THE LIBRARY
OF
THE UNIVERSITY
OF CALIFORNIA
RIVERSIDE
GIFT OF
U.S. Naval Ordnance Laboratory
through
Dr. E.L.Harrington Estate
of Cbicago
A REDETERMINATION OF
THE COEFFICIENT OF VISCOSITY OF AIR
A DISSERTATION
SUBMITTED TO THE FACULTY OF THE OGDEN SCHOOL OF GRADUATE
SCIENCE IN CANDIDACY FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
(DEPARTMENT OF PHYSICS)
BY
ERTLE LESLIE HARRINGTON
A Private Edition
Distributed by
The University of Chicago Libraries
Reprinted from THE PHYSICAL REVIEW, N. S., Vol. VIII, No. 6,
December, 1916
A REDETERMINATION OF THE ABSOLUTE VALUE OF THE
COEFFICIENT OF VISCOSITY OF AIR.
[Reprinted from the PHYSICAL REVIEW, N.S.. Vol. VIII, No. 6, December, 1916.]
A REDETERM I NATION OF THE ABSOLUTE VALUE OF THE
COEFFICIENT OF VISCOSITY OF AIR.
BY ERTLE LESLIE HARRINGTON.
IN view of certain results on the "Coefficients of Slip" obtained by him
from his study of the laws of fall of small spheres in gases, Prof.
Millikan suggested to me that I attempt to check these results by making
some measurements on the coefficients of viscosity at very low pressures
with the constant deflection apparatus designed by himself and Dr.
Gilchrist. In the course of this work the method was so perfected as to
make it capable of a precision apparently unapproached heretofore in
measurements on the viscosity of gases.
Since the knowledge of the exact value of this coefficient for air is of
such fundamental importance in many lines of physical research it
seemed worth while to turn aside for the sake of attempting again to
fix its value with all the precision possible. This was especially needed
since some question 1 has been raised recently relative to the reliability
of the estimate made by Prof. Millikan as to the most probable value of
this constant. From determinations made under his direction by Gil-
christ 2 by the constant deflection method, and by Rapp 3 by an improved
capillary tube method, taken in connection with reliable determinations
made elsewhere and by other methods, Prof. Millikan published 4 as the
most probable value of this constant, ?;, at 23 C., .0001824, and estimated
its uncertainty at not more than one tenth per cent.
In connection with another problem, Vogel, 5 in 1914, was led to make
and publish a summary of the results of nearly all the determinations of
this constant that had ever been made, but he arrived at a value about
.8 per cent, higher than the above. In this, however, he includes deter-
minations which certainly involve gross error and which are at least very
far from what is now known to be the approximate value of rj. In fact,
he lists values with a total range of nearly n per cent., even in the same
method, while observers with present laboratory methods and facilities,
1 A. Gille, Ann. der Phys., 48, p. 799, 1915.
2 PHYS. REV., I., p. 124, 1913.
3 PHYS. REV., 2, p. 363, 1913.
4 Ann. der Phys., 41, p. 759, 1913.
6 Ann. der Phys., 43, p. 1235, 1914.
VOL. VIII.l
No. 6.
COEFFICIENT OF VISCOSITY OF AIR.
739
using any method whatsoever, would admit no greater error than a
fraction of a per cent., hence the inclusion of such widely variant values
of 77 renders more doubtful the validity of any mean so obtained.
The determination by Gilchrist appears also in this summary by
Vogel, and although he gives it relatively more weight than any other
single determination by any method, he nevertheless expresses the feeling
that the method and theory had not as yet received full development.
Since the publication of the Gilchrist article the method has been used
by Timiriazeff, 1 but his experimental arrangements were such as to make
it better suited to relative than to absolute determinations. It therefore
seemed important to subject the method itself to as critical a study as
possible while making the new determination of 77.
APPARATUS.
The apparatus is the same as that used by Gilchrist save for
pension and for certain other features which
it was necessary to modify in order to adapt
it to running in vacuum. The diagram shows
the general arrangement of the apparatus.
The outer, O, of two concentric brass cylin-
ders is made to rotate with constant velocity
about the inner, 7, which is suitably hung by
an elastic suspension. The viscosity of the
air produces a drag upon the inner cylinder
causing it to be deflected from its position of
rest to such an angle that the restoring couple
of the suspension brought into play exactly
counteracts the drag of the air, the angle of
deflection being measured by the usual mirror
telescope and scale method.
The cylinder frame, F, is mounted upon a
heavy steel plate, P, accurately machined and
having a raised rim in order to provide a
mercury seal for the large glass jar, /, which
covers the whole. To one side of the plate is
drilled a hole to provide pump connections,
and in the bottom is screwed a steel pipe, Q,
enclosing the driving shaft, and of such length
as to serve as a barometer column, and the
lower ends of pipe, rod, and gearing were suit-
ably constructed to dip into a mercury cup,
1 Ann. der Phys., 40, p. 971, 1913.
the sus-
Fig. 1.
74 ERTLE LESLIE HARRINGTON. FSECOND
[SERIES.
C, through which the motion must be transmitted. The cylinders are
carried by a heavy brass frame, F, provided with leveling screws and levels.
The outer cylinder is supported on a slightly conical bearing, B, at the bot-
tom, and at the top rotates on the tube, T, as an axle. The tube, T, at
the same time supports the suspension head, A, and is rigidly held by
the main frame. To eliminate end effects the inner cylinder is provided
at either end with a guard cylinder, G, of the same radius, and rigidly
held at a small distance from it and in perfect alignment with it by three
brass posts which connect the two guard cylinders and suitable end disks.
The upper part of this guard system and the suspension tube above
mentioned are constructed as one piece so that the upper guard cylinder
is held accurately centered with respect to the upper part of the rotating
cylinder. The lower part rests on a conical bearing which is a part of,
and accurately centered with respect to the bottom of the outer cylinder.
The suspension head, A, is so constructed as to permit considerable
translatory motion in any direction, as well as rotatory, thus making
possible accurate adjustment. The above apparatus was constructed
with great care and precision by Wm. Gaertner & Co., and to them
much credit is therefore due for the success of the experiment. Such
accuracy in construction made it possible to have the guard cylinders
very close (.025 cm.) to the suspended cylinder, and thereby completely
eliminate the need of end effect corrections.
A chronograph, K, provided with extra weights serves the double
purpose of driving the apparatus, and also of leaving a permanent record
of the speed of rotation. It was so arranged that the cylinder could be
instantly thrown out of gear and the chronograph used independently
of the other apparatus, this being done each time the period of vibration
of the inner cylinder was determined.
A Beckmann thermometer, calibrated with a standard Baudin ther-
mometer, and reading directly to .01 degree was hung beside the outer
cylinder. A high-grade Centigrade thermometer graduated to .1 degree
was hung beside it to serve as a check. The room itself was one of the
constant temperature rooms of the Ryerson laboratory; a basement
room, with no windows, lined on. all sides, top and bottom, with a 5-in.
layer of cork, and provided with heavy inner and outer doors. The
room was efficiently heated by a new style electric heater (furnished by
the Lee Electric Radiator Co.) used in connection with a sensitive thermo-
static control, and the air was constantly stirred by a large fan. The
temperature control was such that during any run the temperature
variation recorded by the Beckmann thermometer was never more than
a few hundred ths of a degree, and often no change at all was detected.
VOL. VIII.1
No. 6. J
COEFFICIENT OF VISCOSITY OF AIR.
741
The air in the apparatus was kept dry by an enclosed dish of phos-
phorus pentoxide.
