The advantages of this method of subdivision and comparison, in
addition to its extreme simplicity, are (1) that there is only one
possible combination to represent any given weight within the range of
the series; (2) that the least possible number of weights is required
to cover any given range; (3) that the smallest number of
substitutions is required for the complete calibration. These
advantages are important in cases where the accuracy of calibration is
limited by the constancy of the conditions of observation, as in the
case of an electrical resistance-box, but the reverse may be the case
when it is a question of accuracy of estimation by an observer.
In the majority of cases the ease of numeration afforded by
familiarity with the decimal system is the most important
consideration. The most convenient arrangement on the decimal system
for purposes of calibration is to have the units, tens, hundreds, &c.,
arranged in groups of four adjusted in the proportion of the numbers
1, 2, 3, 4. The relative values of the weights in each group of four
can then be determined by substitution independently of the others,
and the total of each group of four, making ten times the unit of the
group, can be compared with the smallest weight in the group above.
This gives a sufficient number of equations to determine the errors of
all the weights by the method of substitution in a very simple manner.
A number of other equations can be obtained by combining the different
groups in other ways, and the whole system of equations may then be
solved by the method of least squares; but the equations so obtained
are not all of equal value, and it may be doubted whether any real
advantage is gained in many cases by the multiplication of
comparisons, since it is not possible in this manner to eliminate
constant errors or personal equation, which are generally aggravated
by prolonging the observations. A common arrangement of the weights in
each group on the decimal system is 5, 2, 1, 1, or 5, 2, 2, 1. These
do not admit of the independent calibration of each group by
substitution. The arrangement 5, 2, 1, 1, 1, or 5, 2, 2, 1, 1, permits
independent calibration, but involves a larger number of weights and
observations than the 1, 2, 3, 4, grouping. The arrangement of ten
equal weights in each group, which is adopted in "dial"
resistance-boxes, and in some forms of chemical balances where the
weights are mechanically applied by turning a handle, presents great
advantages in point of quickness of manipulation and ease of
numeration, but the complete calibration of such an arrangement is
tedious, and in the case of a resistance-box it is difficult to make
the necessary connexions. In all cases where the same total can be
made up in a variety of ways, it is necessary in accurate work to make
sure that the same weights are always used for a given combination, or
else to record the actual weights used on each occasion. In many
investigations where time enters as one of the factors, this is a
serious drawback, and it is better to avoid the more complicated
arrangements. The accurate adjustment of a set of weights is so simple
a matter that it is often possible to neglect the errors of a
well-made set, and no calibration is of any value without the most
scrupulous attention to details of manipulation, and particularly to
the correction for the air displaced in comparing weights of different
materials. Electrical resistances are much more difficult to adjust
owing to the change of resistance with temperature, and the
calibration of a resistance-box can seldom be neglected on account of
the changes of resistance which are liable to occur after adjustment
from imperfect annealing. It is also necessary to remember that the
order of accuracy required, and the actual values of the smaller
resistances, depend to some extent on the method of connexion, and
that the box must be calibrated with due regard to the conditions
under which it is to be used. Otherwise the method of procedure is
much the same as in the case of a box of weights, but it is necessary
to pay more attention to the constancy and uniformity of the
temperature conditions of the observing-room.
_Method of Equal Steps_.--In calibrating a continuous scale divided
into a number of divisions of equal length, such as a metre scale
divided in millimetres, or a thermometer tube divided in degrees of
temperature, or an electrical slide-wire, it is usual to proceed by a
method of equal steps. The simplest method is that known as the method
of Gay Lussac in the calibration of mercurial thermometers or tubes of
small bore. It is essentially a method of substitution employing a
column of mercury of constant volume as the gauge for comparing the
capacities of different parts of the tube. A precisely similar method,
employing a pair of microscopes at a fixed distance apart as a
standard of length, is applicable to the calibration of a divided
scale. The interval to be calibrated is divided into a whole number of
equal steps or sections, the points of division at which the
corrections are to be determined are called _points of calibration._
_Calibration of a Mercury Thermometer_.--To facilitate description, we
will take the case of a fine-bore tube, such as that of a thermometer,
to be calibrated with a thread of mercury. The bore of such a tube
will generally vary considerably even in the best standard
instruments, the tubes of which have been specially drawn and
selected. The correction for inequality of bore may amount to a
quarter or half a degree, and is seldom less than a tenth. In ordinary
chemical thermometers it is usual to make allowance for variations of
bore in graduating the scale, but such instruments present
discontinuities of division, and cannot be used for accurate work, in
which a finely-divided scale of equal parts is essential. The
calibration of a mercury thermometer intended for work of precision is
best effected after it has been sealed. A thread of mercury of the
desired length is separated from the column. The exact adjustment of
the length of the thread requires a little manipulation. The
thermometer is inverted and tapped to make the mercury run down to the
top of the tube, thus collecting a trace of residual gas at the end of
the bulb. By quickly reversing the thermometer the bubble passes to
the neck of the bulb. If the instrument is again inverted and tapped,
the thread will probably break off at the neck of the bulb, which
should be previously cooled or warmed so as to obtain in this manner,
if possible, a thread of the desired length. If the thread so obtained
is too long or not accurate enough, it is removed to the other end of
the tube, and the bulb further warmed till the mercury reaches some
easily recognized division. At this point the broken thread is
rejoined to the mercury column from the bulb, and a microscopic bubble
of gas is condensed which generally suffices to determine the
subsequent breaking of the mercury column at the same point of the
tube. The bulb is then allowed to cool till the length of the thread
above the point of separation is equal to the desired length, when a
slight tap suffices to separate the thread. This method is difficult
to work with short threads owing to deficient inertia, especially if
the tube is very perfectly evacuated. A thread can always be separated
by local heating with a small flame, but this is dangerous to the
thermometer, it is difficult to adjust the thread exactly to the
required length, and the mercury does not run easily past a point of
the tube which has been locally heated in this manner.
