ON THE
MECHANICAL EQUIVALENT OF HEAT,
WITH SUBSIDIARY RESEARCHES ON
THE VARIATION OF THE MERCURIAL FROM THE AIR
THERMOMETER, AND ON THE VARIATION OF
THE SPECIFIC HEAT OF WATER.
BY
HENRY A. ROWLAND,
PROFESSOR OF PHYSIOS IN THE JOHNS HOPKINS UNIVERSITY.
Presented June llth, 1879.
[REPRINTED FROM THE PROCEEDINGS OF THE AMERICAN ACADEMY OF
ARTS AND SCIENCES.]
CAMBRIDGE:
UNIVERSITY PRESS: JOHN WILSON & SON.
1880.
OP ARTS AND SCIENCES.
INVESTIGATIONS ON LIGHT AND HEAT, made and published wholly or in part with
appropriation from the KUMFOUD FUND.
Geol.
75 Lib.
QC
V.
ON THE MECHANICAL EQUIVALENT OF HEAT, WITH SUB-
SIDIARY RESEARCHES ON THE VARIATION OF THE
MERCURIAL FROM THE AIR THERMOMETER, AND ON
THE VARIATION OF THE SPECIFIC HEAT OF WATER.
BY HENRY A. ROWLAND,*
Professor of Physics in the Johns Hopkins University.
Presented June llth, 1879.
CONTENTS.
I. Introductory Remarks ...
. Thermometry ....
(a.) General View of Thermome-
try
(6.) The Mercurial Thermometer 78
(c.) Relation of the Mercurial and
Air Thermometers . . 83
1. General and Historical
Remarks . . 88
2. Description of Apparatus 90
3. Results of Comparison . 97
(d.) Reduction to the Absolute
Scale .... 112
Appendix to Thermometry . . 116
III. Calorimetry ..... 119
(a.) Specific Heat of Water . . 119
(6.) Heat Capacity of the Calo-
rimeter .... 131
IV. Determination of Equivalent . . 137
(a.) Historical Remarks . . 137
1. General Review of Meth-
ods .... 137
2. Results of Best Deter-
minations . . .140
(6.) Description of Apparatus . 165
1. Preliminary Remarks . 155
2. General Description . 157
3. Details . . . .168
(c. ) Theory of the Experiment . 168
1. Estimation of Work done 163
2. Radiation . . . .168
3. Corrections to Thermom-
eters, etc. ... in
(d.) Results 173
1. Constant Data . . 173
2. Experimental Data and
Tables of Results . .174
V. Concluding Remarks, and Criticism
of Results and Methods . . 197
I. INTRODUCTORY REMARKS.
AMONG the more important constants of nature, the ratio of the
heat unit to the unit of mechanical work stands forth prominent, and
is used almost daily by the physicist. Yet, when we come to consider
* This research was originally to have been performed in connection with
Professor Pickering, but the plan was frustrated by the great distance between
our residences. An appropriation for this experiment was made by the Ameri-
can Academy of Arts and Sciences at Boston, from the fund which was insti-
tuted by Count Rumford, and liberal aid was also given by the Trustees of the
Johns Hopkins University, who are desirous, as far as they can, to promote
original scientific investigation.
76 PROCEEDINGS OF THE AMERICAN ACADEMY
the history of the subject carefully, we find that the only experimenter
tho has made the determination with anything like the accuracy
demanded by modern science, and by a method capable of giving
good results, is Joule, whose determination of thirty years ago, con-
firmed by some recent results, to-day stands almost, if not quite, alone
among accurate results on the subject.
But Joule experimented on water of one temperature only, ai
did not reduce his results to the air thermometer; so that we are still
left in doubt, even to the extent of one per cent, as to the value
equivalent on the air thermometer.
The reduction of the mercurial to the air thermometer, and thence
to the absolute scale, has generally been neglected between and
100 by most physicists, though it is known that they differ several
tenths of a degree at the 45 point. In calorimetric researches this
may produce an error of over one, and even approaching two per cent,
especially when a Geissler thermometer is used, which is the worst in
this respect of any that I have experimented on ; and small intervals
on the mercurial thermometers differ among themselves more than
one per cent from the difference of the glass used in them.
Again, as water is necessarily the liquid used in calorimeters, its
variation of specific heat with the temperature is a very important
factor in the determination of the equivalent. Strange as it may
appear, we may be said to know almost nothing about the variation
of the specific heat of water with the temperature between and
100 C.
Regnault experimented only above 100 C. The experiments of /
Him, and of Jamin and Amaury, are absurd, from the amount of
variation which they give. Pfaundler and Plattner confined them-
selves to points between and 13. Munchausen seems to have
made the best experiments, but they must be rejected because he did
not reduce to the air thermometer.
In the present series of researches, I have sought, firstly, a method
of measuring temperatures on the perfect gas thermometer with an
accuracy scarcely hitherto attempted, and to this end have made an
extended study of the deviation of ordinary thermometers from the
air thermometer ; and, secondly, I have sought a method of determin-
ing the mechanical equivalent of heat so accurate, and of so extended
a range, that the variation of the specific heat of water should follow
from the experiments alone.
As to whether or not these have been accomplished, the following
pages will show. The curious result that the specific heat of water
OP ARTS AND SCIENCES. 77
on the air thermometer decreases from to about 30 or 35, after
which it increases, seems to be an entirely unique fact in nature, seeing
that there is apparently no other substance hitherto experimented upon
whose specific heat decreases on rise of temperature without change of
state. From a thermodynamic point of view, however, it is of the
Bame nature as the decrease of specific heat which takes place after
the vaporization of a liquid.
The close agreement of my result at 15.7 C. with the old result of
Joule, after approximately reducing his to the air thermometer and
latitude of Baltimore, and correcting the specific heat of copper, is
very satisfactory to us both, as the difference is not greater than 1 in
400, and is probably less.
I hope at some future time to make a comparison with Joule's
thermometers, when the difference can be accurately stated.
II. THERMOMETRY.
(a.) General View.
The science of thermometry, as ordinarily studied, is based upon
the changes produced in bodies by heat. Among these we may men-
tion change in volume, pressure, state of aggregation, dissociation,
amount and color of light reflected, transmitted, or emitted, hardness,
pyro-electric and thermo-electric properties, electric conductivity or
specific induction capacity, magnetic properties, thermo-dynamic prop-
erties, &c. ; and on each of these may be based a system of ther-
mometry, each one of which is perfect in itself, but which differs from
all the others widely. Indeed, each method may be applied to nearly
all the bodies in nature, and hundreds or thousands of thermometric
scales may be produced, which may be made to agree at two fixed
points, such as the freezing and boiling points of water, but which
will in general differ at nearly, if not all, other points.
But from the way in which the science has advanced, it has come
to pass that all methods of thermometry in general use to the present
time have been reduced to two or three, based respectively on the
apparent expansion of mercury in glass and on the absolute expan-
sion of some gas, and more lately on the second law of thermo-
dynamics.
Each of these systems is perfectly correct in itself, and we have no
right to designate either of them as incorrect. We must decide
a priori on some system, and then express all our results in that
system : the accuracy of science demands that there should be no
78 PROCEEDINGS OF THE AMERICAN ACADEMY
ambiguity on that subject. In deciding among the three systems, we
should be guided by the following rules :
1st. The system should be perfectly definite, so that the same
temperature should be indicated, whatever the thermometer.
2d. The system should lead to the most simple laws in nature. ^
Sir William Thomson's absolute system of thermometry, coinciding
with that based on the expansion of a perfect gas, satisfies these most
nearly. The mercurial thermometer is not definite unless the kind of
glass is given, and even then it may vary according to the way the
bulb is blown. The gas thermometer, unless the kind of gas is given,
is not definite. And, further, if the temperature. as given by either of
these thermometers was introduced into the equations of thermo-
dynamics, the simplest of them would immediately become compli-
cated.
Throughout a small range of temperature, these systems agree
more or less Completely, and it is the habit even with many eminent
physicists to regard them as coincident between the freezing and boil-
ing points of water. We shall see, however, that the difference
between them is of the highest importance in thermometry, especially
where differences of temperature are to be used.
For these reasons I have reduced all my measures to the absolute
system.
The relation between the absolute system and the system based on
the expansion of gases has been determined by Joule and Thomson
in their experiments on the flow of gases through porous plugs
(Philosophical Transactions for 1862, p. 579). Air was one of the
most important substances they experimented upon.
To measure temperature on the absolute scale, we have thus only
to determine the temperature on the air thermometer, and then reduce
to the absolute scale. But as the air thermometer is very inconvenient
to use, it is generally more convenient to use a mercurial thermometer
which has been compared with the air thermometer. Also, for small
changes of temperature the air thermometer is not sufficiently sensi-
tive, and a mercurial thermometer is necessary for interpolation.
1 shall occupy myself first with a careful study of the mercurial
thermometer.
(6.) The Mercurial Thermometer.
Of the two kinds of mercurial thermometers, the weight ther-
mometer is of little importance to our subject. I shall therefore con-
fine myself principally to that form having a graduated stem. For
OF ARTS AND SCIENCES. 79
convenience in use and in calibration, the principal bulb should be
elongated, and another small bulb should be blown at the top. This
latter is also of the utmost importance to the accuracy of the instru-
ment, and is placed there by nearly all makers of standards.* It is
used to place some of the mercury in while calibrating, as well as
when a high temperature is to be measured ; also, the mercury in the
larger bulb can be made free from air-bubbles by its means.
Most standard thermometers are graduated to degrees; but Reg-
nault preferred to have his thermometers graduated to parts of equal
capacity whose value was arbitrary, and others have used a single
millimeter division. 'As thermometers change with age, the last two
methods are the best ; and of the two I prefer the latter where the
highest accuracy is desired, seeing that it leaves less to the maker and
more to the scientist. The cross-section of the tube changes continu-
ously from point to point, and therefore the distribution of marks
on the tube should be continuous, which would involve a change of the
dividing engine for each division. But as the maker divides his tube,
he only changes the length of his divisions every now and then, so as
to average his errors. This gives a sufficiently exact graduation for
large ranges of temperature ; but for small, great errors may be intro-
duced. Where there is an arbitrary scale of millimeters, I believe it
is possible to calibrate the tube so that the errors shall be less than
can be seen with the naked eye, and that the table found shall repre-
sent very exactly the gradual variation of the tube.
In the calibration of my thermometers with the millimetric scale, I
have used several methods, all of which are based upon some graphical
method. The first, which gives all the irregularities of the tube with
great exactness, is as follows.
A portion of the mercury having been put in the upper bulb, so as
to leave the tube free, a column about l;) mm - long is separated off.
This is moved from point to point of the tube, and its length carefully
measured on the dividing engine. It is not generally necessary to
move the column its own length every time, but it may be moved
20"""- or 25 nim- , a record of the position of its centre being kept. To
eliminate any errors of division or of the dividing engine, readings
were then taken on the scale, and the "lengths reduced to their value
in scale divisions. The area of the tube at every point is inversely as
the length of the column. We shall thus have a series of figures
nearly equal to each other, if the tube is good. By subtracting the
* Geissler and Casella omit it, which should condemn their thermometers.
80 PROCEEDINGS OP THE AMERICAN ACADEMY
smallest from each of the others, and plotting the results as ordinates,
with the thermometer scale as abscissas, and drawing a curve through
the points so found, we have means of finding the area at any point.
The curve should not he drawn exactly through the points, but rather
around them, seeing they are the average areas for some distance each
side of the point. With good judgment, the curve can be drawn with
great accuracy. I then draw ordinates every 10 mra> , and estimate the
average area of the tube for that distance, which I set down in a table.
As the lengths are uniform, the volume of the tube to any point is
found by adding up the areas to that point.
But it would be unwise to trust such a method for very long tubes,
seeing the mercury column is so short, and the columns are not end to
end. Hence I use it only as supplementary to one where the column
is about 50 ram - long, and is always moved its own length. This estab-
lishes the volumes to a series of points about 50"' m- apart, and the
other table is only used to interpolate in this one. There seems to be
no practical object in using columns longer than this.
Having finally constructed the arbitrary table of volumes, I then
test it by reading with the eye the length of a long mercury column.
No certain error was thus found at any point of any of the ther-
mometers which I have used in these experiments.
While measuring the column, great care must be taken to preserve
all parts of the tube at a uniform temperature, and only the extreme
ends must be touched with the hands, which should be covered with
cloth.
If V is the volume on this arbitrary scale, the temperature on the
mercurial thermometer is found from the formula T =.O V < ,
where C and f. Q are constants to be determined. If the thermometer
contains the and 100 points, we have simply
G _ 100
- i o v ~ V
Otherwise C is found by comparison with some other thermometer,
which must be of the same kind of glass.
It is to be carefully noted that the temperature on the mercurial
thermometer, as I have defined it, is proportional to the apparent
expansion of mercury as measured on the stem. By defining it as
proportional to the true volume of mercury in the stem, we have to
introduce a correction to ordinary thermometers, as Poggendorf has
shown. As I ouly use the mercurial thermometer to compare with
the air thermometer, and as either definition is equally correct, I will
OF ARTS AND SCIENCES. 81
not further discuss the matter, but will use the first definition, as
being the simplest.
In the above formula I have implicitly assumed that the apparent
expansion is only a function of the temperature ; but in solid bodies
like glass there seems to be a progressive change in the volume as
time advances, and especially after it has been heated. And hence in
mercurial and alcohol thermometers, and probably in general in all
thermometers which depend more or less on the expansion of solid
bodies, we firyi that the reading of the thermometer depends, not only
on its present temperature, but also on that to which it has been sub-
jected within a short time ; so that, on heating a thermometer up to a
certain temperature, it does not stand at the same point as if it had
been cooled from a higher temperature to the given temperature. As
these effects are without doubt due to the glass envelope, we might
greatly diminish them by using thermometers filled with liquids which
expand more than* mercury : there are many of these which expand
six or eight times as much, and so the irregularity might be dimin-
ished in this ratio. But in this case we should find that the correction
for that part of the stem which was outside the vessel whose tem-
perature we were determining would be increased in the same propor-
tion ; and besides, as all the liquids are quite volatile, or at least wet
the glass, there would be an irregularity introduced on that account.
A thermometer with liquid in the bulb and mercury in the stem would
obviate these inconveniences ; but even in this case the stem would
have to be calibrated before the thermometer was made. By a com-
parison with the air-thermometer, a proper formula could be obtained
for finding the temperature.
But I hardly believe that any thermometer superior to the mer-
curial can at present be made, that is, any thermometer within the
same compass as a mercurial thermometer, and I think that the
best result for small ranges of temperature can be obtained with it by
studying and avoiding all its sources of error.
To judge somewhat of the laws of the change of zero within the
limits of temperature which I wished to use, I took thermometer
No. G1G3, which had lain in its case during four months at an average
temperature of about 20 or 25 C., and observed the zero point, after
heating to various temperatures, with the following result. The time
of heating was only a few minutes, and the zero point was taken
immediately after ; some fifteen minutes, however, being necessary for
the thermometer to entirely cool.
VOL. xv. (N. s. vn.) 6
82
PROCEEDINGS OF THE AMERICAN ACADEMY
TABLE I. SHOWING CHANGE OF ZERO POINT.
Temperature
of Bulb
before finding
the Point.
Change of
Point.
Temperature
of Bulb
before finding
the Point.
Change of
Point.
22.5
700
.115
300
.016
81.0
.170
405
.033
90.0
.281
' 51.0
.0:59
100.0
.313
CO.O
.105
100.0
.347
The second 100 reading was taken after boiling for some time.
It is seen that the zero point is always lower after heating, and that
in the limits of the table the lowering of the zero is about propor-
tional to the square of the increase of temperature above 25 C.
This law is not true much above 100, and above a certain tempera-
ture the phenomenon is reversed, and the zero point is higher after
heating ; but for the given range it seems quite exact.
It is not my purpose to make a complete study of this phenomenon
with a view to correcting the thermometer, although this has been
undertaken by others. But we see from the table that the error can-
not exceed certain limits. The range of temperature which I have
used in each experiment is from 20 to 30 C., and the temperature
rarely rose above 40 C. The change of zero in this range only
amounts to 0.03 C.
The exact distribution of the error from this cause throughout the
scale has never been determined, and it affects my results so little that
I have not considered it worth investigating. It seems probable, how-
ever, that the error is distributed throughout the scale. IE it were
uniformly distributed, the value of each division would be less than
before by the ratio of the lowering at zero to the temperature to
which the thermometer was heated.
The maximum errors produced in my thermometers by this cause
would thus amount to 1 in 1300 nearly for the 40 thermometer, and
to about 1 in 2000 for the others. Rather than allow for this, it is
better to allow time for the thermometer to resume its original state.
Only a few observations were made upon the rapidity with which
the zero returned to its original position. After heating to 81, the*
zero returned from OM70 to 0.148 in two hours and a half.
After heating to 100, the zero returned from 0.347 to 110
in nine days, and to -0.022 in one month. Reasoning from this, I
OF ARTS AND SCIENCES. 83
should say that in one week thermometers which had not been heated
above 40 should be ready for use again, the error being then supposed
to be less than 1 in 4000, and this would be partially eliminated by
comparing with the air thermometer at the same intervals as the ther-
mometer is used, or at least heating to 40 one week before comparing
with the air thermometer.
As stated before, when a thermometer is heated to a very high
point, its zero point is raised instead of lowered, and it seems probable
that at some higher point the direction of change is reversed again ;
for, after the instrument comes from the maker, the zero point con-
stantly rises until it may be 0.6 above the mark on the tube. This
gradual change is of no importance in my experiments, as I only
use differences of temperature, and also as it Was almost inappreciable
in my thermometers.
Another source of error in thermometers is that due to the pressure
on the bulb. In determining the freezing point, large errors may be
made, amounting to several hundredths of a degree, by the pressure of
pieces of -ice. In my experiments, the zero point was determined in
ice, and then the thermometer was immersed in the water of the com-
parator at a depth of about G0 cm . The pressure of this water affected
the thermometer to the extent of about 0.01, and a correction was
accordingly made. As differences of temperature were only needed,
no correction was made for variation in pressure of the air.
It does 'not seem to me well to use thermometers with too small a
stem, as I have no doubt that they are subject to much greater irreg-
ularities than those with a coarse bore. For the capillary action
always exerts a pressure on the bulb. Hence, when the mercury rises,
the pressure is due to a rising meniscus which causes greater pressure
than the falling meniscus. Hence, an apparent friction of the mercu-
rial column. Also, the capillary constant of mercury seems to depend
on the electric potential of its surface, which may not be constant,
and would thus cause an irregularity.
My own thermometers did not show any apparent action of this
kind, but Pfaundler and Plattner mention such an'action, though they
give another reason for it.
(c.) Relation of the Mercurial and Air Thermometers.
1. GENERAL AND HISTORICAL REMARKS.
Since the time of Dulong and Petit, many experiments have been
made on the difference between the mercurial and the air thermometer,
84 PROCEEDINGS OF THE AMERICAN ACADEMY
but unfortunately most of them have been at high temperatures. As
weight thermometers have been used by some of the best experi-
menters, I shall commence by proving that the weight thermometer
and stem thermometer give the same temperature ; at the same time,
however, obtaining a convenient formula for the comparison of the air
thermometer with the mercurial.
For the expansion of mercury and of glass the following formulae
must hold :
For mercury, V = V (1 + a t -f J * 2 + &c.) ;
glass, V = V' Q (1 + a t 4- (3 1 2 4- &c.).
In both the weight and stem thermometers we must have V '= V.
where V and F are the volumes of the glass and of the mercury
reduced to zero, and t is the temperature on the air thermometer.
The temperature by the weight thermometer is
P
where P , P t , &c. are the weights of mercury in the bulb at C.,
t C., &c.
Now these weights are directly as the volumes of the mercury at 0.
seeing that V is constant.
.-. T 100
In the stem thermometers we have F , the volume of mercury at 0,
constant, and the volume of the glass that the mercury fills, reduced
to 0, variable. As the volume of the glass V' n is the volume reduced
to 0, it will be proportional to the volume of bulb plus the volume
of the tube as read off on the scale which should be on the tube.
OF ARTS AND SCIENCES. 85
At
.-. r=ioo
which is the same as for the weight thermometer.
If the fixed points are and t* instead of and 100, we can
write
T _ t ,At + Bfl^-Cfl + & c .
r=<{i + (<-<<) [f + fl' + fc+o] +&<>}
As T and t are nearly equal, and as we shall determine the con-
stants experimentally, we may write
t = T a t (t 1 t) (b t) + &c.,
where t is the temperature on the air thermometer, and T that on the
mercurial thermometer, and a and b are constants to be determined for
each thermometer.
The formula might be expanded still further, but I think there are
few cases which it will not represent as it is. Considering b as equal
to 0, a formula is obtained which has been used by others, and from
which some very wrong conclusions have been drawn. In some kinds
of glass there are three points which coincide with the air thermome-
ter, and it requires at least an equation of the third degree to repre-
sent this.
The three points in which the two thermometers coincide are given
by the roots of the equation
t(*-t) (6-0 = 0,
and are, therefore,
In the following discussion of the historical results, I shall take
and 100 as the fixed points. Hence, t' = 100. To obtain a and b,
two observations are needed at some points at a distance from and
100. That we may get some idea of the values of the constants in
the formula for different kinds of glass, I will discuss some of the
experimental results of Regnault and others with this in view.
86 PROCEEDINGS OF THE AMERICAN ACADEMY
Renault's results are embodied, for the most part, in tables given on
p 239 of the first volume of his Relation des Esper^ces. The
Lres given there are obtained from curves drawn to represent he
mean of his experiments, and do not contain any theorefcal results.
The direct application of my formula to his experiments could hardly
be made without immense labor in finding the most probable value
the constants.
But the following seem to satisfy the experiments quite
' Cristal de Choisy-le-Roi i = 0, = -000 000 82.
Verre Ordinaire 6-245, = .000 000 34.
VerreVert * = 270, a = .000 000 095.
Verrede Suede i = +10, a = .000 000 14.
From these values I have calculated the following :
TABLE II. REGNAULT'S RESULTS COMPARED WITH THE FORMULA.
1
Choisy-le-Roi.
Verre Ordinaire.
Verre Vert.
Verre de Suede.
1
|
1
1
1
1
1
1
*5
i
1
|
1
I
3
1
1
5
1
1
5
| .2
1 o
5
i
5
100
o
o
o
120
140
160
180
200
120.12
140 29
160.52
180.80
201 25
120.09
14025
KiO.49
18083
201 .28
+.03
+.04
+.03
-.03
-.03
119.95
139. W 5
159.74
179.63
199.70
119.90
139 80
159.72
179.68
19!l.fi9
+.05
+.05
+.02
.05
+ .01
12007
140.21
16(1.40
180.60
20080
120.09 .01
140 22 ! .01
160.391 +.01
180.62 i .02
200.-S9 .09
120.04 120.04
140.11 ! 140.10
160 20 i 160.21
1MI.33 18034
20 ( 50 ! 200 53
+.01
.01
.01
.03
220
221 KJ
221.86
.04
2l9.HOl219.78
+ .02
221.20 221.23
.03
220 75 ! 220 78
.03
240
242.55
242.50
239.90! 239 96
-.06
241.60 241.63
-.03
241.16
2*1.08
+ .08
200
263 44
203.4H
.02
260.20
2C021
-.01
262.15 262.09
+.07
WO
2H4.48
2S452
-.04
280.58 2SO.O>l
02
282.85 262.63
+.22
300
3?0
3(15.72
::27.25
3115.70
327.20
-.04
-.05
301.08! 301. 12
321.80! 321.80
.04
00
340
349.30
348.88
+.42
434.00
34204
+.36
The formula, as we see from the table, represents all Regnault's
curves with great accuracy, and if we turn to his experimental results
we shall find that the deviation is far within the limits of the experi-
mental errors. The greatest deviation happens at 340, and may be
accounted for by an error in drawing the curve, as there are few ex-
perimental results so high as this, and the formula seems to agree
with them almost as well as Regnault's own curve.
The object of comparing the formula with Regnault's results at
temperatures so much higher than I need, is simply to test the formula
through as great a range of temperatures, and for as many kinds of
Corrected from 280.52 in Regnault's table.
OF ARTS AND SCIENCES.
87
glass, as possible. If it agrees reasonably well throughout a great
range, it will probably be very accurate for a small range, provided
we obtain the constants to represent that small range the best.
Having obtained a formula to represent any 'series of experiments,
we can hardly expect it to hold for points outside our series, or even
for interpolating between experiments too far apart, as, very often, a
small change in one of the constants may affect the part we have not
experimented on in a very marked manner. ' Thus in applying the
formula to points between and 100 the value of b will affect the
result very much. In the case of the glass Choisy-le-Roi many
values of b will satisfy the observations besides 6 = 0. For the
ordinary glass, however, b is well determined, and the formula is of
more value between and 100.
The following table gives the results of the calculation.
TABLE III. REGNAOI/T'S RESULTS COMPARED WITH THE FORMULA.
Calculated
Calculated
Calculated
.
ft = .00000032
a = .00000031
Observed.
= .00000044
Thermome-
ft = 0.
b = 245.
A
6 = 260.
J
ter.
Clioisy-le-Roi.
Verre
Ordinaire.
Verre
Ordinaire.
Verre
Ordinaire.
o
10
10.00
10.07
....
10.10
20
19.99
20.12
....
20 17
. . .
30
29.98
30.15
30.12
+.03
30.21
+.09
40
39.97,
40.17
40.23
-.06
40.23
60
49.96
60.17
50.23
.06
60.23
60
5:>.'.>5
60.15
60.24
.09
60.21
.03
70
09.95
70.12
70.22
.10
70.18
.04
80
79.96
80.09
80.10
.01
80.11
+.01
90
89.97
90.05
90.07
100
100
100
100 '
....
100
Regnault does not seem to have published any experiments on
Choisy-le-Roi glass between and 100, but in the table between
pp. 226, 227, there are some results for ordinary glass. The separate
observations do not seem to have been very good, but by combining
the total number of observations I have found the results given
above. The numbers in the fourth column are found by taking the
mean of Regnault's results for points as near the given temperature
as possible. The agreement is only fair, but we must remember that
the same specimens of glass were not used in this experiment as in the
others, and that for these specimens the agreement is also poor above
100. The values a = .000 000 44 and b = 260 are much better
PROCEEDINGS OP THE AMERICAN ACADEMY
above 100 for the given specimens.
The table seems to show that between and 100 a thermometer of
Choisy-le-Koi almost exactly agrees with the air, thermometer But
this is not at all conclusive. Regnault, however, remarks * that be-
tween and 100 thermometers of this glass agree more nearly with
the air thermometer than those of ordinary glass, though he stat
the difference to amount to .1 to .2 of a degree, the mercurial t
mometer standing below the air thermometer. With the exception of
this remark of Regnault's, no experiments have ever been publi.
in which the direction of the deviation was similar to tins. All (
percenters have found the mercurial thermometer to stand above the
air thermometer between and 100, and my own expenments agree
with this. However, no general rule for all kinds of glass can b<
laid down.
Boscha has given an excellent study of Regnault's results
subject, though I cannot agree with all his conclusions on this subject.
In discussing the difference between and 100 he uses a formula of
the form
and deduces from it the erroneous conclusion that the difference is
greatest at 50 C., instead of between 40 and 50. His results for
T t at 50 are
Choisy-le-Roi ...... .22
Verre Ordinaire ...... -J-.25
VerreVert ....... +.14
Verre de Suede ...... -f- 56
and these are probably somewhat nearly correct, except the negative
value for Choisy-le-Roi.
With the exception of Regnault, very few observers have taken up
this subject. Among these, however, we may mention Recknagel,
who has made the determination for common glass between and
100. I have found approximately the constants for my formula in
this case, and have calculated the values in the fourth column of the
following table.
* Comptes Rendus, Ixix.
OF ARTS AND SCIENCES.