CRITICAL STUDY OF APPARATUS AND DETERMINATION OF DIMENSIONS.
Inasmuch as one of the objects of the experiment was to study the
possibilities of the method as well as to find the absolute value of 17,
considerable time was spent on this phase of the work, and w enever
Fig. 2.
possible various methods of measurement were used for the purpose of
cross checking.
i. The Inner Cylinder. Three methods were used to measure the
diameter of the inner cylinder. In order to insure getting different
diameters and to entitle each different one to equal weight in the computa-
tion of the mean a large number of points evenly distributed over the
surface of the cylinder were systematically numbered and the measure-
ments were taken at these points. The first method was to place the
cylinder vertically on the bed of a dividing engine having mounted at one
742 ERTLE LESLIE HARRINGTON.
side with its axes perpendicular to the direction of motion a short focus
telescope provided with cross hairs. The distance between tangent
lines was then determined from the screw readings. The second method
was to adjust the jaws of a large-size vernier caliper to the diameter of
the cylinder and then determine the perpendicular distance between the
locked jaws by placing the same on the bed of the dividing engine, and
measuring by the usual method. The third method was to use a
micrometer screw caliper of sufficiently large size to measure directly
the diameter. The last two methods were by far the more convenient
and yielded as accurate results. These two results differed by only I
part in 13,000, and their mean agreed to Gilchrist's value to within I
part in 6,000, which is a liberal estimate of the possible error in this
dimension. The length was determined by cathetometer methods, and
the value found to agree exactly with that given by Gilchrist, and is
probably correct to within I part in 6,000.
2. The Outer Cylinder. The most satisfactory method of measuring
the diameter of the outer cylinder was found to be by filling it with dis-
tilled water observing the temperature and depth, and weighing the water
used. Results thus obtained differed by only I part in 15,000, and the
mean differed from that by Gilchrist by exactly the same amount,
so the error is perhaps no more than I part in 8,000.
The accuracy in the inner surface was studied in this way : The inner
cylinder, the lower guard ring, and the posts were removed, leaving the
upper part of the guard cylinder system, which, as above described, form
the axle for the outer cylinder. A heavy rod with one end hollowed
out to fit the conical bearing in the bottom of the outer cylinder was
inserted in the opening and brought to rest on the bearing. This rod
carried a lever arm suitably curved to rest against the wall of the cylinder,
capable of being adjusted to different heights, and bearing at the fulcrum
a mirror. A suitable telescope with vertical scale placed at a distance
of nearly three meters made it possible to detect the slightest deviation
from constancy in the radius or perfection in symmetry. In doing this
the lever was held lightly against the wall by a weight, and the cylinder
slowly rotated at a constant speed. From the readings of the observer
at the telescope and a measurement of all distances concerned, any
variations could be quantitatively determined. It is important to note
that this method not only tests the accuracy of the inner surface but
tests also the symmetry of this surface about the same bearing that is to
carry and hold in position the inner cylinder system. The results of this
test were very satisfactory, inasmuch as the variations were of the order
of i part in 2,500 and of such nature as to be self-counteracting in their
No L 6 VIIL ] COEFFICIENT OF VISCOSITY OF AIR. 743
effect on the final results, which were therefore affected probably less
than i part in 5,000.
3. Moment of Inertia of the Inner Cylinder. This was found as usual
by determining the period of vibration of the cylinder alone, and then
when a known moment of inertia was added. The cylinder was sus-
pended by piano wire and the cylindrical surface made plumb and
symmetrical about the axis by suitable adjustment of the three support
bars which connect the cylinder with the suspension clamp, and by use
of small weights fastened to the upper supporting vane of the cylinder.
The heavy, double support of the clamp from which the cylinder was
suspended rested on a stone bench, and although the room was apparently
free from air currents the cylinder was surrounded by a much larger one
in order to insure entire absence of them. The passage through the zero
position of the cylinder was indicated by the flash of a light into a tele-
scope at the opposite side of the room, and an electric key enabled the
operator to accurately register this time on the chronograph. The time
divisions were marked by impulses from the standard laboratory clock
and involved no appreciable error whatever, and the divisions could be
read to one one hundredth of a second. By taking the period from a
rather long run thus measured, the individual runs varied from the mean
for any suspension by an amount of the order of I part in 10,000, which
may be considered the probable error from this source. The added
known inertia consisted of a bar and ring, accurately machined, and
placed on the cylinder symmetrically. The dimensions of the ring and
bar were determined on the dividing engine, and their weights by the
analytical balance. The inertia of the combined ring and bar was
computed by the usual formula derived for such bodies, and the result
agreed to I part in 8,000 with that obtained by Gilchrist, this ratio
probably representing the accuracy of this determination. No faulty
adjustment to symmetry about the axis of suspension could have been
constant to the different series of runs, since the ring and bar were
frequently removed and replaced.
Having the two periods and the one known inertia, the moment of
inertia of the inner cylinder was computed from the usual formula,
_
r 2 /
The average departure of the various results from the mean for the
moment of inertia of the inner cylinder obtained from the different
suspensions used was I part in 5,000 and the mean differed from that
obtained by Gilchrist by exactly the same amount.
744 ERTLE LESLIE HARRINGTON.
DIMENSIONS.
Dimensions that are not involved in the computation of the results are
here included for descriptive purposes, but only approximate values for
such are given.
Vertical distance from base of chronograph to extreme top of cover 185 cm.
Diameter and height of glass jar, respectively 28 and 62 cm.
Length of outer cylinder 46
Length of suspension 23.5
Length of each guard cylinder 10
Length of inner cylinder 24.88
Distance between guard cylinders 24.93
Weight of inner cylinder 321 g.
Radius of outer cylinder 6.06317 cm.
Radius of inner cylinder 5.341 16
Moment of inertia of inner cylinder (experimentally determined) . . 7,617.3
ADJUSTMENT OF INSTRUMENT.
Before being placed in the instrument the inner cylinder was again
tested for symmetry and perpendicularity with the same methods and
precautions used preliminary to the determination of its moment of
inertia. In this three plumb bobs instead of one were used for the
purpose of expediting matters and permitting simultaneous observations
in three directions without in any way disturbing the cylinder or plumb
bobs.
The lower end of the outer cylinder being in place, the inner cylinder
and the guard cylinder system were put into position, fastened rigidly
by the top brace, and after suspending the inner cylinder the suspension
head screws and the leveling screws of the base of the instrument were so
manipulated as to bring the surface of the guard cylinders into alignment
with the inner cylinder as perfectly as possible without the use of a
telescope. The outer cylinder was then put into position, the suspension
tube again clamped, and the deflection of the inner cylinder noted as the
outer cylinder was rotated at the speed to be used later in determinations.
The suspension head was then rotated until the zero position was brought
as much to one side as the deflection position was to the other side of
the telescope which was in the middle of, and perpendicular to the
scale. Three small support screws in the base of the instrument were
then brought just to touch the bottom of the outer cylinder in order to
hold it in its exact position during the final adjustment. The outer
cylinder was loosened from its bottom and removed and the top brace
replaced and fastened firmly in position. The final, and more careful
adjustment of the guard cylinder system to alignment with the inner
cylinder was made by means of a short focus telescope. With a suitable
JJ L - 6 y iIL ] COEFFICIENT OF VISCOSITY OF AIR. 745
clamp system attached to the frame the outer cylinder was replaced
without at any stage of the process inclining or otherwise disturbing the
inner cylinder. The preliminary adjustment to zero position made un-
necessary the turning of either the guard cylinder system or the suspen-
sion head after this final adjustment, and thereby eliminated any error
that might have come from a lack of perfect coincidence of the suspension
with the axis of the suspension head collar. The use of the support screws
for the base of the outer cylinder holds it, and therefore the lower end of
the inner cylinder system, in precisely the position they occupy during
a run. Keeping the guard cylinder system in place while replacing the
outer cylinder eliminates the question as to whether it returns to the
exact position it had during adjustment. With these precautions it
would seem unlikely that any appreciable error could arise from faulty
adjustment, and the results later given involving results obtained before
and after dissembling and readjusting furnish convincing evidence of
the absence of such error.
