Compton's formula is actually incorrect by a factor of 2. In order ...
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The author of this paper is the Nobel laureate physicist Arthur Com...
In the case of Foucault's pendulum, the plane of oscillation rotate...
It's interesting that we don't even need the latitude of the place ...
WY
23,
19131
SCIENCE
803
timber, such, for instance, as making alcohol
from wood waste; in addition, Products is
gathering much statistical information of use
not only to the
For& Service, but to all
wood-using industries. Produc~ comes in
closer contact with the lumber industry than
any other branch of the service and has al-
ready secured results of great value to lum-
bermen. Under Silviculture, the
Review
gives in some detail the important problems
on which the service is working.
It
describes
briefly the establishment and purpose of the
experiment stations; under each head (for-
estation, forest influences, management, etc.)
it not only gives the problem8 to be studied,
but show8 their importance and their relation
to each other. The experiment being con-
ducted at Wagon Wheel Gap to determine the
influence of forest cover on run off
and ero-
sion is given rather fully. This is probably
the most complete and far-reaching experi-
ment of its kind in the world.
At the end of the
Review
is the investiga-
tive program for
1912.
A
study of this pro-
gram will show the thoroughness with which
the field is being covered.
BARRINGTONNOORE
WASHINGTON,
D.
C.
SPECIAL
ARTICLES
A
LABORATORY
METHOD
OF
DEMONSTRATING
THE
THE two laboratory methods in general use
for proving the rotation of the earth are
Foucault's pendulum and gyroscope experi-
ments. The first is inapplicable in many
laboratories, because there is no convenient
place to hang a sufficiently long and heavy
pendulum, while the apparatus for the second
is necessarily expensive. The following ex-
periment is designed to provide a simple and
convenient means by which the earth's rota-
tion may be demonstrated in a small labora-
tory.
The demonstration depends upon the
fact that, if a circular tube filled with water
is placed in a plane perpendicular to the
earth's axis, the upper part of the tube with
the water in it is moving toward the east with
respect to the lower part. If the tube ia
quickly rotated through
180
degrees about its
east and west diameter as an axis, the part of
the tube which was on the upper side attains
a
relatively westward motion as it is turned
downwards (since
it
is drawing nearer the
earth's axis). But the water in this part of
the tube retains
a
large part of its original
eastward motioa, and this can be detected by
suitable means.
Since the east and west axis itself is ro-
tating with the earth, only that component of
the water's momentum which is parallel to
this axis will have an effect in producing
a
relative motion when the tube is turned.
If
then
a
is the angular velocity of the earth's
rotation,
r
the radius of the circle into which*
the tube is bent, and
6
the angular distance
of any small portion of the tube from the east
and west axis, the relative velocity between
the water and the tube when
it
is quickly
turned from a position perpendicular to the
earth's axis through
180
degrees is
Velocity
=
V
=
sin'
B&
=
w.
In order
td
prevent convection currents, it
is best to hold the ring normally in a hori-
zontal position, in which case the relative
motion is of course
aT
sin
cp,
where
+
is the
latitude of the experimenter.
To perform the experiment, glass tubing
1.3
cm. inside diameter was bent into a cir-
cular ring
99.3
cm. in radius, and a short
glass tube closedi with a rubber tube and screw
clamp was sealed into it to allow for the ex-.
pansion of the water and to provide a place
for filling. The ring was fastened with tape,
into notches in the wooden rod
A
(Fig.
I),.
which served as the horizontal axis, and was,
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-
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-
--
--
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-
--
-
-
-
--
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--
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-
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-
-
--
-
--
-
--
-
SCIENCE
IN.
s.
VOL.
XXXVII.
NO.
960
supported by wires from the extremities of
the cross rod
B.
The ends of the rod
R
were
made adjustable perpendicularly to the plane
of the ring, so that the ring might be made to
swing on
an
axis parallel to its plane. The
ends of the rod were swung in solid supports,
adjustable to make the axis horizontal. In
order that the motion of the water might be
detected, a mixture of linseed oil and oil of
cloves of the same density as water was pre-
pared, and a fow drops of the mixture were
shaken up with the water with which the tube
was to be filled. The globules of oil were
observed at a point
C,
between the ends of the
axis, through a micrometer microscope.