89
TABLE IV. RECKNAGEL'S RESULTS COMPARED WITH THE FORMULA.
Air
Thermometer.
Mercurial Thermometer.
Difference.
Observed.
Calculated.
10
10.08
10.08
20
20.14
20.14
30
30.18
30.18
40
40.20
40.20
60
50.20
50.20
60
60.18
60.18
70
70.14
70.15
+.01
80
80.10
80.11
+.01
90
90.05
90.06
+.01
100
100.00
b = 290
a = .000 000 33
(100 t)(b t)
It will be seen that the values of the constants are not very different
from those which satisfy Regnault's experiments.
There seems to be no doubt, from all the experiments we have now
discussed, that the point of maximum difference is not at 50, but at
some less temperature, as 40 to 45, and this agrees with my own
experiments, and a recent statement by Ellis in the Philosophical
Magazine. And I think the discussion has proved beyond doubt
that the formula is sufficiently accurate to express the difference of
the mercurial and air thermometers throughout at least a range of
200, and hence is probably very accurate for the range of only
100 between and 100.
Hence it is only necessary to find the constants for my thermom-
eters. But before doing this it will be well to see how exact the
comparison must be. As the thermometers are to be used in a
calorimetric research in which differences of temperature enter, the
error of the mercurial compared with the air thermometer will be
which for the constants used in Recknagel's table becomes
Error = ^ 1 = .000 000 33 j 29000. 780 1 -f- 3 1* I .
This amounts to nearly one per cent at 0, and thence decreases to
45, after which it increases again. As only 0.2 at the 40 point
90 PROCEEDINGS OP THE AMERICAN ACADEMY
produces this large error at 0, it follows that an error of only 0.02
at 40 will produce an error of TT ^ T at 0. At other points the
errors will be less.
Hence extreme care must be taken in the comparison and the most
accurate apparatus must be constructed for the purpose.
2. DESCRIPTION or APPARATUS.
The Air Thermometer.
In designing the apparatus, I have have had in view the production
of a uniform temperature combined with ease of reading the ther-
mometers, which must be totally immersed in the water. The uni-
formity, however, needed only to apply to the air thermometer and to
the bulbs of the mercurial thermometer, as a slight variation in the
temperature of the stems is of no consequence. A uniform tempera-
ture for the air thermometer is important, because it must take time
for a mass of air to heat up to a given temperature within U.01 or
less.
Fig. 1 gives a section of the apparatus. This consists of a large
copper vessel, nickel-plated on the outside, with double walls an inch
apart, and made in two parts, so that it could be put together Avater-
tight along the line a b. As seen from the dimensions, it required
about 28 kilogi'ammes v of water to fill it. Inside of this was the vessel
m d efg h k I n, which could be separated along the line c/. In the
upper part of this vessel, a piston, q, worked, and could draw the
water from the vessel. The top was closed by a loose piece of metal,
o p, which fell down and acted as a valve. The bottom of this
inner vessel had a false bottom, c /, above which was a row of large
holes ; above these was a perforated diaphragm, . The bulb of the
air thermometer was at <, with the bulbs of the mercurial thermometers
almost touching it. The air thermometer bulb was very much elon-
gated, being about 18 cm - long and 3 to 5 cra - in diameter. Although
the bulbs of the thermometers were in the inner vessel, the stems
were in the outer one, and the reading was accomplished through the
thick glass window u v.
The change of the temperature was effected by means of a Bunsen
burner under the vessel w.
The working of the apparatus was as follows. The temperature
having been raised to the required point, the piston q was worked to
stir up the water ; this it did by drawing the water through the holes
at cl and the perforated diaphragm s, and thence up through the
OP ARTS AND SCIENCES.
91
apparatus to return on the outside^ When the whole of the water is
at a nearly uniform temperature the stirring is stopped, the valve op
falls into place, and the connection of the water in the outer and
inner vessels is practically closed as far as currents are concerned, and
Fig.2.
before the water inside can cool a little the outer water must have
cooled considerably.
So effective was this arrangement that, although some of the ther-
mometers read to 0.007 C., yet they would remain perfectly station-
ary for several minutes, even when at 40 C. At very high tempera-
tures, such as 80 or 90 C., the burner was kept under the vessel w
all the time, and supplied the loss of the outer vessel by radiation.
The inner vessel would under these circumstances remain at a very
92 PROCEEDINGS OP THE AMERICAN ACADEMY
constant temperature. The water in the outer vessel never.differed
by more than a small fraction of a degree from that in the inner one.
To get the and 100 points the upper parts of the vessel above
the line a b were removed, and ice placed around the bulb of the air
thermometer, and left for several hours, until no further lowering took
place. For the 100 point the copper vessel shown in Fig. 3 was
used. The portion y of this vessel fitted directly over the bulb of the
air thermometer. On boiling water in x, the steam passed through
the tube to the air thermometer. It is with considerable difficulty
that the 100 point is accurately reached, and, unless care be taken,
the bulb will be at a slightly lower temperature. Not only must the
bulb be in the steam, but the walls of the cavity must also be at 100.
To accomplish this in this case, a large mass of cloth was heaped over
the instrument, and then the water in x vigorously boiled for an hour
or so. After fifteen minutes there was generally no perceptible in-
crease of temperature, though an hour was allowed so as to make
certain.
The external appearance of the apparatus is seen in Fig. 2. The
method of measuring the pressure was in some respects similar to that
used in the air thermometer of Jolly, except that the reading was
taken by a cathetometer rather than by a scale on a mirror. .The
capillary stem of the air thermometer leaves the water vessel at a, and
passes to the tube b, which is joined to the three-way cock c. The
lower part of the cock is joined by a rubber tube to another glass tube
at d, which can be raised and lowered to any extent, and has also a fine
adjustment. These tubes were about 1.5 cm - diameter on the inside,
so that there should be little or no error from capillarity. Both tubes
were exactly of the same size, and for a similar reason.
The three-way cock is used to fill the apparatus with dry air, and
also to determine the capacity of the tube above a given mark. In
filling the bulb, the air was pumped out about twenty times, and
allowed to enter through tubes containing chloride of calcium, sul-
phuric acid, and caustic soda, so as to absorb the water and the car-
bonic acid.
The Cathetometer.
The cathetometer was one made by Meyerstein, and was selected
because of the form of slide used. The support was round, and the
telescope was attached to a sleeve which exactly fitted the support.
The greatest error of cathetometers arises from the upright support
not being exactly true, so that the telescope will not remain in level
OP ARTS AND SCIENCES.
93
at all heights. It is true that the level should be constantly adjusted,
but it is also true that an instrument can be made where such an ad-
justment is not necessary. And where time is an element in the
accuracy, such an instrument should be used. In the present case it
was absolutely necessary to read as quickly as possible, so as not to
leave time for' the column to change. In the first place the round
column, when made, was turned in a lathe to nearly its final dimen-
sions. The line joining the centres of the sections must then have
been very accurately straight. In the subsequent fitting some slight
irregularities must have been introduced, but they could not have been
great with good workmanship.* The upright column was fixed, and
the telescope moved around it by a sleeve on the other sleeve. Where
the objects to be measured are not situated at a very wide angle from
each other, this is a good arrangement, and has the advantage that any
side of the column can be turned toward the object, and so, even if it
were crooked, we could yet turn it into such a position as to nearly
eliminate error.
It was used at a distance of about HO"" 1 - from the object, and
no difficulty was found after practice in setting it on the column to
fo mm - at least. The cross hairs made an angle of 45 with the
horizontal, as this was found to be the most sensitive arrangement.
The scale was carefully calibrated, and the relative errors f for the
* The change of level along the portion generally used did not amount to
more than .1 of a division, or about .Ol mm - at the mercury column, as this is
about the smallest quantity which could be observed on the level.
t These amounted to less than .016 mm - at any part.
94 PROCEEDINGS OF THE AMERICAN ACADEMY
denoted by the subfix. Then approximat
As the height of the barometer varies only very slightly during an
experiment, the value of this expression is very nearly
which does not depend on the absolute value of the scale divisions
But the best manner of testing a cathetometer is to take readings
upon an accurate scale placed near the mercury columns to b
measured. I tried this with my instrument, and found that it agree<
with the scale to within two or three one-hundredths of a millimeter,
which was as near as I could read on such an object.
In conclusion, every care was taken to eliminate the errors of this
instrument, as the possibility of such errors was constantly present in
my mind; and it is supposed that the instrumental errors did not
amount to more than one or two one-hundredths of a millimeter on the
mercury column. The proof of this will be shown in the results
obtained.
The Barometer.
This was of the form designed by Fortin, and was made by
James Green of New York. The tube was 2.0 cm diameter nearly
on the outside, and about 1.7 cm - on the inside. The correction for
capillarity is therefore almost inappreciable, especially as, when it
remains constant, it is exactly eliminated from the equation. The
depression for this diameter is about .08 mm -, but depends upon the
height of the meniscus. The height of the meniscus was generally
about l.S 1 ""' ; but according as it was a rising or falling meniscus,
it varied from 1.4 to 1.2 mm -. These are the practical values of the
variation, and would have been greater if the barometer had not been
attached to the wall a little loosely, so as to have a slight motion when
handled. Also in use the instrument was slightly tapped before read-
OF ARTS AND SCIENCES. 95
ing. The variation of the height of the meniscus from 1.2 to 1.4 mm -
would affect the reading only to the extent of .01 to ,02 mm -.
The only case where any correction for capillarity is needed is in
finding the temperatures of the steam at the 100 point, and will then
affect that temperature only to the extent of about 0.005.
The scale of the instrument was very nearly standard at C., and
was on brass.
At the centre of the brass tube which surrounded the barometer, a
thermometer was fixed, the bulb being surrounded by brass, and there-
fore indicating the temperature of the brass tube.
In order that it should also indicate the temperature of the barome-
ter, the whole tube and thermometer were wrapped in cloth until a
thickness of about 5 or 6 cm ' was laid over the tube, a portion being
displaced to read the thermometers. This wrapping of the barometer
was very important, and only poor results were obtained before its
use; and this'is seen from the fact that 1 on the thermometer indi-
cates a correction of ,12 mm- on the barometer, and hence makes a
difference of 0.04 on the air thermometer.
As this is one of the most important sources of error, I have now
devised means of almost entirely eliminating it, and making continual
reading of the barometer unnecessary. This I intend doing by an
artificial atmosphere, consisting of a large vessel of air in ice, and
attached to the open tube of the manometer of the air thermometer.
The Thermometers.
The standard thermometers used in my experiments are given in
the following table.
96
PROCEEDINGS OP THE AMERICAN ACADEMY
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OF ARTS AND SCIENCES. 97
The calibration of the first four thermometers has been described.
The calibration of the Kew standard was almost perfect, and no cor-
rection was thought necessary. The scale divided on the tube was to
half-degrees Fahrenheit; but as the 32 and 212 points were not
correct, it was in practice used as a thermometer with arbitrary
divisions. The interval between the and 100 points, as Welsh
found it, was 180.12, using barometer at 30 inches, or 180.05 as cor-
rected to 760 mm - of mercury.* At the present time it is 179.68,f
showing a change of 1 part in 486 in twenty-five years. This fact
shows that the ordinary method of correcting for change of zero is not
correct, and that the coefficient of expansion of glass changes with
time4
I have not been able to find any reference to the kind of glass used
in this thermometer. But in a report by Mr. Welsh we find a com-
parison, made on March 19, 1852, of some of his thermometers with
two other thermometers, one by Pastre", examined and approved by
Regnault, and the other by Troughton and Simms. The thermometer
which I used was made a little more than a year after this ; and it is
reasonable to suppose that the glass was from the same source as the
standards Nos. 4 and 14 there used. We also know that Regnault
was consulted as to the methods, and that the apparatus for calibration
was obtained under his direction.
I reproduce the table here with some alterations, the principal one
of which is the correction of the Troughton and Simms thermometers,
so as to read correctly at 32 and 212, the calibration being assumed
correct, but the divisions arbitrary.
* Boiling point, Welsh, Aug. 17, 1853, 212.17 ; barometer 30* .
Freezing point, " " " 82.05.
Boiling point, Rowland, June 22, 1878, 2 12.46; barometer 760 mm ..
Freezing point, " " " 32.78.
The freezing point was taken before the boiling point in either case,
t 179.70, as determined again in January, 1879.
t The increase shown here is 1 in 80 nearly ! It is evidently connected with
the change of zero; for when glass has been heated to 100, the mean coefficient
of expansion between and 100 often changes as much, as 1 in 50. Hence it
is not strange that it should change 1 in 80 in twenty-five years. I believe this
fact has been noticed in the case of standards of length.
VOL. xv. (N. s. vn.)
98 PROCEEDINGS OP THE AMERICAN ACADEMY
TABLE VI. COMPARISON BY WELSH, 1852.
Mean of
Kew Standards
Nos. 4 and 14.
Fastre231,
Kegnault.
A
Kew.
Troughton and Simms
(Royal Society).
A
Kew.
32.00
32.00
32.00
38!? 1
45.04
38.72
45.03
+.01
-.01
38.70
45.03
.01
.01
49.96
49.96
.00
49.96
.00
55.34
55.37
+.03
55.34
.00
60.07
60.05
.02
60.06
.01
65.39
65.41
+.02
65.36
.03
69.93
69.95
+.02
69.93
.00
74.69
74.69
.00
74.72
+.03
80.05
80.06
+.01
80.14
+.09
85.30
85.33
+.03
85.44
+.14
90.60
90.51
+.01
90.56
+.06
95.26
101.77
95.24
101.77
.02
.00
95.40
101.94
4-.14
+.15
. 109.16
109.15
.01
109.25
+.08
212.00
212.00
.00
212.00
.00
It is seeruhat the Kew standards and the Fastre agree perfectly, but
that the Troughton and Simms standard stands above the Kew ther-
mometers at 100 F.
The Geissler standard was made by Geissler of Bonn, and its scale
was on a piece of milk glass, enclosed in a tube with the stem. The
calibration was fair, the greatest error being about 0.015 C., at
50 C. ; but no correction for calibration was made, as the instrument
was only used as a check for the other thermometers.
3. RESULTS OF COMPARISON.
Calculation of Air Thermometer.
This has already been described, and it only remains to discuss the
formula and constants, and the accuracy with which the different
quantities must be known.
The well-known formula for the air thermometer is
H-h +
T=
l + at> _
Solving with reference to T, and placing in a more convenient form,
we have
H-h'
nearly,
OF ARTS AND SCIENCES.
99
where
and
For the first bulb,
For the second bulb,
= .0058.
To discuss the error of T due to errors in the constants, we must
replace a by its experimental value, seeing that it was determined
with the same apparatus as that by which T was found. As it does
not change very much, we may write approximately
T= 100
H h
V l + yt
From this formula we can obtain by differentiation the error in
each of the quantities, which would make an error of one tenth of
one per cent in T. The values are for T = 40 nearly ; t = 20 ;
ff lw h = 270""" ; and h = 750 mm -. If x is the variable,
A x =
dx
dT
' *JL _L_ - ru d JL
d T looo - u * d T
TABLE VII. ERRORS PRODUCING AN ERROR IN T OF 1 IN 1000 AT 40 C.
H
H 100 orA.
V
T
ha
a
- constant,
a
b
a
^5 constant.
1,90
k^/con*.
b ino -b
a
-^ constant.
Absolute
ralue,
Ax
n mm.
27 mm.
.005
.00074
.00087
.0047
.00087
Relative
value,
Az
X
...
...
0.9
.10
.12
.62
...
From this table it would seem that there should be no difficulty in
determining the 40 point on the air thermometer to at least 1 in 2000 ;
and experience has justified this result. The principal difficulty is in
the determination of H, seeing that this includes errors in reading the
barometer as well as the cathetometer. For this reason, as men-
tioned before, I have designed another instrument for future use, in
which the barometer is nearly dispensed with by use of an artificial
atmosphere of constant pressure.
The value of ^ does not seem to affect the result to any great
extent; and if it was omitted altogether, the error would be only
100
PROCEEDINGS OF THE AMERICAN ACADEMY
about 1 in 1,000, assuming that the temperature t was the same at the
determination of the zero point, the 40 point, and the 100 point.
It seldom varied much.
The coefficient of expansion of the glass influences the result very
slightly, especially if we know the difference of the mean coefficients
between and 100, and say 10 and -f 10. This difference I at
first determined from Regnault's tables, but afterwards made a deter-
mination of it, and have applied the correction.*
The table given by Regnault is for one specimen of glass only ; and
I sought to better it by taking the expansion at 100 from the mean
of the five specimens given by Regnault on p. 231 of the first volume
of his Relation des Experiences, and reducing ' the numbers on
page 237 in the same proportion. I thus found the values given in
the second column of the following table.
TABLE VIII. COEFFICIENT OF EXPANSION OF THE GLASS OF THE AIR
THERMOMETER, ACCORDING TO THE AIR THERMOMETER.
Tempera-
ture ac-
cording to
Air Ther-
mometer.
Values of 6
used for a first
Calculation
b from
Regnault's
Table,
Glass No. 5.
Experimental Results.
Apparent
Coefficient of
Expansion of
Mercury.
b, using
Regnault's
Value for
Mercury .t
6, using
Recknagel'g
Value for
Mercury 4
6, using
Wullnel's
Value for
Mercury .
20
40
60
80
100
.0000252
.0000253
.0000256
.0000259
.0000262
.0000264
.0000263
.0000264
.0000267
.0000270
.0000273
.0000276
.00015410
.00015395
.00015391
.00015381
.0000254
.0000258
.0000261
.0000277
.0000264
.0000266
.0000267
.0000277
.0000273
.0000276
.0000278
.0000287
The second column contains the values which I have used, and one
of the last three columns contains my experimental results, the last
being probably the best. The errors by the use of the second column
compared with the last are as follows :
TTT ^ from using b m 6 40 = .0000008 instead of .0000011 ;
TuW from using b m = .0000264 instead of .0000287 ;
or, ffa-Q for both together.
* This was determined by means of a large weight thermometer in which
the mercury had been carefully boiled. The glass was from the same tube as
that of the air thermometer, and they were cut from it within a few inches of
each other.
t Relation des Experiences, i. 328.
t Pogg. Ann., cxxiii. 135. Experimental Physik, Wullner, i. 67.
OP ARTS AND SCIENCES.
101
As the error is so small, I have not thought it worth while to entirely
recalculate the tables, but have calculated a table of corrections as
follows, and have so corrected them :
TABLE IX. TABLE OP CORRECTIONS.
Tf
T
Calculated
Corrected
Correction.
Temperature.
Temperature.
8
10
9.9971
.0029
20
19.9946
.0054
30
29.9924
0076
40
39.9907
.0093
50
49.9894
.0106
60
69.9865
.0135
80
79.9880
.0120
100
100.
T = T' {1 + 373 (b' m - b m ) - (273 + T) (V - &)}.
T T 1 {1 .000858 -f (273 -f T'} (b b')}.
T = .99975 T' approximately between and 40. This last is
true within less than T ^ 7 of a degree.
The two bulbs of the air thermometer used were from the same
piece of glass tubing, and consequently had nearly, if not quite, the
same coefficient of expansion.
In the reduction of the barometer and other mercurial columns to
zero, the coefficient .000162 was used, seeing that all the scales were
of brass.
In the tables the readings of the thermometers are reduced to
volumes of the tube from the tables of calibration, and they are cor-
rected for the pressure of water, which increased their reading,
except at 0, by about 0.01 C.
The order of the readings was as follows in each observation :
1st, barometer; 2d, cathetometer; 3d, thermometers forward and
backward ; 4th, cathetometer ; 5th, barometer, &c., repeating the
same once or twice at each temperature. In the later observations,
two series like the above were taken, and the water stirred between
them.
The following results were obtained at various times for the value
of a with the first bulb :
102
PROCEEDINGS OP THE AMERICAN ACADEMY
Mean
.0036664
.0036670
.0036658
.0036664
.0036676
.00366664
obtained by using the coefficient of expansion of glass .0000264 at
100 or a = 0036698, using the coefficient .0000287.
The thermometers Nos. 6163, 6165, 6166, were always taken out
of the bath when the temperature of 40 was reached, except on
November 14, when they remained in throughout the whole experi-
The thermometer readings are reduced to volumes by the tables of
calibration.
TABLE X. IST SERIES, Nov. 14, 1877.
Relative
Weight.
Air
Thermometer.
V
6163.
V
6166.
7
6167.
Temperature
by 6167.
A
4
o
115.33
21.25
6.147
4
17.1425
422.84
255.80
15.685
17.661
.236
4
23.793
634.71
341.05
19.157
24.089
.296
5
30.582
653.49
431.71
22.833
80.896
.814
2
38.569
793.18
27.175
38.935
.366
2
61.040
33.864
61.320
.280
4
59.137
38.256
69.452
.816
The first four series, Tables X. to XIII., were made with one bulb
to the air thermometer. A new bulb was now made, whose capacity
was 192.0 ccm -, that of the old being 201.98 c - cm -. The value of -p for
the new bulb was .0058. The values of h' and a were obtained as
follows :
a
hf
June 8th
.00366790
753.876
June 22d
.00366977
753.805
June 25th
.00366779
753.837
Mean
.0036685
753.84
This value of a is calculated with the old coefficient for glass,
new would have given .0036717.
The
OF ARTS AND SCIENCES.
103
W
53 5 gg 8
Jf
fc-|7 '"*'*'*^-* 1 <1 ;i *'*'*T>50Ou3o
Jih
ooioo
s^ i as ^ s s ^ s
104 ' PROCEEDINGS OF THE AMERICAN ACADEMY
fi-
J
^
iii
s s s s
T* t-- t 1 * CO
OS 00 -H rH
g 35 3
f S S5
a? s
CO CO ^tl
00 g O W N
c$ t- as i-i
S 8
oo ec oj
to to to
S S
OF ARTS AND SCIENCES.
105
*> it-
Difference
Column
reduced
~ 8 S
i
s s s
10 cd ca t^
S S
& &
If 5
S 2 -N
S S
3 g
8 S
5! c
*Q CO CC
^ CO l~ t^ O
00 O r-!
2 3 $
co rh co
S S 58
!o lo S
11
ft
!J
S at
11
P CO
cc cc -^ 10
106 PROCEEDINGS OF THE AMERICAN ACADEMY
5 q q
i + I
o o
5
So o co 3" e
S 3 TP eo to
&
S
S 5
ll
S S
r- O
CO t^-
Nil-
Ili
ll 2
8 S I .8 I
-f S ^ S S?
x
8 9 S
2 5! S
S S 8 8
o> oo 10 -j(
O CD
5
*-l (N (N (M
OP ARTS AND SCIENCES.
107
+ + + + + + I" I
t-< (N t-.
^ S So
? S S
co Oi I-* I-H o i>- < 100
T*t-COiOiOOOcO
S S ^ S S g
S I S 88
I S 3 I S g s
ji
fe o'
i
|U
I
J_I_
f: S S
5 S <p
1 S;
l-sl
Jt^ O ?O ^O CO CD ^?
S5 Sg S 9 ^ ri
I S I 1 I 1
108 PROCEEDINGS OF THE AMERICAN ACADEMY
It now remains to determine from these experiments the most
probable values of the constants in the formula, comparing the air
with the mercurial thermometer. The formula is, as we have i
t=T-at(t'-t) (6-0:
but I have generally used it in the following form :
t=CV-t - m (100-0 (1-n (100 + 0),
t = C' V t' - m t (40 - n ( 40 + '))'
And the following relations hold among the constants :
_ Q, (i + m (60 8400 n)), nearly,
a = mn,
I = JL 100,
T GVt M
In these formulae t is the temperature on the air thermometer ; Vis
the volume of the stem of the mercurial thermometer, as determined
from the calibration and measured from any arbitrary point; and
C", f' , m, and n are constants to be determined.
The best way of finding these is by the method of least squares.
C" must be found very exactly ; t a is only to be eliminated from the
equations ; m must be found within say ten per cent, and n need only
be determined roughly. To find them only within these limits is a
very difficult matter.
* Determination of n.
As this constant needs a wide range of temperatures to produce
much effect, it can only be determined from thermometer No. 6167,
which was of the same glass as 61 63, 61 65, and 6166. It is unfortunate
that it was broken on November 2 1 , and so we only have the experi-
ments of the first and second series. From these I have found
n = .003 nearly. This makes b = 233*, which is not very far from
the values found before from experiments above 100 by Regnault
on ordinary glass.*
* Some experiments with Baudin thermometers at high temperatures have
given me about 240, a remarkable agreement, as the point must be uncertain
to 10 or more.
OP ARTS AND SCIENCES. 109
Determination of C and m.
I shall first discuss the determination of these for thermometers
Nos. 6163, 6165, and 6166, as these were the principal ones used.
As No. 61 63 extended from to 40, and the others only from
to 30, it was thought best to determine the constants for this one
first, and then find those for 6165 and 6166 by comparison. As this
comparison is deduced from the same experiments as those from which
we determine the constants of 6163, very nearly the same result is
found as if we obtained the constants directly by comparison with the
air thermometer.
The constants of 6163 can be found either by comparison with
6167, or by direct comparison with the air thermometer. I shall first
determine the constants for No. 6167.
The constants C and t for this thermometer were found directly
by observation of the and 100 points; and we might assume these,
and so seek only for m. In other words, we might seek only to
express the difference of the thermometers from the air thermometer .
by a formula. But this is evidently incorrect, seeing that we thus
give an infinite weight to the observations at the and 100 points.
The true way is obviously to form an equation for each temperature,
giving each its proper weight. Thus from the first series we find for
No. 6167,
Weight. Equations of condition .
4 == 6.147 C 1 ,
4 17.427 = 15.685 C t 930 m,
4 23.793 = 19.157 0t 1140 m,
&c. &c. &c.
5 100 = 60.156 C t ,
which can be solved by the method of least squares. As t is un-
important, we simply eliminate it from the equations. I have thus
found,
Weight.
1 Nov. 14 (7=1.85171 m = .000217
2 Nov. 20, 21 G= 1.85127 m = .000172
Mean 0= 1.85142 m = .000187
The difference in the values of m is due to the observations not being
so good as were afterwards obtained. However, the difference only
signifies about 0.03 difference from the mean at the 50 point. After
November 20 the errors are seldom half of this, on account of the
greater experience gained in observation.
110 PROCEEDINGS OF THE AMERICAN ACADEMY
The ratio of C for 6167 and 6163 is found in the same way.
Weight.
1 Nov. 14 .0310091
2 Nov. 20 .0309846
Mean .0309928
Hence for 6163 we have in this way
G = .057381 C' = .056995 m = .000187.
By direct comparison of No. 6163 with the air thermometer, we
find the following.
Date. Weight <X. m.
Nov. 14 1 .056920 .000239
Nov. 20 2 .056985 .000166
Jan. 25 3 .056986 .000226
Feb. 11 4 .056997 .000155
June 8 3 .056961 .000071
June 22 2 .056959 .000115
Mean .056976 .000004 .000154 .000010
The values of C' agree with each other with great exactness, and
the probable error is only O.003 C. at the 40 point.
The great differences in the values of m, when we estimate exactly
what they mean in degrees, also show great exactness in the experi-
ments. The mean value of m indicates a difference of only 0.05
between the mercurial and air thermometer at the 20 point, the
and 40 points coinciding. The probable error of m in degrees is
only 0.003 C.
There is one more method of finding m from these experiments ;
and that is by comparing the values of G' with No. 6167, the glass of
6167 being supposed to be the same as that of 6163.
We have the formula
C= G> (1 +34.8 m).
Hence m = ~^ .
We thus obtain the following results :
Date. Weight. Value of m.
