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°, and moving it on by steps of 1°. The observations in the first column are the excess-lengths of the thread of 1° 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.

Table II.—*Complete Calibration of Interval of 10° in 10 Steps.*

Lengths of Threads. | 1° | 2° | 3° | 4° | 5° | 6° | 7° | 8° | 9° | |

Observed excess-lengths | 0° | −28 | −32 | −47 | −62 | −11 | −15 | −48 | − 2 | − 8 |

of threads, in various | 1° | −33 | −21 | −47 | −28 | +14 | − 8 | −22 | +21 | +24 |

positions, the beginning | 2° | −17 | + 2 | − 8 | + 1 | +26 | +23 | + 6 | +58 | |

of the thread being set | 3° | − 9 | +26 | + 5 | − 3 | +41 | +36 | +28 | ||

near the points. | 4° | + 6 | +31 | − 7 | + 4 | +45 | +49 | |||

5° | − 3 | + 5 | −15 | − 6 | +43 | |||||

6° | −20 | + 7 | −16 | + 2 | ||||||

7° | − 1 | +23 | +10 | |||||||

8° | − 4 | +29 | ||||||||

9° | + 5 |

Table III.—*Solution of Complete Calibration.*

Step No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

1 | 0 | − 5 | +11 | +20 | +34 | +25 | + 7 | +26 | +23 | +32 |

2 | + 5 | 0 | +16 | +23 | +39 | +29 | +12 | +31 | +28 | +37 |

3 | −11 | −16 | 0 | + 8 | +24 | +13 | − 4 | +15 | +13 | +22 |

4 | −20 | −23 | − 8 | 0 | +15 | + 5 | −12 | + 7 | + 4 | +13 |

5 | −34 | −39 | −24 | −15 | 0 | − 9 | −26 | − 8 | −10 | − 2 |

6 | −25 | −29 | −13 | − 5 | + 9 | 0 | −17 | + 2 | − 1 | + 8 |

7 | − 7 | −12 | + 4 | +12 | +26 | +17 | 0 | +19 | +16 | +26 |

8 | −26 | −31 | −15 | − 7 | + 8 | − 2 | −19 | 0 | − 3 | + 6 |

9 | −23 | −28 | −13 | − 4 | +10 | + 1 | −16 | + 3 | 0 | + 9 |

10 | −32 | −37 | −22 | −13 | + 2 | − 8 | −26 | − 6 | − 9 | 0 |

Error of step. | −17.3 | −22.0 | − 6.4 | + 1.9 | +16.7 | + 7.1 | −10.1 | + 8.9 | + 6.1 | +15.1 |

Corrections. | +17.3 | +39.3 | +45.7 | +43.8 | +27.1 | +20.0 | +30.1 | +21.2 | +15.1 | 0 |

The advantages of this method are the simplicity and symmetry of the work of reduction, and the accuracy of the result, which exceeds that of the Gay Lussac method in consequence of the much larger number of independent observations. It may be noticed, for instance, that the correction at point 5 is 27.1 thousandths by the complete calibration, which is 2 thousandths less than the value 29 obtained by the Gay Lussac method, but agrees well with the value 27 thousandths obtained by taking only the first and last observations with the thread of 5°. The disadvantage of the method lies in the great number of observations required, and in the labour of adjusting so many different threads to suitable lengths. It is probable that sufficiently good results may be obtained with much less trouble by using fewer threads, especially if more care is taken in the micrometric determination of their errors.

The method adopted for dividing up the fundamental interval of any thermometer into sections and steps for calibration may be widely varied, and is necessarily modified in cases where auxiliary bulbs or “ampoules” are employed. The Paris mercury-standards, which read continuously from 0° to 100° C., without intermediate ampoules, were calibrated by Chappuis in five sections of 20° each, to determine the corrections at the points 20°, 40°, 60°, 80°, which may be called the “principal points” of the calibration, in terms of the fundamental interval. Each section of 20° was subsequently calibrated in steps of 2°, the corrections being at first referred, as in the example already given, to the mean degree of the section itself, and being afterwards expressed, by a simple transformation, in terms of the fundamental interval, by means of the corrections already found for the ends of the section. Supposing, for instance, that the corrections at the points 0° and 10° of Table III. are not zero, but *C*° and *C*′ respectively, the correction Cn at any intermediate point n will evidently be given by the formula,

*C*

_{n}=

*C*° + c

_{n}+ (

*C*′ −

*C*°)

*n*/10 . . (3)

where *c*_{n} is the correction already given in the table.

If the corrections are required to the thousandth of a degree, it is necessary to tabulate the results of the calibration at much more frequent intervals than 2°, since the correction, even of a good thermometer, may change by as much as 20 or 30 thousandths in 2°. To save the labour and difficulty of calibrating with shorter threads, the corrections at intermediate points are usually calculated by a formula of interpolation. This leaves much to be desired, as the section of a tube often changes very suddenly and capriciously. It is probable that the graphic method gives equally good results with less labour.

*Slide-Wire.*—The calibration of an electrical slide-wire into parts of equal resistance is precisely analogous to that of a capillary tube into parts of equal volume. The Carey Foster method, employing short steps of equal resistance, effected by transferring a suitable small resistance from one side of the slide-wire to the other, is exactly analogous to the Gay Lussac method, and suffers from the same defect of the accumulation of small errors unless steps of several different lengths are used. The calibration of a slide-wire, however, is much less troublesome than that of a thermometer tube for several reasons. It is easy to obtain a wire uniform to one part in 500 or even less, and the section is not liable to capricious variations. In all work of precision the slide-wire is supplemented by auxiliary resistances by which the scale may be indefinitely extended. In accurate electrical thermometry, for example, the slide-wire itself would correspond to only 1°, or less, of the whole scale, which is less than a single step in the calibration of a mercury thermometer, so that an accuracy of a thousandth of a degree can generally be obtained without any calibration of the slide-wire. In the rare cases in which it is necessary to employ a long slide-wire, such as the cylinder potentiometer of Latimer Clark, the calibration is best effected by comparison with a standard, such as a Thomson-Varley slide-box.

*Graphic Representation of Results.*—The results of a calibration are often best represented by means of a correction curve, such as that illustrated in the diagram, which is plotted to represent the corrections found in Table III. The abscissa of such a curve is the reading of the instrument to be corrected. The ordinate is the correction to be added to the observed reading to reduce to a uniform scale. The corrections are plotted in the figure in terms of the whole section, taking the correction to be zero at the beginning and end. As a matter of fact the corrections at these points in terms of the fundamental interval were found to be −29 and −9 thousandths respectively. The correction curve is transformed to give corrections in terms of the fundamental interval by ruling a straight line joining the points +29 and +9 respectively, and reckoning the ordinates from this line instead of from the base-line. Or the curve may be replotted with the new ordinates thus obtained. In drawing the curve from the corrections obtained at the points of calibration, the exact form of the curve is to some extent a matter of taste, but the curve should generally be drawn as smoothly as possible on the assumption that the changes are gradual and continuous.

The ruling of the straight line across the curve to express the corrections in terms of the fundamental interval, corresponds to the first part of the process of calibration mentioned above under the term “Standardization.” It effects the reduction of the