were to be lived independently of their home life. No
confidences were to be expected or demanded as a matter of
course. If confidences were made they were to be free and
spontaneous, at the wish or whim of each.
The contract had been loyally observed. Ethel never had any secrets.
Rita had had several during the year of their association, but they
had proved only minor little secrets after all. Sooner or later she had
told them, and they had been food for virginal laughter for them
both.
But now, during the last few weeks?—Ethel's glance flitted uneasily
from the big photograph upon the piano to a little round table of
bamboo work in one corner of the sitting room.
Upon this table lay a huge bunch of dark red roses. The stalks were
fitted into a holder of finely-woven white grass—as delicate in
texture as a panama hat—and the bouquet was tied with graceful
bows and streamers of purple satin—broad, expensive ribbon.
A boy messenger, most unusual visitor, had brought them an hour
ago. "For Miss Rita Wallace."

The quiet mind, the crystal soul of this girl, dimly discerned
something alien and disturbing.
The door of the sitting-room opened and Rita came in.
She was radiant. Her one evening dress was not an expensive affair,
a simple, girl's frock of olive-green crêpe de chene in the Empire
fashion, but the girl and her clothes were one.
The high "waist," coming just under the curve of the breast, was
edged with an embroidery of dull silver thread, and the gleam of this
upon its olive setting threw up the fair column of the throat the
rounded arms, the whiteness of the girlish bosom, with a most
striking and arresting lustre.
Round her neck the girl wore a riband of dark green velvet, and as a
pendant from it hung a little star of amethysts and olivines set in a
filigree of platinum, no rare nor costly jewel, but a beautiful one. She
was pulling her long white gloves up to the elbow as she entered the
room.
Ethel loved Rita dearly. Rita was her romance, the art and colour of
her life. She was always saying or doing astonishing things, she was
always beautiful. To-night, though the frock was an old friend, the
pendant quite familiar, Ethel thought that she had never seen her
friend so lovely. The nut-brown hair was shining, the young, brown
eyes lit up with excitement and joy, the tints of rose and pearl upon
Rita's cheeks came and went as her heart beat.
"A Duke might be glad to marry her," the plain girl thought without a
throb of envy.
She was perfectly right. If Rita had been in society or on the stage
she probably would have married a peer—not a Duke though, that
was Ethel's inexperience. There are so few dukes that they have not
the same liberty of action as other noblemen. The Beauty Market is

badly organised—curious fact in an age when to purvey cats' meat is
a specialised industry. But the fact remains. The prettiest girls in
England don't have their pictures in the papers and advertise no
dentrifice or musical comedy on the one hand, nor St. Peter and St.
George, their fashionable West End temples, on the other. Buyers of
Beauty have but a limited choice, and on the whole it is a salutary
thing, though doubtless hard upon loveliness that perforce throws
itself away upon men without rank or fortune for want of proper
opportunity!
"How do I look, Wog dear?" Rita asked.
"Splendid, darling," Ethel answered eagerly—a pretty junior typist in
Ethel's office, who had been snubbed, had once sent her homely
senior a golliwog doll, and since then the good-humoured Ethel was
"Wog" to her friends.
"I'm so glad. I want to look my best to-night."
"Well, then, you do," Ethel replied, and with an heroic effort forbore
further questioning.
She always kept loyally to the compact of silence and non-
interference with what went on outside the flat.
Rita chuckled and darted one of her naughty, provocative glances.
"Wog! You're dying to know where I'm going!"
Some girls would have affected indifference immediately. Not so the
simple Wog.
"Of course I am, Cupid," she said.
"I'm going to dine with Gilbert."
"Gilbert?"