Nov. 14 1 .000236
Nov. 20 2 .000218
Jan. 25 3 .000217
Feb. 11 4 .000197
June 8 3 .000215
June 22 2 .000216
Mean .000213
OF ARTS AND SCIENCES. Ill
The results for m are then as follows :
From direct comparison of No. 6167 with the air thermometer .000187
From " " " No. 6163 " " .000154
From comparison of No. 6163 with No. 6167 .000213
The first and last are undoubtedly the most exact numerically, but
they apply to No. 6167, and are also, especially the first, derived
from somewhat higher temperatures than the 20 point, where the
correction is the most important. The value of m, as determined in
either of these ways, depends upon the determination of a difference of
temperature amounting to 0.30, and hence should be quite exact.
The value of m, as obtained from the direct comparison of No. 6163
with the air thermometer, depends upon the determination of a differ-
ence of about 0.05 between the mercurial and the air thermometer.
At the same time, the comparison is direct, the temperatures are the
same as we wish to use, and the glass is the same. I have combined
the results as follows :
m from No. 6167 .000200
m " 6163 .000154
Mean .00018*
It now remains to deduce from the tables the ratios of the constants
for the different thermometers.
The proper method of forming the equations of condition are as
follows, applying the method to the first series:
Weight.
4 21.25 C tll = 115.33 C, v
4 255.80 C UI = 422.84 C, v
4 341.05 C in = 534.71 G t v
5 431.71 C UI = 653.49 C, v
where G UI is the constant for No. 6166, C, is that for No. 6163,
and v is a constant to be eliminated. Dividing by (7/, the equations
S~i
can be solved for --. The following table gives the results.
* See Appendix to Thermometry, where it is finally thought best to reject
the value from No. 6167 altogether.
112
PROCEEDINGS OP THE AMERICAN ACADEMY
TABLE XVI. RATIOS OF CONSTANTS.
6163
6166
6166
6165
6165
Date.
Weight.
6167
6167
6163
6163
6166
Nov. 14
1
.031009
.040058
1.3111
Nov. 20
2
.030985
.040670
1.3128
Jan. 25
3
....
....
1.3122
Feb. 11
4
1.3115
8.6588
6.1449
June 8
3
....
1.3108
8.0605
6.1469
June 22
2
1.3122
8.0588
6.1428
Mean j
.030993
'.00005
.040666
.000003
1.31175
.0004
8.0594
.0002
6.1451
.0004
From these we have the following, as the final most probable
results :
' . C n =8.0601 G t ,
. C UI = 1.31175 (7,,
C, = .031003 (7 iT ,
C,, .24991 C iv ,
C UI = .040661 <7 iv ,
of which the last three are only used to calculate the temperatures on
the mercurial thermometer, and hence are of little importance in the
remainder of this paper.
The value of C" which we have found for the old value of the
coefficient of expansion of glass was
6^ = . 056976;
and hence, corrected to the new coefficient, it is, as I have shown,
C, =.056962.
Hence,
<7 ;/ = .45912,
<?, = .074720.
And we have finally the three following equations to reduce the ther-
mometers to temperatures on the air thermometer:
Thermometer No. 6163:
T=. 056962 V-t '-.miST(4Q-T) (l - .003 (7*+ 40)).
Thermometer No. 6165 :
^=.45912 V"- C __. 00018 T(T- 40) (l _ .003 (7+40)).
OF ARTS AND SCIENCES. 113
Thermometer No. 6166 :
= . 074720 r'"*,/" .00018 T(T 40) (l .
where V, V", and V" are the volumes of the tube obtained by
calibration ; t ', t ", and t '" are constants depending on the zero point,
and of little importance where a difference of temperature is to be
measured ; and T is the temperature on the air thermometer.
On the mercurial thermometer, using the and 100 points as fixed,
we have the following by comparison with No. 6167 :
Thermometer No. 6163 ; t = .057400 V t ;
Thermometer No. 6165 ; t = .46265 V t Q ;
Thermometer No. 6166 ; t = .4)75281 V t v
The Kew Standard.
The Kew standard must be treated separately from the above, as
the glass is not the same. This thermometer has been treated as if
its scale was arbitrary.
In order to have variety, I have merely plotted all the results
with this thermometer, including those given in the Appendix, and
drawn a curve through them. Owing to the thermometer being- only
divided to F., the readings could not be taken with great accuracy,
and so the results are not very accordant ; but I have done the best I
could, and the result probably represents the correction to at least
0.02 or 0.03 at every point.
('/) Reduction to the Absolute Scale.
The correction to the air thermometer to reduce to the absolute
scale has been given by Joule and Thomson, in the Philosophical
Transactions for 1854 ; but as the formula there used is not correct,
I have recalculated a table from the new formula used by them in
their paper of 1862.
That equation, which originated with Rankine, can be placed in
the form &i/Vl~ '
where p, v, and p. are the pressure, volume, and absolute temperature
of a given weight of the air ; D is its density referred to air at C.
and 760 mm - pressure ; fj, is the absolute temperature of the freezing
point; and m is a constant which for air is 0.33 C.
VOL. xv. (N. s. vn.) 8
14 PROCEEDINGS OP THE AMERICAN ACADEMY
For the air thermometer with constant volume
T = 100 pt ~ Po -,
PlGO PO
or, since D = 1,
,,- ^ = T- . 00088 T =,
from which I have calculated the following table of corrections.
TABLE XVII. REDUCTION OF AIR THERMOMETER TO ABSOLUTE SCALE.
T
Air Thermometer.
A* Mo
Absolute Temperature.
A
or Correction to
Air Thermometer.
10
9.9972
.0028
20
19.9952
.0048
30
29.9939
.0061
40
39.9933
.0067
50
49.9932
.0068
60
. 59.9937
.0063
70
69.9946
.0054
80
79.9956
.0044
90
89.9978
.0022
100
100.000
200
200.037
+.037
300
300.092
+.092
400
400.157
+.157
500
500.228
+.228
It is a curious circumstance, that the point of maximum difference
occurs at about the same point as in the comparison of the mercurial
and air thermometers.
From the previous formula, and from this table of corrections, the
following tables were constructed.
^ 7 '
OF ARTS AND SCIENCES.
115
TABLE XVIII. THERMOMETKB No. 6163.
= Q
11
i
li^
111
H
i!
II
If!
ill!
ll
iS
1 Reading in
meters on
Temperature
curiiil Them
and 100
Temperature
curial Them
and 40 fi
Air Thermc
1
Temperate
Absolute Set
0C
1 Reading in
meters on
Temperature
curial Them
and 100
Temperature
curial Them
and 40 fi
Air Thermc
II
Temperatu
Absolute Sea
0C.
50
.923
.917
-.911
.911
240
20.557
20.409
20.350
20.345
58.1
250
21.670
21.515
21.457
21.452
60
+.217
+.215
+.214
+.214
260
22.776
22.616
22.559
22.564
70
1.356
1.336
1.328
1.328
270
23.884
23.713
23.657
23.652
80
2.494
2.475
2.461
2.460
280
24.989
24.810
24.755
24.750
90
3.631
3.604
3.584
3.583
290
26.093
25.907
25.854
25.848
100
4.767
4.733
4.707
4.706
300
27.200
27.006
26.956
26.950
110
5.903
5.860
5.829
5.827
310
28.311
28.108
28.060
28.056
120
7.036
6.986
6.950
6.948
320
29.425
29.214
29.169
29.163
130
8.170
8.111
8.071
8.069
330
30.541
30.324
30.282
30.276
140
9.304
9.237
9.193
9.190
340
31.662
31.436
31.398
31.392
150
10.436
10.361
10.314
10.311
350
32.782
32.548
32.514
32.508
160
11.568
11.485
11.435
11.432
360
33.903
33.660
33.630
33.624
170
12.700
12.608
12.556
12.553
370
35.023
34.773
34.748
34.742
180
13.829
13.730
13.676
13.672
380
36.143
35.884
35.864
35.857
190
14.957
14.850
14.794
14.790
390
37.261
36.994
36.979
36.972
200
16.081
15.906
15.909
15.905
400
38.377
38.103
38.094
38.087
210
17.203
17.080
17.022
17.018
410
39.492
39.210
39.206
39.199
220
18322
18.191
18.132
18.127
420
40.604
40.314
40.316
40.309
230
19.440
19.301
19.242
19.237
TABLE XIX. THERMOMETER No. 6165.
s
1
!fi
lilt
|
si
B-3
21
ifj
11!
|
si
EJ
.S o
sPa,
5-S2
2
SSd
a -
So
g Q tea 2
3
3 d
ii
Temperatu
curial Thei
and 101
Temperatu
curial Thei
and 40
Air Then
1
Tempera
Absolute S
bo _
-
Temperatu
curial Thei
and 1(K
Temperatu
curial Thei
and 40
Air Then
II
-
po
'!
30
-.464
-.460
-.457
-457
230
17.198
17.067
17.009
17.005
35
240
18.056
17.920
17.861
17.857
40
+.463
+.460
+.457
+.457
250
18.917
18.773
18.714
18.709
50
1.387
1.376
1.368
1.368
260
19.771
19.621
19.562
19.557
60
2.307
2.290
2.276
2.275
270
20.621
20.465
20.406
20.401
70
3.216
3.192
3.174
3.173
280
21.469
21.306
21.247
21.242
80
4.122
4.092
4.069
4.068
290
22.308
22.139
22.081
22.076
90
5.022
4.984
4.957
4.955
300
23.144
22.969
22.912
22.907
100
5.916
5.872
5.841
5.839
310
23.974
23.792
23.736
23.731
110
6.804
6.753
6.714
6.712
320
24.796
24.607
24.552
24.547
120
7.685
7.628
7.590
7.588
330
25.618
25.424 1 25.370
25.365
130
8.564
8.500
8.459
8.456
340
26.433
26.232
46.180
26.174
140
9.439
9.368
9.324
9.321
350
27.245
27.038
16.987
26.981
150
10.309
10.232
10.186
10.183
360
28.049
27.837
27.788
27.782
160
11.174
11.091
11.042
11.039
370
28.856
28.637
28.590
28.584
170
12.038
11.947
11.896
11.893
380
29.651
29.426
29.382
29.376
180
12.900
12.802
12.749
12.746
390
30.449
30.218
30.176
30.170
190
13.760
13.655
13.601
13.598
400
31.249
31.011
30.971
30.965
200
14.619
14.508
14.453
14.450
410
32.073
31.829
31.782
31.786
210
15.479
15.362
15.305
15.302
420
32.861
32.611
32.577
32.581
220
16.340
16.215
16.157
16.153
116
PROCEEDINGS OF THE AMERICAN ACADEMY
TABLE XX. THERMOMETER No. 6166.
1*
111
ifi
1
?. a
2
il
ill
If*
13
1
11
*tfl
H
Ji
Temperature <
curial Therm(
and 100 i
Temperature <
curial Therm <
and 40 f
Temperatui
Air Thermoi
Temperatu
Absolute Sea
0C.
2 'ia
J?B
Temperature
curial Therm
and 100
Temperature
curial Therm
and 40
1
Temperatu
Absolute Sot
0C.
20
.036
-.036
.034
-.034
230
16.478
16.356
16.298
16.294
30
40
+.770
1.574
+.764
1.562
+.759
1.553
+.759
1.553
240
250
17.259
18.042
17.132
17.908
17.074
17.849
17.070
17.845
50
2.368
2.350
2.336
2.335
260
18.825
18.686
18.627
18.622
60
3.156
3.133
3.115
3.114
270
19.609
19.464
19.405
19.400
70
3.941
3.911
3.889
3.888
280
20.392
20.241
20.182
20.177
80
4.726
4.691
4.665
4.664
290
21.170
21019
20.960
20.955
90
5.509
5.468
5.438
5.436
300
21.735
21.793
21.735
21.730
100
6.293
6.246
6.212
6.210
310
22.511
22.569
22.511
22.506
110
7.076
7024
6.988
6.986
320
23.292
23.349
23.292
23.287
120
7.862
7.804
7.765
7.763
330
24.075
24.131
24.075
24.070
130
8.649
8.585
8.544
8.542
340
24.855
24.910
24.855
24.850
140
9.437
9.367
9.323
9.321
350
25.634
25.687
25634
25.628
150
10.228
10.151
10.105
10.102
360
26.415
26.466
26.412
26.406
160
11017
10.935
10.887
10.884
370
27.441
27.245
27.195
27.189
170
11.805
11.717
11.667
11.664
380
28.240
28.030
27.982
27.976
180
12.589
12.496
12.444
12.441
390
29.030
28.814
28.768
28.762
190
13.370
13.271
13.217
13.214
400
29.819
29.597
29.550
29.544
200
14.148
14.043
13.988
13.984
410
30.608
30.381
30.339
30.333
210
14.923
14.812
14.756
14.752
420
31.396
31.162
31.123
31.117
220
15.699
15.583
15.526
15.522
430
32.189
31.950
31.914
31.908
In using these tables a correction is of course to be made should
the zero point change.
TABLE XXI. CORRECTION OF KEW STANDARD TO THE ABSOLUTE SCALE.
Temperature C.
Correction in
Degrees C.
10
.03
20
.05
30
.06
40 .07
50
.07
60
.06
70
-.04
80
.02
90
.01
100
OF ARTS AND SCIENCES.
117
Appendix to Thermometry.
The last of January, 1879, Mr. S. W. Holraan, of the Massachusetts
Institute of Technology, came to Baltimore to compare some ther-
mometers with the air thermometer ; and by his kindness I will give
here the results of the comparison which we then made together.
As in this comparison some thermometers made by Fastre in 1851
were used, the results are of the greatest interest.
'The tables are calculated with the newest value for the coefficient
of expansion of glass. The calibration of all the thermometers, except
the two by Casella, has been examined, and found good. The Casella
thermometers had no reservoir at the top, and could not thus be readily
calibrated after being made. The Geissler also had none, but I suc-
ceeded in separating a column.
The absence of a reservoir at the top should immediately condemn
a standard, for there is no certainty in the work done with it.
TABLE XXII. SEVENTH SERIES.
Original Readings.
Reduced Readings.
Air
6163
Ther-
Kew
Reduced
Kew
mome-
ter.
6163.
7334
Baudin.
Stand-
ard
32374
Casella.
Geissler.
to Air
Ther-
7334
Baudin.
Stand-
ard
32374
Casella.
Geissler.
No. 104.
mome-
No.104.
ter.
*58.83
.11
32.68
+.20
+.69
6
a
t.43
63.5
3360
.71
.52
.52
.51
6.08
113.0
43.65
6.33
6.08
6.11
6.13
12.68
171.55
12.59
55.47
12.91
13.42
12.65
12.73
12.68
1270
12.82
20.49
242.0
20.48
69.56
20.77
21.29
20.49
20.63
20.57
20.56
20.74
24.55
278.8
24.50
76.90
24.80
25.33
24.54
24.66
24.61
24.59
24.81
29.51
323.9
29.49
85.88
29.80
30.32
29.52
29.66
29.61
29.58
29.83
39.45
4131
39.43
103.72
39.76
40.22
39.47
39.62
39.53
39.54
39.80
39.15
4107
39.15
103.23
39.48
39.08
39.20
39.34
39.26
39.26
39.56
51.17
51.10
124.84
51.49
51.83
....
51:32
51.29
51.26
51.49
61.12
61.05
142.73
61.47
61.69
61.29
61.24
61.23
61.41
70.74
' |
70.57
159.87
71.00
71.14
.....
70.83
70.78
70.76
70.92
80.09
79.74
176.50
80.31
80.25
80.02
80.04
80.06
80.10
80.39
80.15
177.23
80.74
.80.66
.
80.43
80.44
80.49
80.51
89.95
89.63
194.35
90.22
90.11
89.93
89.97
89.97
90.03
89.92
89.59
194.22
90.18
90.06
.
89.89
89.90
89.93
89.98
100.00
99.69
212.37
100.06
99.32
. . .
100.00
100.00
100.00
100.00
* The original readings in ice were 58.68 and 58 45, to which .15 was added
to allow for the pressure of water in the comparator. This, of course, gives
the same final result as if .15 were subtracted from eacli of the other tempera-
tures. No correction was made to the others.
t Probably some error of reading.
118
PROCEEDINGS OP THE AMERICAN- ACADEMY
TABLE XXIII EIGHTH SERIES.
Original Readings.
Reduced Readings.
Air
Ther-
mome-
6163
376
7316
Baudin.
368
Fastr<5.
3236
Casella.
6163
Reduced
to Air
Ther-
376
Fastrd.
7316
Baudin
868.
Fastr<5.
3235
Casella.
ter.
Fastre.
mome-
ter.
*5860
111.3
.23
87.6
32.80
5
ft
3.67
90.7
130.0
106.25
39.35
3.61
3.64
3.64
3.65
11.55
161.6
170.9
11.40
147.2
53.70
11.56
11.60
il.64
11.62
11. Q3
20.72
243.7
217.9
20.59
194.2
70.15
20.70
20.75
20.84
20.80
20.79
32.19
347.4
276.9
32.09
253.2
90.80
32.17
32.24
32.34
32.28
32.29
39.36
411.85
313.85
39.26
290.1
103.68
39.36
39.43
39.52
39.48
39.45
50.71
372.0
50.57
248.2
123.65
i
50.75
50.84
50.80
50.57
60.10
. '.
420.0
59.92
39(5.45
140.80
.
60.10
60.19
60.21
60.12
73.82
490.6
73.59
466.85
165.68
73.84
73.87
73.93
73.97
86.50
655.25
86.16
531.22
188.20
.
86.48
86.51
86.56
86.56
550.2
85.21
525.95
186.42
86.45
85.60
85.45
85.51
100.66
624.93
99.70
600.58
212.45
. . .
100.00
100.00
100.00
100.00
From these tables we would draw the inference that No. 6163
represents the air thermometer with considerable accuracy. At the
same time, both tables would give a smaller value of m than I have
used, and not very far from the value found before by direct compari-
son, namely, .00015.
The difference from using m = .00018 would be a little over
0.01 C. at the 20 point.
All the other thermometers stand above the air thermometer,
between and 100, by amounts ranging between about 0.05 and
0.35 C., none standing below. Indeed, no table has ever been
published showing any thermometer standing below the air ther-
mometer between and 100. By inference from experiments above
100 on crystal glass by Regnault, thermometers of this glass should
stand below, but it never seems to have been proved by direct experi-
ment. The Fastre thermometers are probably made of this glass,
and my Baudins certainly contain lead ; and yet these stand above,
though only to a small amount, in the case of the Fastre's.
The Geissler still seems to retain its pre-eminence as having the
greatest error of the lot.
The Baudin thermometers agree well together, but are evidently
made from another lot of glass from the No. 6 1 67 used before. These
last two depart less from the air thermometer. The explanation is
plain, as Baudin had manufactured more than one thousand ther-
* See note on preceding page.
OF ARTS AND SCIENCES.
119
mometers between the two, and so had probably used up the first
stock of glass. And even glass of the same lot differs, especially as
Regnault has shown that the method of working it before the blow-
pipe affects it very greatly.
It is very easy to test whether the calorimeter thermometers are of
the same glass as any of the others, by testing whether they agree
with No. 6163 throughout the whole range of 40. The difference
in the values of m for the two kinds of glass will then be about
.003 of the difference between them at 20, the and 40 points
agreeing. The only difficulty is in calibrating or reading the 100
thermometers accurately enough.
The Baudin thermometers were very well calibrated, and were
graduated to -fa C., and so were best adapted to this kind of work.
Hence I have constructed the following tables, making the and 40
points agree.
TABLE XXIV. COMPARISON OF 6163 AND THE BAUDIN STANDARDS.
6163
Mercurial
and 40 fixed.
7334*
Difference.
6163
Mercurial
and 40 fixed.
7316.*
Difference.
12.699
20.547
24.604
29.564
39.337
12.673
20.553
24.567
29.550
39.337
+.026
.006
+.037
+ 014
11.609
20.762
32.203
39.358
11.584
20.746
32.211
39.358
+.025
+.016
.008
Taking the average of the two, it would seem that No. 6163 stood
about .015 higher than the mean of 7334 and 7316 at the 20 point,
or 6163 has a higher value of m by .000045 than the others.
These differ about .17 from the air thermometer at 40, which gives
the value of m about .000104. Whence m for 6163 is .00015, as we
have found before by direct comparison with the air thermometer.
I am inclined to think that the former value, .00018, is too large,
and to take .00015, which is the value found by direct comparison, as
the true value. As the change, however, only makes at most a differ-
ence of 0.01 at any one point, and as I have already used the previous
value in all calculations, I have not thought it worth while to go over
all my work again, but will refer to the matter again in the final
results, and then reduce the final results to this value.
* A correction of 0.01 was made to the zero points of these thermometers
on account of the pressure of the water.
120 PROCEEDINGS OP THE AMERICAN ACADEMY
III. CALORIMETRY.
(a) Specific Heat of Water.
The first observers on the specific heat of water, such as De Luc,
completed the experiment with a view of testing the thermometer ;
and it is curious to note that both De Luc and Flaugergues found the
temperature of the mixture less than the mean of the two equal por-
tions of which it was composed, and hence the specific heat of cold
water higher than that of warm.
The experiments of Flaugergues were apparently the best, and he
found as follows : *
3 parts of water at and 1 part at 80 R. gave 19.86 R.
2 parts of " " 2 parts " " 39.81 R.
1 part of " " 3 parts " " 59.87 R.
But it is not at all certain that any correction was made for the
specific heat of the vessel, or whether the loss by evaporation or
radiation was guarded against.
The first experiments of any accuracy on .this subject seem to have
been made by F. E. Newmann in 1831.f He finds that the specific
heat of water at the boiling point is 1.0127 times that at about 28 C.
(22 R.).
The next observer seems to have been Regnault,t who, in 1840,
found the mean specific heat between 100 C. and 16 C. to be
1.00709 and 1.00890 times that at about 14.
But the principal experiments on the subject were published by
Regnault in 1850, and these have been accepted to the present time.
It is unfortunate that these experiments were all made by mixing
water above 100 with water at ordinary temperatures, it being
assumed that water at ordinary temperatures changed little, if any.
An interpolation formula was then found to represent the results;
and it was assumed that the same formula held at ordinary tempera-
ture, or even as low as C. It is true that Regnault experimented
on the subject at points around 4 C. by determining the specific heat
of lead in water at various temperatures ; but the results were not of
sufficient accuracy to warrant any conclusions except that the variation
was not great.
* Gehler, Phys. Worterbuch, i. 641.
t Pogg. Ann., xxiii. 40. j Ibid., li. 72.
Pogg. Ann., Ixxix. 241 ; also, Rel. d. Exp., i. 729.
OF ARTS AND SCIENCES. 121
Boscha has attempted to correct Regnault's results so as to reduce
them to the air thermometer ; but Regnault, in Comptes Rendus, has
not accepted the correction, as the results were already reduced to the
air thermometer.
Him (Comptes Rendus, Ixx. 592, 831) has given the results of
some experiments on the specific heat of water at low temperatures,
which give the absurd result that the specific heat of water increases
about six or seven per cent between zero and 13! The method of
experiment was to immerse the bulb of a water thermometer in the
water of the calorimeter, until the water had contracted just so much,
when it was withdrawn. The idea of thus giviug equal quanti-
ties of heat to the water was excellent, but could not be carried
into execution without a great amount of error. Indeed, experi-
ments so full of error only confuse the physicist, and are worse than
useless.
The experiments of Jamin and Amaury, by the heating of water by
electricity, were better in principle, and, if carried out with care,
would doubtless give good results. But no particular care seems to
have been taken to determine the variatiorfof the resistance of the
wire with accuracy, and the measurement of the temperature is
passed over as if it were a very simple, instead of an immensely diffi-
cult matter. Their results are thus to be rejected ; and, indeed,
Regnault does not accept them, but believes there is very little change
between 5 and 25.
In PoggendorfFs Annalen for 1870 a paper by Pfaundler and Plat-
ter appeared, giving the results of experiments around 4 C., and
deducing the remarkable result that water from to 10 C. varied as
much as twenty per cent, in specific heat, and in a very irregular man-
ner, first decreasing, then increasing, and again decreasing. But
soon after another paper appeared, showing that the results of the
previous experiments were entirely erroneous.
The new experiments, which extended up to 13 C., seemed to give
an increase of specific heat up to about 6, after which there was
apparently a decrease. It is to be noted that Geissler's thermometers
were used, which I have found to depart more than any other from
the air thermometer.
But as the range of temperature is very small, the reduction to the
air thermometer will not affect the results very much, though it will
somewhat decrease the apparent change of specific heat.
In the Journal de Physique for November, 1878, there is a notice of
some experiments of M. von Munchausen on the specific heat of
122 PROCEEDINGS OF THE AMERICAN ACADEMY
water. The method was that of mixture in an open vessel, where
evaporation might interfere very much with the experiment. No
reference is made to the thermometer, but it seems not improbable
that it was one from Geissler ; in which case the error would be very
great, as the range was large, and reached even up to 70 C. The
error of the Geissler would be in the direction of making the specific
heat increase more rapidly than it should. The formula he gives for
the specific heat of water at the temperature t is
1 _|_ .000302 t.
Assuming that the thermometer was from Geissler, the formula,
reduced to the air thermometer, would become approximately
1 _ .00009 t + .0000015 t 2 .
Had the thermometer been similar to that of Recknagel, it would
have been 1 -f .000045 t + .000001 t\
It is to be noted that the first formula would actually give a decrease
of specific heat at first, and then an increase.
As all these results vary so very much from each other, we can
hardly say that we know anything about the specific heat of water
between and 100, though Regnault's results above that temperature
are probably very nearly correct.
It seems to me probable that my results with the mechanical
equivalent apparatus give the variation of the specific heat of water
with considerable accuracy ; indeed, far surpassing any results which
we can obtain by the method of mixture. It is a curious result of
those experiments, that at low temperatures, or up to about 30 C.,
the specific heat of water is about constant on the mercurial ther-
mometer made by Baudin, but decreases to a minimum at about 30
when the reduction is mvde to the air thermometer or the absolute scale,
or, indeed, the Kew standard.
' As this curious and interesting result depends upon the accurate
comparison of the mercurial with the air thermometer, I have spent
the greater part of a year in the study of the comparison, but have
not been able to find any error, and am now thoroughly convinced of
the truth"of this decrease of the specific heat.' But to make certain,
I have instituted the following independent series of investigations
on the specific heat of water, using, however, the same thermome-
ters.
The apparatus is shown in Fig. 4. A copper vessel, A, about 20 cm -
OP ARTS AND SCIENCES.
123
in diameter and 23 cm high, rests upon a tripod. In its interior is a
three-way stopcock, communicating with the small interior vessel ,
the vessel A, and the vulcanite spout G. By turning it, the vessel B
could be filled with water, and its temperature measured by the ther-
mometer D, after which it could be delivered through the spout into
the calorimeter. As the.vessel B, the stopcock, and most of the spout,
were within the vessel A, and thus surrounded by water, and as the
vulcanite tube was very thin, the water could be delivered into the
calorimeter without appreciable change of temperature. The proof
of this will follow later.
The calorimeter, JE, was of very thin copper, nickel-plated very
thinly. A hole in the back at F allowed the delivery spout to enter,
and two openings on top admitted the thermometers. A wire attached
to a stirrer also passed through the top. The calorimeter had a
capacity of about three litres, and weighed complete about 388.3
grammes. Its calorific capacity was estimated at 35.4 grammes. It
rested on three vulcanite pieces, to prevent conduction to the jacket.
124 PROCEEDINGS OP THE AMERICAN ACADEMY
Around the calorimeter on all sides was a water-jacket, nickel-
plated on its interior, to make the radiation perfectly definite.
The calorific capacity of the thermometers, including the immersed
stem and the mercury of the bulb, was estimated as follows: 14 cm - of
stem weighed about 3.8*% and had a capacity of .8^; 10"- of
mercury had a capacity of .3**- ; total, I.!* 1 -.
Often the vessel B was removed, and the water allowed to flow
directly into the calorimeter.