TRIAL RUNS AND A STUDY OF FACTORS AFFECTING RESULTS.
As will appear later, the value of rj is computed from the time of rota-
tion, the period of vibration of the inner cylinder, the deflection, the
distance to the scale, and the temperature observations. The method of
taking these observations was this: The zero position was read when the
cylinder was at rest and checked by causing the cylinder to make small
vibrations (about I cm. as seen on the scale) and determining the zero
position as is done in using balances. This was done as a precaution
against any error due to sticking, although experience showed this step
really unnecessary. The outer cylinder was then put into rotation, the
speed at first being modified by a brake attachment to the chronograph
which could be operated from the position of the telescope in order to
bring the inner cylinder quickly to near rest in its deflected position.
As soon as a steady state was attained the stylus was lowered on to the
waxed paper of the chronograph drum and the rotation continued until
the stylus had traveled the length of the drum, which meant, with the
speed used, an interval of about 13 min. In practice the inner cylinder
was allowed to vibrate through two or three millimeters as seen on the
scale, readings being taken at short intervals; this plan giving a large
number of independent readings, eliminating the barely possible source
of error due to sticking, and affecting a great saving of time that would
otherwise be required to bring the cylinder to absolute rest, inasmuch as
the period of vibration was very long, and the damping very small.
At frequent intervals during this period the temperature was read on
746 ERTLE LESLIE HARRINGTON. [SEMES!
the Beckmann thermometer. At the close of the run the cylinder was
slowly let back and its zero position again checked. Now the outer
cylinder was given a rotation sufficient to set the inner cylinder in vibra-
tion, and then thrown out of gear, thus allowing the chronograph to be
used merely as such, whereupon the period of vibration was taken over a
period of about 45 min. To do this an electric key at the position of
the telescope enabled the operator to make accurate chronograph record
of the passage through the zero position, the first five and the last five
being recorded in order to provide means of cross checking and thus
insure absence of appreciable error from this source. The periods thus
obtained were probably accurate to within I part in 8,000.
The scale on which the deflections were read was a carefully selected
straight meter stick tested with a standard metric steel scale. The
magnification of the telescope was such that .1 mm. could be read easily
and since the deflections were of the order of 600 mm. the readings were
probably correct to within I part in 6,000. A steel tape was used to set
the ends of the scale equidistant fiom the mirror and to determine
its perpendicular distance from the mirror. The error in this was prob-
ably not greater than I part in 6,oob.
Irregularities in the speed of the chronograph might offer a source of
error, not on account of uncertainty as to what the speed is, for that is
obtainable directly from the record, but on account of the resulting
unsteadiness of the deflection. Fortunately it was found that the chrono-
graph drove the apparatus at a very constant speed, though of course
not perfectly so, owing perhaps to the irregularities in the friction, or a
lack of perfection in the gearing. However, a small auxiliary weight,
at the side of the operator, brought into series with the main weight by
two pulleys, enabled him after some experience to almost completely
neutralize any such irregularities, thereby reducing the error from this
source to probably I part in 5,000.
The high consistency in the early trials in the results for any suspension
indicated a satisfactory control in all the above sources of error, but it
was found that variations in the suspension produced rather great
variations in the results. In view of the experience of Gilchrist the
bifilar form of suspension was tried at first, but on account of the rather
great weight (321 g.) of the inner cylinder, and its small distance (.25
mm.) from the guard cylinders which permitted no sag, it was not possible
to use silk fibers which permit, perhaps, the closest approach to the true
bifilar type. Metallic ribbons were therefore used, but after two months
of experience with them they were wholly discarded, since it was not
found possible to entirely eliminate the effect of a change in the ribbon
No"6 VIIL ] COEFFICIENT OF VISCOSITY OF AIR. 747
or even of a mere change in the separation of the strands or in the manner
of clamping, upon the results obtained. The trouble no doubt lies par-
tially in the rather great discrepancy between such a bifilar and those
ordinarily treated theoretically since the deflection brings into play not
only a restoring couple due to the slight raise in the weight, but also a
restoring couple due to the twist in the strands themselves. Such a
suspension is therefore a sort of hybrid between the bifilar and the
ordinary elastic unifilar suspension. The greatest error, however, prob-
ably comes from the fact that the ribbons have widths of the same order
of magnitude as the separation of the strands which makes it quite likely
that as deflection occurs the two edges of either strand may assume
varying portions of the load, thereby causing a virtual change in the
distance between the strands. If the strands be clamped at both ends
it is unlikely that the load is equalized between the two strands, and if
instead the strand be simply looped about a pin at one end in order to
permit constant equalization of the load there is the possibility of error
due to a rolling of the strands about the pin as the deflection occurs.
Unrolled phosphor brojize wire in place of the ribbon was even less
satisfactory owing to the residual coil and the consequent drift, nor was
either found satisfactory later as a unifilar suspension. In fact the
writer was led to conclude that little dependence could be placed on phos-
phor bronze where precise results are expected. Quartz fibers were tried,
but it was not found possible to obtain fibers coarse enough to support
the rather great weight and at the same time fine enough to give sufficient
deflections. The smallest piano steel wire obtainable was much too stiff,
but it was found possible by reducing the size of the smallest obtained,
by very carefully rubbing with fine emery paper, to secure a sufficiently
large deflection and yet retain adequate tensile strength. After the
adoption of this plan no further suspension troubles were experienced.
A number of such were made, and in all cases there was a good zero
return and a satisfactory absence of drift. Moreover, different suspen-
sions, though differing greatly in stiffness, yielded entirely concordant
results. Later some samples of the uncoiled stock from which the hair
springs of watches are made were furnished by the Elgin Watch Company
and found to have the proper range of stiffness.
As will be seen later the torsion constant of the suspension is expressed
in terms of the inertia of the inner cylinder and the period of vibration.
From the observed damping, and the theoretical relation between the
magnitude of damping and the effect on the period as given, for example,
by Helmholtz, it was calculated that the total effect due to the damping
factors would be of the order of I part in 10,000, or quite inappreciable.
748 ERTLE LESLIE HARRINGTON.
Moreover, by experiment no effect on the period could be observed when
the outer cylinder was removed and the guard cylinders were separated
many times as far from the vibrating cylinder. But, although such
factors as viscosity involved above do not appreciably affect the period,
the fact must not be overlooked that the vibrating cylinder does carry
the air with it, and the moment of inertia of this air must be taken into
account. This point was first called to my attention by Dr. Lunn.
The periods taken in vacua were actually found to be i part in 750 less
than the periods taken at ordinary pressures, so the data given make due
allowance for this effect.
COMPUTATION OF RESULTS.
