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CALIBRATION 499 and so on, we obtain at once the value of each weight in terms of be omitted. (1) The determination of the value of the preceding, so that all may be expressed in terms of the largest, unit to which the measurements are referred by comparison the which is most conveniently taken as the standard with a standard unit of the same kind. This is often B=A/2 + (b-a)/2, C=B/2 + (c-b)2, kc. . (2). described as the Standardisation of the instrument, or the The advantages of this method of subdivision and comparison, in determination of the Reduction factor. (2) The verificato its extreme simplicity, are (1) that there is only one tion of the accuracy of the subdivision of the scale of the addition possible combination to represent any given weight within the instrument. This may be termed calibration of the scale, range of the series ; (2) that the least possible number of weights and does not necessarily involve the comparison of the is required to cover any given range ; (3) that the smallest instrument with any independent standard, but merely the number of substitutions is required for the complete calibration. verification of the accuracy of the relative values of its These advantages are important in cases where the accuracy of is limited by the constancy of the conditions of observaindications. In many cases the process of calibration calibration tion, as in the case of an electrical resistance-box, but the reverse adopted consists in the comparison of the instrument to be may be the case Avhen it is a question of accuracy of estimation by tested with a standard over the whole range of its indica- an observer. In the majority of cases the ease of numeration afforded by tions, the relative values of the subdivisions of the standard with the decimal system is the most important conitself having been previously tested. In this case the familiarity sideration. The most convenient arrangement on the decimal distinction of two parts in the process is unnecessary, and system for purposes of calibration is to have the units, tens, the term calibration is for this reason frequently employed hundreds, &c., arranged in groups of four adjusted in the proporto include both. In some cases it is employed to denote tion of the numbers 1, 2, 3, 4. The relative values of the weights each group of four can then be determined by substitution inthe first part only, but for greater clearness and convenience in dependently of the others, and the total of each group of four, of description we shall restrict the term as far as possible making ten times the unit of the group, can be compared with the to the second meaning. smallest weight in the group above. This gives a sufficient number of equations to determine the errors of all the weights by The methods of standardization or calibration employed have the method of substitution in a very simple manner. A number much in common even in the cases that appear most diverse. of other equations can be obtained by combining the different They are all founded on the axiom that “things which are equal groups in other ways, and the whole system of equations may then to the same thing are equal to one another.” Whether it is a be solved by the method of least squares ; but the equations so question of comparing a scale with a standard, or of testing the obtained are not all of equal value, and it may be doubted whether equality of two parts of the same scale, the process is essentially any real advantage is gained in many cases by the multiplication one of interchanging, or substituting one for the other, the two of comparisons, since it is not possible in this manner to eliminate things to be compared. In addition to the things to be tested constant errors or personal equation, which are generally aggrathere is usually required some form of balance, or comparator, or vated by prolonging the observations. A common arrangement of gauge, by which the equality may be tested. The simplest of such the weights in each group on the decimal system is 5, 2, 1, 1, or comparators is the instrument known as the callipers, from the 5, 2, 2, 1. These do not admit of the independent calibration of same root as calibre, which is in constant use in the workshop for each group by substitution. The arrangement 5, 2, 1, 1, 1, or testing equality of linear dimensions, or uniformity of diameter of 5, 2, 2, 1,1, permits independent calibration, but involves a larger tubes or rods. The more complicated forms of optical comparators * number of weights and observations than the 1, 2, 3, 4, grouping. or measuring machines with scales and screw adjustments, are The arrangement of ten equal weights in each group, which is essentially similar in principle, being finely adjustable gauges to adopted in “dial ” resistance-boxes, and in some forms of chemical which the things to be compared can be successively fitted. A balances where the weights are mechanically applied by turning still simpler and more accurate comparison is that of volume or a handle, presents great advantages in point of quickness of capacity, using a given mass of liquid as the gauge or test of manipulation and