"Gilbert Lothian I mean, of course. We are absolute friends, Wog
dear—he and I. I haven't told you before, but I will now. You
remember that night I was home so late, nearly a month ago? Yes?
—well I had been motoring to Brighton with Gilbert. I met him for
the first time at the Amberleys'—but that you know. Since then we
have become friends—such a strange and wonderful friendship it is,
Ethel! It's made things so different for me."
"But how friends? Have you seen him often, then? But you can't
have?"
Rita shook her head, impatiently for a moment, and then she smiled
gently. How could poor old Wog know or understand!
"No!" she cried, with a little tap of her shoe upon the carpet. "But
there are such things as letters aren't there?"
"Has he been writing to you, then?"
"Writing! I have had four of the most beautiful letters that a poet
ever wrote. It took him days to write each one. He chose every
word, over and over again. Every sentence is music, every word a
note in a chord!"
Ethel went up to her friend and kissed her. "Dear old Cupid," she
said, "I'm so glad, so very glad. I don't understand his poems
myself, but Father simply loves them. I am sure you will be very
happy. Only I do hope he is a good man—really worthy of my dear!
And so"—she continued, with a struggle to get down to
commonplace brightness of manner—"And so he's coming for you
to-night! Now I know why you look so beautiful and are so happy."
Two tears gathered in the kind green eyes, tears of joy at her dear
girl's happiness, but with a tincture of sadness too. With a somewhat
unaccustomed flash of imagination, she looked into the future and

saw herself lonely in the flat, or with another girl who could never be
to her what Rita was.
She looked up at Rita again, trying to smile through her tears.
What she saw astounded her.
Rita's face was flushed. A knot of wrinkles had sprung between her
eyebrows. Her mouth was mutinous, her brown eyes lit with an
angry and puzzled light.
"I don't understand you, Ethel," she said in a voice which was so
cold and unusual that the other girl was dumb.—"What on earth do
you mean?"
"Mean, dear," Ethel faltered. "I don't quite understand. I thought you
meant—I thought . . ."
"What did you think?"
"I thought you meant that you were engaged to him, Cupid darling!"
"Engaged!—Why Gilbert is married."
Ethel glanced quickly at the flowers, at the photograph upon the
piano. Things seemed going round and round her—the heat, that
was it—"But the letters!" she managed to say at length, "and, and—
oh, Cupid, what are you doing? He can't be a good man. I'm certain
of it, dear! I'm older than you are. I know more about things. You
don't realise,—but how should you poor darling! He can't be a good
man! Rita, does his wife know?"
The girl frowned impatiently. "How limited and narrow you are,
Ethel," she said. "Have you such low ideals that you think friendship
between a man and a woman impossible? Are you entirely fettered
by convention and silly old puritanical nonsense? Wouldn't you be
glad and proud to have a man with a wonderful mind for your friend

—a man who is all chivalry and kindness, who pours out the
treasures of his intellect for one?"
Ethel did not answer. She did not, in truth, know what to say. There
was no reason she could adduce why Rita should not have a man
friend. She knew that many singular and fine natures despised
conventionality or ordinary rules and seemed to have the right to do
so. And then—honi soit! Yet, inarticulate as she was, she felt by
some instinct that there was something wrong. Mr. Gilbert Lothian
was married. That meant everything. A married man, and a poet
too! oughtn't to have any secret and very intimate friendships with
beautiful, wilful and unprotected girls.
. . . "You have nothing to say! Of course! There is nothing that any
wide-minded person could say. Ethel, you're a dear old stupe!"—she
crossed the room and kissed her friend.
And Ethel was so glad to hear the customary affection return to
Rita's voice, the soft lips upon her cheek set her gentle and loving
heart in so warm a glow, that her fears and objections dissolved and
she said no more.
The electric bell at the front door whirred.
Rita tore herself from Ethel's embrace. There was a mirror over the
mantel-shelf. She gazed into it for a few seconds and then hurried
away into the little hall.
There was the click of the latch as it was drawn back, a moment of
silence, and then Ethel heard a voice with a peculiar vibration and
timbre—an altogether unforgettable voice—say two words.
"At last!"
Then there was a murmur of conversation, the words of which she
could not catch, interrupted once by Rita's happy laughter.

Finally she heard Rita hurry into the bedroom, no doubt for her
cloak, and return with an excited word. Then the door closed and
there was an instant of footsteps upon the stone stairs outside.
Ethel was left alone.
She went to her bookshelf—she did not seem to want to think just
now—and after a moment's hesitation took down "Sesame and
Lilies." Then she sat at the table with a sigh and looked without
much interest at the bananas, the sardines and the brown bread.
Ethel was left alone.