For the calculation of the results the well-known and very simple
formula 1 was used in this form :
- a 2 )
where 77 is the coefficient of viscosity, I the moment of inertia of the inner
cylinder, a and b the radii of the inner and outer cylinders respectively,
I the length of the inner cylinder, <f> the angular deflection of the inner
cylinder, T the period of vibration of the inner cylinder, and o> the
constant angular velocity of the outer cylinder. If the period of rotation
of the outer cylinder, /, be substituted for 27r/w, and a constant, K, for
the product [I(b 2 a 2 )]/(2a 2 & 2 /) (having here a numerical value of
1.20188) the formula becomes:
tK<f> tK s
77 = or i, = tan l ,
where 5 is the deflection as read on a straight scale, and d the distance
of the scale from the mirror. The latter form was used in all computa-
tion. A development of the above formula involving a more general
treatment which considers the coefficient of slip will be given in a paper,
following this, which will consider the problem of viscosity at the low
pressures where the effect of slip becomes appreciable. In apparatus
of the dimensions here used the correction for slip at ordinary pressures
amounts to about 2 parts in 100,000.
All determinations were made at temperatures in the neighborhood of
23 C., and the values for 77 reduced to that temperature by the use of
the formula suggested by Prof. Millikan (1. c.),
1723 = 779 + .000000493(23 - 0),
where 779 is the value of the viscosity coefficient obtained at C. This
1 See Poynting and Thompson, Properties of Matter, p. 213.
VOL. VIII.l
No. 6.
COEFFICIENT OF VISCOSITY OF AIR.
749
simple formula, found satisfactory through the range mentioned by him
must certainly hold in the small ranges here involved, which are, with
but two exceptions, less than one degree.
RESULTS.
The following data show the results of thirty-one determinations of
77 made at various times during about three months, and involve the use
of six different suspensions. Suspensions C and D were really the same
suspension under different physical conditions. In most cases during
the series of runs for a given suspension the apparatus was taken apart
and readjusted in order to make sure there was no error of adjustment.
Sus.
/(Cm.).
j.
e.
t.
T.
1>H X 1C*.
A
200.0
40.21
23.97
30.059
140.70
1,823.7
A
200.0
40.26
24.45
30.020
140.61
1,823.5
A
200.0
40.06
23.06
30.018
140.62
1,820.9
A
200.0
40.05
23.31
30.085
140.62
1,823.3
A
199.7
40.025
23.18
30.000
140.58
1,821.4
B
200.2
35.81
23.61
30.014
132.81
1,821.2
B
200.2
35.665
22.88
30.114
132.79
1,824.1
B
200.2
35.73
23.14
30.031
132.79
1,821.0
B
200.2
35.837
23.43
29.971
132.80
1,821.2
B
200.2
35.80
23.46
30.032
132.80
1,822.8
C
200.8
60.51
22.88
30.090
172.13
1,825.9
C
200.8
60.67
23.055
29.940
172.14
1,820.5
C
200.8
61.064
22.93
29.840
172.21
1,825.1
C
200.8
60.624
23.165
30.014
172.18
1,822.2
C
200.8
60.604
22.84
29.962
172.13
1,821.1
c
200.8
60.68
23.09
29.936
172.13
1,820.8
C
200.8
60.73
22.91
29.940
172.13
1,823.1
C
200.8
60.77
23.19
29.907
172.14
1,821.0
C
200.8
60.75
22.90
29.895
172.14
1,820.9
C
200.6
60.905
24.10
29.912
172.13
1,822.7
C
200.6
60.54
23.00
29.982
172.10
1,822.0
D
200.7
- 61.154
22.97
30.014
172.98
1,822.9
D
200.7
61.216
22.99
29.954
172.95
1,821.6
D
200.7
61.07
23.22
30.075
172.96
1,823.2
E
200.85
51.66
23.27
30.178
159.39
1,824.8
E
200.85
51.168
23.28
29.888
159.39
1,824.7
F
200.9
63.34
23.16
29.946
175.62
1,823.8
F
200.9
62.988
23.26
30.087
175.60
1,822.3
F
200.9
62.957
23.11
30.103
175.56
1,824.0
F
200.9
62.974
23.10
30.066
175.58
1,821.9
F
200.9
63.093
23.19
30.008
175.55
1,821.9
Mean value of r, at 23 C. = 1822.6 X 10^.
750 ERTLE LESLIE HARRINGTON.
Moreover, the runs show such variations in the various factors involved
as to make each determination an independent one, and no determination
made with satisfactory control in the manipulation of all steps was
discarded.
The strongest evidence of the advantages of this method for the deter-
mination of 77 lies in the remarkable consistency of the results here shown.
If from the individual deviations from the above mean one computes
the probable error by the usual least square method the result is found
to be .19 or i part in 9,600. Moreover, it should be emphasized that this
really includes every source of error except those involved in the instru-
ment constant, K. The probable error for each of the various determina-
tions involved in this constant has been given above in detail and if the
probable error in this constant be computed by the same method as
above it is found to be 1.9 parts in 5,000. The combined or total error
would be by this method of calculation only I part in 2,500 or .04 per
cent. Moreover, if the means for the different suspensions be compared
it is found that the maximum variation from the above mean is less than
.03 per cent, with the one exception of suspension , a watch-spring
suspension the use of which was discontinued on account of its tendency
to drift. Considering these two striking results, and making any reason-
able allowance for any unreliability in the least square method of com-
puting errors or of the estimates made of any individual probable error
it seems entirely justifiable to claim that the above mean is correct to
within less than .1 per cent, of the true value of t\ at the temperature
considered.
A comparison of the consistency of these results with the lack of con-
sistency of the results obtained by any other method, more especially
by the capillary tube method, shows the marked superiority of this
method. The uncertainties of the capillary tube method need not be
considered here inasmuch as they are well known and have been dis-
cussed by Prof. Millikan, 1 by Fisher, 2 Vogel, 3 and others. Only a few
points of contrast need be mentioned; the capillary tube method is
based on incomplete theory since it does not consider the effect of the
radial component of the velocity and other uncertainties due to the
increase in the volume of the gas as it passes along the tube, the general
end effects in addition to the question of the effect of irregularities in the
bore upon the stream lines, the difficulties in getting the exact pressures,
and above all, the impossibility of getting perfect tubes of the small radii
usually employed, and the great difficulty of subjecting any tube selected
to accurate examination as to uniformity, circularity, and even as to
1 L. c. * PHYS. REV., 29, p. 147, 1909. * L. c.
No L 6 ] COEFFICIENT OF VISCOSITY OF AIR. 751
the value of the radius itself, which enters, it should be recalled, in the
fourth power. The wide variation in the results obtained by this method
by different observers, and even by the same observer with different
tubes, is ample proof that these uncertainties exist. On the other hand,
with the method described above the theory is complete, there are no
end corrections involved, no expansion takes place, every step in the
construction of the cylinders is subject to control, the dimensions are so
great that they may be determined with great accuracy, and every portion
of the apparatus is subject to minute study for irregularities, and even
should such exist, their effect would be far less serious than in the case
of the other method. As to consistency with other observers we are
essentially limited to the value obtained by Gilchrist who first used the
method and made the claim that his result contained not more than .2
per cent, error. The value here obtained differs from his by less than
that amount. That his values fluctuate through a greater range than
the range here obtained is without doubt due principally to the suspension
troubles mentioned above which were here so largely eliminated, for
every determination of his of any instrument constant that could at this
time be checked was most critically examined, and no one of them found
to differ by more than i part in 6,000 from the value here given.