Having separated a thread of the required length, the thermometer is
mounted in a horizontal position on a suitable support, preferably
with a screw adjustment in the direction of its length. By tilting or
tapping the instrument the thread is brought into position
corresponding to the steps of the calibration successively, and its
length in each position is carefully observed with a pair of reading
microscopes fixed at a suitable distance apart. Assuming that the
temperature remains constant, the variations of length of the thread
are inversely as the variations of cross-section of the tube. If the
length of the thread is very nearly equal to one step, and if the tube
is nearly uniform, the average of the observed lengths of the thread,
taking all the steps throughout the interval, is equal to the length
which the thread should have occupied in each position had the bore
been uniform throughout and all the divisions equal. The error of each
step is therefore found by subtracting the average length from the
observed length in each position. Assuming that the ends of the
interval itself are correct, the correction to be applied at any point
of calibration to reduce the readings to a uniform tube and scale, is
found by taking the sum of the errors of the steps up to the point
considered with the sign reversed.
Table I.--_Calibration by Method of Gay Lussac_.
+----------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
| No. of | | | | | | | | | | |
| Step. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
+----------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
| Ends of |+.010|-.016|-.020|-.031|+.016|+.008|+.013|+.017|+.004|-.088|
| thread. |+.038|+.017|-.003|-.022|+.010|+.005|+.033|+.018|+.013|-.003|
| | | | | | | | | | | |
| Excess- |-.028|-.033|-.017|-.009|+.006|-.003|-.020|-.001|-.004|+.005|
| Length | | | | | | | | | | |
| Error of |-17.6|-22.6|- 6.6|+ 1.4|+16.4|+ 7.4|- 9.6|+ 9.4|+ 6.4|+15.4|
| step. | | | | | | | | | | |
| Correc- |+17.6|+40.2|+46.8|+45.4|+29.0|+21.6|+31.2|+21.8|+15.4| 0 |
| tion. | | | | | | | | | | |
+----------+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+
In the preceding example of the method an interval of ten degrees is
taken, divided into ten steps of 1 deg. each. The distances of the
ends of the thread from the nearest degree divisions are estimated by
the aid of micrometers to the thousandth of a degree. The error of any
one of these readings probably does not exceed half a thousandth, but
they are given to the nearest thousandth only. The excess length of
the thread in each position over the corresponding degree is obtained
by subtracting the second reading from the first. Taking the average
of the numbers in this line, the mean excess-length is -10.4
thousandths. The error of each step is found by subtracting this mean
from each of the numbers in the previous line. Finally, the
corrections at each degree are obtained by adding up the errors of the
steps and changing the sign. The errors and corrections are given in
thousandths of 1 deg.
_Complete Calibration._--The simple method of Gay Lussac does very
well for short intervals when the number of steps is not excessive,
but it would not be satisfactory for a large range owing to the
accumulation of small errors of estimation, and the variation of the
personal equation. The observer might, for instance, consistently
over-estimate the length of the thread in one half of the tube, and
under-estimate it in the other. The errors near the middle of the
range would probably be large. It is evident that the correction at
the middle point of the interval could be much more accurately
determined by using a thread equal to half the length of the interval.
To minimize the effect of these errors of estimation, it is usual to
employ threads of different lengths in calibrating the same interval,
and to divide up the fundamental interval of the thermometer into a
number of subsidiary sections for the purpose of calibration, each of
these sections being treated as a step in the calibration of the
fundamental interval. The most symmetrical method of calibrating a
section, called by C.E. Guillaume a "Complete Calibration," is to use
threads of all possible lengths which are integral multiples of the
calibration step. In the example already given nine different threads
were used, and the length of each was observed in as many positions as
possible. Proceeding in this manner the following numbers were
obtained for the excess-length of each thread in thousandths of a
degree in different positions, starting in each case with the
beginning of the thread at 0 deg., and moving it on by steps of 1 deg.
The observations in the first column are the excess-lengths of the
thread of 1 deg. already given in illustration of the method of Gay
Lussac. The other columns give the corresponding observations with the
longer threads. The simplest and most symmetrical method of solving
these observations, so as to find the errors of each step in terms of
the whole interval, is to obtain the differences of the steps in pairs
by subtracting each observation from the one above it. This method
eliminates the unknown lengths of the threads, and gives each
observation approximately its due weight. Subtracting the observations
in the second line from those in the first, we obtain a series of
numbers, entered in column 1 of the next table, representing the
excess of step (1) over each of the other steps. The sum of these
differences is ten times the error of the first step, since by
hypothesis the sum of the errors of all the steps is zero in terms of
the whole interval. The numbers in the second column of Table III. are
similarly obtained by subtracting the third line from the second in
Table II., each difference being inserted in its appropriate place in
the table. Proceeding in this way we find the excess of each interval
over those which follow it. The table is completed by a diagonal row
of zeros representing the difference of each step from itself, and by
repeating the numbers already found in symmetrical positions with
their signs changed, since the excess of any step, say 6 over 3, is
evidently equal to that of 3 over 6 with the sign changed. The errors
of each step having been found by adding the columns, and dividing by
10, the corrections at each point of the calibration are deduced as
before.