The following is the process followed during one experiment at low
temperatures. The vessel A was filled with clean broken ice, the open-
ing into the stopcock being covered with fine gauze to prevent any
small particles of ice from flowing out. The whole was then covered
with cloth, to prevent melting. The vessel was then filled with water,
and the two thermometers immersed to get the zero points. The
calorimeter being about two thirds filled with water, and having been
weighed, was then put in position, the holes corked up, and one ther-
mometer placed in it, the other being in the melting ice. An obser-
vation of its temperature was then taken every minute, it being
frequently stirred.
When enough observations had been obtained in this way, the cork
was taken out of the aperture .Fand the spout inserted, and the water
allowed to run for a given time, or until the calorimeter was full. It
was then removed, the cork replaced, and the second thermometer
removed from the ice to the calorimeter. Observations were then
taken as before, and the vessel again weighed.
Two thermometers were used in the way specified, so that one
might approach the final temperature from above and the other from
below. But no regular difference was ever observed, and so some
experiments were made with both thermometers in the calorimeter
during the whole experiment.
The principal sources of error are as follows :
1st. Thermometers lag behind their true reading. This was not
noticed, and would probably be greater in thermometers with very
fine stems like Geissler's. At any rate, it was almost eliminated in
the experiment by using two thermometers.
2d. The water may be changed in temperature in passing through
the spout. This was eliminated by allowing the water to run some
time before it went into the calorimeter. The spout being very thin,
and made of vulcanite, covered on the outside with cloth, it is not
thought that there was any appreciable error. It will be discussed
more at length below, and an experiment given to prove this.
OP ARTS AND SCIENCES.
125
3d. The top of the calorimeter not being in contact with the water
rt. temperature may be uncertain. To eliminate this, the
was often at the temperature of the air to commence with. Also the
water was sometimes violently agitated just before taking the final
leading previous to letting in the cold water. Even if the tempera
ture of this part was take* as that of the air, the error would scarcely
ever be of sufficient importance to vitiate the conclusions
5th Some water might remain in the spout whose temperature
might be different from the rest. This was guarded against.
6 h. Evaporation. Impossible, as the calorimeter was closed.
7th. The introduction of cold water may cause dew to be depos-
octrred rimeter ' The ex P eri <*^ were rejected where this
The corrections for the protruding thermometer stem, for radiation
&c, were made as usual, the radiation being estimated by a series of
observations before and after the experiment, as is usual in determin-
ing the specific heat of solids.
June 14, 1878. Wr* Experiment.
Time.
41
42
43
44
Ther. 6168.
296.75
296.7
296.7
296.65
Points.
6163, 57.9 Air, 21 C.
6165, 34.8 Jacket about 25 C.
6166, 20.5
44-44f Water running.
*6i 218.7 251.7
4 H 218.8 251.8
*8 218.9 252.0
Calorimeter before
after
Water at added
Thermometer
2043.0
2853.3
810.3
1.1
Total' at
811.4
Temperature before 296.6
Correction for -}- .2
Calorimeter before
Weight of vessel
Water
Capacity of calorim.
thermom
2043.0
388.3
TesI?
35.4
1.1
296.8 = 26.597
Correction for stem -{- .019
Initial temp, of calorimeter 26.616
Total capacity
1691.2
126 PROCEEDINGS OP THE AMERICAN ACADEMY
218 . 6 + .2 = 218.8 = 17.994 251.6 - 1 = 251.5 = 17.962
Correction for stem 006 Correction for stem 006
1^988 17 ' 956
Mean temperature of mixture, 17.972.
Mean specific heat 18 1691.2 X 8.644 __ l m5
Me^n specific heat 18 27 811.4 X 17-972
June 14. Second Experiment.
Calorimeter before 2016.3; temperature 361.4 by No. 6163.
after 3047.0; " 244.5 and 288.7.
Air, 21 C.; jacket about 27.
361.4 + .2 = 361.6 = 33.803, or 33.863 when' corrected for stem.
244.5 -j- .2 = 244.7 = 20.865 ; no correction for stem.
288.7 .1 = 288.6 = 20.846 ;
Mean, 20.855.
Mean specific heat between and 21 ^ ^^gg
Mean specific heat between 21 and 34
June 14. Third Experiment.
Calorimeter before 1961.8; temperature 293.6 by No. 6166.
" ' after 3044.6; " 243.7 and 213.0.
Air and jacket, about 18 C.
393.6 .1 = 393.5 = 29.036, or 29.077 when corrected for stem.
243.7 .1 = 243.6 = 17.349 ; no correction for stem.
213.0 4- .2 = 213.2 = 17.374 ;
Mean, 17.361.
Mean specific heat between and 17 _ QQOI
Mean specific heat between 17 and 29 ~
It is to be observed that thermometer No. 6166 in all cases gave
temperatures about 0.02 or 0.03 below No. 6163. This difference
is undoubtedly in the determination of the zero points, as on June 15
the zero points were found to be 20.4 and 58.0. As one has gone up
and the other down, the mean of the temperatures needs no cor-
rection.
OP ARTS AND SCIENCES. 127
i
June 15.
Calorimeter before 2068.2 ; temperature 364.6 by No. 6166.
after 2929.2 ; " 249.7 and 217.7.
Air and jacket at about 22 C.
264.6 == 26.766, or 26.782 when Corrected for stem.
249.7 = 17.822, or 17.812 "
217.7 + .1 =217.8 = 17.884, or 17.874 when corrected for stem.
Rejected on account of great difference in final temperatures by the
two thermometers, which was probably due to some error in reading.
June 21.
Calorimeter before 2002.7 ; temperature 330.3 by No. 6163.
" after 3075.2; " 221.9 and 256.6.
.-
Air and jacket, 21 C.
330.3 -f- .1 = 330.4 = 30.321, or 30.359 when corrected for stem.
221.9 + - 1 == 222 - = 18.349, or 18.343
256.6 -j- .0 = 256.6 = 18.358, or 18.352 " "
Mean, 18.347.
Specific heat between and 18
Specific heat between 18 and 30
June 21.
Calorimeter before 2073.8 ; temperature 347.8 by No. 6166.
" after 2986.8 ; " 234.5 and 206.6.
Air and jacket, about 21 C.
347.8 -|-.0 = 347.8 = 25.457, or 25.471 when corrected for stem.
234.5 -f -0 = 234.5 = 16.643, or 16.636 " "
206.6 -j-.l = 206.7 = 16.651, or 16.644
Mean, 16.640.
Specific heat between and 17 99971
Specific heat between 17. and 25
Rejected because dew was formed on the calorimeter.
A series was now tried with both thermometers in the calorimeter
from the beginning.
128 ' PROCEEDINGS OF THE AMERICAN ACADEMY
June 25.
Calor. before 2220.3 ; temperat. 325.6 by No. 6166 ; 309.9 by No. 6165.
after 3031.4; 233.4 224.6
Air, 24.2 C. ; jacket, 23.5.
325 6 4- .0 = 325.6'= 23.725, or 23.72'6 when corrected for stem.
309'.9 + '.2 = 310.1 = 23.739, or 23.740
233.4 -f -0 = 233.4 = 16.558, or 16.545
224.6 + .2 = 224.8 = 16.562, or 16.549
Means, 28.733 and 16.547.
Specific heat between and 16 _ 1 QQ ^
Specific heat between 16 and 24
June 25.
Calor. before 2278.6; temperat. 340.35 by No. 6166; 324.1 by No. 6165.
after 3130.2; " 242.5 " 232.8 " "
Air, 23.5 C. ; jacket, 22.5.
340.35 + .0 = 340.35 = 24 .877, or 24.881 when corrected for stem.
324.1 +.2=324.3 = 24.899, or 24.903 " "
242.5 -f .0 = 242.5 = 17.264, or 17.253 " "
232.8 +.2 = 233.0 = 17.261, or 17.250 " "
Specific heat between and 17 ^7
Specific heat between 17 and 25
, June 25.
Calor. before 2316.35 ; temperat. 386.1 by No. 6166 ; 368.4 by No. 61 65.
" after 2966.90; " 295.4 " " 281.7 " "
Air, 23.5 C.; -jacket, 22.5.
386.1 -f .0 = 386.1 = 28.455, or 28.465 when corrected for stem.
368.4 -f- .2 = 368.6 = 28.472, or 28.482 "
295.4 -f -0 = 295.4 = 21.374, or 21.368
281.7 -j- '-2 = 281.9 = 21.400, or 21.394 " "
Means, 28.473 and 21.381.
Specific heat between and 21
Specific heat between 21 and 28 *
Two experiments were made on June 23 with warm water in
vessel A, readings being taken of the temperature of the water, as it
OF ARTS AND SCIENCES. 129
flowed out, by one thermojneter, which was then transferred to the
calorimeter as before.
June 23.
Water in A while running, 314.15 by No. 6163'.
Calor. before 1530.9 ; temperat. 281.1 by No. 6166.
" after 2996.3; % " 328.4byNo.6166; 272.7byNo.6163.
314.15 + .1 = 314.25 = 28.526, or 28.552 when corrected for stem.
281.1 -{-.0 = 281.1 = 20.262, or 20.258 " " "
328.4 4- -0 = 328.4 = 23.945, or 23.950
272.7 -f- 1 = 272.8 = 23.960, or 23.966
Specific heat between 20 and 24 _
Specific heat between 24 and 29 ~
June 23.
Water in A while running, 383.9 by No. 6163.
Calor. before 1624.9 ; temperat. 286.75 by 6166.
after 3048.2 ; 392.45 by 6166, and 318.1 by 6163.
383.9 -f- .1 = 384.0 = 36.303, or 36.357 when corrected for stem.
286.75 -j- .0 = 286.75 = 20.702, or 20.700
392.45 -f -0 = 392.45 = 28.954, or 28.980
318.1 -f .1 = 318.2 = 28.9.64, or 28.992
Specific heat between 21 and 29
Specific heat between 29 and 36
To test the apparatus, and also to check the estimated specific heat
of the calorimeter, the water was almost entirely poured out of the
calorimeter, and warm water placed in the vessel A, which was then
allowed to flow into the calorimeter.
Water in A while running, 309.0 by No. 6163.
Calor. before 391.3; temperat. 314.5 by 6166.
" after 3129.0; " 308.3 by 6166, and 378.5 by 6163.
Air about 21 C.
Therefore, water lost 0.078, and calorimeter gained 5. Hence
the capacity of the calorimeter is 39.
Another experiment, more carefully made, in which the range was
greater, gave 35.
VOL. xv. (N. s. vn.) 9
130 PROCEEDINGS OP THE AMERICAN ACADEMY
The close agreement of these with the estimated amount is, of
course, only accidental, for they depend upon an estimation of only
0.08 and 0.12 respectively. But they at least show that the
water is delivered into the calorimeter without much change of
temperature.
A few experiments were made as follows between ordinary tempera-
tures and 100, seeing that this has already been determined by Reg-
nault.
Two thermometers were placed in the calorimeter, the temperature
of which was about 5 below that of the atmosphere. The vessel B
was then filled, and the water let into the calorimeter, by which the
temperature was nearly brought to that of the atmosphere ; the opera-
tion was then immediately repeated, by which the temperature rose
about 5 above the atmosphere. The temperature of the boiling
water was given by a thermometer whose 100 was taken several
times.
As only the rise of temperature is needed, the zero points of the
thermometers in the calorimeter are unnecessary, except to know that
they are within 0.02 of correct.
Jujie 18.
Temperature of boiling water, 99.9.
Calor. before 2684.7; temperat. 259.2 by 6166, and 248.3 by 6165.
" after 2993.2; " 381.0 363.4 "
259.3 = 18.568, or 18.555 when corrected for stem.
248.3 = 18.564, or 18.551
381.0 = 28.054, or 28.065
363.4 = 28.055, or 28.066
Specific heat 28 100
"Specific heal 18 ~^28" =
Other experiments gave 1.0015 and 1.0060, the mean of all of
which is 1.0033. Regnault's formula gives 1.005 ; but going directly
to his experiments, we get about 1.004, the other quantity being for
The agreement is very satisfactory, though one would expect my
small apparatus to lose more of the heat of the boiling water than
Regnault's. Indeed, for high temperatures my apparatus is much
r to Regnault's, and so I have not attempted any further
experiments at high temperatures.
OP ARTS AND SCIENCES. 131
My only object was to confirm by this method the results deduced
from the experiments on the mechanical equivalent ; and this I have
done, for the experiments nearly all show that the specific heat of
water decreases to about 30, after which it increases. But the
mechanical equivalent experiments give by far the most accurate
solution of the problem ; and, indeed, give it with an accuracy hitherto
unattempted in experiments of this nature.
But whether water increases or decreases in specific heat from to
30 depends upon the determination of the reduction to the air ther-
mometer. According to the mercurial thermometers Nos. 6163, 6165,
and 6166, treating them only as mercurial thermometers, the specific heat
of water up to 30 is nearly constant, but by the air thermometer, or by
the Kew standard or Fastre, it decreases.
Full and complete tables of comparison are published, and from
them any one can satisfy himself of the facts in the case.
I am myself satisfied that I have obtained a very near approxi-
mation to absolute temperatures, and accept them as the standard.
And by this standard the specific heat of water undoubtedly decreases
from to about 30.
To show that I have not arrived at this result rashly, I may men-
tion that I fought against a conclusion so much at variance with my
preconceived notions, but was forced at last to accept it, after studying
it for more than a year, and making frequent comparisons of ther-
mometers, and examinations of all other sources of error.
However remarkable this fact may be, being the first instance of
the decrease of the specific heat with rise of temperature, it is no
more remarkable than the contraction of water to 4. Indeed, in
both cases the water hardly seems to have recovered from freezing.
The specific heat of melting ice is infinite. Why is it necessary that
the specific heat should instantly fall, and then recover as the tempera-
ture rises ? Is it not more natural to suppose that it continues to fall
even after the ice is melted, and then to rise again as the specific heat
approaches infinity at the boiling point? And of all the bodies which
we should select as probably exhibiting this property, water is cer-
tainly the first.
(J.) Heat Capacity of Calorimeter.
During the construction of the calorimeter, pieces of all the material
were saved in order to obtain the specific heat. The calorimeter which
Joule used was put together with screws, and with little or no solder.
But in my calorimeter it was necessary to use solder, as it was of a
132 PROCEEDINGS OF THE AMERICAN ACADEMY
much more complicated pattern. The total capacity of the solder
used was only about ^ of the total capacity including the water;
and if we should neglect the whole, and call it copper, the error would
be only about T sW Hence H was considered 8ufficient to wei S h the
solder before and after use, being careful to weigh the scraps. The
error in the weight of solder could not possibly have been as great as
ten per cent, which would affect the capacity only 1 part in 12,000.
To determine the nickel used in plating, the calorimeter was weighed
before and after plating; but it weighed less after than before, owing
to the polishing of the copper. But I estimated the amount from the
thickness of a loose portion of the plating. I thus found the approxi-
mate weight of nickel, but as it was so small, I counted it as copper.
The following are the constituents of the calorimeter :
Thick sheet copper .
25.1 per cent.
Thin .
45.7 "
Cast brass
17.9 "
Rolled or drawn brass
5.7 "
Solder . . ' .
4.0 "
Steel . .. ' .
1.6 "
100.0
Nickel . . . . .3
To determine the mean specific heat, the basket of a Regnault's
apparatus was filled with the scraps in the above proportion, allowing
the basket of brass gauze, which was very light, to count toward the
drawn brass. The specific heat was then determined between 20
and 100, and between about 10 and 40. Between 20 and 100
the ordinary steam apparatus was used, but between 10 and 40 a
special apparatus filled with water was used, the water being around
the tube containing the basket, in the same manner as the steam is in
the original apparatus. In the calorimeter a stirrer was used, so that
the basket and water should rapidly attain the same temperature.
The water was weighed before and after the experiment, to allow for
evaporation. A correction of about 1 part in 1,000 was made, on
account of the heat lost by the basket in passing from the apparatus
to the calorimeter, in the 100 series, but no correction was made in
the other series. The thermometers in the calorimeter were Nos. 6163
and 6166 in the different experiments.
The principal difficulty in the determination is in the correction for
radiation, and for the heat which still remains in the basket after some
OF ARTS AND SCIENCES. 133
time. After the basket has descended into the water, it commences
to give out heat to the water ; this, in turn, radiates heat ; and the
temperature we measure is dependent upon both these quantities.
Let T = temperature of the basket at the time t
" T' = " " " "
" T" = " " " " oo
" 6 " water " t
u Ql ,
0" = " C
0" = T"
We may then put approximately
T T" = (T' T") e~7,
where c is a constant. But
T' T" TIT
hence t
0' = (d 6') (1 fi-^T).
To find c we have
1 0" tf
where 0" can be estimated sufficiently accurately to find C' approxi-
mately.
These formulae apply when there is no radiation. "When radiation
takes place, we may write, therefore, when t is not too small,
where C is a coefficient of radiation, and t is a quantity which must be
subtracted from t, as the temperature of the calorimeter does not
rise instantaneously. To estimate < , T a being the temperature of the
air, we have, according to Newton's law of cooling,
C(t < ) = jr=jr f( d ~ Ta) dt nearly '
6" 6'
- D y>
where it is to be noted that _ , is nearly a constant for all values
of 0" T a according to Newton's law of cooling.
134 PROCEEDINGS OF THE AMERICAN ACADEMY
The temperature reaches a maximum nearly at the time
and if O m is the maximum temperature, we have the value of 6" as
follows :
6" = T" = 6 m + (t m + c * );
and this is the final temperature provided there was no loss of heat.
When the final temperature of the water is nearly equal to that
of the air, G will be small, but the time t m of reaching the maximum
will be great. If a is a constant, we can put G = a (0" T a ), and
O(t m _j- c t ) will be a minimum, when
That is, the temperature of the air must be lower than the tempera-
ture of the water, so that T a = 0" as nearly as possible ; but the for-
mula shows that this method makes the corrections greater than if we
make T a = 6', the reason being that the maximum temperature is
not reached until after an infinite time. It will in practice, however,
be found best to make the temperature of the water at the beginning
about that of the air. It is by far the best and easiest method to
make all the corrections graphically, and I have constructed the fol-
lowing graphical method from the formulae.
First make a series of measurements of the temperature of the
water of the calorimeter, before and after the basket is dipped, together
with the times. Then plot them on a piece of paper as in Fig. 5,
making the scale sufficiently large to insure accuracy. Five or ten
centimeters to a degree are sufficient.
n abed is the plot of the. temperature of the water of the calo-
rimeter, the time being indicated by the horizontal line. Continue
the line dc until it meets the line la. Draw a horizontal line
through the point /. At any point, b, of the curve, draw a tangent
and also a vertical line b g ; the distance e g will be nearly the value of
the constant c in the formulas. Lay off //equal to c, and draw the
line fh k through the point h, which indicates the temperature of the
atmosphere or of the vessel surrounding the calorimeter. Draw a
vertical line, j k, through the point k. From the point of maximum,
OF ARTS AND SCIENCES.
135
c, draw a line, j c, parallel to dm, and where it meets kj will be the
required point, and will give the value of d". Hence, the rise of tem-
perature, corrected for all errors, will be kj.
This method, of course, only applies to cases where the final tem-
perature of the calorimeter is^ greater than that of the air ; otherwise
there will be no maximum.
In practice, the line dm is not straight, but becomes more and more
nearly parallel to the base line. This is partly due to the constant
decrease of the difference of temperature between the calorimeter and
the air, but is too great for that to account for it. I have traced it to
the thin metal jacket surrounding the calorimeter, and I must con-
demn, in the strongest possible manner, all such arrangements of calo-
rimeters as have such a thin metal jacket around them. The jacket is
of an uncertain temperature, between that of the calorimeter and the
air. When the calorimeter changes in temperature, the jacket follows
it, but only after some time ; hence, the heat lost in radiation is uncer-
tain. The true method is to have a water jacket of constant tempera-
ture, and then the rate of decrease of temperature will be nearly
constant for a long time.
The following results have been obtained by Mr. Jacques, Fellow
of the University, though the first was obtained by myself. Correc-
tions' were, of course, made for the amount of thermometer stem in
the air.
Temperature. Mean Specific Heat.
24 to 100 .0915
26 " 100 .0915
25 " 100 .0896 ,-'
13 " 39 .0895
14 " 38 .0885
9 " 41 .0910
136 PROCEEDINGS OP THE AMERICAN ACADEMY
To reduce these to the mean temperature of to 40, I have used
the rate of increase foundry B^de for copper. They then become., .
for the mean from to 40,
.0897
.0897 .
.0878
.0893
.0883
.0906
Mean .0892 dc .00027
As the capacity of the calorimeter is about four per cent of that of
the total capacity, including the water, this probable error is about
J fo T of the total capacity, and may thus be considered as satisfactory.
I have also computed the mean specific heat as follows, from other
observers :
Copper between 20 and 100 nearly.
.0949 Dulong.
.0935 ) ^
>0952 }Regnault.
.0933 Be-de.
.0930 Kopp.
' .0940
This reduced to between and 40 by Bede's formula gives .0922.
Hence we have the following for the calorimeter: *
Per cent. Specific Heat between and 40 0.
Copper 91.4 .0922
Zinc .7 .0896
Tin 3.6 .0550
Lead 2.7 .0310
Steel 1.6 .1110
Mean .0895
The close agreement of this number with the experimental result
can only be accidental, as the reduction to the air thermometer would
decrease it somewhat, and so make it even lower than mine. How-
* The cast brass was composed of 28 parts of copper, 2 of tin, 1 of zinc, and
1 of lead. The rolled brass was assumed to have the same composition. The
solder was assumed to be made of equal parts of tin and lead.
OF ARTS AND SCIENCES. 137
ever, the difference could not at most amount to more than 0.5 per
cent, which is very satisfactory.
The total capacity of the calorimeter is reckoned as follows :
Weight of calorimeter 3.8712 kilogrammes.
" screws ' .0016 "
" part of suspending wires .0052 "
Total weight 3.8780 "
Capacity = 3.878 X -0892 = .3459 kilogrammes.
To this must be added the capacity of the thermometer bulb and sev-
eral inches of the stem, and of a tube used as a safety valve, and we
must subtract the capacity of a part of the shaft which was joined to
the shaft turning the paddles. Hence,
.3459
+ .0011 "
-f .0010
- .0010 Jtyr
Capacity = .3470
As this is only about four per cent of the total capacity, it is not
necessary to consider the variation of this quantity with the tempera-
ture through the range from to 40 which I have used.
IV. DETERMINATION OF EQUIVALENT.
(a.) Historical Remarks.
The history of the determination of the mechanical equivalent of
heat is that of thermodynamics, and as such it is impossible to give
it at length here.
I shall simply refer to the few experiments which a priori seem to
possess the greatest value, and which have been made rather for the
determination of the quantity than for the illustration of a method,
and shall criticise them to the best of my ability, to find, if possible,
the cause of the great discrepancies.
1. GEXERAL REVIEW OF METHODS.
Whenever heat and mechanical energy are converted the one into
the other, we are able by measuring the amounts of each to obtain
the ratio. Every equation of thermodynamics proper is an equation
138 PROCEEDINGS OF THE AMERICAN ACADEMY
between mechanical energy and heat, and so should be able to give us
the mechanical equivalent. Besides this, we are able to measure a
certain amount of electrical energy in both mechanical and heat units,
and thus to also get the ratio. Chemical energy can be measured in
heat units, and can also be made to produce an electric current of
known mechanical energy. Indeed, we may sum up as follows the
different kinds of energy whose conversion into one another may fur-
nish us with the mechanical equivalent of heat. And the problem in
general would be the ratio by which each kind of energy may be con-
verted into each of the others, or into mechanical or absolute units.
a. Mechanical energy.
b. Heat.
c. Electrical energy.
d. Magnetic "
e. Gravitation u
/. Radiant "
g. Chemical "
h. Capillary "
Of these different kinds of energy, only the first five can be meas-
ured other than by their conversion into other forms of energy,
although Sir William Thomson, by the introduction of such terms as
" cubic mile of sunlight," has made some progress in the case of radia-
tion. Hence for these five only can the ratio be known.
Mechanical energy is measured by the force multiplied by the dis-
tance through which the force acts, and also by the mass of a body
multiplied by half the square of its velocity. Heat is usually referred
to the quantity required to raise a certain amount of water so many
degrees, though hitherto the temperature of the water and the reduc-
tion to the air thermometer have been almost neglected.
The energy of electricity at rest is the quantity multiplied by half
the potential ; or of a current, it is the strength of current multiplied
by the electro-motive force, and by the time; or for all attractive
forces varying inversely as the square of the distance, Sir William
Thomson has given the expression
where It is the resultant force at any point in space, and the integral
is taken throughout space.
These last three kinds of energy are already measured in absolute
OP ARTS AND SCIENCES. 139
measure, and hence their ratios are accurately known. The only
ratio, then, that remains is that of heat to one of the others, and this
must be determined by experiment alone.
But although we cannot measure /, g, h in general, yet we can
often measure off equal amounts of energy of these kinds. Thus,
although we cannot predict what quantities of heat are produced when
two atoms of different substances unite, yet, when the same quantities
of the same substances unite to produce the same compound, we are
safe in assuming that the same quantity of chemical energy comes
into play.
According to these principles, I have divided the methods into direct
and indirect.
Direct methods are those where b is converted directly or indirectly
into a, c, d, or e, or vice versa.
Indirect methods are those where some kind of energy, as g, is con-
verted into 6, and also into a, c, d, or e.
In this classification I have made the arrangement with respect to
the kinds of energy which are measured, and not to the intermediate
steps. Thus Joule's method with the magneto-electric machine would
be classed as mechanical energy into heat, although it is first con-
verted into electrical energy. The table does not pretend to be com-
plete, but gives, as it were, a bird's-eye view of the subject. It could
be extended by including more complicated transformations ; and, in-
deed, the symmetrical form in which it is placed suggests many other
transformations. As it stands, however, it includes all methods so far
used, besides many more.
In the table of indirect methods, the kind of energy mentioned first
is to be eliminated from the result by measuring it both in terms of
heat and one of the other kinds of energy, whose value is known in
absolute or mechanical units.
It is to be noted that, although it is theoretically possible to measure
magnetic energy in absolute units, yet it cannot be done practically
with any great accuracy, and is thus useless in the determination of
the equivalent. It could be thus left out from the direct methods
without harm, as also out of the next to last term in the indirect
methods.
140
PROCEEDINGS OF THE AMERICAN ACADEMY
TABLE XXV. SYNOPSIS OP METHODS FOB OBTAINING THE MECHANICAL
' EQUIVALENT OF HEAT.
re. Expansion or compression ac-
cording to adiabatic curve.
b. Expansion or compression ac
cording to isothermal curve.
1. Reversible Process
c. Expansion or compression ac-
cording to any curve with re-
generator.
f-Heat| M -So a n E " rgy -
d. Electro-magnetic engine driven
by thermo-electric pile in a cir-
cuit of no resistance.
( n. Friction, percussion, etc.
*
2. Irreversible Process \ b. Heat from magneto-electric cur-
( rents, or electric machine.
f
a Thermo-electric currents
{1. Reversible Process
b, Pyro-electric phenomena (prob-
2
ably).
a. Heating of wire by current, or
<4
2. Irreversible Process
heat produced by discharge of
electric battery.
I y. Heat, Magnetic Energy
1. Reversible Process
o. Thermo-electric current magnet-
izing a magnet in a circuit of
{a. Mechanical Energy.
b. Electrical "
M *
c. Magnetic
d. Gravitation
Crooke's radiometer.
Thermo-electric pile.
Thermo-electric pile with electro-
magnet in circuit.
a Mechanical Energy
( 1. Cannon.
< 2. Electro-magnetic machine run by
( galv. battery..
. Chemical Energy, Heat . .
(Combustion, etc.)
b. Electrical "
c. Magnetic
Current from battery.
( Electro-magnet, magnetized by a
I battery current.
d. Gravitation "
a. Mechanical Energy.
Movement of liquid by capillarity.
y. Capillary energy, Heat . . .
6. Electrical
{Electrical currents from capillary
action at surface of mercury.