It is interesting to note that Rapp, 1 who perhaps came more nearly
completely eliminating the errors in the capillary tube method than
anyone else, and who used a large number of tubes, is practically identical
with the result here obtained, as is also that obtained by Hogg, 2 who used
an oscillation method. The result obtained by Grindley and Gibson, 3
using still a different plan differs only by about .03 per cent. More
significant still is the fact that the value published by Prof. Millikan as
correct to within .1 per cent, is less than .08 per cent, above the value
here obtained, and as his value was based on perhaps the most accurate
determinations by five different methods, it would seem that the above
claim that the result here obtained is within .1 per cent, of the true value
is well founded, since the above mentioned values all lie well within
this limit.
In conclusion the writer wishes to acknowledge his gratitude to Prof.
Millikan, who suggested the problem and maintained such constant and
helpful interest in the research during its progress, and to Prof. Michelson,
the head of the department, for various helpful suggestions.
RYERSON PHYSICAL LABORATORY,
UNIVERSITY OF CHICAGO,
June, 1916.
iL. c.
2 Am. Acad. Proc., 40, 18, p. 611, 1905.
* Proc. Roy. Soc., A, 80, p. 114, 1908.
[Reprinted from THE PHYSICAL REVIEW, Vol. 21, No. 3, March, 1923.]
A DETERMINATION BY THE CONSTANT DEFLECTION
METHOD OF THE VALUE OF THE COEFFICIENT OF
SLIP FOR ROUGH AND FOR SMOOTH
SURFACES IN AIR.
BY LELAND JOHNSON STACY.
ABSTRACT.
Coefficient of slip for rough and smooth surfaces in air, determined by the
constant deflection method. The apparent coefficient of viscosity measured
by this exceptionally precise method does not come out constant for the lower
pressures unless correction is made for the slip at the surfaces. The relation
7p'(i + kp) = rj = constant, between rj p ', the apparent viscosity coefficient,
and f p , the coefficient of slip, enables f p to be determined from measurements of
r) p f at pressures of 0.2 mm or lower. The apparatus included a vacuum-
tight chamber inside which a cylinder of radius 5.341 cm was suspended by a
steel wire concentric with a cylinder of radius 6.063 cm which could be rotated
at a constant slow rate so as to cause a steady deflection of the inner cylinder.
The accuracy of this method of measuring rj p ' is so great that the values of
frs = fp/76 all lie within 4 per cent of the mean. The chief difficulty
was in keeping the air pure because of the gradual evolution of gas, probably
hydrogen, inside the apparatus; but by taking observations only shortly
after evacuation this effect was avoided. For brass surfaces, f, reduced to
23 and 76 cm, came out 66.15 X io~ 7 which is practically the theoretical
minimum deduced by Millikan for a completely diffusing surface, 65.9 X io~ 7 .
For surfaces coated with shellac, the coefficient was found to be 97 X io~ 7 for a
fresh surface, in agreement with the value obtained by Lee from droplet
measurements, but it decreased steadily with time, presumably because of a
roughening due to oxidation, falling in two months to within 3 per cent of the
theoretical minimum. The early part of this work was done in collaboration
with E. L. Harrington.
Coefficient of viscosity of air at o.i mm is the same as at atmospheric pres-
sure when correction is made for the slip effect. The constancy of the values
obtained for f provides new evidence that the coefficient is independent of
the pressure.
T
I. HISTORICAL DEVELOPMENT OF THE IDEA OF SLIP.
HE theory that the coefficient of viscosity of a gas should be inde-
pendent of the pressure was first deduced by Maxwell * from a
consideration of the internal friction of molecules assumed to be rigid
spheres. He first deduced the relation
(i) 77 = pel,
where 17 = coefficient of viscosity; p = density of the gas; c = mean
1 Phil. Mag., 1860, Vol. 19, p. 31.
240 LELAND JOHNSON STACY.
, molecular velocity; / = mean free path of gas molecule. Since the
density is directly proportional to the pressure, while the mean free path
is inversely proportional to the same quantity, the product pi, and there-
fore 17, should be independent of the pressure. 1 In a later paper, 2 Maxwell
reported some experimental tests of his theory for pressures ranging
from 30 in. to 0.5 in. of mercury. His apparatus consisted of a torsion
pendulum of three plane-parallel plates suspended by an elastic fiber
between four fixed plane-parallel plates. The whole system was enclosed
in an airtight vessel and the logarithmic decrement of the oscillations
was observed for different pressures. He found no observable change in the
decrement, which is a measure of the viscosity, within the pressure range
studied.
The same relation (i) was derived later by O. E. Meyer. 3 By an
experimental arrangement similar to Maxwell's he checked the theoretical
deductions for the same range of pressures. Results at pressures below
1/60 atmosphere showed a falling off of the viscosity coefficient, which
he later ascribed to the fact that, in the theory of the experimental
method, the external friction (e) has been considered infinitely large in
comparison with the internal friction or viscosity. Thus he introduced
into the Kinetic Theory of Gases the slip coefficient f = rj/e, which
Helmholtz 4 had previously defined for liquids.
II. DETERMINATIONS OF THE COEFFICIENT OF SLIP IN AIR.
The first experimental determination of the value of the coefficient of
slip was made by Kundt and Warburg 5 who used the capillary tube
method. Their results were:
Tube No. j Pressure. Temperature.
I
33.8 mm
15 c
.00017
76 X io~ 7
2
39-0 '
.00016
82 X io~ 7
2
33-8 "
.00018
80 X io~ 7
Mean
79 X io~ 7
_.
The mean of their values at 15 C and 76 cm is about 79 X io~ 7 .
Millikan in a preceding article has reduced this to 23 C, getting f 76 at
23 C = 82 X io- 7 .
1 Using Stokes' value Vij/p = .116 for air, Maxwell made the first calculation of the
mean free path of a gas molecule.
2 Phil. Trans., 1866, Vol. 156, p. 249.
3 Pogg. Ann., 1865, Vol. 125, p. 177.
4 Wiener Sitzung., 1860, Vol. 40, p. 607.
6 Pogg. Ann., 1876, Vol. 159, p. 399.
COEFFICIENT OF SLIP IN AIR. 241
In the determination of the elementary electrical charge by the falling
drop method Millikan * found that very small droplets of oil did not
follow Stokes' equation
(2) v = 2 ~^( ~ P)-
From his observations he made an empirical correction of the above
equation, writing the corrected law in the form
(3) . : -?<.-.
where Al was determined from the curve of his observations. Millikan
pointed out that the correction of Stokes' Law for Slip z gave
which, for small values of f/o, reduces to
(5) *=-^(<r-p)(i + r/a).
9 1
Thus, the Al determined by Millikan was really the coefficient of slip
for oil and air. His value for 23 C was f 76 = 77 X io~ 7 . In a later
determination 3 a more detailed study of the failure of Stokes' Law for
small oil drops gave the value 76 at 23 C = 82.2 X io~ 7 .
For small drops of shellac, Lee 4 observed Al = fr 6 at 76 cm and 23 C
to be 100 X io~ 7 .
These experimental results led Millikan to a theoretical study of slip
for different boundary conditions. He concluded that when no gas
molecules were regularly reflected after impact upon the walls, the
maximum of external friction, and hence the minimum of slip, would
result. For a mechanically rough surface, which would cause such
diffuse reflection of all gas molecules, he calculated the minimum value
of f at 23 C and 76 cm to be 65.9 X lO" 7 .
III. THEORY FOR SLIP DETERMINATIONS BY THE CONSTANT DEFLECTION
METHOD.
The theory of the Constant Deflection Method of determining viscosity
coefficients gives
1 PHYS. REV., 1911, 32, 382.
2 Bassett, Hydrodynamics, Vol. II., p. 271.
J PHYS. REV., 1913, Vol. II., p. 139.
