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 dif