(Heat produced when a liq-
c Magnetic "
uid is absorbed by a po-
it. Gravitation "
Raising of liquid by capillarity.
rous solid.)
(a. Mechanical Energy
{Magneto-electric or electro-magnetic
machine. Electric attraction.
S. Electrical energy, Heat . . (
(Heat generated in a wire by
6. Magnetic "
c. Gravitation "
Electro-magnet.
an electrical current.)
ia. Mechanical Energy
{Armature attracted by a permanent
magnet.
t
Magnetic Energy, Heat . .
(Heat generated on demag-
6. Electrical "
{Induced current on demagnetizing
a magnet.
netizing a magnet.)
c. Gravitation "
(. Gravitation Energy, Heat .
(Heat generated by a fall-
a. Mechanical Energy.
6. Electrical
c. Magnetic "
Velocity imparted to a falling body.
ing.body.)
2. RESULTS OF BEST DETERMINATIONS.
On the basis of this table of methods I have arranged the following
table, showing the principal results so far obtained.
In giving the indirect results, many persons have only measured
one of the transformations required ; and as it would lengthen out the
OP ARTS AND SCIENCES.
141
TABLE XXVL-H,.*,.,
OF Exr ERIMENTAL RE8ULm
Me
0<
thod in
neral. Method in Particular.
.
Observe
Date
Result.
Compression of air
Expansion " . ! .
Joule 2
184
443.8
Theory of gases (see below) .
Joule 2
184
437.8
' vapors (see below) .
Experiments on steam-engine
Him 7
185
413.0
Him 7
186
420-432
Expansion and contraction of meta
Edlund 8
1865
443.6
4301
428.3
Boring of cannon ....
Friction of water in tubea .
in calorimeter
in calorimeter
. . in calorimeter
Junction of mercury in calonmete
plates of iron . . .
Rumforc
Joule 3
Joule*
Joule 5
Joule 6
Joule 6
Joule 6
1798
1843
1845
1847
1850
1850
1850
940ft.lbs.
424.6
488.3
428.9
423.9
424.7
4252
metals
metals in mercury calo
Him 7
Favre 9
1857
1858
371.6
413.2
metals ....
Joring of metals
Vater in balance a frottement . '.
Flow of liquids under strong pressur
Crushing of lead .
Him 7
Him 7
Him?
Him 7
TT- _T
1858
1858
1860
1860-
400-450
425.0
4320
432.0
Friction of metals .
-TUttl'
1860-
425.0
Water in calorimeter ....
Joule
1876
1878
426.6
423.9
a
Heating by magneto electric cur- )
rents . .' t
Joule 8
1843
460.0
Heat generated in a disc between )
the poles of a magnet . . . j
vww.
1870
435~.2
434.9
435.8
437.4
A ft
Heat developed in wire of known (
absolute resistance . . . . "
Quintus
1857
399.7
(
so Weber
D . do. do. J
.<enz, also
1859 .
396.4
_ |
Weber
478.2
ft
Do. do. do.
Do. do. do.
Diminishing of the heat produced )
Joule 13
F.Weber 1 *
1867
1878
29.5
28.15
in a battery circuit when the >
current produces work . . )
Joule 3
1843
99.0
Do. do. do.
Favre 15
1858
43.0
Heat due to electrical current, ]
electro-chemical equivalent of
Weber,
water = .009379, absolute re- 1
sistance electro-motive force of f
Daniell cell, heat developed by
action of zinc on sul. of copper I
Boscha,
ivre, and
jermann
857
32.1
eat developed in Daniell cell . .
Electro-motive force of Daniell cell
Joule
oscha 12
859
19.5
142
PROCEEDINGS OF THE AMERICAN ACADEMY
table very much to give the complete calculation of the equivalent
from these selected two by two, I have sometimes given tables of these
parts. As the labor of looking up and reducing these is very great,
it is very possible that there have been some omissions.
I have taken the table published by the Physical Society of Berlin, 1
as the basis "down to 1857, though many changes have been made
even within this limit.
I shall now lake up some of the principal methods, and discuss them
somewhat in detail.
Method from Theory of Gases.
As the different constants used in this method have been obtained
by many observers, I shall first give their results.
TABLE XXVII. SPECIFIC HEAT OF GASES.
Limit to
Temperature.
Approximate
Temperature
of Water
Temperature
reduced to
Specific Heat.
Air
|
Mercurial
1 2669 I
Delaroche and
20 to 210
" I
*14.2 \
(
Thermometer
Air
Thermometer
r i
| .23751"
Be"rard.
Regnault.
20 to 100
20 {
Mercurial
Thermometer
I .2389"
E.Wiedemann.
Hydrogen
|
Mercurial
I 3 2936 \
Delaroche and
15 to 200
(
12.2 |
Thermometer
Air
Thermometer
j 3.4090' 6
Be'rard.
Regnault.
21 to 100
21 j
Mercurial
Thermometer
| 3.410"
E.Wiedemann.
TABLE XXVIII. COEFFICIENT OF EXPANSION OF Am UNDER CONSTANT
VOLUME.
Taking Expansion of Mercury
according to Regnault.
Taking Expansion of Mercury
according to Wiillner's Re-
calculation of Ragnault's Ex-
periments.
Regnault .....
Magnus ....
Jolly
Rowland
.0036655
.0036678
.0036695
.0036676
.0036687
.0036710
.0036727
.0036707
Mean
.0036676
.0036708
* Taking mean of results on page 101 of Rel. des Exp., torn. ii.
OP ARTS AND SCIENCES. 143
TABLE XXIX. RATIO OF SPECIFIC HEATS OF AIB.
Method.
Observer.
Date.
Ratio
of Specific
Heats.
Method of Clement & De'- )
sormes, globe 20 litres . . )
Never fully published . . .
Method of Clement & De'- I
sormes J
Using Breguet thermometer .
Cle'ment & Desormes, globe )
Cle'ment & D^sormes' 8 )
Gay-Lussac& Welter 19 .
Delaroche & Be'rard*' .
Favre & Silbermann 23 .
Masson 20
1812
Publishe'd in
1819
1853
1858
( 1.354
1.3748
1.249
1.421
1.4196
Clement & De'sormes ....
Clement & Desormes, globe )
Weisbach.2i
Him 22
1859
1861
1.4025
1.3845
Passage of gas from one ves- )
Cazin 2 *
1862
141
litres )
Pressure in globe changed by )
aspirator, globe 25 litres . J
Heating of gas by electric \
Dupre' 25
Jamin & Richard 28 . .
1863
1864
1.41
Clement & De'sormes . . .
Barometer under air-pump /
receiver of 6 litres . . . )
Compression and expansion )
Tresca et Laboulaye 29 .
Kohlrausch 26 ....
1864
1869
1871 <
1.302
Results lost
of gas by piston . . . . J
Clement & Desormes with )
Rontgen 27 . . ,
1878
of Paris.
14053
70 litres )
Compression of gas by piston .
Amagat 30
1874
1.397
References. (Tables XXVI. to XXX.)
1 Physical Society of Berlin, Fort, der Phys., 1858.
2 Joule, Phil. Mag., ser. 3, vol. xxvi. See also Mec. Warmeaquivalent,
Gesammelte Abhandlungen von J. P. Joule, Braunschweig, 1872.
3 Joule, Phil. Mag., ser. 3, vol. xxiii. See also 2 above.
* " " " " " xxvi.
6 xxvji.
6 " " " xxxi.
7 Him, The"orie Me'c. de la Chaleur, ser. 1, 3 me ed.
8 Edlund, Pogg. Ann., cxiv. 1, 1865.
9 Favre, Comptes Rend., Feb. 15, 1858; also Phil. Mag., xv. 406.
i Violle, Ann. de Chim., ser. 4, xxii. 64.
11 Quintus Icilius, Pogg. Ann., ci. 69.
12 Boscha, Pogg. Ann., cviii. 162.
13 Joule, Report of the Committee on Electrical Standards of the B. A., Lon-
don, 1873, p. 175.
i* H. F. Weber, Phil. Mag., ser. 5, v. 30.
15 Favre, Comptes Rend., xlvii. 599.
16 Regnault, Rel. des Expe'riences, torn. ii.
E Wiedemann, Pogg. Ann., clvii. 1.
144 PROCEEDINGS OF THE AMERICAN ACADEMY
*!.,-
CO t-
CO
NOD o ; .' o
:
!JB!
S S
CO CO
CO CO [ _' CO
Si :
ff!|
| j
CO CO
; g
S
^2
ea> -<
1
Velocity
reduced to
0and
Ordinary Air.
a s
OJ t-
s<i co
cS co
t333.0cm.
t329.36 m.
s s
CO S S S
CO CO CO CO CO
CO CO CO CO C0_
c<i co
^ Si
vi
H
~
S ;
;
|
<N
, ! ? "? I
iq q
h
H
H
|
I'll'
I? Si
>
2 .
2-2
d :
lO
fc |*
o S g +
&
c3
t.
O
Tempera
Observ
6 to 7
o^ 2
00 *
o? ^ 2 P o
-^-1
2 5
2 "
2
D
*t
5
<
>
| J3.2
IH
"-
J
s
5 ^
cw
M
s
y
O
s
iS
1
1
11
.2 .2
'O 13
c c
8 -S 1 S eq
I III s
PC, < K a P*
'
<
%
1
1 S
i g
<N C, CO =0 3
1 S
s
H
hJ
-
s
j
c
H
s S
1 1 :
. s
I 4 .
T3 w
fcP
< 2
1
1 s i
o > B
>-> og a - S
'o u > PH
. S
S
=a
;
bo
s *" og
1 -a
3
~- c
(S
1
3 2 "0 S
B co S " (2
( B
1
M
1
^-v-w
N-*^-^X
a
rH (M
CO
^ CD t-
QO C>
S
OP ARTS AND SCIENCES. 145
Estimating th<
follows:
3 weight
No.
1
rather arbitrarily,
Velocity at 0. C.
Dry Air.
332.6
I have combined them as
Estimated Weight
of Observation.
2
2
332.7
2
3
330.9
2
4
330.8
4
5
332.5
3
6
332.8
7
7
332.0
1
8
331.8
1
9
332.4
4
10
330.7
10
Mean 331.75
Or, corrected for the normal carbonic acid in the atmosphere, it be-
comes 331.78 meters per second in dry pure air at C.
18 Clement et Desdrmes, Journal de Physique, Ixxxix. 333, 1819.
is Laplace, Mec. Celeste, v. 125.
20 Masson, Ann. de China, et de Phys., ser. 3, torn. liii.
21 Weisbach, Der Civilingenieur, Neue Folge, Bd. v., 1859.
22 Him, The'orie Mec. de la Chaleur, i. 111.
23 Favre et Silbermann, Ann. de Chim., ser. 3, xxxvii. 1851.
z * Cazin, Ann. de Chim., ser. 3, torn. Ixvi.
25 Dupre, Ann. de Chim., 3 me ser., Ixvii. 359, 1863.
26 Kohlrausch, Pogg. Ann., cxxxvi. 618.
Rontgen, Pogg. Ann., cxlviii. 603.
28 Jamin and Richard, Comptes Rend., Ixxi. 336.
29 Tresca and Laboulaye, Comptes Rend., Iviii. 358. Ann. du Conserv. des
Arts et Me'tiers, vi. 365.
80 Amagat, Comptes Rend., Ixxvii. 1326.
Me'm. de 1'Acad. des Sei., 1738, p. 128.
82 Benzenberg, Gilbert's Annalen, xlii. 1.
38 Goldingham, Phil. Trans., 1823, p. 96.
8 * Ann. de Chim., 1822, xx. 210 ; also, CEuvres de Arago, Me'm. Sci., ii. 1.
85 Stampfer and Von Myrbach, Pogg. Ann., v. 496.
S6 Moll and Van Beek, Phil. Trans., 1824, p. 424. See also Shroder van der
Kolk, Phil. Mag., 1865.
37 Parry and Foster, Journal of the Third Voyage, 1824-5, Appendix, p. 86.
Phil. Trans., 1828, p. 97.
38 Savart, Ann. de Chim., ser. 2, Ixxi. 20. Recalculated.
89 Bravais and Martins, Ann. de Chim., ser. 3, xiii. 6.
Regnault, Rel. des Exp., iii. 533.
41 Delaroche and Berard, Ann. de Chim., Ixxxv. 72 and 113.
42 Puluj, Pogg. Ann., clvii. 656.
VOL. xv. (N. 8. vn.) 10
146 PROCEEDINGS OP THE AMERICAN ACADEMY
From Regnault's experiments on the velocity in pipes I find by
graphical means 331. 4 m - in free air, which is very similar to the above.
Calculation from Properties of Gases.
K= specific heat of gas at constant pressure.
k = " " " volume.
p = pressure in absolute units of a unit of mass.
v = volume " " " "
p = absolute temperature.
J = Joule's equivalent in absolute measure.
y ~-
7 ~~k'
General formula for all bodies :
1
7 =
JK \dft) v \dn) p
. J= L _ JL y
K *
/dvy
Also, J= ^- \ dfl ' p
K d
-
\d P V 2
Application to gases; Rankine's formula is, _
If a, is the coefficient of expansion between and 100, then
Mo = (l-|- .00635m);
r PH , . / 7 >
^ F a " a "(^i,
OF ARTS AND SCIENCES. 147
where a' p and a' p are the true coefficients of expansion at the given
temperature ;
/=
According to Thomson and Joule's experiments m = 0.33 C. for
air and about 2.0 for CO 2 . Hence /i - 272.99.
The equations should be applied to the observations directly at the
given temperature, but it will generally be sufficient to use them after
reduction to C. Using K= .2375 according to Regnault for air,
we have for the latitude of Baltimore,
From Rb'ntgen's value y = 1.4053
" Amagat's " 1.397
9
" velocity of sound 33 US- per sec. = 429.6.
Using Wiedemann's value for K, .2389, these become
= 427.8 ; = 434.0 ; L = 427.1.
9 y 9
As Wiedemann, however, used the mercurial thermometer, and as
the reduction to the air thermometer would increase these figures
from .2 to .8 per cent., it is evident that Regnault's value for K is
the more nearly correct. I take the weights rather arbitrarily as
follows :
Weight. J.
Rontgen 3 430.3
Amagat 1 436.6
Velocity of sound 4 429.6
Mean 430.7
And this is of course the value referred to water at 1 4 C. and in the
latitude of Baltimore. My value at this point is 427.7.
* Rontgen gives the value 428.1 for the latitude of Paris as calculated by a
formula of Shroder v. d. Kolk, and 427.3 from the formula for a perfect gas,
and these both agree more nearly with my result than that calculated from my
own formula.
148 PROCEEDINGS OP THE AMERICAN ACADEMY
This determination of the mechanical equivalent from the proper-
ties of air is at most very imperfect, as a very slight change in either
y or the velocity of sound- will produce a great change in the mechan-
ical equivalent.
From Theory of Vapors.
Another important method of calculating the mechanical equivalent
of heat is from the equation for a body at its change of state, as for
instance in vaporization. Let v be the volume of the vapor, and
v 1 the volume of the liquid, and H the heat required to vaporize a
unit of mass of the water ; also let p be the pressure in absolute
units, and ^ the absolute temperature. Then
JH
The quantity Hand, the relation of p to p. have been determined with
considerable accuracy by Regnault. To determine J it is only
required to measure the volume of saturated steam from a given
weight of water ; and the principal difficulty of the process lies in
this determination, though the other quantities are also difficult of
determination.
This volume can be calculated from the density of the vapor, but
this is generally taken in the superheated -state.
The experiments of Fairbairn and Tate * are probably the best
direct experiments on the density of saturated vapor, but even those
do not pretend to a greater accuracy than about 1 in 100. With
Regnault's values of the other quantities, they give about Joule's value
for the equivalent, namely 425. Him, Herwig, and others have
also made the determination, but the results do not agree very well.
Herwig even used a Giessler standard thermometer, which I have
shown to depart very much from the air thermometer.
Indeed, the experiments on this subject are so uncertain, that
physicists have about concluded to use this method rather for the de-
termination of the volume of saturated vapors than for the mechanical
equivalent of heat.
From the Steam-Engine and Expansion of Metals.
The experiments of Him on the steam-engine and of Edlund on
the expansion and contraction of metals, are very excellent as illustrat-
* Phil. Mag., ser. 4, xxi. 230.
OP ARTS AND SCIENCES. 149
ing the theory of the subject, but cannot have ajiy weight as accurate
determinations of the equivalent.
From Friction Experiments.
Experiments of this nature, that is, irreversible processes for con-
verting mechanical energy into heat, give by far the best methods for
the determination of the equivalent.
Rumford's experiment of 1798 is only valuable from an historical
point of view. Joule's results since .1843 undoubtedly give the best
data we yet have for the determination of the equivalent. The mean
of all his friction experiments of 1847 and 1850 which are given in
the table is 425.8, though he prefers the smallest number, 423.9, of
1850. This last number is at present accepted throughout the civil-
ized world, though there is at present a tendency to consider the
number too small. But this value and his recent result of 1878 have
undoubtedly as much weight as all other results put together.
As sources of error in these determinations I would suggest, first,
the use of the mercurial instead of the air thermometer. Joule com-
pared his thermometers with one made by Fastre. In the Appendix
to Thermometry I give the comparison of two thermometers made by
Fastre in 1850, with the air thermometer, as well as of a large number
of others. From this it seems that all thermometers as far as measured
stand above the air thermometer between and 1 00, and that the aver-
age for the Fastre* at 40 is about 0.l C. Using the formula given
in Thermometry this would produce an error of about 3 parts in 1,000
at 15 C., the temperature Joule used.
The specific heat of copper which Joule uses, namely, .09515, is
undoubtedly too large. Using the value deduced from more recent
experiments in calculating the capacity of my calorimeter, .0922,
Joule's number would again be increased 13 parts in 10,000, so that
we have,
Joule's value 423.9, water at 15.7 C.
Reduction to air thermometer . . -{-1.3
Correction for specific heat of copper -f- .5
" to latitude of Baltimore -\- .5
426.2
It does not seem improbable that this should be still further in-
creased, seeing that the reduction to the air thermometer is the small-
est admissible, as most other thermometers which I have measured
give greater correction, and some even more than three times as great
150 PROCEEDINGS OF THE AMERICAN ACADEMY
as the one here used, .and would thus bring the value even as high
as 429.
One very serious defect in Joule's experiments is the small range
of temperature used, this being only about half a degree Fahrenheit,
or about six divisions on, his thermometer. It would seem almost im-
possible to calibrate a thermometer so accurately that six divisions
should be accurate to one per cent, and it would certainly need a very
skilful observer to read to that degree of accuracy. Further, the
same thermometer " A " was used throughout the whole experiment
with water, and so the error of calibration was hardly eliminated, the
temperature of the water being nearly the same. In the experiment
on quicksilver another thermometer was used, and he then finds a
higher result, 424.7, which, reduced as above, gives 427.0 at Baltimore.
The experiments on the friction of iron should be probably re-
jected on account of the large and uncertain correction for the energy
given out in sound.
The recent experiments of 1878 give a value of 772.55, which re-
duced gives at Baltimore 426.2, the same as the other experiment.
The agreement of these reduced values with my value at the
same temperature, namely 427.3, is certainly very remarkable, and
shows what an accurate experimenter Joule must be to get with his
simple apparatus results so near those from my elaborate apparatus,
which almost grinds out accurate results without labor except in re-
duction. Indeed, the quantity is the same as I find at about 20 C.
The experiments of Him of 1860-61 seem to point to a value of
the equivalent higher than that found by Joule, but the details of the
experiment do not seem to have been published, and they certainly
were not reduced to the air thermometer.
The method used by Violle in 1870 does not seem capable of
accuracy, seeing that the heat lost by a disc in rapid rotation, and while
carried to the calorimeter, must have been uncertain.
The experiments of Him are of much interest from the methods
used, but can hardly have weight as accurate determinations. Some
of the methods will be again referred to when I come to the descrip-
tion of apparatus.
Method by Heat generated by Electric Current.
The old experiments of Quintus Icilius or Lenz do not have any
sxcept historical value, seeing that Weber's measure of absolute
resistance was certainly incorrect, and we now have no means of find-
ing its error.
OP ARTS AND SCIENCES.
151
The theory of the process is as follows. The energy of electricity
being the product of the potential by the quantity, the energy ex-
pended by forcing the quantity of electricity, Q, along a wire of re-
sistance, R, in a second of time, must be Q*R, and as this must equal
the mechanical equivalent of the heat generated, we must have
JH= Q 2 Rt, where H is the heat generated and t is the time the
current Q flows.
The principal difficulty about the determination by this method
seems to be that of finding R in absolute measure. A table of the
values of the ohm as obtained by different observers, was published by
me in my paper on the " Absolute Unit of Electrical Resistance,"
in the American Journal of Science, Vol. XV., and I here give it
with some changes.
TABLE XXXI.
Date.
Observer.
Value of
Ohm.
REMARKS.
1849
Kirehoff
.88 to .90
Approximately.
1851
Weber
.95 to .97
Approximately.
1862
Weber
j 1.088
| 1.075
From Thomson's unit.
From Weber's value of Siemens unit.
1863-4
B. A. Committee
( 1.0000
\ .993
Mean of all results.
Corrected by Rowland to zero velocity
of coil.
1870
Kohlrausch
1.0193
1873
Lorenz
.975
Approximately.
1876
Rowland
.9911*
From a preliminary comparison with
the B. A. unit.
1878
H. F. Weber
1.0014
Using ratio of Siemens unit to ohm,
.9536.
The ratio of the Siemens unit to the ohm is now generally taken at
.9536, though previous to 1864 there seems to have been some doubt
as to the value of the Siemens unit.
Since 1863-4, when units of resistance first began to be made with
great accuracy, two determinations of fche heat generated have been
made. The first by Joule with the ohm, and the second by H. F.
Weber, of Zurich, with the Siemens unit.
Each determination of resistance with each of these experiments
gives one value of the mechanical equivalent. As Lorenz's result was
only in illustration of a method, I have not included it among the
exact determinations.
The result found by Joule was /= 25187 in absolute measure
-* Given .9912 by mistake in the other tables.
152
PROCEEDINGS OF THE AMERICAN ACADEMY
using feet and degrees F., which becomes 429.9 in degrees C. on a
mercurial thermometer and in the latitude of Baltimore, compared
with water at 18. 6 C.
TABLE XXXII. EXPERIMENTS OF JOULE.
Observer.
Value of
B. A. Unit.
Mechanical Equivalent
from Joule's Exp.
Mechanical Equivalent
reduced to Air Ther-
mometer and cor-
rected for Sp. Ht. of
Copper.
B. A. Committee
Ditto corrected by Rowland
1.0000
.993
1.0193
429.9
426.9
438.2
431.4
428.4
439.7
.9911
426.1
427.6
H. F. Weber
1.0014
430.5
432.0
The experiments of H. F. Weber* gave 428.15 in the latitude of
Zurich and for 1 C. on the air thermometer and at a temperature of
18 C. This reduced to the latitude of Baltimore gives 428.45.
TABLE XXXIII.
EXPERIMENTS o
F H. F WEBEB
Mean of Joule and
Weber, giving Joule
twice the Weight of
Weber.
Observer.
Value of
B. A. Unit.
Mechanical Equivalent
of Heat from Weber's
Experiments.
Mean Equivalent re-
duced toAirThennom-
eter In the Latitude of
Baltimore.
B. A. Committee
Ditto corrected by Rowland
Kohlrausch
Rowland
1.000
.993
1.0193
9911
427.9
424.9
436.2
424 1
430.2
427.2
439.1
426 4
II . F. Weber. . .
1 0014
428 5
401 A
My own value at this temperatuj-e is 426.8, which agrees almost
exactly with the fourth value from my own determination of the ab-
solute unit.f
There can be no doubt that Joule's result is most exact, and hence
I have given his results twice the weight of Weber's. Weber used a
wire of about 14 ohms' resistance, and a small calorimeter holding only
250 grammes of water. This wire was apparently placed in the water
without any insulating coating, and yet current enough was sent through
* Phil. Mag, 1878, 5th ser, v. 135.
t The value of the ohm found by reversing the calculation would be .992,
almost exactly my value.
OP ARTS AND SCIENCES.
153
it to heat the water 15 during the experiment. No precaution
seems to have been taken as to the current passing into the water,
which Joule accurately investigated. Again, the water does not seem
to have been continuously stirred, which Joule found necessary. And
further, Newton's law of cooling does not apply to so great a range
as 15, though the error from this source was probably small. Further-
more, I know of no platinum which Jias an increase of coefficient of
.001054 for 1 C., but it is usually given at about .003.
There can be no doubt that experiments depending on the heating
of a wire give too small value of the equivalent, seeing that the
temperature of the wire during the heating must always be higher
than that of the water surrounding it, and hence more heat will be
generated than there should be. Hence the numbers should be
slightly increased. Joule used wire of platinum-silver alloy, and
Weber platinum wire, which may account for Weber's finding a
smaller value than Joule, and Weber's value would be more in error
than Joule's. Undoubtedly this is a serious source of error, and I am
about to repeat an experiment of this kind in which it is entirely
avoided. Considering this source of error, these experiments confirm
both my value of the ohm and of the mechanical equivalent, and
unquestionably show a large error in Kohlrausch's absolute value of
the Siemens unit or ohm.
The experiments of Joule and Favre, where the heat generated by
a current, both when it does mechanical work and when it does not,
are very interesting, but can hardly have any weight in an estimation
of the true value of the equivalent.
The method of calculating the equivalent from the chemical action
in a battery, or the electro-motive force required to decompose any
substance, such as water, is as follows.
Let E be such electro-motive force and c be the quantity of chemi-
cal substance formed in battery or decomposed in voltameter per
second. Then total energy of current of energy per second is E Q,
where Q is the current, or cQ ffj, where ffis the heat generated by
unit of c, or required to decompose unit of c. Hence, if the process
is entirely reversible, we must have in either case
CHJ E.
But the process is not always reversible, seeing that it requires more
electro-motive force to decompose water than is given by a gas
battery. This is probably due to the formation at first of some un-
stable compound like ozone. The process with a battery seems to be
154 PROCEEDINGS OP THE AMERICAN ACADEMY
best, and we can thus apply it to the Daniell cell. The following
quantities are mostly taken from Kohlrausch.
The quantity c has been found by various observers, and Kohl-
rausch* gives the mean value as .009421 for water according to his
units (mg., mm., second system). Therefore for hydrogen it is
.001047.
The quantity H can be observed directly by short-circuiting the
battery, or can be found from experiments like those of Favre and
Silbermann.
The electro-motive force E can be made- to depend either upon the
absolute measure of resistance, or can be determined, as Thomson has
done, in electro-static units. In electro-magnetic units it is
Absolute Measure
Siemens. Ohms. according to my
Determination.
After Waltenhofen 11.43 10.90 10.80 X 10 10
Kohlrausch f H-71 11.17 11.07 X 10 10
After Favre, 1 equivalent of. zinc develops in the Daniell cell
23993 heat units ;
On the mg., mm., second system, we have E = 10.935 X 10 10 ,
c = .001047, H= 23993, g = 9800.5 at Baltimore.
.-. y = 444160 mm - = 444.2 meters.
Using Kohlrausch's value for absolute resistance, he finds 456.5,
which is much more in error than that from my determination. I do
not give the calculation from the Grove battery, because the Grove
battery is not reversible, and action takes place in it even when no
current flows.
Thomson finds the difference of potential between the poles of a
Daniell cell in electro-static measure to be .OQ374 on the cm., grm.,
second system. $ Using the ratio 29,900 000 000 cm> per second, as I
have recently found, but not yet published, we have 111 800 000 011
the electro-magnetic system or 11.18 X 10 10 on the mm., mg., second
system. This gives
= 474.3 meters.
* Pogg. Ann., cxlix. 179.
t Given by Kohlrausch, Pogg. Ann., cxlix. 182.
t Thomson, Papers on Electrostatics and Magnetism, p. 246.