4 PHYS. REV., 1914, IV., 420.
242 LELAND JOHNSON STACY.
I = moment of inertia of suspended cylinder ; <f> angular displacement
of suspended cylinder; a = radius of suspended cylinder; / = length of
suspended cylinder; T = period of oscillation of suspended cylinder;
co = angular velocity of outer cylinder; b = radius of outer cylinder,
when it is assumed that there is no slip at the surfaces of the cylinders.
At ordinary pressures, this assumption involves an error too small to be
observed by this method. For the case where slip becomes appreciable
the complete equation as developed by Millikan is
(7) u = -.
a 2 \/ , 6 3 + a 3 \
2 J\ ^ P ab 3 -a s b)
The slip term at atmospheric pressure may be calculated for this appa-
ratus since a = 5.3412 cm, b = 6.0632 cm, and f 76 = 66 X io~ 7 (from
theory). This gives a value for the term
b 3 + a 3
2{ P -=~ - = .00002
ab 3 a 3 b
which is quite negligible, since the experimental error is about .1 per cent.
Hereafter the equation (6) will be written
and tip denned as the apparent viscosity coefficient at the pressure p.
The value 1776' will be taken as the true value of the viscosity coefficient.
Equation (7) may now be written
and solving for f p
(10) fp = (-^--
Thus to determine the slip coefficient by this method it is necessary,
first, to determine the coefficient of viscosity (77 = T7 76 ') and then ij p at
some other pressure. Application of equation (10) gives f p , and f 76 is
calculated from the formula f 76 = fp(/76). 17' and 77 p must, of course,
be observed at the same temperature or reduced to the same conditions
by the proper formula.
The investigation of slip coefficients by this method was undertaken,
at Prof. Millikan's suggestion, by E. L. Harrington x to determine
whether the slip depended on the nature of the surface as the lack of
1 PHYS. REV., 1916, VIII., p. 738.
COEFFICIENT OF SLIP IN AIR.
agreement between the values for oil and shellac drops had indicated.
Harrington devoted his time chiefly, however, to the improvement of
the precision of the method of determining the viscosity coefficient,
leaving the writer to carry on the slip determinations. The writer
assisted Harrington for a short time in the first determinations of the
slip coefficient. Some of Harrington's results will be in-
cluded in this paper.
IV. ADAPTATION OF THE APPARATUS FOR SLIP DETER-
MINATIONS.
The Constant Deflection Apparatus consists essen-
tially of two concentric brass cylinders, the inner, /,
being suspended on an elastic suspension, s, so that,
when the outer cylinder, O, is driven at a constant speed
by a clock driving mechanism, K, a constant torque due
to viscosity will cause a constant deflection of the sus-
pended cylinder from its equilibrium position. To elim-
inate end effects the inner cylinder is suspended between
two guard rings, G, of the same diameter and less than
.3 mm from it. A small mirror mounted at the base of
the suspension wire makes it possible to observe the de-
Fig, i.
flection by a telescope and scale. The period of rotation (t = 27r/o>) of
the outer cylinder was determined by a chronograph attached to the
driving mechanism.
The constants of the apparatus as determined by Harrington were used.
Moment of inertia of inner cylinder / = 7617.3
Radius of inner cylinder a = 5.3410 cm
Length of inner cylinder / = 24.88 cm
Radius of outer cylinder b = 6.0632 cm
For work at low pressures, the cylinders were set upon a steel plate,
P, about 30 cm in diameter. A large-mouthed glass bottle, /, about
28 cm in diameter and 62 cm high was found and the mouth ground
plane to fit tightly a rubber gasket laid on the steel base-plate. An
opening was ground in the bottom of the bottle for the suspension head,
T, which projected some 25 cm above the large bottle. A glass tube, A,
sealed at the top was fitted by a ground glass joint into the bottom of the
large bottle thus closing the apparatus. A plate glass observing window
was sealed over a small hole in the side of the suspension cover to prevent
distortion of the image seen in the observing mirror. A discharge tube
for spectroscopic work was also sealed into the side of the suspension cover.
The driving shaft for the rotating cylinder was led through an iron pipe,
244 LELAND JOHNSON STACY.
Q, sealed into the steel base-plate. The lower end of this pipe was
immersed in a vessel of mercury, C, thus sealing it from the outside.
A rim around the edge of the base-plate made it possible to use a mercury
seal at this joint and, the bottom of the bottle being somewhat concave,
a mercury seal was also used at the upper joint. Thus the apparatus
was enclosed tightly enough to permit evacuation to about .001 mm
pressure.
A Gaede mercury pump backed by a rotary oil pump made it possible
to reduce the pressure within the apparatus to .1 mm in a little over
two hours. The volume of the apparatus was about 150 liters. Pres-
sures were read by a McLeod gauge calibrated to read .0001 mm. Tem-
peratures within the apparatus were read from a Beckmann thermometer
which had been calibrated by comparison with a standard Baudin
instrument. The observing telescope and scale were mounted about
200 cm distant from the suspended mirror and the deflection could be
read to .1 mm. The steel suspension wire used throughout this work
gave an observed scale deflection of 62 to 64 cm when the period of rota-
tion of the.outer cylinder was about 30 sec.
V. EXPERIMENTAL METHODS AND ELIMINATION OF ERRORS.
The determination of the viscosity coefficient was made from the
formula
/ -.-,- t , S < 2?T - S
rj p = K tan" 1 where / = ; $ = tan" 1 >
T 2 2# co 2a
The period T and the scale distance d being determined beforehand, it
was only necessary to observe 5 and / so that r) p ' could be calculated.
Temperatures and pressures were read as already described.
Having determined f]^'(= >/), it was necessary to determine -rj p ' at
some low pressure and calculate fre from the equations
*:,\tt.!t .*:
For atmospheric pressure the observed scale deflection 5 was about 63
cm with an error of .3 mm. At 0.12 mm pressure and t = 30 sec. the
scale deflection was found to be about 56 cm. Thus a difference of 7 cm
with an error of .3 mm in reading would give an apparent error of not
more than I part in 200. The gauge reading to .0001 mm, this error
should be only i part in 1,200. Errors in observing all the other factors
were much less than these, so they may be neglected. It was found,
COEFFICIENT OF SLIP IN AIR. 245
however, that the principal source of inaccuracy was due to change of
the pressure during observations. This rise of pressure was considerable
in the first few hours after evacuation was stopped, but reached a steady
value after 24 hours or so. It was found by experiment that during the
lo-minute interval necessary for one complete observation the pressure
change was from .001 to .003 mm. After one or two days the pressure
rose less than that in 24 hours. This was explained by supposing the
increase in pressure to be due to gases released from the glass and metal
surfaces. By taking pressure readings before and after each observation
a fairly accurate mean value was obtained. The temperature was also
read at frequent intervals and the mean value used. The apparatus
being set up in a constant temperature room, the observations were made
within a very narrow range (22 to 24 C). Variations from 23 C were
corrected for by Millikan's formula (Ann. der Physik, 1913, Vol. 41, p.
759)-
-no = >?23 - .000000493(23 - 0).
In the early work on slip determinations, a series of observations was
made after a single evacuation. The values of fre calculated from these
observations gave an initial value of about 70 X io~ 7 but rose steadily
until a value of 200 X io~ 7 was found about two weeks later. The
pressure change was from .1238 to .1448 mm during this interval. Since
f 7 e should be independent of the pressure, this indicates that the increase
in pressure must be due to the presence of some gas of a lower viscosity
than air. This suggested that hydrogen (viscosity about one half that
of air) was being released from occlusion by the metal parts of the
apparatus. Spectroscopic examination of the discharge tube showed a
definite increase in the intensity of the hydrogen lines when the appa-
ratus was allowed to stand several days at a low pressure.