CHAPTER II
OVER THE RUBICON
"Inside the Horsel here the air is hot;
Right little peace one hath for it, God wot;
The scented dusty daylight burns the air,
And my heart chokes me till I hear it not."
—Swinburne.
Gilbert and Rita said hardly anything to each other as the motor-cab
drove them to the restaurant where they were to dine.
There was a sort of constraint between them. It was not
awkwardness, it was not shyness. Nevertheless, they had little to say
to each other—yet.
They had become extraordinarily intimate during the last weeks by
means of the letters that had passed between them. In all his life
Lothian had never written anything like these letters. Those already
written, and those that were to be written before the end, would
catch the imagination of Europe and America could they ever be
published. In prose of a subtle beauty, which was at the same time
virile and with the organ-note of a big, revealing mind, he had
poured his thoughts upon the girl.
She was the inspiration, the raison d'être, of these letters. That
"friendship" which his heated brain had created and imposed upon
hers, he had set up before him like a picture and had woven fervent
and critical rhapsodies about it. The joy that he had experienced in
the making of these letters was more real and utterly satisfying than

any he had ever known. He was filled and exalted by a sense of high
power as he wrote the lovely words. He knew how she would read,
understand and be thrilled by them. Paragraph after paragraph,
sentence after sentence, were designed to play upon some part of
the girl's mind and temperament—to flatter her own opinion at a
definite point, and to flatter it with a flattery so subtle and delicate,
so instinct with knowledge, that it came to her as a discovery of
herself. He would please her—since she was steeped in books and
their appeal, utterly ignorant of Life itself—with a pleasure that he
alone could give. He would wrap her round with the force and power
of his mind, make her his utterly in the bonds of a high intellectual
friendship, dominate her, achieve her—through the mind.
He had set himself to do this thing and he had done it.
Her letters to him, in their innocent, unskilful, but real and vivid
response had shown him everything. From each one he gathered
new material for his reply.
He had lived of late in a new world, where, neglecting everything
else, he sat Jove-like upon the Olympus of his own erection and
drew a young and supremely beautiful girl nearer and nearer to him
by his pen.
He had fallen into many mortal sins during his life. Until now he had
not known the one by which the angels fell, the last sin of Pride
which burns with a fierce, white consuming flame.
All these wonderful letters had been wrought under the influence of
alcohol. He would go to his study tired in body and so wearied in
brain that he felt as if his skull were literally packed with grey wool.
"I must write to Rita," he would think, and sit down with the blank
sheet before him. There would not be an idea. The books upon the
walls called to him to lose himself in noble company. The Dog Trust
gambolling with Tumpany in the garden invited him to play. The

sight of Mary with her basket on her arm setting out upon some
errand of mercy in the village, spoke of the pleasant, gracious hours
he might spend with her, watching how sweet and wise she was with
the poor people and how she was beloved.
But no, he must write to Rita. He felt chained by the necessity. And
then the fat cut-glass bottle from the tantalus would make an
appearance, the syphon of soda-water in its holder of silver filigree.
The first drink would have little or no effect—a faint stirring of the
pulses, a sort of dull opening of tired mental eyes, perhaps. Yet even
that was enough to create the desire for the moment when the brain
should leap up to full power. Another drink—the letter begun.
Another, and images, sentences which rang and chimed, gossamer
points of view, mosaics and vignettes glowing with color, merry
sunlight laughter, compliments and devoirs of exquisite grace and
refinement, all flowed from him with steady, uninterrupted progress.
. . . But now, as he sat beside Rita, touching her, with the fragrance
of her hair athwart his face, all ideas and thoughts had to be
readjusted.
The dream was over. The dream personality, created and worshipped
by his Art in those long, drugged reveries, was a thing of the past.
He had never realised Rita to himself as being quite a human girl. No
grossness had ever entered into his thoughts about her. He was not
gross. The temper of his mind was refined and high. The steady
progress of the Fiend Alcohol had not progressed thus far as yet.
Sex was a live fact in this strangely-coloured "friendship" which he
had created, but, as yet, in his wildest imaginings it had always been
chivalrous, abstract and pure. Passion had never soiled it even in
thought. It had all been mystical, not Swinburnian.
And the fact had been as a salve to his Conscience. His Conscience
told him from the first—when, after the excursion to Brighton he had
taken up his pen to continue the association—that he was doing