OF ARTS AND SCIENCES. 155
General Criticism.
All the results so far obtained, except those of Joule, seem to be of
the crudest description ; and even when care was apparently taken in
the experiment, the method seems to be defective, or the determination
is made to rest upon the determination of some other constant whose
value is not accurately known. Again, only one or two observers have
compared their thermometers with the air thermometer, although I
have shown in " Thermometry " that an error of more than one per
cent may be made by this method. The range of temperatures is
also small as a general rule and the specific heat of water is assumed
constant.
Hence a new determination, avoiding these sources of error, seems
to be imperatively demanded.
(b.) Description of Apparatus.
1. PRELIMINARY REMARKS.
As we have seen in the historical portion, the only experiments of
a high degree of accuracy to the present time are those of Joule.
Looked at from a general point of view, the principal defects of
his method were the use of the mercurial instead of the air thermom-
eter, and the small rate at which the temperature of his calorimeter
rose. ,
In devising a new method a great rise of temperature in a short time
was considered to be the great point, combined, of course, with an
accurate measurement of the work done. For a great rise of tem-
perature great work must be done, which necessitates the use of a
steam-engine or other motive power. For the measurement of the
work done, there is only one principle in use at present, which is,
that the work transmitted by any shaft in a given time is equal to 2 w
times the product of the moment of the force by the number of revo-
lutions of the shaft in that time.
In mechanics it is common to measure the amount of the force
twisting the shaft by breaking it at the given point, and attaching the
two ends together by some arrangement of springs whose stretching
gives the moment. Morin's dynamometer is an example. Hirn*
gives a method which he seems to consider new, but which is immedi-
ately recognized as Huyghens's arrangement for winding clocks with-
* Exposition de la Theorie Mecanique de la Chaleur, 3 me ed., p. 18.
156 PROCEEDINGS OF THE AMERICAN ACADEMY
out stopping them. As cords and pulleys are used which may slip on
each other, it cannot possess much accuracy. I have devised a method
by cog-wheels which is more accurate, but which is better adapted for
use in the machine-shop than for scientific experimentation.
But the most accurate method known to engineers for measuring
the work of an engine is that of White's friction brake, and on this I
have based my apparatus. Him was the first to use this principle in
determining the mechanical equivalent of heat. In his experiment a
horizontal axis was turned by a steam-engine. On the axis was a
pulley with a flat surface, on which rested a piece of bronze which was
to be heated by the friction. The moment of the force with which
the friction tended to turn the piece of bronze was measured, together
with the velocity of revolution. This experiment, which Him calls a
balance de frottement, was first constructed by him to test the quality
of oils used' in the industrial arts. He experimented by passing a
current of water through the apparatus and observing the tempera-
ture of the water before and after passing through. He thus ob-
tained a rough approximation to Joule's equivalent.
He afterwards constructed an apparatus consisting of two cylinders
about 30- in diameter and lOO 01 "- long, turning one within the other,
the annular space between which could be filled with water, or through
which a stream of water could be made to flow whose temperature
could be measured before and after. The work was measured by the
same method as before.
But in neither of these methods does Hirn seem to have recognized
the principle of the work transmitted by a shaft being equal to the
moment of the force multiplied by the angle of rotation of the shaft.
In designing his apparatus, he evidently had in view the reproduction
in circular motion of the case of friction between two planes in linear
motion.
Since I designed my apparatus, Puluj * has designed an instrument
to be worked by hand, and based on the principle used by Hirn. He
places the revolving axis vertical, and the friction part consists of two
cones rubbing together. But no new principle is involved in his
apparatus further than in that used by Hirn.f
* Fogg. Ann., clvii. 437.
t Joule's latest results were published after this was written, and I was not
aware that he had made this improvement until lately. The result of his
experiment, however, reached me soon after, and I have referred to it in the
paper, but I did not see the complete paper until much later.
OF AETS AND SCIENCES. 157
In my apparatus one of the new features has. been the introduction
of the Joule calorimeter in the place of the friction cylinders of Him
or the cones of Puluj. At first sight the currents and whirlpools in
such a calorimeter might be supposed to have some effect ; but when
the motion is steady, it is readily seen that the torsion of the calorim-
eter is equal to that of the shaft, and hence the principle must apply.
This change, together with the other new features in the experi-
ments and apparatus, has at once made the method one of extreme
accuracy, surpassing all others very many fold.
2. GENERAL DESCRIPTION.
The apparatus was situated in a small building, entirely separate
from the other University buildings, and where it was free from dis-
turbances.
Fig. 6 gives a general view of the apparatus. To a movable
axis, a b, a calorimeter similar to Joule's is attached, and the whole is
suspended by a torsion wire, c. The shaft of the calorimeter comes
out from the bottom, and is attached to a shaft, e f, which receives a
uniform motion from the engine by means of the bevel wheels g and h.
To the axis, a b, an accurately turned wheel, k I, was attached, and the
moment of the force tending to turn the calorimeter was measured by
the weights o and p, attached to silk tapes passing around the circum-
ference of this wheel in combination with the torsion of the suspend-
ing wire. To this axis was also attached a long arm, having two
sliding weights, q and r, by which the moment of inertia could be
varied or determined.
The number of revolutions was determined by a chronograph, which
received motion by a screw on the shaft ef, and which made one
revolution for 102 of the shaft. On this chronograph was recorded
the transit of the mercury over the divisions of the thermometer.
Around the calorimeter a water jacket, t M, made in halves, was
placed, so that the radiation could be estimated. A wooden box sur-
rounded the whole, to shield the observer from the calorimeter.
The action of the apparatus is in general as follows. As the inner
paddles revolve, the water strikes against the outer paddles, and so
tends to turn the calorimeter. When this force is balanced by the
weights o p, the whole will be in equilibrium, which is rendered stable
by the torsion of the wire c d. Should any slight change take place
in the velocity, the calorimeter will revolve in one direction or the
other until the torsion brings it into equilibrium again. The amount
158 PROCEEDINGS OF THE AMERICAN ACADEMY
of torsion read off on a scale on the edge of k I gives the correction to
be added to or. subtracted from the weights op.
One observer constantly reads the circle k I, and the other con-
stantly records the transits of the mercury over the divisions of the
thermometer.
A series extending over from one half to a whole hour, and record-
ing a rise of 15 C. to perhaps 25 C., and in which a record was
made for perhaps each tenth of a degree, would thus contain several
hundred observations, from any two of which the equivalent of heat
could b,e determined, though they would not all be independent.
Such a series would evidently have immense weight ; and, in fact, I
estimate that, neglecting constant errors, a single series has more
weight than all of Joule's experiments of 1849, on water, put
together.*
The correction for radiation is inversely proportional to the ratio of
the rate of work generated to the rate at which the heat is lost ; and
this for equal ranges of temperature is only -fa as great in my
measures as in Joule's ; for Joule's rate of increase was about 0.62 C.
per hour, while mine is about 35 C. in the same time, and can be
increased to over 45 C. per hour.
3. DETAILS,
The Calorimeter.
Joule's calorimeter was made in a very simple manner, with few
paddles, and without reference to the production of currents to mix
up the water. Hence the paddles were made without solder, and
were screwed together. Indeed, there was no solder about the
apparatus.
But, for my purpose, the number of paddles must be multiplied, so
that there shall be no jerk in the motion, and that the resistance
may be great: they must be stronger, to resist the force from the
engine, and they must be light, so as not to add an uncertain quantity
to the calorific capacity. Besides this, the shape must be such as to
cause the whole of the water to run in a constant stream past the
thermometer, and to cause constant exchange between the water at
the top and at the bottom.
l^? 1 "? expertments > with an average rise of temperature of 0.56 F., equal
' -31 C, gives a total rise of 12.4 C., which is only about two thirds the
erage of one of my experiments. As my work is measured with equal accu-
racy, and my radiation with greater, the statement seems to be correct.
OP ARTS AND SCIENCES.
159
160
PROCEEDINGS OP THE AMERICAN ACADEMY
Fig. 7 shows a section of the calorimeter, and Fig. 8 a per-
spective view of the revolving paddles removed from the appa-
ratus, and with the exterior paddles removed from around it; which
could not, however, be accomplished physically without destroying
them.
To the axis e b, Fig. 7, which was of steel, and 6 mm - in diame-
ter, a copper cylinder, a d, was attached, by means of four stout wires
at e, and four more at/. To this cylinder four rings, g, //, i,j, were
attached, which supported the paddles. Each one had eight paddles,
but each ring was displaced through a small angle with reference to
Fig. 7.
Fig. 8.
the one below it, so that no one paddle came over another. This was
to make the resistance continuous, and not periodical. The lower
row of paddles were turned backwards, so that they had a tendency
to throw the water outwards and make the circulation, as I shall
show afterwards.
Around these movable paddles were the stationary paddles, consist-
ing of five rows of ten each. These were attached to the movable
paddles by bearings, at the points c and k, of the shaft, and were
removed with the latter w^ien this was taken from the calorimeter.
"When the whole was placed in the calorimeter, these outer paddles
were attached to it by means of four screws, I and m, so as to be
immovable.
The cover of the calorimeter was attached to a brass ring, which
was nicely ground to another brass ring on the calorimeter, and which
OF ARTS AND SCIENCES. 161
could be made perfectly tight by means of a little white-lead paint.
The shaft passed through a stuffing-box at the bottom, which was
entirely within the outer surface of, the calorimeter, so that the heat
generated should all go to the water. The upper end of the shaft
rested in a bearing in a piece of brass attached to the cover. In the
cover there were two openings, one for the thermometer, and the
other for filling the calorimeter with water.
From the opening for the thermometer, a tube of copper, perforated
with large holes, descended nearly to the centre of the calorimeter.
The thermometer was in this sieve-like tube at only a short distance
from the centre of the calorimeter, with the revolving paddles outside
of it, and in the stream of water, which circulated as shown by the
arrows.
This circulation of water took place as follows. The lower paddles
threw the water violently outwards, while the upper paddles were
Fig. 9.
prevented from doing so by a cylinder surrounding the fixed paddles.
The consequence was, that the water flowed up in the space between
the outer shell and the fixed paddles, and down through the central
tube of the revolving paddles. As there was always a little air at the
top to allow for expansion, it would also aid in the same direction.
These currents, which were very violent, could be observed through
the openings.
The calorimeter was attached to a wheel, fixed to the shaft a b, by
the method shown in Fig. 9. At the edge of the wheel, which was of
the exact diameter of the calorimeter, two screws were attached, from
which wires descended to a single screw in the edge of the calorimeter.
Through the wheel, a screw armed with^ a vulcanite point pressed
upon the calorimeter, and held it firmly. Three of these arrange-
ments, at distances of 120, were used. To centre the calorimeter,
a piece of vulcanite at the centre was used. By this method of
suspension very little heat could escape, and the amount could be
allowed for by the radiation experiments.
VOL. xv. (N. s. vii.) 11
162 PROCEEDINGS OP THE AMERICAN ACADEMY
The Torsion System.
The torsion wire was of such strength that one millimeter on the
scale at the edge of the wheel signified 11.8 grammes, or about 7 ^ of
the weights o p generally used. There were stops on the wheel, so
that it could not move through more than a small angle. The weights
were suspended by very flexible silk tapes, 6 mm or 8 mm broad and
0.3 mra - thick. They varied from 4.5 k - to 8.5 k - taken together. The
shaft, a b, was of uniform size throughout, so that the wire c sus-
pended the whole system, and no "weight rested on the bearings.
The pulleys, m, n, Fig. 6, were very exactly turned and balanced,
and the whole suspended system was so free as to vibrate for a con-
siderable time. However, as will be shown hereafter, its freedom is
of little consequence.
The Water Jacket.
Around the calorimeter, a watef jacket, t u, was placed, so that
the radiation should be perfectly definite. During the preliminary
experiments a simple tin jacket was used, whose temperature was
determined by two thermometers, one above and the other below,
inserted in tubes attached to the jacket.
The Driving Gear.
The cog-wheels, g, h, were made by Messrs-. Brown and Sharpe, of
Providence, and were so well cut that the motion transmitted to the
calorimeter must have been very uniform.
The Chronograph.
The cylinder of the chronograph was turned by a screw on the shaft
ef, and received one revolution for 102 of the paddles ; 155 revolutions
of the cylinder, or 15,810 of the paddles, could be recorded, though,
when necessary, the paper could be changed without stopping, and the
experiment thus continued without interruption.
The Frame and Foundation.
The frame was very massive and strong, so as to prevent oscillation;
and the whole instrument weighed about 500 pounds as nearly as
could be estimated. It was placed on a solid brick pier, with a firm
foundation in the ground. The trembling was barely perceptible to
the hand when running the fasteot.
OP ARTS AND SCIENCES. 163
The Engine.
The driving power was a petroleum engine, which was very efficient
in driving the apparatus with uniformity.
The Balance.
For weighing the calorimeter, a balance capable of showing the
presence of less than ^ gramme with 15,000 grammes was used.
The weights, however, by Schickert, of Dresden, were accurate among
themselves to at least o"* for the larger weights, and in proportion for
the smaller. A more accurate balance would have been useless, as
will be seen further on.
Adjustments.
There are few adjustments, and they were principally made in the
construction.
In the first place, the shafts a b and ef must be on line. Secondly,
the wheels m n must be so adjusted that their planes are vertical, and
that the tapes shall pass over them symmetrically, and that their edges
shall be in the plane of the wheel k I.
Deviation from these adjustments only produced small error.
(c.) Theory of the Experiment.
1. ESTIMATION OF WORK DONE.
The calorimeter is constantly receiving heat from the friction, and
is giving out heat by radiation and conduction. Now, at any given
instant of time, the temperature of the whole of the calorimeter is not
the same. Owing to the violent stirring, the water is undoubtedly at
a very uniform temperature throughout. But the solid parts of the
calorimeter cannot be so. The greatest difference of temperature is
evidently soon after the commencement of the operation. But after
some time the apparatus reaches a stationary state, in which, but for
the radiation, the rise of temperature at all points would be the same.
This steady state will be theoretically reached only after an infinite
time ; but as most of the metal is copper, and quite thin, and as the
whole capacity of the metal work is only about four per cent of the
total capacity, I have thought that one or two minutes was enough to
allow, though, if others do not think this time sufficient, they can
readily reject the first few observations of each series. When there
is radiation, the stationary state will never be reached theoretically,
164 PROCEEDINGS OF THE AMERICAN ACADEMY
though practically there is little difference from the case where there
is no radiation.
The measurement of the work done can be computed as follows.
Let M be the moment of the force tending to turn the calorimeter,
and d 9 the angle moved by tHe shaft. The work done in the time i,
will be f M d 6. If the moment of the force is constant, the integral
is simply M 9 ; but it is impossible to obtain an engine which runs
with perfect steadiness, and although we may be able to calculate the
integral, as far as long periods are concerned, by observation of the
torsion circle, yet we are not thus able to allow for the irregularity
during one revolution of the engine. Hence I have devised the follow-
ing theory. I have found, by experiments with the instrument, that
the moment of the force 1 is very nearly, for high velocities at least,
proportional to the square of the velocity. For rapid changes of the
velocity, this is not exactly true, but as the paddles are very numerous
in the calorimeter, it is probably very nearly true. We have then
where C is a constant. Hence the work done becomes
Yrfi,
As we allow for irregularities of long period by readings of the
torsion circle, we can assume in this investigation that the mean
velocity is constant, and equal to v . The form of the variation of
the velocity must be assumed, and I shall put, without further dis-
cussion,
' d / . 2 IT t\
-T = v ( 1 4- c cos ) .
V a /
We then find, on integrating from a to 0,
w= Cv s a (1 -|- f c 8 ),
which is the work on the calorimeter during one revolution of the
engine.
^The equation of the motion of the calorimeter, supposing it to be
nearly stationary, and neglecting the change of torsion of the sus-
pending wire, is
where m is the moment of inertia of the calorimeter and its attach-
ments, f is the angular position of the calorimeter, W is the sum of
OF AETS AND SCIENCES. 165
the torsion weights, and D is the diameter of the torsion wheel.
Hence,
When W D = 2 C v 2 (1 -f- c 2 ), the calorimeter will merely oscil-
late around a given position, and will reach its maximum at the times
t = 0, | a, a, &c.
The total amplitude of each oscillation will be very nearly
, _ ,/ __ C v^ga^c _ TF > gr a z c
V Y ~ * m 2ir* m '
If a; is the amplitude of each oscillation, as measured in millimeters,
on the edge of the wheel of diameter D, we have \jr i// = -^.
where n is the number of revolutions of the engine per second.
Having found c in this way, the work will be, during any time,
W = TT WD N(l +c 2 ),
where N is the total number of revolutions of the paddles.
A variation of the velocity of ten per cent from the- mean, or
twenty per cent total, would thus only cause an error of one per cent
in the equivalent.
Hence, although the engine was only single acting, yet it ran easily,
had great excess of power, and was very, constant as far as long
periods were concerned. The engine ran very fast, making from
200 to 250 revolutions per minute. The fly-wheel weighed about 220
pounds, and had a radius of lj feet. At four turns per second, this
gives an energy of about 3400 foot pounds stored in the wheel. The
calorimeter required about one-half horse-power to drive it ; and,
assuming the same for the engine friction, we have about 140 foot
pounds of work required per revolution. Taking the most unfavorable
case, where all the power is given to the engine at one point, the
velocity changes during the revolution about four per cent, or c would
nearly equal .02, causing an error of 1 part in 2500 nearly. By
means of the shaking of the calorimeter, I have estimated c as follows,
the value of m being changed by changing the weight on the inertia
bar, or taking it off altogether. The estimate of the shaking was
made by two persons independently.
166 PROCEEDINGS OP THE AMERICAN ACADEMY
m x observed. calculated.
2,200,000 grins, cm. 2 .6 mm. .016
3,100,000 " -36 " -013
11,800,000 " .13 " jO!7
Mean, c = .015
causing a correction of 1 part in 5000.
Another method of estimating the irregularity of running is to put
on or take off weights until the calorimeter rests so firmly against the
stops that the vibration ceases. Estimated in this way, I have found
a little larger value of c, namely, about .017.
But as one cannot be too careful about such sources of error, I
have experimented on the equivalent with different velocities and with
very different ways of running the engine, by which c was greatly
changed, and so have satisfied myself that the correction from this
source is inappreciable in the present state of the science of heat.
Hence I shall simply put for the work
w = -trNWD,
in gravitation measure at Baltimore. To reduce to absolute measure,
we must multiply by the force of gravity given by the formula
g = 9.78009 -f- .0508 sin 2 <,
which gives 9.8005 meters per second at Baltimore. If the calo-
rimeter moved without friction, no work would be required to cause it
to vibrate back and forth, as I have described; but when it moves
with friction, some work is required. When I designed the apparatus,
I thus had an idea that it would be best to make it as immovable as
possible by adding to its moment of inertia by means of the inertia
bar and weights. But on considering the subject further, I see that
only the excess of energy represented by c z wN W D can be used in
this way. For, when the calorimeter is rendered nearly immovable
by its great moment of inertia, the work done on it is, as we have
seen, TT N W D (1 -\- c 2 ) ; but if it had no inertia, it is evident that
the work would be only TrNWD. If, therefore, the calorimeter is
made partially stationary, either by its moment of inertia or by fric-
tion, the work will be somewhere between these two, and the work
spent in friction will be only so much taken from the error. Hence
in the latter experiments the inertia bar was taken off, and then the
calorimeter constantly vibrated through about half a millimeter on the
torsion scale.
Besides this quick vibration, the calorimeter is constantly moving to
OF ARTS AND SCIENCES. 167
the extent of a few millimeters back and forth, according to the vary-
ing velocity of the engine. As frequent readings were taken, these
changes were eliminated. In very rare cases the weights had to be
changed during the experiment ; but this was very seldom.
The vibration and irregular motion of the calorimeter back and
forth served a very useful purpose, inasmuch as it caused the friction
of the torsion apparatus to act first in one direction and then in the
other, so that it was finally eliminated. The torsion apparatus moved
very freely when the calorimeter was not in position, and would keep
vibrating for some minutes by itself, but with the calorimeter there
was necessarily some binding. But the vibration made it so free that
it would return quickly to its exact 'position of equilibrium when drawn
aside, and would also quickly show any small addition to the weights.
This was tried in each experiment.
To measure the heat generated, we require to know the calorific
capacity of the whole calorimeter, and the rise of temperature which
would have taken place provided no heat had been lost by radiation.
The capacity of the calorimeter alone I have discussed elsewhere,
finding the total amount equal to .347 k ' of water at ordinary tempera-
tures. The total capacity of the calorimeter is then A -f- -347, where
A is the weight of water. Hence Joule's equivalent in absolute
measure is
~~ (4 + .347) (t t>) y \
where n is the number of revolutions of the chronograph, it making
one revolution to 102 of the paddles.
The corrections needed are as follows :
1st. Correction for weighing in air. This must be made to JF, the
cast-iron weights, and to A -j- .347, the water and copper of the calo-
rimeter. If A is the density of the air under the given conditions, the
correction is .835 A.
2d. For the weight of the tape by which the weights are hung.
This is '-rp;
3d. For the expansion of torsion wheel, D' being the diameter at
20 C. This is .000018 (t" 20). Hence,
nnofi
(1 -j_ .000018 (f '-20) 4- jp .835 A)/^.a
where t t 1 is the rise of the temperature corrected for radia-
tion.
. Oo
.
168 PROCEEDINGS OP THE AMERICAN ACADEMY
2. RADIATION.
The correction for radiation varies, of course, with the difference of
temperature between the calorimeter and jacket; but, owing to the
rapid generation of heat, the correction is generally small in propor-
tion. The temperature , generated was generally about 0.6 per
minute. The loss of temperature per minute by radiation was approxi-
mately .0014 6 per minute, where 6 is the difference of the tempera-
ture. This is one per cent for 10.7, and four per cent for 14.2.
Generally, the calorimeter was cooler than the jacket to start with,
and so a rise of about 20 could be accomplished without a rate of
correction at any point of more than fpur per cent, and an average
correction of less than two per cenl. An error of ten per cent is thus
required in the estimation of the radiation to produce an average
error of 1 in 500, or 1 in 250 at a single point. The coefficients
never differ from the mean more than about two per cent. The
observations on the equivalent, being at a great variety of tempera-
tures, check each other as to any error in the radiation.
The losses of heat which I place under the head of radiation include
conduction and convection as well. I divide the losses of heat into the
following parts: 1st. Conduction down the shaft; 2d. Conduction by
means of the suspending wires or vulcanite points to the wheel above ;
3d. True radiation ; 4th, Convection by the air. To get some idea
of the relative amounts lost in this way, we can calculate the loss by
conduction from the known coefficients of conduction, and we can get
some idea of the relative loss from a polished surface from the experi-
ments of Mr. Nichol. In this way I suppose the total coefficient of
radiation to be made up approximately as follows :
Conduction along shaft . . .00011
' " suspending wires .00006
True radiation 00017
Convection .00106
Total . . . .00140
The conduction through the vulcanite only amounts to .0000002.
From this it would seem that three fourths of the loss is due to
radiation and convection combined.
The last two 'losses depend upon the difference of temperature
between the calorimeter and the jacket, but the first two upon the
difference between the calorimeter and frame of the machine and the
wheel respectively. The frame was always of very nearly the same
OF ARTS AND SCIENCES. 169
temperature as the water jacket, but the wheel was usually slightly
above it. At first its temperature was noted by a thermometer, and
the loss to it computed separately; but it was found to be unnecessary,
and finally the whole was assumed to be a function of the tempera-
ture of the calorimeter and of the jacket only.
At first sight it might seem that there was a source of error in
having a journal so near the bottom of the calorimeter, and joined to
it by a shaft. But if we consider it a moment, we shall see that the
error is inappreciable ; for even if there was friction enough in the
journal to heat it as fast as the calorimeter, it would decrease the
radiation only seven per cent, or make an average error in the experi-
ment of only 1 in 700. But, in fact, the journal was very perfectly
made, and there was no strain on it to produce friction ; besides which,
it was connected to a large mass of cast-iron which was attached to
the base. Hence, as a matter of fact, the journal was not appreciably
warmer after running than before, although tested by a thermometer.
The difference could not have been more than a degree or so at most.
The warming of the wheel by conduction and of the journal by
friction would tend to neutralize each other, as the wheel would be
warmer and the journal cooler during the radiation experiment than
the friction experiment.
The usual method of obtaining the coefficient of radiation would be
to stop the engine while the calorimeter was hot, and observe the
cooling, stirring the water occasionally when the temperature was
read. This method I used at first, reading the temperature at inter-
vals of about a half to a whole hour. But on thinking the matter
over, it became apparent that the coefficient found in this way would
be too small, especially at small differences of temperature ; for the
layer next to the outside would be cooled lower than the mean tem-
perature, and the heat could only get to the outside by conduction
through the water or by convection currents.
Hence I arranged the engine so as to run the paddles very slowly,
so as to stir the water constantly, taking account of the number of
the revolutions and the torsion, so as to compute the work. As I
had foreseen, the results in this case were higher than by the other
method. At low temperatures the error of the first method was
fifteen per cent ; but at high, it did not amount to more than about
three to five per cent, and probably at very high temperatures it would
almost vanish.
I do not consider it necessary to give all the details of the radiation
experiments, but will merely remark that, as the calorimeter was
170
PROCEEDINGS OP THE AMERICAN ACADEMY
nickel-plated, and as seventy-five per cent of the so-called radiation is due
to convection by the air, the coefficients of radiation were found to be
very constant under similar conditions, even after long intervals of time.
The experiments were divided into two groups ; one when the
temperature of the jacket was about 5 C., and the other when it
averaged about 20 C.
The results were then plotted, and the mean curve drawn through
them, from which the following coefficients were obtained. These
coefficients are the loss of temperature per minute, and per degree
difference of temperature.
TABLE XXXV. COEFFICIENTS OF RADIATION.
Difference
between Jacket
Jacket 5.
Jacket 20.
and Calorimeter.
-5
.00138
.00134
.00135
.00130
+5
.00137
.00132
10
.00142
.00138
15
.00148
.00144
20
.00154
.00150
25
.00158
.00154
As the quantity of water in the calorimeter sometimes varied
slightly, the numbers should be modified to suit, they being true when
the total capacity of the calorimeter was 8.75 kil. The total surface
of the calorimeter was about 2350 sq. cm., and the unit of time
one minute. To compare my results with those of McFarlane and
of Nichol given in the Proc. R. S. and Proc. R. S. E., I will reduce
my results so that they can be compared with the tables given by
Professor Everett in his " Illustrations of the Centimeter-Gramme-
Second System of Units," pp. 50, 51.
The reducing factor is .0621, and hence the last results for the
jacket at 20 C. become :
TABLE XXXVI.
Difference of
Temperature.
Coefficient of Radiation
on the C. G. S. System.
McFarlane's Value.
K&tio.
3
.000081
.000168
2.07
6
.000082
. .000178
2.17
10
.OOOOSff
.000186
2.16
15
.000089
.000193
2.17
20
.000093
.000201
2.16
25
.000096
.000207
2.16
OF ARTS AND SCIENCES. 171
The variation which I find is almost exactly that given by McFar-
lane, as is shown by the constancy of the column of ratios. But my
coefficients are less than half those of McFarlane. This may possibly
be due to the fact that the walls of McFarlane's enclosure were
blackened, and to his surface being of polished copper and mine of
polished nickel : his surface may also have been better adapted by its
form to the loss of heat by convection. The results of Nichol are
also much lower than those of McFarlane.
The fact that the coefficients of radiation are less with increased
temperature of jacket is just contrary to what Dulong and Petit found
for radiation. But as I have shown that convection is the principal
factor, I am at a loss to check my result with any other observer.