ease of numeration, but the complete calibraequality, which is the basis of many of the most accurate and most tion of such an arrangement is tedious, and in the case of a important methods of calibration. The common balance, for test- resistance-box it is difficult to make the necessary connexions. ing equality of mass or weight, is so delicate and so easily tested, In all cases where the same total can be made up in a variety of that the process of calibration may frequently with advantage be it is necessary in accurate work to make sure that the reduced to a series of weighings, as for instance in the calibration ways, same weights are always used for a given combination, or else to of a burette or measure-glass by weighing the quantities of mercury record the actual weights used on each occasion. In many inrequired to fill it to different marks. The balance may, however, vestigations where time enters as one of the factors, this is a serious be regarded more broadly as the type of a general method capable drawback, and it is better to avoid the more complicated arrangeof the widest application in accurate testing. It is possible, for ments. The accurate adjustment of a set of weights is so simple instance, to balance two electromotive forces or two electrical a matter that it is often possible to neglect the errors of a wellresistances against each other, or to measure the refractivity of a made set, and no calibration any value without the most gas by balancing it against a column of air adjusted to produce scrupulous attention to details isof ofmanipulation, and particularly the same retardation in a beam of light. These ‘ ‘ equilibrium, ” or to the correction for the air displaced in comparing weights of “null,” or “balance” methods of comparison afford the most different materials. Electrical resistances are much more difficult accurate measurements, and are generally selected if possible as to adjust owing to the change of resistance with temperature, the basis of any process of calibration. In spite of the great the calibration of a resistance-box can seldom be neglectedand on diversity in the nature of the things to be compared, the funda- account of the changes of resistance which are liable to occur after mental principles of the methods employed are so essentially adjustment from imperfect annealing. It is also necessary to similar that it is possible, for instance, to describe the testing of a remember that the order of accuracy required and the actual values set of weights, or the calibration of an electrical resistance-box, in of the smaller resistances depend to some extent on the method of almost the same terms, and to represent the calibration correction of connexion, and that the box must be calibrated with due regard a mercury thermometer or of an ammeter by precisely similar curves. to the conditions under which it is to be used. Otherwise the Method of Substitution.—In comparing two units of the same of procedure is much the same as in the case of a box of kind and of nearly equal magnitude, some variety of the general method but it is necessary to pay more attention to the constancy method of substitution is invariably adopted. The same method weights, and uniformity of the temperature conditions of the observingin a more elaborate form is employed in the calibration of a series room. of multiples or submultiples of any unit. The details of the Method of Equal Steps. — In calibrating a continuous scale method depend on the system of subdivision adopted, which is to divided into a number of divisions of equal length, such as a metre some extent a matter of taste. The simplest method of subdivision scale divided in millimetres, or a thermometer tube divided in is that on the binary scale, proceeding by multiples of 2. With a degrees of temperature, or an electrical slide-wire, it is usual to pair of submultiples of the smallest denomination and one of each proceed by a method of equal steps. The simplest method is that of the rest, thus 1, 1, 2, 4, 8, 16, &c., each weight or multiple is known as the method of Gay-Lussac calibration of mercurial equal to the sum of all the smaller weights, which may be substi- thermometers or tubes of small the It is essentially a method tuted for it, and the small difference, if any, observed. If we call of substitution employing a column of mercury of constant volume the weights A, B, C, &c., whei'e each is approximately double the as the gauge for comparing the capacities of different parts of the following weight, and if we write a for observed excess of A over tube. A precisely similar method, employing a pair of microscopes the rest of the weights, b for that of i?over C+ D + kc., and soon, at a fixed distance apart as a standard of length, is applicable to the observations by the method of substitution give the series of the calibration of a divided scale. The interval to be calibrated equations, is divided into a whole number of equal steps or sections, the ^-rest =a, i?-rest = &, C'-rest = c, &c. . (1). points of division at which the corrections are to be determined Subtracting the second from the first, the third from the second, i are called points of calibration.