wrong. He knew it with all the more poignancy because he had
never done sweet Mary a treachery in allegiance before. She had
always been the perfect and utterly satisfying woman to him. His
"fountain was blessed; and he rejoiced with the wife of his youth."
But the inhabiting Devil had found a speedy answer. It had told him
that such a man as he was might well have a pure and intellectual
friendship with such a girl as Rita was. It harmed none, it was of
mutual and uplifting benefit.
Who of the world could point an accusing finger, utter a word of
censure upon this delightful meeting of minds and temperaments
through the medium of paper and pen?
"No one at all," came the satisfactory answer.
Lothian at the prompting of Alcohol was content to entertain and
welcome a low material standard of conduct, a debased ideal, which
he would have scorned in any other department of life.
And as for Rita, she hadn't thought about such things at all. She had
been content with the music which irradiated everything.
It was only now, with a flesh and blood man by her side in the little
box of the taxi-cab, that she glanced curiously at the Musician and
felt—also—that revision and re-statement were at hand.
So they said very little until they were seated at the table which had
been reserved for them at a celebrated restaurant in the Strand.
Rita looked round her and gave a deep sigh of pleasure. They sat in
a long high hall with a painted ceiling. At the side opposite to them
and at the end were galleries with gilded latticework. At the other
end, in the gilded cage which hid the performers from view, was an
orchestra which discoursed sweet music—a little orchestra of artists.
The walls of the white and gold hall were covered with brilliantly
painted frescoes of scenes in that Italy from where the first

proprietor had come. The blue seas, the little white towns clustering
round the base of some volcanic mountain, the sunlight and gaiety
of Italy were there, in these paintings so cunningly drawn and
coloured by a great scenic artist. A soft, white and bright light
pervaded everything. There was not a sound of service as the
waiters moved over the thick carpets.
The innumerable tables, for two or four, set with finest crystal and
silver and fair linen had little electric lamps of silver with red shades
upon them. Beautiful, radiant women with white arms and shining
jewels sat with perfectly dressed men at the tables covered with
flowers. It was a succession of little dinner parties; it seemed as if
no one could come here without election or choice. The ordinary
world did not exist in this kingdom of luxury, ease and wealth.
She leant over the little table against the wall. "It's marvellous," she
said. "The whole atmosphere is new. I did not think such a place as
this existed."
"And the Metropole at Brighton?"
"It was like a bathing machine is to Buckingham Palace, compared to
this. How exquisite the band is! Oh, I am so happy!"
"That makes me happy, Cupid. This is the night of your initiation.
Our wonderful weeks have begun. I have thought out a whole series
of delights and contrasts. Every night shall be a surprise. You will
never know what we are going to do. London is a magic city and you
have known nothing of it."
"How could the 'Girl from Podley's' know?—That's what I am, the Girl
from Podley's. I feel like Cinderella must have felt when she went to
the ball. Oh, I am so happy!"
He smiled at her. Something had taken ten years from his age to-
night. Youth shone out upon his face, the beauty of his twenties had

come back. "Lalage!" he murmured, more to himself than to her
—"dulce ridentem, dulce loquentem!"
"What—Gilbert?"
"I was quoting some Latin to myself, Cupid dear."
"And it was all Greek to me!" she said in a flash. "Oh! who ever saw
so many hors d'œuvres all at one time! I love hors d'œuvres, advise
me, don't let me have too many different sorts, Gilbert, or I shan't
be able to eat anything afterwards."
How extraordinarily fresh and innocent she was! She possessed in
perfection that light, reckless and freakish humour which was so
strong a side of his own temperament.
She had stepped from her dingy little flat, from a common cab,
straight into the Dance of the Hours, taking her place with instant
grace in the gay and stately minuet.
For it was stately. All this quintessence of ordered luxury and
splendour had a most powerful influence upon the mind. It might
have made Caliban outwardly courteous and debonnair.
Yes, she was marvellously fresh! He had never met any one like her.
And it was innocence, it must be. Yet she was very conscious of the
power of her beauty and her sex—over him at any rate. She
obviously knew nothing of the furtive attention she was exciting in a
place where so many jaded experts came to look at the flowers. It
was the naïve and innocent Aspasia in every young girl bubbling up
with entire frankness. She was amazed and half frightened at herself
—he could see that.
Well! he was very content to be Pericles for a space, to join hands
and tread a measure with her and the rosy-bosomed hours in their
dance.