Dulong and Petit make the loss from convection dependent only upon
the difference of temperature, and approximately upon the square root
of the pressure of the gas. Theoretically it would seem that the loss
should be less as the mean temperature rises, seeing that the air be-
comes less dense and its viscosity increases./ Should we substitute
density for pressure in Dulong's law, we should have the loss by con-
vection inversely as the square root of the mean absolute tempera-
ture, or approximately the absolute temperature of the jacket. This
would give a decrease of one per cent in the radiation for about 6,
which is not far from what I have found.
To estimate the accuracy with which the radiation has been obtained
is a very difficult matter, for the circumstances in the experiment are
not the same as when the radiation was obtained. In the first place,
although the water is stirred during the radiation, yet it is not stirred
so violently as during the experiment. Further, the wheel above
the calorimeter is warmer during radiation than during the experi-
ment. Both these sources of error tend to give too small coefficients
of radiation, and this is confirmed by looking over the final tables.
But I have not felt at liberty to make any corrections based on the
final results, as that would destroy the independence of the observa-
tions. But we are able thus to get the limits of the error 'produced.
During the preliminary experiments a water jacket was not used,
but only a tin case, whose temperature was noted by a thermometer
above and below. The radiation under these circumstances was
larger, as the case was not entirely closed at the bottom, and so per-
mitted more circulation of air.
3. CORRECTIONS TO THERMOMETERS, ETC.
Among the other corrections to the temperature as read off from
the thermometers, the correction for the stem at the temperature of
172 PEOCEEDINGS OP THE AMEEICAN ACADEMY
the air is the greatest. The ordinary formula for the correction is
.000156 n (t ") But, in applying this correction, it is difficult to
estimate n, the number of degrees of thermometer outside the calo-
rimeter and at the temperature of the air, seeing that part of the stem
is heated by conduction. The uncertainty vanishes as the thermometer
becomes longer and longer, or rather as it is more and more sensitive.
But even then some of the uncertainty remains. I have sought to
avoid this uncertainty by placing a short tube filled with water about
the lower part of the thermometer as it comes out of the calorimeter.
The temperature of this was indicated by a thermometer, by aid of
which also the heat lost to the water by conduction through the ther-
mometer stem could be computed ; this, however, was very minute
compared with the whole heat generated, say 1 in 10,000.
The water being very nearly at the temperature of the air, the stem
above it could be assumed to be at the temperature of the air indicated
by a thermometer hung within an inch or two of it. The correction
for stem would thus have to be divided into two parts, and calculated
separately. Calculated in this way, I suppose the correction is per-
fectly certain to much less than one hundredth of a degree : the total
amount was seldom over one tenth of a degree.
Among the uncertain errors to which the measurement of temper-
ature is subjected, I may mention the following :
1. Pressure on bulb. A pressure of 60 cm - of water produced a
change of about 0.01 in the thermometers. When the calorimeter
was entirely closed there was soon some pressure generated. Hence
the introduction of the safety-tube, a tube of thin glass about
lO 6 - long, extending through a cprk in the top of the calorimeter.
The top of the safety-tube was nearly closed by a cork to prevent .
evaporation. Had the tube been shorter, water would have been
forced out, as well as air.
2. Conduction along stem from outside to thermometer bulb. To
avoid this, not only was the bulb immersed, but also quite a length of
stem. As this portion of the stem, as also the bulb, was surrounded
by water in violent motion, there could have been no large error from
this source. The immersed stem to the top of the bulb was generally
about 5 cm - or more, and the stem only about .8- in diameter.
3. The thermometer is never at the temperature of the water, be-
cause the latter is constantly rising ; but we do not assume that it is
so in the experiment. We only assume that it lags behind the water
to the same amount at all parts of the experiment, -and this is doubt-
less true.
OF ARTS AND SCIENCES. 173
To see if the amount was appreciable, I suddenly threw the appa-
ratus out of gear, thus stopping it. The temperature was observed to
continue rising about 0.02 C. Allowing 0.01 for the rise due to
motion after the word " Stop " was given, we have about 0.01 C. as
the amount the thermometer lagged behind the water.
4. Evaporation. A possible source of error exists in the cooling of
the calorimeter by evaporation of water leaking out from it.
The water was always weighed before and after the experiment in
a balance giving ^ gramme with accuracy. The normal amount of
loss from removal of thermometer, wet corks, &c. was about 1
gramme. The calorimeter was perfectly tight, and had no leakage at
any point in its normal state. Once or twice the screws of the stuffing-
box worked loose, but these experiments were rejected.
The evaporation of 1 gramme of water requires about 600 heat
units, which is sufficient to depress the temperature of the calorimeter
about 0.07 C. As the only point at which evaporation could take
place was through a hole less than I" 1 - diameter in the safety-tube, I
think it is reasonable to assume that the error from this source is in-
appreciable. But to be doubly certain, I observed the time which
drops of water of known weight and area, placed on the warm calo-
rimeter, took to dry. From these experiments it was evident that it
would require a considerable area of wet surface to produce an ap-
preciable effect. This wet surface never existed unless the calo-
rimeter was wet by dew deposited on the cool surface. To guard
against this error, the calorimeter was never cooled so low that dew
formed ; it was carefully rubbed with a towel, and placed in the appa-
ratus half an hour to an hour before the experiment, exposed freely to
the air. The surface being polished, the slightest deposit of dew was
readily visible. The greatest care was taken to guard against this
source of error, and I think the experiment is free from it.
(d.) Results.
1. CONSTANT DATA.
Joule's equivalent in gravitation measure is of the dimensions of
length only, being the height which water would have to fall to be
heated one degree. Or let water flow downward with uniform velocity
through a capillary tube impervious to heat ; assuming the viscosity
constant, the rate of variation of height with temperature will be
Joule's equivalent.
Hence, besides the force of gravity the only thing required in ab-
174 PROCEEDINGS OP THE AMERICAN ACADEMY
solute measure is some length. The length that enters the equation
is the diameter of the torsion wheel. This was determined under a
microscope comparator by comparison with a standard meter belong-
ing to Professor Rogers of Harvard Observatory, which had been
compared at Washington with the Coast Survey standards, as well as
by comparison with one of our own meter scales which had also been
so compared. The result was .26908 meter at 20 C.
To this must be added the thickness of the silk tape suspending the
weights. This thickness was carefully determined by a micrometer
screw while the tape was stretched, the screw having a flat end. The
result was .0003 l m -.
So that, finally, J7 = .26939 meter at 20 C. Separating the
constant from the variable parts, the formula now becomes
f=
g = 9.8005 at Baltimore.
It is unnecessary to have the weights exact to standard, provided
they are relatively correct, or to make double weighings, provided the
same scale of the balance is always used. For both numerator and
denominator of the fraction contain a weight.
2. EXPERIMENTAL DATA AND TABLES OF RESULTS.
In exhibiting the results of the experiments, it is much more sat-
isfactory to compute at once from the observations the work neces-
sary to raise 1 UL of the water from the first temperature observed to
each succeeding temperature. By interpolation in such a table we
can then reduce to even degrees. To compare the different results I
have then added to each table such a quantity as to bring the result
at 20 about equal to 10,000 kilogramme meters.
The process for each experiment may be described as follows.
The calorimeter was first filled with distilled water a little cooler
than the atmosphere, but not so cool as to cause a deposit of dew.
It was then placed in the machine and adjusted to its position,
though the outer half of the jacket was left off for some time, so that
the calorimeter should become perfectly dry ; to aid which the calo-
rimeter was polished with a cloth. The thermometer and safety-
tube were also inserted at this time.
After half an hour or so, the chronograph was adjusted, the outer
half of the jacket put in place, the wooden screen fixed in position,
and all was ready to start. The engine, which had been running
OP ARTS AND SCIENCES. 175
quietly for some time, was now attached, and the experiment com-
menced. First the weights had to be adjusted so as to produce equi-
librium as nearly as possible.
The observers then took their positions. One observer constantly
recorded the. transit of the mercury over the divisions of the ther-
mometer, making other suitable marks, so that* the divisions could be
afterwards recognized. He also read the thermometers giving the
temperatures of the air, the bottom of the calorimeter thermometer,
and of the wheel just above the calorimeter ; and sometimes another,
giving that of the cast-iron frame of the instrument.
The other observer read the torsion wheel once every revolution of
the chronograph cylinder, recording the time by his watch. He also
recorded on the chronograph every five minutes by his watch, and
likewise stirred the water in the jacket at intervals, and read its tem-
perature.
The recording of the time was for the purpose of giving the con-
necting link between the readings of the torsion circle and of the ther-
mometer. This, however, as the readings were quite constant, had
only to be done roughly, say to half a minute of time, though the
records of time on the chronograph were true to about a second.
The thermometers to read the temperature of the water in the
jacket were graduated to 0.2 C., but were generally read to 0.l C.,
'and had been compared with the standards. There was no object in
using more delicate thermometers.
After the experiment had continued long enough, the engine was
stopped and a radiation experiment begun. The last operation was
to weigh the calorimeter again, after removing the thermometer and
safety-tube, and also the weights which had been used.
The chronograph sheet, having then been removed from the cylin-
der, had the time records identified and marked, as well as the ther-
mometer records. Each line of the chronograph record was then
numbered arbitrarily, and a table made indicating the stand of the
thermometer and the number of the revolutions and fractions of a
revolution as recorded on the chronograph sheet. The times at which
these temperatures were reached was also found by interpolation, and
recorded in another column.
From the column of times the readings of the torsion circle could
be identified, and so all the necessary data would be at hand for cal-
culating the work required to raise the temperature of one kilo-
gramme of the water from the first recorded temperature to any
succeeding temperature.
' 176 PROCEEDINGS OP THE AMERICAN ACADEMY
As these temperatures usually contained fractions, the amount of
work necessary to raise one kilogramme of the water to the even
degrees could then be found from this table by interpolation. Joule's
equivalent at any point would then be merely the difference of any
two succeeding numbers ; or, better, one tenth the difference of two
numbers situated 10 apart, or, in general, the difference of the num-
bers divided by the difference of the temperatures.
It would be a perfectly simple matter to make the record of the
torsion circle entirely automatic, and I think- I shall modify the
apparatus in that manner in the future.
It would take too much space to give the details of each experiment ;
but, to show the process of calculation, I will give the experiment of
Dec. 17, 1878 as a specimen. The chronograph sheet, of course, I
cannot give. The computation is at first in gravitation measure, but
afterwards reduced to absolute measure.
The calorimeter before the experiment weighed 12.2733 kil.
" " after " " " 12.2716 "
Mean 12.2720
Weight of calorimeter alone 3.8721 "
.-. Water alone 'weighed 8.3999 "
.3470
Total capacity 8.7469 "
The correction for weighing in air was .835 X = .00106.
The total term containing the correction is therefore .99878.
log 86.324 =1.9361316
log .99878 = 1.9994698
1.9*356014
log 8.7469" = .9418542
log const, factor = .9937472 = log 9.85706.
Hence the work per kilogramme is 9.85706 2JFn in gravitation
measure, the term 2 W n being used to denote the sum of products
similar to Wn as obtained by simultaneous readings of torsion circle
and records on chronograph sheet.
Zero of torsion wheel, 79.3 mm -.
Value of l mm - on torsion wheel .0118 kQ -.
The following were the records of time on the chronograph sheet :
OP ARTS AND SCIENCES. 177
Time observed.
Revolutions of Chronograph.
Time calculated.
15
8.74
15.2
20
25.32
20.1
25
42.10
25.0
30
59.05
30.0
35
76.00
35.0
40
93.03
40.0
45
109.97
45.0
50
126.92
50.0
55
144.14
55.0
The times were calculated by the formula
Time = .294 X Revolutions -f 12.66,
which assumes that the engine moves with uniform velocity. As the
principal error in using an incorrect interpolation formula comes from
the calculation of the radiation, and as this formula is correct within
a few seconds for all the higher temperatures, we can use it in the
calculation of the times.
The records of the transits of the mercury over the divisions of
the thermometer were nearly always made for each division, but it is
useless to calculate for each. I usually select the even centimeters,
and take the mean of the records for several divisions on each side.
While the mercury was rising l cm on No. 6163, there would be
about seven revolutions of the chronograph, and consequently seven
readings of the torsion circle, each one of which was the average for
a little time as estimated by the eye.
I have obtained more than thirty series of results, but have thus
far reduced only fourteen, five of which are preliminary, or were made
with the simple jacket instead of the water jacket, the radiation to
which was much greater, as there was a hole at the bottom which
allowed more circulation of the air. The mean of the preliminary
results agrees so closely with the mean of the final results, that I have
in the end given them equal weight.
On March 24th, the same thermometer was used for a second ex-
periment directly after the first, seeing that the chronograph failed to
work in the first experiment until 8 was reached. The error from
this cause was small, as the first experiment only reached to 26 C., and
hence there could have been no change of zero, as this is very nearly
the temperature at which the thermometer was generally kept.
Having thus calculated the work in conjunction with the tempera-
ture, I have next interpolated so as to obtain the -work at the even
VOL. xv. (N. s. vn.) 12
178 PROCEEDINGS OF THE AMERICAN ACADEMY
degrees. The tables so formed I have combined in two ways : first,
I have added to the column of* work in each table an arbitrary number,
such as to make the work at 20 about 10,000, and have then combined
them as seen in . Table LI. ; and, secondly, I have subtracted each
number from the one 1 farther down the table, and divided the num-
bers so found by 10, thus obtaining the mechanical equivalent of heat.
In these tables four thermometers have been used, and yet they
were so accurate that little difference can be observed in" the experi-
ments which can be traced to an error of the thermometer, although
the Kew standard has some local irregularities. The greatest difference
between any column of Table LI. and the general mean is only 10
kilogramme-meters, or 0.023 degree, and this includes all errors of
calibration of thermometers, radiation, &c. This seems to me to be a
very remarkable result, and demonstrates the surpassing accuracy of
the method. Indeed, the limit of accuracy in thermometry is the only
limit which we can at present give to this method of experiment.
Hence the large proportional time spent on that subject.
The accuracy of the radiation is demonstrated, to some extent, by
the agreement of the results obtained even with different temperatures
of the jacket. But on close observation it seems apparent that the
coefficients of radiation should be further increased as there is a ten-
dency of the end figures in each series to become too high. This is
exactly what we should suppose, as we have seen that nearly all
sources of error tend in the direction of making the radiation too
small. For instance, an error came from not stirring the water dur-
ing the radiation, and there must be a small residual error from not
stirring so fast during radiation as during the experiment. Besides
this, some parts around the calorimeter were warm during the radiation
which were 'cool during the experiment. And both of these make the
correction for radiation too small. However, the error from this
source is small, and cannot possibly affect the general conclusions. In
each column of Tables LI. and LII. a dash is placed at the tem-
perature of the jacket, and for fifteen degrees below this point the
error in the radiation must produce only an inappreciable error
in the equivalent: taking the observations within this limit as the
standards, and rejecting the others, we should still arrive at very
nearly the same conclusions as if we accepted the whole.
Most of the experiments are made with a weight of about 7.3 kiL as
everything seemed to work best with this weight. But for the sake
of a test I have run the weight up to 8.6 and down to 4.4 kil - by which
the mte of generation of the heat was changed nearly three times.
OF ARTS AND SCIENCES. 179
By this the correction for the radiation and the error due to the
irregularity of the engine are changed, and yet scarcely an appreciable
difference in the results can be observed.
The tables explain themselves very well, but some remarks may be
in order. Tables XXXVII. to L. inclusive are the results of fourteen
experiments selected from the total of about thirty, the others not
having been worked up yet, though I propose to do so at my leisure.
Table LI. gives the collected results. At the top of each column
the date of the experiment and number of the thermometer are
given, together with Xhe .approximate torsion weight and the rate of
rise of temperature per hour. The dash in each column gives
approximately the temperature of the jacket, and hence of the air.
There are four columns of mean values, but the last, produced from
the combination of the table by parts, is the best.
Table LII. gives the mechanical equivalent of heat as deduced from
intervals of 10 on Table LI. The selection of intervals of 10 tends
to screen the variation of the specific, heat of water from view, but a
smaller interval gives too many local irregularities. In taking the
mean I have given all the observations equal weight, but as the Kew
standard was only graduated to ? F. it was impossible to calibrate.it
BO accurately as to avoid irregularities of 0.02 C. which would affect
the quantities 1 in 500. , Hence, in drawing a curve through the
results, as given in the last column, I have almost neglected the Kew,
and have otherwise sought to draw a regular curve without points of
inflection. The figures in the last column I consider the best.
Table LIII. takes the mean values as found in Tables LL and LII.,
and exhibits them with respect to the temperatures on the different
thermometers, to the different parts of the earth, and also gives the
reduction to the absolute scale. I am inclined to favor the absolute
scale, using m = .00015, as given in the Appendix to Thermometry,
rather than .00018, as ufeed throughout the paper.
Table LIV. gives what I consider the final result of the experiment.
It is based on the result m = .00015 for the thermometers, and is
corrected for the irregularity of the engine by adding 1 in 4000.
The minor irregularities are also corrected so that the results signify
a smooth curve, without irregularity or points of contrary flexure.
But the curve for the work does not differ more than three kilo-
gramme-meters from the actual experiment at any point, and generally
coincides with it to about one kilogramme-meter. These differences
signify 0.007 C. and 0.002 C., respectively. The mechanical equiv-
alent is for single degrees rather than for ten degrees, as in the
other tables.
180
PROCEEDINGS OP THE AMERICAN ACADEMY
TABLE XXXVII. FIRST SERIES. Preliminary.
January 16, 1878. Jacket and Air about 14 C.
ji
Ja
H
140 '
160
180
203
220
240
259
289
|
S
Correction.
Corrected
Temperature.
Revolutions of
Chronograph
Mean Weight
W.
Work per
Kilogramme.
I
Work per
Kilogramme.
IWork per
Kilogramme
+ 6380
1
1
52.0
'56.0
59.2
63.4
66.5
70.2
74.0
80.0
.005
.003
+.006
-j-.Oll
+.020
+.028
+.045
.017
.022
.015
.001
+.027
+.067
+.161
9.185
11.412
13.650
16.230
18.137
20.392
22.538
25.943
5.485
18.023
30.652
45.329
56.241
69.158
81.484
101.214
7.509
7.478
7.442
7.394
7.364
7.354
7.292
951
1906
3010
3825
4786
5702
7156
o
io
n
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
348
775
1202
1629
2056
2484
2012
3340
3767
4193
4619
5048
5472
5899
6326
6753
7180
5728
6155
6582
7009
7436
7864
8292
8720
9147
9573
9999
10428
10852
11279
11706
12133
12560
TABLE XXXVIII. SECOND SERIES. Preliminary.
March 7, 1878. Jacket 18.5 to 22.5. Air about 21 C.
j
Correction.
$1
If
i
MS
j
J
jj
SI
fijfj
P.S
o
-
ifc
IS"
0^
&te
1
S&
Je
Ji
|
1
1
n
Is
E
* S 1
1
g
^ +
170
19.9
.016
12.537
6.03
13
198
7010
180
190
13.646
14 755
11.12
17 22
7.710
474
947
14
15
625
1052
7437
200
210
26.8
15.863
16.972
23.36
29.55
7.666
7.642
1421
1897
16
17
1480
1909
8292
8721
.010
.036
220
230
....
18.085
19.196
35.70
41.90
7.630
2369
2845
18
19
2333
2761
9145
9573
24.0
250
260
270
280
290
33.8
+.003
.036
20.305
21.419
22.533
23.642
24.754
25.867
48.09
64.30
66.69
72.92
79.16
7.600
7.596
7.582
7.552
7.547
3319
3794
4740
5213
5687
20
21
22
23
24
25
3189
3615
4041
4467
4892
6318
10001
10427
10853
11279
11704
12130
*"*
40.8
+.020
.001
300
26.990
85.42
6164
26
6744
12556
* In the calculation of this column, more exact data were used than given in the other
two columns, seeing that the original calculation was made every 6 mm. of the thermometer.
Hence the last figure may not always agree with the rest of the data.
OP ARTS AND SCIENCES.
181
TABLE XXXVIII. Continued.
ji
1
Correction.
Corrected
Temperature.
1
Mean Weight
TV.
8
1
Work per
Kilogramme.
1
1
I
310
320
330
340
350
360
370
3.80
390
4V.8
61.4
+.044
+.073
28.119
29.253
30.393
31.540
32.689
33.842
34.998
36.158
37.321
91.67
97.98
104.28
110.67
117.12
123.54
130.04
136.56
143.08
7.611
7.604
7.611
7.617
7.602
7.592
7.576
7.550
7.550
6643
7125
7608.
8097
8590
9081
9576
10071
10567
28
29
30
31
33
34
35
36
37
6168
6593
7017
7441
7867
8294
8722
9149
9577
10004
10430
12980
13405
13829
14253
14679
15106
15534
15961
16389
16816
17242
65.0
58.7
+.072
+588
+.184
+.261
TABLE XXXIX. THIRD SERIES. Preliminary.
March 12, 1878. Jacket 13.2 to 16.6. Air about!5 C.
Is
p
g
p
Correction.
Corrected
Temperature.
Revolutions of
Chronograph
2n.
i-
& |l
S3 g>^?
p S w
II
Temperature.
Work per
Kilogramme.
ijs
*&*
*a+
00
I
205
210
220
230
240
250
260
270
280
290
300
310
320
330
340
350
360
370
380
390
400
410
420
28.0
28.6
29.9
31.1
32.4
33.6
34.9
36.2
37.4
38.7
39.9
41.2
42.5
43.7
45.0
46.3
47.6
48.9
50.1
51.4
52.7
64.0
55.3
+.663
+.002
+.6io
14.368
14.754
15.529
16.307
17.090
17.875
18.662
19.452
20.242
21.029
21.825
22.619
23.418
24.220
25.023
28.825
26.628
27.438
28.253
29.069
29.884
30.703
31.519
3.156
6.334
9.770
14.184
18.642
23.080
27.550
32.014
36.474
40.924
45.424
49.838
54.302
58.844
63.366
67.874
72.403
76.987
81.550
86.100
90.720
95.316
99.920
7.5167
7.5462
7.5668
7.5875
7.5763
I 7.5872
7.5801
164
495
827
1160
1495
1831
2167
2504
2840
3179
3514
3853
4194
4536
4876
5219
5565
5910
6255
6604
6951
7299
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
269
696
1122
1548
1975
2401
2828
3253
3676
4101
4526
4951
5378
5803
6226
6653
7078
7868
8295
8721
9147
9574
10000
10427
10852
11275
11700
12125
12550
12977
13402
13825
14252
14677
+.009
+.021
+.014
+.038
+.019
+.055
+.024
+.030
+.038
+.047
+.056
+.066
+.089
+.i20
+.159
+.202
+.251
+.304
* As this table was originally calculated for every 5 mm. on the thermometer, I have given the
weights which were used to check the more exact calculation.
182
PROCEEDINGS OP THE AMERICAN ACADEMY
TABLE XL. FOURTH SERIES. Preliminary*
March 24, 1878. Jacket 5.4 to 8.2. Air about 6 C.
ft
Correction.
1|
t -
$
fg
f
JL
S.ss
I
c
If
Rag
I s
Jafc
1
03
t
Si*
u
9*
*ly
!
II
lit
130
140
150
160
27.4
29.2
31.0
32.9
+.002
8?071
9.204
10.340
11.480
42.364
48.898
55.438
62.066
7.471
7.446
7.442
485
968
1458
8
9
]0
11
30
398
823
1252
4872
5300
5725
6154
+.010
+.019
170
180
34.7
36.6
+.017
+.050
12.620
13.763
68.669
75.330
7.390
1944
2433
12
13
1680
2107
6582
7009
190
200
38.4
403
+.025
+.093
14.908
16.054
81.973
88.597
7.431
2921
3410
14
15
2534
3960
7436
8862
210
220
42.2
44.2
+.034
+.150
17.202
18.350
95.264
101.941
7.437
3902
4395
16
17
3387
3815
8289
8717
230
46.1
+.046
+.222
19.604
108.588
4886
18
4245
9147
240
!.
19
4672
9574
250
7.4617
''0
5098
10000
260
21
5524
10426
270
280
53.6
55.7
+.073
+.399
24.124
25.288
135.158
141.803
7.509
6855
7350
22
23
5950
6376
10852
11278
290
57.7
+.084
+.524
26.456
148.427
7844
24
6802
11704
....
25
7228
12130
26
7651
12553
TABLE XLI. FIFTH SERIES. Preliminary.
March 24, 1878. Jacket 5.4 to 8.4. Air about 6 C.
1 .
Corre
tion.
4
* M
1
4
1
dj
p
1
I
i
l!
Ei
lf H
Ir
I*
IS?
I
If
f"
75
80
0.9
1.7
.003
L891
2451
3.154
6 118
8.1544
239
2
g
46
477
2296
2727
90
100
110
120
130
140
3.4
5.1
6.8
8.5
10.2
12,0
.002
b
+.003
.012
.61 7
.612
3.569
4.690
5.810
6.936
8.060
9190
12.174
18.172
24.212
30.397
36.621
42 854
8.0900
8.0409
8.0074
7.9170
7.8973
7.8786
723
1200
1677
2161
2647
3132
4
6
6
7
8
g
906
1332
1759
2189
2621
3050
3156
3582
4009
4439
4871
5300
150
160
13.7
15.5
+.007
+.005
10.323
11 459
49.068
55 398
7.8512
7.8061
3614
10
3477
5727
170
180
17.2
190
+.015
+.032
12.600
13 742
61.707
68036
'7.7799
7.7622
4588
12
4333
6583
7009
190
200
210
20.8
22.6
243
+.024
+.028
+.068
+.092
14.882
16.025
17 170
74.358
80.716
87 064
7.7643
7.7807
7.8419
5558
6047
14
15
5183
5608
7433
7858
QOQO
220
230
26.1
27.9
+.039
+.150
18.316
19.467
93.402
99.677
7.8468
7.8579
7030
7518
17
18
6466
6895
8716
9146
* The first part of the experiments were lost, as the pen of the chronograph did not work.
OF ARTS AND SCIENCES.
TABLE XLI. Continued.
183
Thermometer
No. 6163.
,
Correction.
Corrected
Temperature.
Revolutions of
Chronograph
2n.
^'
Work per
Kilogramme
= 2 9.8816 Wn.
Temperature.
Work per
Kilogramme.
1
I
I
240
250
260
270
280
290
300
310
29.6
+.050
+.270
20615
105.950
7.8802
f 7.8980
7.9038
7.9091
7.8979
7.8974
8006
9482
9976
10474
10974
11481
19
20
21
22
24
25
26
27
7320
7745
8170
8597
9024
9451
9878
10305
10733
11160
9570
9995
10420
10847
11274
11701
11128
12555
12983
13410
34'.9
36.7
38.5
40.2
42.1
+.069
+.087
+.109
+.351
+.450
+.583
24.072
25.231
26.395
27.565
28.748
124.863
131.181
137.560
143.972
150.467
TABLE XLIL SIXTH SERIES.
May 14, 1878. Jacket 12.l to 12.4. Air about 13 C.
lo
Correction.
if
^
sg-
%*
e
&
i
5
1.1
!ii
B*
I
5 is
Ii*
2
8
Jl
1
a
1
3J
II
i
*g
I
*I
if +
140
46.4
.002
9319
1.93
9
137
5296
150
47.9
10.178
7.07
7.2291
370
10
293
6726
160
49.4
' '.000
.007
11.032
12.19
.