Admission of air to full atmospheric pressure to flush out the apparatus
and a second evacuation gave the same result, viz., a low value during
the first two or three hours after the pumps were stopped, then a steady
rise in the value of the slip constant fre. To eliminate this variation it
was found advisable to admit air immediately after a set of observations
was completed and to evacuate only a short time before readings were
to be taken. Thus the time during which the "hydrogen effect" might
be present was so short that it did not affect the results appreciably.
It was found that the values of f 76 obtained within three or four hours
after an evacuation were quite consistent. Observations were usually
made within an hour after evacuation. After many evacuations this
"hydrogen effect" was less marked but was always present. By taking
observations shortly after evacuation, it was avoided.
246
LELAND JOHNSON STACY.
VI. TABLE OF OBSERVATIONS AND CALCULATED DATA FOR BRASS
SURFACES IN AIR.
Results on Brass Surfaces in Air.
d = 200.5 cm; T = 175.48; 11 = 1822.6 X io~ 7 .
5 (cm).
* (sec.).
e ( C).
7?p X I0 7 .
r P x io".
p (mm).
fn x io 7 .
56.28. . . .
56.56. . . .
. 30.134
30.040
23.04
23-05
1643.0
1646.0
3884
3811
1303
.1328
66.6
66.6
56.71---.
56.93 -..
30.034
29.989
22.76
22.81
1649.4 3701
l6 53.2 3616
.1361
.1388
66.4
66.0
57-23....
55-95....
56.76. . . .
56.63....
29-500
30.267
29.992
30.124
22.67
22.67
22.72
22.72
1634.7 4044
1640.2 3912
1646.0 3726
1646.2 3721
.1238
.1266
.1340
.1362
65-9
65-2
65-7
66.7
55-77....
56.25....
30.187
29-975
22.81
22.82
1628.2 4166
1630.6 4 IO 7
.1200 65.8
.1209 65.3
57-25 -..
57-02....
29.720
29.880
23-70
23.66
1645.0
1647.1
3856
3797
1253
.1268
63-6
63-4
57-10....
56.68....
29-795
30.032
22.81
22.81
1644.8
1645-9
3764
3738
.1287
-I3II
63-8
64-5
56.71--..
56.86. . . .
30.134
30.079
22.80
22.83
1652.9
1653.8
3570
3556
.1361
1367
63-9
64.0
57-20. . . .
57-27.-..
29.929
29.922
22.82
22.79
1655-1 3520
1656.7 3480
.1405
.1428
65-1
65-4
57-15.-..
57-05-...
30.000
30.053
22.80
22.73
1657.6 3459
1657-7 3451
.1445 65.8
.1459 66.2
57.32-...
57-30. . . .
29.922
29.930
22.79
22.77
1656.2
1656.5
3437
3444
.1467
1475
66.3
66.8
56.51-...
29.952
22.87
1640.5 3836
.1368 69.0
55.77....
55.65..-.
29.917
30.000
22.84
22.78
1614.0
1614.6
4469
4436
.1178 69.3
.1166 68.1
56.47....
29.873
22.90
1631.2 4053
.1300 69.3
56.53-
30.037
22.42
1641.9 3733
.1402 68.9
56.73-.. -
30.043
23-01
1651.4 3405
.1508 67.6
56.62... .
29-915
22.73
1637.0 3886
.1343 68.7
56.68....
30.102
22.84
1649.8 3591
.1349 63.7
56.46. . . .
56.76. . . .
30.100
30.018
22.81
22.79
1647.1 3646
1647.4 3630
.1378 66.1
.1424 68.0
56.68....
57.00. . . .
29.972
29.860
22.87
22.99
1642.6
1645.6
3762
3710
.1308 64.7
.1338 65.3
57-03....
30.158
23-33
1662.9
3338
1558
68.3
56.34-
30.080
23-46
1638.8
3915
.1258
64.8
Mean . . .
66.15
COEFFICIENT OF SLIP IN AIR. 247
The observations were made within a pressure range of .1 to .18 mm
since at lower pressures the error due to change of pressure during an
observation was large, while at pressures above .18 mm the deflection
5 differed so little from the deflection at atmospheric pressure that
the error of observing this difference became considerable.
In general, the experimental conditions were as follows:
Temperature 22 C to 24 C
Pressure .1000 mm to .1800 mm
Period (t) 30 sec. (.2)
Period (r) - 175.5 sec.
Scale deflection, for p = 76 cm -630 mm 1
" p = o.i 8 mm - 580 mm > approximately
" p = o.io mm - 550 mm J
Observations were usually made in pairs within an hour after evacua-
tion.
The mean value (66.15 X io~ 7 ) is very close to Millikan's theoretical
minimum value (65.9 X io 7 ) but is considerably lower than any of the
values found by other methods for oil or for glass surfaces.
VII. DETERMINATION OF SLIP FOR SHELLAC SURFACES IN AIR.
The cylinders were next coated with a thin layer of shellac, dried by
an air blast and replaced in position. From the weight of the cylinders
before and after the shellac was applied and the surface area of the
cylinders, the average thickness of the shellac film was calculated. This
was found to be .01 mm and made only a small correction in the values
of the constants a and b.
After the determination of the viscosity coefficient at atmospheric
pressure, the pressure was reduced and observations made as before.
At first a high value of fre = 97 X io~ 7 was found while the shellac was
fresh. Later, the values of fa dropped steadily toward the minimum
value. This result is shown in the two sets of data here given ; the first
set of observations being due to Harrington and the second to the writer.
In two other trials, accidental experimental difficulties prevented the
writer from making observations while the shellac was fresh but values
between 80 and 90 X io~ 7 were found several days after application
of the shellac. The fall in the observed value of the slip coefficient
indicates a change in the surface due probably to oxidation of the
shellac which seems to produce a rough surface.
248
LELAND JOHNSON STACY.
Data on Fresh Shellac Surfaces in Air.
d = 200.6, T = 176.00.
E. L. Harrington: Fresh Shellac, August i, ipi6.
Date.
S.
t.
e.
ij/ X io 7 .
TP X IO B .
p (mm).
ft. X io 7 .
Aug. 9
55-93
29.719
23-38
1602.8
4911
.1474
95-2
55-72
29.727
23-44
1597-3
5059
.1478
98.5
Aug. 10, A.M. .
53-42
29-715
23.76
I53L3
6842
.1048
94-3
52-55
29.960
23-86
I5I9-4
7188
.1058
IOO.O
Aug. 10, P.M..
53-99
29.936
23.67
1559-0
6080
.1125
90.0
53-19
30.263
23.60
1552.9
6240
1133
93-0
Aug. ii, A.M..
53-43
30.227
23.46
1556.6
6121
.1107
89.0
52.96
30.440
23-50
1553-9
6204 .1112
90.8
Aug. ii, P.M..
55-19
30.100
23-13
1602.0
4886
.1288
82.8
55-H
30.100
23-13
1600.5
4920
.1298
84.1
Aug. 12
56-94
29.827
23.64
1637.0
4099
.1438
77-6
56.75
29.917
23.70
1636.5
4108
.1438
77-7
Aug. 30
57-28
29.780
23.10
1653-1
3538
.1506
70.1
L. J. Stacy: Fresh Shellac, March 7, 1917.
Mar. 9, A.M..