It was as though they had known each other for ever and a day, ere
half the elaborate dinner was over.
She had called him "Gilbert" at once, as if he were her brother, her
lover even. He could have found or forged no words to describe the
extraordinary intimacy that had sprung up between them. It almost
seemed unreal, he had to wonder if this were not a dream.
She became girlishly imperious. When they brought the golden
plovers—king and skipper, as good epicures know, of all birds that fly
—she leant over the table till her perfect face was close to his.
"Oh, Gilbert dear! what is it now!"
He told her how these little birds, with their "trail" upon the toast
and their accompaniment of tiny mushrooms stewed in Sillery, were
said to be the rarest flower in the gourmet's garden, one of the
supreme pleasures that the cycle of the seasons bring to those who
love and live to eat.
"How perfectly sweet! Like the little roast pigling was to Elia! Gilbert,
I'm so happy."
She chattered away to him, as he sat and watched her, with an
entire freedom. She told him all about her life in the flat with Ethel
Harrison. Her brown eyes shone with happiness, he heard the silver
ripple of her voice in a mist of pleasure.
Once he caught a man whom he knew watching them furtively. It
was a very well-known actor, who at the moment was rehearsing his
autumn play.
This celebrated person was, as Gilbert well knew, a monster. He
lived his life with a dreadful callousness which made him capable of
every bestial habit and crime, without fear, without pleasure, without
horror, and without pity.

The poet shuddered as he caught that evil glance, and then,
listening anew to Rita's joyous confidences, he became painfully
aware of the brute that is in every man, in himself too, though as
yet he had never allowed it to be clamant.
The happy girl went on talking. Suddenly Gilbert realised that she
was telling him something, innocent enough in her mouth, but
something that a woman should tell to a woman and not to a man.
The decent gentleman in him became wide awake, the sense of
comeliness and propriety. He wasn't in the least shocked—indeed
there was nothing whatever to be shocked about—but he wanted to
save her, in time, from an after-realisation of a frankness that might
give her moments of confusion.
He did it, as he did everything when he was really sober, really
himself, with a supreme grace and delicacy. "Cupid dear," he said
with his open and boyish smile, "you really oughtn't to tell me that,
you know. I mean—well, think!"
She looked at him with puzzled eyes for a moment and then she
took his meaning. A slight flush came into her cheeks.
"Oh, I see," she replied thoughtfully, and then, with a radiant smile
and the provocative, challenging look—"Gilbert dear, you seem just
like a girl to me. I quite forgot you were a man. So it doesn't matter,
does it?"
Who was to attempt to preserve les convenances with such a
delightful child as this?
"Here is the dessert," he said gaily, as waiters brought ices,
nectarines, and pear-shaped Paris bon-bons filled with Benedictine
and Chartreuse.
A single bottle of champagne had served them for the meal. Gilbert
lit a cigarette and said two words to a waiter. In a minute he was

brought a carafe of whiskey and a big bottle of Perrier in a silver
stand. It was a dreadful thing to do, from a gastronomic or from a
health point of view. Whiskey, now! He saw the look of wonder on
the waiter's face, a pained wonder, as who should say, "Well, I
shouldn't have thought this gentleman would have done such a
thing."
But Lothian didn't care. It was only upon the morning after a
debauch, when with moles' eyes he watched every one with
suspicion and with fear, that he cared twopence what people
thought about anything he did.
He was roused to a high pitch of excitement by his beautiful
companion. Recklessness, an entire abandon to the Dance of the
Hours was mounting up within him. But where there's a conscience,
there's a Rubicon. The little brook stretched before him still, but now
he meant to leap over it into the forbidden, enchanted country
beyond. He ordered "jumping powder."
He drank deeply, dropped his cigarette into the copper bowl of rose
water at his side and lit another.
"Cupid!" he said suddenly, in a voice that was quite changed, "Rita
dear, I'm going to show you something!"
She heard the change in his voice, recognised it instantly, must have
known by instinct, if not by knowledge, what it meant. But there was
no confusion, nor consciousness in her face. She only leant over the
narrow table and blew a spiral of cigarette smoke from her parted
lips.
"What, Gilbert?" she said, and he seemed to hear a caress in her
voice that fired him.
"You shall hear," he said in a low and unsteady voice. He drew a
calling card from the little curved case of thin gold he carried in his