735
11
721
6154
170
50.9
11.886
17.37
} 7.1608
1102
12
1151
6584
180
62.5
+.002
.008
12.740
22.52
1467
13
1579
7012
190
54.0
13.596
27.70
7.1600
1835
14
2007
7440
200
55.5
+.006
.002
14.454
32.88
2201
15
2434
7867
210
57:0
15.314
38.07
7.1512
2568
16
2863
8296
220
68.5
+.6io
+.6ii
16.174
43.29
2938
17
3290
8723
230"
60.0
17.037
48.50
7.1446
8306
18
3716
9149
240
61.6
+.015
+.031
17^093
63.70
3675
19
4142
9575
250
7.1536
20
4567
10000
260
21
4993
10426
270
280
66.2
67.7
+.024
+.075
20.500
21.362
69.27
74.50
7.1230
4778
5148
22
23
6420
5846
10853
11279
290
300
69.2
70.7
+.031
+.113
22220
23.076
79.69
84.84
7.1344
5514
5878
24
25
6271
6696
11704
12129
310
320
72.2
73.7
+.039
+158
23.928
24.774
89.97
95.05
7.1302
6240
6600
26
27
7121
7547
12554
12980
330
340
75.2
76.2
+.047
+.2i2
25.624
26.467
100.19
105.27
7.1117
6962
7319
28
29
7973
8400
13406
13833
350
360
78.2
79.7
+.056
+.272
27.309
28.147
110.39
115.44
7.0958
7680
8035
30
31
8829
9259
14262
14692
370
380
81.2
82.7
+.065
+.341
28.990
29.825
120.57
125.66
7.1076
8396
8754
32
33
9678
10096
15111
15529
390
84.2
+.076
+.417
30.663
130.78
7.1088
9115
....
400
85.7
31.505
135.90
9475
....
410
420
87.2
88.7
+.087
+.504
32.377
33.226
140.98
146.08
7.1064
9833
10192
'.'.'.'.
184 PROCEEDINGS OF THE AMERICAN ACADEMY
TABLE XLIII. SEVENTH SERIES.
May 16, 1878. Jacket 1P.8 to 12. Air about 12 C.
Thermometer 1
No. 6163.
1
Correction.
if
1!
Revolutions of
Chronograph
Mean Weight
W.
1
Temperature.
Work per
Kilogramme.
1
1
i
130
140
30.9
32.2
.004
8538
9.315
5.07
9.73
7.2350
335
9
199
6296
150
160
170
33'.6
350
36.3
.002
.006
10.094
10.875
11.654
14.36
18.98
23.56
7.3011
668
1003
1335
10
11
12
628
1056
1484
5725
6153
6581
.010
180
37.6
12.433
28.16
7.3165
1670
13
1913
7010
190
200
38.9
40.2
+.003
.008
13.209
13.984
32.74
37.31
7.3460
2003
2337
14
15
2344
2770
7441
7867
210
220
41.5
42.8
+.006
.000
14.758
15.536
41.84
46.38
7.3094
2667
2998
16
17
3196
3623
8293
8720
230
240
44^2
45.5
+.010
+.013
16.317
17.103
50.99
55.62
i 7.2846
3332
3667
18
19
4052
4478
9149
9575
250
260
46.9
48.3
+.014
+.032
17.891
18.682
60.29
}
7.2822
4005
20
21
4906
5324
10003
10421
270
280
49.6
50.9
+.619
+.056
19.475
20.269
69.63
74.34
7.2610
468i
. 5021
22
23
6754
6179
10851
11276
290
300
52.3
53.6
+.025
+.090
21.079
21.866
79.01
83.71
7.2504
5358
5697
24
25
6603
7028
11700
12125
310
320
65.0
56.4
+.032
+.127
22.665
23.471
88.42
93.14
7.2893
6037
6379
26
27
7454
7883
12551
12980
330
67.8
+.039
+.172
24.281
97.88
6722
28
8307
13404
340
59.2
25.088
102.61
7.3047
7065
29
872'.)
13826
350
60.5
+.046
+.222
25.896
107.36
7410
30
9157
14254
360
61.9
....
26.706
112.14
7.3389
7759
31
9582 14679
370
63.2
+.055
+.279
27.523
116.88
1 '
8104
32
10009
15106
380
64.6
....
28.346
121.62
\ 7.4109
8454
....
....
390
66.0
+.065
+.345
29.172
126.34
1 . ,
8801
400
67.4
29^996
13L12
7.4356
9155
410
68.8
+.075
+.419
30.827
135.90
9508
420
70.1
+.080
+.456
31.653
140.66
9861
....
TABLE XLIV. EIGHTH SERIES.
May 23, 1878. Jacket 16.2 to 16.5. Air about 20 C.
ii
Ji
1
Correction.
Corrected
Temperature".
Revolutions of
Chronograph
2n.
Mean Weight
,|*
IgjW
i
1
Work per
Kilogramme.
*9*
&
S
1
230
240
250
260
270
280
290
23.9
25.4
26.8
28.3
29.7
31.2
32.7
.007
16287
17.063
39.120
43.982
6.9137
> 6.9358
6.9007
6.9125
333
i338
1673
2010
17
18
19
20
21
22
306
735
1163
1592
2019
2446
8715
9144
9572
10001
10428
10866
.000
+.005
19.405
20.190
20.978
58.602
63503
68.428
....
OP ARTS AND SCIENCES.
TABLE XLIV. Continued.
185
I
Correction.
.
**
|
.
5
ll
1!.-
f
!i|
I
la
ig
jM
i
1
i
l|
r
1*
|||
|
:
Iff
H
3
EH
PSO
s
n
fcH
P M
N
300
34.2
21/765
73.351
6.8878
2346
23
2871
11280
310
320
35.6
37.1
+.008
+.040
22.554
23.350
78.283
83.245
6.8866
6.8594
2682
3020
24
25
3298
3722
11707
12131
330
38.6
24.151
88.314
6.8358
3363
26
4150
12559
340
350
40.1
41.6
+.6i7
+.085
24.952
25.751
93.294
98.275"
6.8748
6.9184
3702
4044
27
28
4574
4999
12983
14408
360
43.1
26.552
103.232
6.9444
4385
29
5423
13832
370
380
44.6
46.0
+.028
" "
+.144
'*"
27.361
28.175
108.216
113.269
6.9291
6.9338
4727
5074
30
31
5851
6275
14260
14684
390
47.5
28.989
118.231
6.9385
5418
....
400
410
49.0
50.6
+.039
+217
29.800
30.624
123.329
128.399
6.9444
6.9467
5766
6115
....
....
420
52.1
+.047
+.281
31.445
133.480
6.9314
6464
....
TABLE XL V. NINTH SERIES.
May 27, 1878. Jacket 19.6 to 20. Air about 23 C.
I
Correction.
g*
J
^
life
4
C 6
II
lii
t
ll
3
M
*l
tii
1
1
fl
(Sw
i
" S ll
1
Mr
200
380
.015
15890
6.33
16
47
8293
210
394
17:000
11.74
8.8108
473
17
473
8719
220
409
.011
.010
18.106
17.17
946
18
901
9147
230
49, 3
19.219
22.62
8.7341
1419
19
1326
9572
240
438
.005
.011
20.329
28.13
1895
20
1754
10000
250
45.3
21.442
33.68
2368
21
2180
10426
260
270
+.002
.004
22.552
23659
[ 8.4800
....
22
23
2606
3031
10852
11277
280
290
49.8
51.3
+.009
+.012
24.771
25.885
60.55
56.25
)
8.4399
3785
4263
24
25
3457
3883
11703
12129
300
310
52.9
544
+.019
+.037
27.006
28.133
61.93
67.63
8.4765
4737
5215
26
27
4312
4734
12558
12980
320
330
56.0
575
+.029
+.072
29.264
30.404
73.36
79.15
8.4552
5697
6182
28
29
5159
5584
13405
13830
340
350
59.1
60 fi
+.042
+.118
31.552
32.702
84.97
90.85
} 8.4015
6669
7159
30
31
6010
6435
14256
14681
360
370
62.2
638
+.056
+.173
33.853
35.011
96.78
102.66
[ 8.4222
7652
8143
32
33
6860
7286
15106
15532
380
390
65.4
670
+.071
+.242
36.170
37.331
108.59
114.45
)
8.4706
8638
9128
34
35
7714
8138
15960
16384
400
410
68.6
70.2
+.088
+.322
38.497
39.664
120.36
126.33
8.4316
9626
10126
36
37
8565
8988
16811
17234
420
71.8
+.105
+.419
40.833
132.26
10620
38
39
9414
9842
17660
18Q88
40
10268
18514
41
10691
18937
186
PROCEEDINGS OP THE AMERICAN ACADEMY
TABLE XL VI. TEHTH SERIES.
June 3, 1878. Jacket 18.l to 18.4. Air about 20 C.
g
b
Correction.
"Sji
^
|P
S
i
if
fl
If"
1
||1
5
g
11
Iff
H
I
1
1
k
II
|v
*$%
H
*B
250
4.1
.007
17838
7.82
18
69
9145
260
7.0
18.617
4.3899
19
496
9572
270
9.9
.003
+.004
19.401
23.'l9
'667
20
925
10001
280
12.8
20.188
3095
4.3919
1005
21
1350
10426
290
15.7
+.003
+.020
20.978
38.70
1341
22
1778
10854
300
18.7
21.763
46.41
i 4.3912
1676
23
2204
11280
310
320
21.6
24.5
+.008
+.037
22.551
23.354
54.21
62.04
)
{ 4.3907
2014
2354
24
25
2627
3054
11703
12130
330
340
27.5
30.5
+.014
+.078
24.162
24 970
69.92
77.92
J
4.3624
2696
3041
26
27
3479
3904
12555
12980
350
360
336
36.6
+.020
+.132
25.780
26.593
85.89
93.94
4.3542
3385
3731
28
29
4332
4852
13408
13828
370
396
+.028
+.198
27.415
102.05
4081*
30
5179
14255
380
42.7
28.246
110.34
4.3362
4437
31
5604
14680
390
45.8
+.036
+.281
29.079
118.49
4786
....
....
400
48.9
29.911
126.66
4.3978
5141
410
52.0
+.044
+.377
30.754
134.89
5499
....
....
TABLE XLVII. ELEVENTH SERIES.
June 19, 1878. Jacket 19.6 to 20. Air about 23 C.
Thermometer
No. 6163.
5
H
Correction.
g
SI
ReTolutions of
Chronograph
2n.
Mean Weight
III
O C OJ
s w
Temperature.
Work per
Kilogramme.
*3 +
1
I
250
260
270
280
290
300
310
320
330
340
350
360
370
380
390
400
410
420
...
.002
+.002
+.006
+.029
+.063
-f.iis
2L450
22.562
24.789
25.007
27.032
28.168
29.307
30.456
31.612
32 774
8.933
16.087
30.281
37.439
44.655
61.848
69.098
66.390
73.724
81.163
88.462
95.734
103.093
110.560
118.121
126.693
133.250
'6.7572
6.7678
6.7749
6.7896
6.7973
| 6.8188
6.9165
6.7876
6.7808
476
i42J
1899
2379
286p
3344
3832
4323
4817
5311
5807
6307
6808
7311
7815
8321
21
22
23
24
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
192
235
602
1087
1511
1939
23*55
2789
3214
3638
4003
4488
4913
5337
6760
6187
6614
7040
7465
7891
8317
10428
10855
11282
11707
12131
12569
12985
13409
13834
14258
14683
15108
15533
15957
16380
10807
17234
17660
18085
18511
18937
+.010
+.019
+.031
+.043
+.177
+.058
+.072
+.257
+.351
33.939
35.110
36.280
37.456
38.637
39.821
41.010
+.087
+.106
+.463
+.596
OF ARTS AND SCIENCES.
187
0961 +
tauiBjSonji
J3d Jt.lO.ii.
~ ~ X ^1 1^ I-H i
work pe
ilogram
CC W CO C3i * CC CO O * rti <N t-
!M (M O5 CC O
^io *^io^
coco 'T v i'M~>j
co co ' co ao co
a>Q5oa>Qajao0Q S
qdBoSouoaqo
jo
SiggggSSSSScSiSSSg g S5 : : :
(6 co cc t-^ t^ CD co to o so d o co t-^ t-^ co t^ 05 *i t^ r I I
jo suonn[OA8H
1 1 1 ++++++++++++++++++
.I._I_._I_._L._L.O --^ -co -
\^ . T^ . l^ . |7 . |^ .(?q CO '-^t* 'i
' "M 7* *
.. 1 .. .. .. .. .
'8919 J
ffl
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CO O lO O O O 'CO -CD
CO 'CO CO 'CO '"^ *O "lO '*O
Ico Ico Ico led Ico" Ico Ico !cd I I I
8919 -ON *q
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oanjK.iadiu.JX *
r-! cs co * vo co r^ oJ o r cc
'S9T9 'ON g ;
J3;3uiouijaqx
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to .
188
PROCEEDINGS OP THE AMERICAN ACADEMY
TABLE XLIX. THIRTEENTH SERIES.
Dee. 19, 1878. Jacket 3.2 to 3.5. Air 4.2 to 5.2 C.
55
Corrections.
^
^
,-i
S
|i
if
IL
I
f|*
I
1
El
1
;
H
I
I
||
If
G**"
* 6 a!
1
1
H
I
70
1248
1.72
Oj
106
1858
8.6610
485.0
80
....
'
2.378
7.38
485.0
2
+323
2287
8.5571
485.1
90
-.003
3.500
13.11
970.1
3
754
2718
8.4325
482.2
100
4.626
18.89
1452.3
4
1184
3148
8.3688
481.1
110
+.001
+.003
5.751
24.70
1933.4
5
1612
3576
8.4155
487.1
120
6.881
30.55
2420.5
6
2041
4005
8.4189
485.6
130
+.005
+.019
8.013
36.38
2906.1
7
2472
4436
8.3953
489.2
140
9.148
42.27
3395.3
g
9001
.Q_
8.4366
486.6
150
+.009
+.044
10.284
48.10
3881.9
9
3331
5295
8.4484
486.5
160
11.424
53.92
AnpQ A
- -.
Q7Pf\
_^ .
8.4189
490.6
&12A
170
+.016
+.080
12.569
59.81
4859.0
11
4187
6151
8.3988
491.1
180
13.713
65.72
5350.1
12
4615
6579
8.4153
487.1
190
+.023
+.126
14.859
71.57
5837.2
13
5045
7009
8.3811
491.7
200
16.005
77.50
6328.9
14
5472
7436
210
+.033
+.183
17.154
83.40
8.3835
489.4
6818.3
15
5898
7862
8.3976
490.2
220
18.300
89.30
7308.5
16
6327
8291
8.4035
493.0
230
+.044
+.251
19.452
95.23
7801.5
17
6753
8717
8.4460
496.4
240
20.604
101.17
8297.9
18
7180
9144
250
+.056
+.332
21.760
f- 8.4555
981.3
19
7608
9572
260
22.912
112.90
9279.2
20
8038
10002
270
+.069
+.424
24.065
118.81
8.4602
494.7
9773.9
21
8465
10429
280
25.221
124.70
8.4779
494.0
10267.9
22
8891
10855
23
9317
11281
25
9746
10173
11710
12137
OF ARTS AND SCIENCES.
189
TABLE L. FOURTEENTH SERIES.
December 20, 1878. Jacket 1.5 to 1.9. Air about 3.4 C.
Temperature
by Kew
Standard.
1
Corrections.
Corrected Tem-
perature, Abso-
lute Scale.
Revolution of
Chronograph
Mean Weight
TV.
Work per
Kilogramme
= 2 9.7832 Wn.
I
Work per
Kilogramme.
Work per Kilo-
gramme -f 2210.
Reduction
to Absolute
Scale.
I
1
36.0
66.0
.00
1.82
8.03
7.3682
2
77
2287
38 5
584
3.23
1637
7.3458
'601
3
* 503
2713
41.0
.9
.01
.00
+.01
4.62
24.78
7.3705
1206
4
936
3146
43.5
3.3
....
....
6.02
33.19
7.4012
1812
6
1370
3580
46.0
5.8
.02
+.01
+.04
7.43
41.48
7.4142
2412
6
1803
4013
485
8 2
884
4981
7 4177
3016
7
2226
4436
51.0
10.7
.03
+.02
+.09
10.26
68.18
7.4390
3624
8
2656
4866
535
13 2
11 68
66 56
7 4107
4234
9
3084
5294
56.0
15.6
.04
+.03
+.16
13.12
74.95
7.3493
4842
10
3513
5723
58.5
18?
14.56
83.56
7.3269
5461
11
3942
6152
61.0
20.7
.04
+.05
+.25
16.01
92.27
7.2335
6085
12
4369
6579
63.5
23.3
....
17.46
100.99
7.1603
6703
13
4790
7000
66.0
25.9
.05
+.06
+.38
18.92
109.95
7.2076
7330
14
5220
7430
685
28 5
2039
118 84
7 1839
7957
15
5650
7860
71.0
31.2
.05
+.08
+.62
21.86
127.83
7.2122
8689
16
6081
8291
73.5
33.8
..:.
23.34
136.75
7.2252
9218
17
6507
8717
76.0
36.5
.06
+.10
+.69
24.84
145.78
7.2134
9857
18
6935
9145
785
392
26 33
15480
10493
19
7364
9574
?0
7791
10001
ftl
8219
10429
9,9:
8648
10858
TO
9074
11284
?4
9499
11709
fln
9925
12135
26
10352
12562
190
PROCEEDINGS OP THE AMERICAN ACADEMY
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191
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196
PROCEEDINGS OP THE AMERICAN ACADEMY
TABLE LIV. FINAL MOST PROBABLE RESULTS.
J.
Work.
Mechanical
Equivalent.
Work.
Mechanical
Equivalent.
|
sf
ft,*
6 a
|| .
h|
B
ftj?
be
^-.
g? II
ill
j2oa
UJ
t*l
I e 1
loi
P a
Ji^K
ll 5
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d
gii
II*
H
Pi
M
PI
Jr
00000.
0000.
00000.
0000.
s
2289
2443
22
10852
10835
426.1
4176
3
2720
2865
23
11278
11253
426.0
4175
4
3150
3286
24
11704
11670
425.9
4174
5
3580
3708
429.8
4212
25
12130
12088
425.8
4173
6
4009
4129
429.5
4209
26
12556
12505
425.7
4172
7
4439
4550
4293
4207
27
12982
12922
425.6
4171
8
4868
4970
429.0
4204
28
13407
13339
425.6
4171
9
6297
5390
428.8
4202
29
13833
13756
425.5
4170
10
5726
6811
428.5
4200
30
14258
14173
425.6
4171
11
6154
6230
428.3
4198
31
14684
14950
425.6
4171
12
6582
6650
428.1
4196
32
16110
15008
425.6
4171
13
7010
7070
427.9
4194
33
15535
15425
425.7
4172
14
7438
7489
427.7
4192
34
15961
15842
425.7
4172
15
7865
7908
427.4
4189)
35
16387
16259
425.8
4173
16
8293
8327
427.2
4187
36
16812
16676
425.8
4173
17
8720
8745
427.0
4185
37
17238
17094
18
9147
9164
426.8
4183
38
17664
17511
19
9574
9582
426.6
4181
89
18091
17930
20
10000
10000
426.4
4179
40
18517
18347
21
10426
10418 426.2
4177
41
18943
18765
TABLE LV. QUANTITY TO ADD TO THE EQUIVALENT AT BALTIMOBE TO
EEDUCE TO ANT LATITUDE.
Latitude.
Addition in
Kilogramme-Meters.
+ 0.89
10
+ 0.82
20
+ 0.63
30
+ 0.34
40
+ 0.08
50
0.41
60
-0.77
70
1.06
80
1.26
90
. 1.33
Manchester 0.5; Paris 0.4; Berlin 0.6.
OF ARTS AND SCIENCES. 197
V. CONCLUDING REMARKS. AND CRITICISM OF RESULTS
AND METHODS.
On looking over the last four columns of Table LIU., which gives
the results of the experiments as expressed in terms of the different
mercurial thermometers, we cannot but be impressed with the unsatis-
factory state of the science of thermometry at the present day, when
nearly all physicists accept the mercurial thermometer as the standard
between and 100. The wide discrepancy in the results of calori-
metric experiments requires no further explanation, especially when
physicists have taken no precaution with respect to the change of zero
after the heating of the thermometer. They show that thermometry
is an immensely difficult subject, and that the results of all physicists
who have not made a special study of their thermometers, and a com-
parison with the air thermometer, must be greatly in error, and should
be rejected in many cases. And this is specially the case where
Geissler thermometers have been used.
The comparison of my own thermometers with the air thermometer
is undoubtedly by far the best so far made, and I have no improve-
ments to offer beyond those P have already mentioned in the " Ap-
pendix to Thermometry." And I now believe that, with the improve-
ment to the air thermometer of an artificial atmosphere of constant
pressure, we could be reasonably certain of obtaining the tempera-
ture at any point up to 50 C. within O.01 C. from the mean of two
or three observations. I believe that my own thermometers scarcely
differ much more than that from the absolute scale at any point up to
40 C., but they represent the mean of eight observations. However,
there is an uncertainty of O.01 C. at the 20 point, owing to the un-
certainty of the value of m. But taking m = .00015, I hardly think
that the point is uncertain to more than that amount for the ther-
mometers Nos. 6163, 6165, and 6166.
As to the comparison of the other thermometers, it is evidently
unsatisfactory, as they do not read accurately enough. However, the
figures given in Table LIU. are probably very nearly correct.
The study of the thermometers from the different makers intro-
duces the question whether there are any thermometers which stand
below the air thermometer between and 100. As far as I can find,
nobody has ever published a table showing such a result, although
Boscha infers that thermometers of " Cristal de Choisy-le-Roi " should
stand below, and his inference has been accepted by Regnault. But
it does not seem to have been proved by direct experiment. My
198 PROCEEDINGS OP THE AMERICAN ACADEMY
Baudin thermometers seem to contain lead as far as one can tell from
the blackening in a gas flame, but they stand very much above the air
thermometer at 40. I have since tried some of the Baudin ther-
mometers up to 300, and find that they stand below the air thermom-
eter between 100 and 240 ; they coincide at about 240, and stand
above between 240 and 300. This is very nearly what Regnault
found for " Verre Ordinaire." It is to be noted that the formula
obtained from experiments below 100 makes them coincide at 233,
which is remarkably close to the result of actual experiment, especially
as it would require a long series of experiments to determine the
point within 10.
The comparison of thermometers also shows that all thermometers
in accurate investigations should be used as thermometers with arbitrary
scales, neither the position of the zero point nor the interval between
the and 100 points being assumed correct. The text books only
give the correction for the zero point, but my observations show that
the interval between the and 100 points is also subject to a secular
change as well as to the temporary change due to heating. Of all
the thermometers used, the Geissler is the worst in this as in other
respects, except accuracy of calibration, in which it is equal to most of
the others.
The experiments on the specific heat of water show an undoubted
decrease as the temperature rises, a fact which will undoubtedly sur-
prise most physicists as much as it surprised me. Indeed, the dis-
covery of this fact put back the completion of this paper many months,
as I wished to make certain of it. There is now no doubt in my mind,
and I put the fact forth as proved. The only way in which an error
accounting for this decrease could have been made appears to me to
be in the determination of m in " Thermometry." The determination
of m rests upon the determination of a difference of only O.05 C. be-
tween the air thermometer and the mercurial, the and 40 points
coinciding, and also upon the comparison of the thermometers with
others whose value of m was known, as in the Appendix. Although
the quantity to be measured is small, yet there can be no doubt at least
that m is larger than zero ; and if so, the specific heat of water
certainly has a minimum at about 30.
One point that might be made against the fact is that the Kew
standard, Table L., gives less change than the others. But the cali-
bration of the Kew standard, although excellent, could hardly be
trusted to 0.02 or 0.03 C., as the graduation was only to J F. In
drawing the curve for the difference between the Kew standard and
OP ARTS AN.D SCIENCES. 199
the air thermometers, I ignored small irregularities and drew a regu-
lar curve. On looking over the observations again, I see that, had I
taken account of the small irregularities, it would have made the ob-
servations agree more nearly with the other thermometers. Hence
the objection vanishes. However, I intend working up some obser-
vations which I have with the Kew standard at a higher temperature,
and shall publish them at a future time.
There is one other error that might produce an apparent decrease
in the specific heat, and that is the slight decrease in the torsion weight
from the beginning to the end of most of the experiments, probably due
to the slowing of the engine. By this means the torsion circle might
lag behind. I made quite an investigation to see if this source of error
existed, and came to the conclusion that it produced no perceptible
effect. An examination of the different experiments shows this also ;
for in some of them the weight increases instead of decreases. See
Tables XXXVII. to L.
The error from the formation of dew might also cause an apparent
decrease ; but I have convinced myself by experiment, and others can
convince themselves from the tables, that this error is also inappre-
ciable.
The observations seem to settle the point with regard to the specific
heat at the 4 point within reasonable limits. There does not seem
to be a change to any great extent at that point, but the specific
heat decreases continuously through that point. It would hardly be
possible to arrive at this so accurately as I have done by any method
of mixture, for Pfaundler and Platter, who examined this point, could
not obtain results within one per cent, while mine show the fact
within a fraction of one per cent.
The point of minimum cannot be said to be known, though I have
placed it provisionally between 30 and 35 P C., but it may vary much
from that.
The method of obtaining the specific heat of the calorimeter seems
to be good. The use of solder introduces an uncertainty, but it is too
small to affect the result appreciably. The different determinations
of the specific heat of the calorimeter do not agree so well as they
might; but the error in the equivalent resulting from this error is very
small, and, besides, the mean result agrees well with the calculated
result. It may be regarded as satisfactory.
The apparatus for determining the equivalent could scarcely be
improved much, although perhaps the record of the torsion might be
made automatic and continuous. The experiment, however, might be
200 PROCEEDINGS OP THE AMERICAN ACADEMY
improved in two ways ; first, by the use of a motive power more
regular in its action ; and, second, by a more exact determination of
the loss due to radiation. The effect of the irregularity of the engine
has been calculated as about 1 in 4,000, and I suppose that the error
due to it cannot be as much as that after applying the correction.
The error due to radiation is nearly neutralized, at least between
and 30, by using the jacket at different temperatures. There may
be an error of a small amount at that point (30) in the direction of
making the mechanical equivalent too great, and the specific heat may
keep on decreasing to even 40.
Between the limits of 15 and 25 I feel almost certain that no
subsequent experiments will change my values of the equivalent so
much as two parts in one thousand, and even outside those limits, say
between 10 and 30, I doubt whether the figures will ever be changed
much more than that amount.
It is my intention to continue the experiments, as well as work up
the remainder of the old ones. I shall also use some liquids in the
calorimeter other than water, and so have the equivalent in terms of
more than one fluid.
BALTIMORE, 1878-79. Finished May 27, 1879.
NOTE FROM PROFESSOR ROWLAND.
[Comparison with Dr. Joule's Thermometer.]
Dr. Joule has kindly sent me the comparison of my thermometer, No.
6166, with his, and the result will be published in full in the " Proceed-
ings of the American Academy of Sciences." In this manner I have
been able to make the exact reduction of his results to the air thermo-
meter. The following are the results:
DATE.
METHOD.
Temperature
of water.
o a
t
Joule's value re-
duced to the air
thermometer,and
to the latitude of
Baltimore.
Kowland's
value.
.
Diflerence.
English
System.
Metric
System.
1847
Friction of water. . .
15.
781.5
787.0
442.8
427.4
+15.4
1850
" ..
14.
772.7
778.0
426.8
427.7
.9
"
" " mercury
9.
772.8
779.2
427.5
428.8
1.3
.
9.
775.4
781.4
428.7
428.8
1
" " iron
9.
776.0
782.2
429.1
428.8
+ .3
"
" " ....
9.
773.9
780.2
428.0
428.8
- .8
1867
Electric heating...
18.6
428.0
426.7
+ 1.8
1878
Friction of water. . .
14.7
772.7
776.1
425.8
427.6
1.8
u _
12.7
774.6
778.5
427.1
428.0
.9
,.
15.5
773.1
776.4,
426.0
427.3
1.3
' . j .< <. .. m
14.5
767.0
770.5
422.7
427.5
4.8
' ..
17.3
774.0
777.0
426.3
426.9
.6
The mean difference of the two amounts to only 1 in 430, almost exactly
the same as I estimated in the body of my paper, an extremely satisfac-
tory result.
H. A. ROWLAND.
BALTIMORE, February 16th, 1880.
lilfiill
AA 001204057 2