56.04
29.789
23-36
1578.8
5588
.1330
97-7
55-92
30.010
23.40
1582.5
5381
1352
Mar. 9, P.M..
57-66
29.800
23-14
1619.7
4580
.1470
88.6
57-26
30.080
23.19
1623.7
4486
.1482
87.5
Mar. io, A.M..
58.15
29-723
23-55
1628.2
4414
.1418
82.3
57-76
29.983
23-59
1632.4
4318
.1441
81.9
Mar. io, P.M..
57.56
30.164
23-53
1636.7
4206
1493
82.6
58.12
29.896
23-50
1637.7
4179
1543
84.8
57-30
30.322
23.24
1637-9
4152
1545
84.4
Mar. ii
56.83
29.985
22.87
1 606.6
4876
I3I3
84.2
56.06
30-432
22.86
1608.7
4820
1323
83-9
Mar. 12
57.38
30.334
23.12
1640.8
4066
.1487
79-5
57-77
30.188
23.21
1643.8
3999
1495
78.7
Mar. 13
57-75
30-144
23-52
1640.9
4102 .1443
77-9
58-05
30.031
23-52
1643.1
4048 .1473
78.4
Mar. 16
58-16
30.169
23.07
1653-7
3747
.1518
74.8
Mar. 17
57-42
30.084
22.85
1628.4
4242
.1332
74-3
The following results were obtained using the same experimental
method as with brass surfaces. The shellac surfaces were about two
months old when the first readings were taken.
The results of these experiments furnish independent evidence of the
fact that the viscosity of a gas is independent of the pressure, since foe
turns out to be a constant for a given surface in air. The value of fre
for rough surfaces checks Millikan's theoretically deduced values within
the limit of error of the experiment. The variation in slip for different
surfaces has been checked and the result for fresh shellac, f 76 = 96.8X io~ 7 ,
is very close to the result Lee obtained, fre = 100 X io~ 7 , from the
correction of Stokes' Law for falling shellac drops.
In conclusion the writer wishes to express his indebtedness to Professor
R. A. Millikan who suggested the problem and directed the experimental
COEFFICIENT OF SLIP IN AIR.
249
Data for Old Shellac Surfaces in Air.
d = 200.6 cm; T = 176.00 sec.
5.
t.
e.
r, p ' X I0 7 .
f, X 10
P (mm).
fte X io 7 .
55-39 ....
30.027
23.67
1605.1
4833
.1088
69.2
57.49. . . .
57.06
29.767
30.076
24.28
24-34
1648.5
1653-3
3875
3767
.1338
.1370
68.2
67.9
5749.
30.023
23.11
1661.5
3443
1535 69.5
57.02....
57.20. . . .
30.058
30.123
22.76
22.87
1653.8
1662.5
3639
3431
.1444
.1478
69.1
66.7
57.38-
57-28....
30.023
30.074
22.90
22.95
1661.7
I662.I
3447
3444
.1488 67.5
.1493 67.7
57.64- -
57-72. . . .
29.885
29.930
22.92
22.97
1659.2
1664.4
3507
3392
.1478 68.2
.1495 66.7
57-44....
57-60....
30.144
30.050
23.08
23.11
1668.9
1668.2
3299
3317
1573 68.3
.1585 69.2
56.75 ..
56.90. . . .
30.111
30.112
23.24
23.28
1647.3
1651.1
3830
3744
1351
.1373
68.1
67.6
57-I3--
30.067
23.21
1656.0
3613
1437 68.3
55.19..-.
55.23...-
57.15....
57.27....
30.014
30.087
30.210
30.271
23.06
23.10
23-57
23.66
1597-2
1602.3
1664.0
1670.8
5032
4910
3464
33i6
1057
.1076
.1498
1533
70.0
69-5
68.3
66.9
56.91..-.
57.I7-...
30.211
30.174
23-21
23-31
1657.2
1662.6
3585
3469
.1411 66.6
.1445 66.0
57.36. . . .
57.37...-
29.963
30.019
22.55
22.58
1650.5
1652.8
3823
3793
1353
.1380
68.1
68.6
56.74. . . .
30.300
22.51
1644.5
3780
.1288
64-1
57.33..-.
57.25...-
30.161
30.160
22.87
22.86
1655.3
1652.6
3742
3806
1356
.1364
66.8
68.3
58.38.-.-
57.23..-.
29.508
30-152
21.94
22.12
1647-3
1655.5
3835
3777
1363
-1373
68.8
68.2
Mean . . .
67.7
This result is 2\ per cent higher than the value found for brass surfaces in air.
work ; to the other members of the Physics Department of the University
of Chicago for their interest and assistance throughout the investigation ;
and, in particular, to Dr. E. L. Harrington with whom the author
worked in the first determination of slip by this method.
RYERSON PHYSICAL LABORATORY,
UNIVERSITY OF CHICAGO,
September 22, I922. 1
*The work described in this paper was completed in February 1919.
Pftys
DATE
! ' ca ' Sciences
DUE
M&raiy
Phys.Sci.
> QC189 Harrington, E.
37J
H37
A redetermination of the
coefficient of viscosity
of air.
1 1 III! LI 1 II II 1 II 1
lillH
3 W*JC^ee524221
Gty
Phys . Sci
QC189 Harrington, E
- H37
A redetermination of the
GAYLORD
PR.NTEO.NU.S.A.
coefficient of viscosity
of air
3ATEtOANED| ISSUED TO } DATE DUE
Physical Sciences Library
University of California
Riverside
TIP
eprinted from NATURE, Vol. 136, page 682, October 26, 1935.)
Viscosity of Air and the Electronic Charge
'
THE greatest uncertainty in determining the
electronic charge e by the oil drop method of
Millikan is introduced by the uncertainty in the
assumed value of the coefficient of viscosity of air, TJ.
The value adopted by Millikan in 1917
T] 23 = (1822-6l-2) x 10- 7
is probably too low, and its accuracy overestimated,
as is pointed out by Shiba 1 .
Considering the fundamental importance of the
constant e, I have undertaken a new determination
of v), using the rotating cylinder method also em-
ployed by Millikan and his co-workers 2 > 3 : An inner
cylinder of electron metal, suspended vertically by
a fine phosphor-bronze wire between two guard
cylinders of equal diameter is deviated from its
equilibrium position through an angle 9 by a con-
centric outer cylinder, rotating with constant velocity,
Y) being calculated from the equation
/ (ft 2 - a 2 ). 9 *
73 = 2 a*6* Z T 2 ' where
a = the radius of the inner cylinder = 2-81767 cm.
at 20 ;
6 = the radius of the outer cylinder = 3-26628 cm.
at 20 or = 3-18328 cm. at 20 (two different
cylinders) ;
I = the length of the inner cylinder = 9-9981 cm.
at 20;
t = the time of revolution of the outer cylinder
(20-150 sec.);
T = the period of oscillation of the suspended system
(53 128 sec., using different suspensions) ;
/ = the moment of inertia of the suspended system
about the line of suspension = 423-22 gm.cm 2 .
The mean value of 51 determinations of TJ for dry
air at temperatures between 18-9 and 20-9 is
7] 20 = (1820-0 3-0) x 10- 7 corresponding to
7] 23 = (1834-8 3-0) x 10- 7 .
From this we get
/1834-8\ 3 / 2
e = (ig22^6J x 4 ' 770 x 10- 10 = (4-818 0-012) x
the uncertainty stated being due only to the viscosity,
other sources of error not being considered here.
I am, therefore, of the opinion that the discrepancy