Difficulties from the astigmatic refraction of
the light by the water in the cylindrical glass
tub were overcome by sealing a tubular par-
affine cap, closed with a cover-glass and filled
with water, on the part of the glass tube under
the microscope, thus presenting a plane sur-
face through which
to
make the observation.
One side of the ring was woighted, so that on
releasing a catch at the side of the observer
the tube swung around through
180
degrees
in a definite time, and was "held again by the
catch just under the microscope.
In
taking a reading, the
microscope
was
focused as nearly
as
possible on the center of
the tube, and the ring was left in position
until the oil globules had no appreciable mo-
tion. As soon as the catch which held the
ring in position was released, the time was
counted, with the aid of a metronome ticking
half-seconds, until the tube had turned and
an oil globule had been fixed upon to follow.
The globule was followed through a measured
length of time by turning the micrometer
screw, and the distance through which it
moved was recorded. Examples of these ob-
servations are given in the first three columns
of Table
I.
Variations in the readings arose frorn the
fact that the part of the ring toward the east
was near a cold wall, so that convection cur-
rents were produced as soon as the tube left
the horizontal position in making a turn.
This effect was made as small as possible by
stirring the air with an electric fan. Other
variations came from the fact that it was
found impossible to adjust the horizontal axis
so nearly parallel to the
plane of the ring as
to prevent a slighk eEect from turning the
TABLE
I
Time from Time of
Time on Initial
Releasing
Curves
Veloc-
Catch
to
lowing
of Com-
ity,
I;
Following
Water's,
pletion
Mm.
Water's
Motion
of Turn
See.-'
iifotion see.
'1
Case
I.
Weight on side
1).
Change
from
heavy to
light side.
-
-.
-
--
-
--
--
7.6secs.
22.5
+
.40 +21.2 +.041
/
1
/
2:
/
7.0 23.0
+
.37 +22.1 +.033
-
.
-
--
--
--
-
Case
11.
Weight on side
D.
Change from light to
heavy side.
7.5 22.5 +1.57 1.0 +.l60
8.0
(
22.0
/
f1.35
1
1
5
.5
/
+.I55
--
.
--
-
Case
111.
Weight on side
F.
Change from heavy
to
light side.
-
8.0 22.0
-
.59 5.0 f13.4 F.067
7.5
/
22.5
1
-
.70
/
4.5
/
+11.4
1
-.075
Case
IV.
Weight on side
F.
Change from light
to
heavy side.
+19.9
/
+.045
+11.5 +.075
i
---
-
-
-
-
-
.
-
-
Average
Y:
Case
I.
=
.0434
;
Case
TI.
=
.I580
;
Case
111.
=
-
.0633
;
Case
IV.
=
.0671.
tube.
Errors from the first cause were cor-
rected by reversing the direction of turning
in alternate readings. Those from the latter
cause were nullified by taking readings with
one side of the ring weighted and then shift-
ing the weight to the other side. In this
manner ten readings of each of four different
kinds were taken (Cases I., II.,
111.
and IV.),
and the fact that the predominant motion is
positive, or toward the west as observed on the
south side, shows that the earth is turning
frorn the west to the east.
Calculation
of
Ihe Initial Velocitg
In
order to make an accurate estimate of
the velocity corresponding to any given read-
ing, the rate of decrease of velocity of the
water in the ring must be determined.
If
the
--
805
MAY
23,
19131
SCIENCE
retardation
r
is taken to be proportional to
the velocity
V
for this low velocity,
dV
V
--
cat,
and
log
V=Ct+
E
will express the value of the velocity at dif-
ferent times.
In
order to determine the
constants
C
and
E,
the ring was held in a
vertical position until the colder water near
the east wall produced a considerable motion.
It
was then
brought back to the horizontal
-
and the time observed which was required
to
move successive quarter millimeters.
A
few
such readings are given in Table
11.
From a
large number of such observations an average
curve was drawn, showing the relation of the
distance covered to the time (Fig. 2, Curve
A).