waistcoat pocket, and wrote a sentence or two upon the back in
French.
A waiter took the card and hurried away.
"Oh, Gilbert dear, what is the surprise?"
"Music, sweetheart. I've sent up to the band to play something.
Something special, Cupid, just for you and me alone on the first of
our Arabian Nights!"
She waited for a minute, following his eyes to the gilded gallery of
the musicians which bulged out into the end of the room.
There was a white card with a great black "7" upon it, hanging to
the rail. And then a sallow man with a moustache of ink came to the
balcony and removed the card, substituting another for it on which
was printed in staring sable letters—"BY DESIRE."
It was all quite new to Rita. She was awed at Gilbert's almost
magical control of everything! She understood what was imminent,
though.
"What's it going to be, Gilbert?" she whispered.
Her hand was stretched over the table. He took its cool virginal ivory
into his for a moment. "The 'Salut d'Amour' of Elgar," he answered
her in a low voice, "just for you and me."
The haunting music began.
To the end of her life Rita Wallace never heard the melody without a
stab of pain and a dreadful catch of horror at her heart.
Perhaps the thing had not been played lately, perhaps the hour was
ripe for it in the great restaurant. But as the violins and 'cello sobbed
out the first movement, a hush fell over the place.

It was the after-dinner hour. The smoke from a hundred cigarettes
curled upwards in delicate spirals like a drawing of Flaxman's. Bright
eyes were languorous and spoke, voices sank to silence. The very
waiters were congregated in little groups round the walls and service
tables.
Salut d'Amour!
The melody wailed out into the great room with all the exquisite
appeal of its rose-leaf sadness, its strange autumnal charm. It was
perfectly rendered. And many brazen-beautiful faces softened for a
moment, many pleasure-sodden hearts had a diastole of
unaccustomed tenderness as the music pulsed to its close.
Gilbert's acquaintance, the well-known actor, who was personified
animal passion clothed in flesh if ever a man was, felt the jews-harp
which he called his heart vibrate within him.
He was a luxurious pariah-dog in his emotions as in everything else.
The last sob of the violins trembled into silence. There was a loud
spontaneous burst of applause, and a slim foreign man, grasping his
fiddle by the neck, came from behind the gilded screen and looked
down into the hall below with patient eyes.
Lothian rose from his chair and bowed to the distant gallery. The
musician's face lighted up and he bowed twice to Lothian. Monsieur
Toché had recognised the name upon the card. And the request,
written in perfect, idiomatic French, had commenced, "Cher Maitre
et Confrère." The lasting hunger of the obscure artist for recognition
by another and greater one was satisfied. Poor Toché went to his
bed that night in Soho feeling as if he had been decorated with the
Order of Merit. And though, during the supper hours from eleven to
half past twelve he had to play "selections" from the Musical Comedy
of the moment, he never lost the sense of bien être conferred upon
him by Gilbert Lothian at dinner.

Gilbert was trembling a little as the music ended. Rita sat back in her
chair with downcast eyes and lips slightly parted. Neither of them
spoke.
Gilbert suddenly experienced a sense of immense sorrow, of infinite
regret too deep for speech or tears. "This is the moment of
realisation," he thought, "the first real moment in my life, perhaps. I
know what I have missed. Of all women this was the one for me, as
I for her. We were made for each other. Too late! too late!"
He struggled for mastery over his emotion. "How well they play," he
said.
She made a slight motion with her hand. "Don't let's talk for a
minute," she answered.
He was thrilled through and through. Did she also, then, feel and
know . . . ? Surely that could not be. His youth was so nearly over,
the keen æsthetic vision of the poet showed him so remorselessly
how changed he was physically from what he had been in years
gone by, for ever.
Mechanically, without thinking, obeying the order given by the new
half-self, that spawn of poison which was his master and which he
mistook for himself, he filled his glass once more and drank.
In forty seconds after, triumph and pride flared up within him like a
sheet of thin paper lit suddenly with a match. Yes! She was his, part
of him—it was true! He, the great poet, had woven his winged words
around her. He had bent the power of his Mind upon her—utterly
desirable, unsoiled and perfect—and she was his.
The blaze passed through him and upwards, a thing from below.
Then it ended and only a curl of grey ash floated in the air.
The most poignant and almost physical sorrow returned to him. His
heart seemed to ache like a tooth. Yet it wasn't dull, hopeless

depression. It was, he thought, a high tragic sorrow ennobling in its
strength; a sorrow such as only the supreme soul-wounded artists of
the world could know—had known.
"She was for me!" his heart cried out. "Ah, if only I had met her
first!" Yes! he was fain of all the tragic sorrows of the Great Ones to
whom he was brother, of whose blood he was.
In a single flash of time—as the drowning man is said to experience
all the events of his life at the penultimate moment of dissolution—
he felt that he knew the secrets of all sorrow, the pangs of all
tragedy.
The inevitable thought of his wife passed like a blur across the fire-lit
heights of his false agony.
"I cannot love her," he said in his mind. "I have never loved her. I
have been blind until this moment." A tear of sentiment welled into
his eye at the thought of poor Mary, bereft of his love. "How sad life
was!"
Nearly every man, at some time or other, has found a faint reflection
of this black thought assail him and has put it from him with a
prayer and the "vade retro Sathanas."
Few men would have chosen their present wives if they had met—let
us assume—fifty other women before they married. And when the
ordinary, normal, decent man meets a woman better, clever, more
desirable than the one he has, it is perfectly natural that he should
admire her. He would be insensible if he did not. But with the normal
man it stops there. He is obliged to be satisfied with his own wife.
The chaos that riotous and unbalanced minds desire has not come
yet. And if a man says that he cannot love a wife who is virtuous
and good, then Satan is in him.

"I cannot love her," Lothian thought of his wife, and in the surveyal
of this fine brain and noble mind poisoned by alcohol it is proper to
remember that two hours before he could not have thought this
thing. It would have been utterly impossible.
Was it then the few recent administrations of poison that had
changed him so terribly, brought him to this?
The Fiend Alcohol has a myriad dominations. A lad from the
University gets drunk in honour on boat-race night—for the first time
in his life—and tries to fight with a policeman. But he is only
temporarily insane, becomes ashamed and wiser in the morning, and
never does such a thing again.
Lothian had been poisoning himself, slowly, gradually, certainly for
years. The disease which was latent in his blood, and for which he
was in no way personally responsible, had been steadily
undermining the forces of his nature.
He had injured his health and was coming near to gravely
endangering his reputation. His work, rendered more brilliant and
appealing at first by the unfair and unnatural stimulus of Alcohol,
was trembling upon the brink of a débâcle. He had inflicted
hundreds of hours of misery and despair upon the woman he had
married.
This, all this, was grave and disastrous enough.
But the awful thing that he was feeding and breeding within him—
the "false Ego," to use the cold, scientific, and appallingly accurate
definition of the doctors—had not achieved supreme power. Even
during the last year of the three or four years of the poisoning
process it had not become all-powerful. It had kept him from
Church; it had kept him from the Eucharist; it had drawn one thick
grey blanket after another between the eye of his soul and the vision
of God. But kindly human instinct had remained unimpaired, and he

had done many things sub specie Crucis—under the influence of,
and for the sake of that Cross which was so surely and steadily
receding from his days and passing away to a dim and far horizon.
But there arrives a time when the pitcher that is filled drop by drop
becomes full. The liquid trembles for a moment upon the brim and
trickles over.
And there comes a sure moment in the life of the alcoholic when the
fiend within waxes strong enough finally to strangle the old self, fills
all the house and reigns supreme.
It is always something of relatively small importance that hastens
the end—ensures the final plunge.
It was the last few whiskeys that sent honour and conscience flying
away with scared faces from this man's soul. But they acted upon
the poison of years, now risen to the very brim of the cup.
One more drop . . .
People were getting up from the tables and leaving the restaurant.
The band was resting, there was no more music at the moment, and
the remaining diners were leaning over the tables and talking to
each other in low, confidential tones.
Rita looked up suddenly. "What are we going to do now?" she said
with her quick bright smile.
"When we went to Brighton together," Gilbert answered, "you told
me that you had never been to a Music Hall. A box at the Empire is
waiting for us. Let us go and see how you like it. If you don't, we
can come away and go for a drive round London in a taxi. The air
will be cooler now, and in the suburbs we may see the moon. But

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Physics Principles With Applications 6th Edition Giancoli Solutions Manual

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