The slope of this curve was taken at two
TABLE
I1
of the
.
most definite points,
t
=,
12.5 and
t
=
30,
and these values were substituted in
equation
(1)
to determine the constants
C
and
K.
The curve in
Fig.
-
3
was then drawn
from the resulting formula, showing the
velocity at any time. Curve
B,
Fig. 2,
was
then constructed by integrating this curve
graphically with respect to
t.
The water in the ring has its maximum
velocity just before the turn is completed.
The time required to make a complete turn
was three seconds,
and if this is subtracted
from the time in column
1,
Table
I.,
it
gives
the length of time between the completion of
the turn and the first observation of the
motion (column
4,
Table
I.).
Now if a por-
tion of Curve
B
(Fig. 2) be taken, such that
the distance represented on the curve in the
time of any particular reading is the same as
the distance in that reading, the beginning of
that portion of the curve will correspond to
the time at which the motion of the globules
was first observed (column 5, Table
I.).
So
if the number of seconds in column four is
subtracted from the time corresponding to the
beginning of the reading, the time correspond-
ing to the completion of the turn is obtained,
and the velocity at that time can be read from
the curve in Fig.
3.
This value is given in
column six, and is the velocity at the time of
completing the turn. The velocities in each
of the four cases are averaged separately, and
the average of the four averages
is
taken as
the true motion due to the earth's rotation.
The average of the velocities in these four
cases is
.0513
mm.
per second. From the
formula
V=,ar
sin
.+
derived above, we ob-
SCIENCE
[N.
S. VOL. XXXVII. No. 960
tain
V=
.0484,
a difference of
5
per cent.
As a check upon the accuracy of the readings,
it
will be seen that the differences between the
velocities in Cases
I.
and
11.
and between
those in
111.
and IV., representing double the
velocity due to the difference in density of the
water in different parts of the tube, are about
equal; also the differences between Cases
I.
and
III.,
and
11.
and
IT.,
representing the
variation due to imperfect adjustment of the
axis, are approximately the same. In order
to show that there was no appreciable effect
from convection currents while the ring was
in
a
horizontal position, several readings were
talcen after the tube had remained at rest for
same time, none of which showed a motion
larger than
.015
mm. per second.
In order to obtain the best possible results,
the ring should be mounted as rigidly as pos-
sible in a room of equal temperature through-
out, and the axis should be capable of accu-
rate adjustment parallel to the ring. If the
radius of the ring were made smaller, although
the effect of the earth's rotation would be less,
it would be easier to keep all parts of the tube
at an equal Bemperature, and the ring could
be turned more quickly.
Moreover, since the
motion would not be so great, the velocity of
the water would diminish less rapidly, so that
more accurate readings could be obtained.
With a more mobile liquid the motion would
of course continue longer. Even with the
comparatively crude apparatus described
above, however,
it
is not difficult to show
that thie earth revolves.
ARTIIUR
HOLLY
COMPTON
PHYSICALLABORATORX,
UNIVERSITY WOOSTER,
OP
January 13, 1913
CROSSOPTERYGIAN
ANOESTRY
OF
THE
AMPHIBIA
FOR
many years evidence has been accumu-
lating for the view that the Amphibia have
been derived not from Dipnoi but from Cros-
sopterygians of some sort. Pollard' held that
the Amphibia were remotely related to the
On the Anatomy and Phylogenetic Position
of Polypterus,"
2001.
Jahrb. Abt.
f.
Anat.
u.
Ont.
(Spengel), V. Bd., Jena, 1892, pp. 387-428, Taf.
27-30.
living Polypterm and Baur2 was able to
strengthen the evidence, to some extent, from
the Stegocephalian side. More recently
Th6veninS has expressed similar views, while
Moodie: correcting Baur's observations on the
lateral line grooves in the slrull has seemingly
demonstrated the general homology of the
skull top of Polypterus with that of Stego-
cephalia. Gegenbaur5 supported the homology
of the Stegocephalian cleithrum with the
"
clavicle
)'
of Polypterus and other fishes,
while .Klaatscha showed that the pectoral
limbs of Polypierus both in musculature and
ostcology in many respects remotely suggest
Amphibian conditions. On the other hand,
Goodrich's' studies on the scales of fishes, to-
gether with the evidence
offered especially by
the brain of Polypterus, tend to remove that
genus widely from genetic relationship with
the Amphibia.
The Paleozoic Crossopterygii have hitherto
yielded a few, though significant, hints of
Amphibian relationship. The Texas Permian
Crossopterygian fish named by Cope Ecto-
sfeorhachis nitidus and recently figured by
EIussakof\s Negalichthys nitidus, suggests
remote Stegocephalian affinities in the skull
and the same
is
true of Rhisodopsis, as figured
by Traquaira and of Osteolepis, as figured by
'
"
Les Plus Aneiens Quadrupedes de France,
"
Annales
de
Pal. (Boule), tonie
V.,
1910, pp. 1-64,
pl. I.-1X.
"The Lateral Lie System of Extinct Am-
phibia," Journ.
of
Morpliol., Vol. XIX., No.
2,
1908, pp. 511-540;
1
pl.
6"Clavicula und Cleithrum," Morphol. Jal~rb.,
XXIII. Bd., Leipzig, 1895, pp.
1-21.
8"Die Brustflosse der Crossopterygier," Fesl-
schr. fiir Gegenbaur,
I.
Bd., 1896, pp. 259-391,
Taf. I.-IV.
"'The Stegocephali.
A
Phylogenetic Study,"
Anat. Anz.,
Xl.
Bd., 1896, No.
22,
pp.
657-673.
Cf.
Lankester's "Treatise on Zoology,
"
Part
IX., first fascicle.
"
Cyelo&omes and Fishes,
"
by
E.
S. Goodrich, 1909, especially pp.
217-219,
290-
300.
'
The Permian Fishes of North America,"
Publ. No. 146 Cmegie Institution of Washington,
pp. 168 and pls. 30,
31.
"On the Cranial Osteology of Rhizodopsis,
"
Trans. Roy. Soc. Edinburgh, Vol. XXX., 1881.
In the case of Foucault's pendulum, the plane of oscillation rotates because of Earth's rotation at an angular speed of
$$
\Phi=360^\circ\sin\varphi\ /\mathrm{day}
$$
where $\varphi$ is the latitude of the place.
It's interesting that we don't even need the latitude of the place to determine Earth's rotation. If we do a second set of measurements flipping the aparatus around the North-South axis, the resulting mean velocity is ${\bar v = 2\Omega R\cos\phi}$. So $\bar u / \bar v = \tan\phi$ which allows us to determine the latitude.
This is another advantage over Foucault’s pendulum, which only allows us to determine one of these quantities.
The author of this paper is the Nobel laureate physicist Arthur Compton. He wrote it in 1913 when he was a 21 year old undegraduate student at the University of Wooster.
In this paper Compton described an experiment to demonstrate the rotation of the Earth using a simple laboratory apparatus. 10 years later Compton would end up discovering the Compton effect, confirming the particle nature of light and receiving the Nobel Prize in Physics for that.
Compton's formula is actually incorrect by a factor of 2. In order to derive the expression we start by defining the angular velocity of Earth
$$
\Omega = (0,\Omega\cos \phi, \Omega\sin \phi)=\Omega \cos \phi j + \Omega \sin \phi k
$$
The Coriolis acceleration in the frame of reference fixed to Earth and rotating with it is
$$
a = -2\Omega \times V
$$
If we now consider a slice of fluid located at an angle $\alpha$ from the north and assuming the tube is rotated through an angle $\beta$ from the horizontal about the $i$-axis, the acceleration of the slice of fluid due to the Coriolis force is
$$
2Rw\Omega \cos \alpha \cos (\phi+\beta)
$$
where $w=\dot{\beta}$ is the rotation rate of the tube. If we now want to have the component of the acceleration tangential to the tube we need to project the vector by multplying by $\cos \alpha$.
$$
a_t= 2Rw\Omega \cos^2 \alpha \cos (\phi+\beta)
$$
Finally if we integrate over $\beta$ and average over $\alpha$ (around the tube) we get the mean drift velocity
$$
\overline{u} = 2\Omega R \sin \phi
$$
Knowing the latitude $\phi$, we can easily calculate $\Omega$.