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leads have substantial terminals and the resistances of these leads, in thousandths of an ohm, is stamped on the terminals. Their values are always included in making up the corrected resistances of the bridge. The measurements shown in Tables I and II, made March 23, illustrate the results obtained in the determination of inductances of one henry and one-tenth henry. An electromotive force of about 50 volts was employed on the bridge.

Fig. 6 shows how the commutator was connected to the bridge so as to reverse P and R, which are equal. Formula (2) in this case

(QS) reduces to

L=CS (2r+P).

Columns 4 and 5 give the nominal values of r in the two positions of the commutator, and column 5 the mean value, corrected from the

results of the latest comparison of the resistances with standard resistances.

Column 9 gives the capacity of the condenser C. Where two inductances are measured in series the measured sum is given in column 10, and the sum of the separate values are given in the next column. The last column gives the differences between these measured values and the sums of the separate values. While these differences are very small, averaging less than one in ten thousand, they are appreciable and always positive. This indicates that there may be some constant source of error in the bridge.

5. SOURCES OF ERROR.

The results given above show that measurements of inductance of very great precision can be made by Anderson's method, provided there are no constant errors of appreciable magnitude entering into

the results. Such errors might be due to any one of the following

causes:

(a) The residual inductance or capacity of the resistance r and of the arms of the bridge (not including, of course, the inductance of the coil in which is being measured) may introduce a constant error in L. As stated above, we have always made the arms P and R equal in value and reversed them by a commutator, in order to eliminate any difference there may be in their resistances or in their inductances or capacities. But the differences between Q and S can not thus be eliminated. The total resistance of Q is, of course, equal to S, since PR (except for a small change due to residual inductances, to be discussed later), but part of this is in the inductive coil itself. Residual inductance in the noninductive part of Q makes the measured value of L too large, and, conversely, capacity would make it too small. The effect of inductance or capacity in S is of opposite sign to that of Q, and hence if the resistances of Q and S are similar, that is, made up as far as possible of the same kind of coils, then their effects will balance except for that part of S which equals the resistance of the inductive coil. In our work & was fixed in any given S case, and was varied to secure a balance; thus usually contains a number of small resistances in addition to the slide wire, and these can not be counterbalanced exactly by S.

(b) The inductive coils must be removed some distance from the bridge and from each other when two or more are measured at once. This requires leads of one to three meters in length (for the larger inductances), and these leads may affect the measured value of the inductance. If they are close together, so as to be noninductive (as twisted lamp cord, for instance), they possess an appreciable capacity; and if far enough apart to be free from capacity, they possess measurable inductance. In measuring small inductance coils the capacity effect is small, and it is better to have the leads close together and as short as is safe. With large inductances the capacity of the leads is more important, and it is better to have them farther apart, to reduce it to a minimum. The inductance of the leads can then be calculated (or separately measured) and applied as a correction, if desired, or the same leads may always be employed with a given coil and considered as a part of the coil. The inductance of the wires joining the condenser to the bridge tends to reduce the capacity in the ratio of pl to or p❜le to unity, where 7 is the small inductance of the leads; on the other hand, if these leads are close together, their capacity is added to that of the condenser. In our experiments, where the leads

1

pc

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were short and wide apart, both these effects were inappreciable. But if currents of high frequency are used, particularly with large capacity, the error due to the inductance of the leads may be appreciable; on the other hand, with small condenser capacity, the error due to the capacity of lead wires near together (as a twisted lamp cord) may be considerable and of opposite sign to the other. In precision measurements, therefore, care should be taken that no error is introduced in this manner.

(c) The inductive coil itself has a certain electrostatic capacity which modifies its measured inductance by an amount depending on the frequency of the current and the inductance of the coil, as well as its capacity." The approximate value of the expression for the measured inductance L' in terms of the true inductance Lis

where c is the electrostatic capacity of the coil and p=27 times the frequency. In practice, e is found by measuring L' at two different frequencies; it is too small to be important for the smaller values of inductance at low frequencies. One of our inductance coils, having an inductance of 1 henry, has a capacity of 1×10-10 farads. For a frequency of 112, this value makes the correction term p2 CL in the above expression .00005, a quantity which can be detected, but which is not a large error. If, however, the frequency were ten times as great, this term would become .005, a very important correction.

The electrostatic capacity of a coil can be made relatively small by winding it in a deep channel, so that there are many layers and comparatively few turns in a layer. This, however, reduces its inductance, and in practice it is better not to depart very far from the form giving maximum inductance. The electrostatic capacity of the cord, as already pointed out, increases the value of this correction term.

(d) The capacity of the condenser, as already stated, can be determined with very great accuracy, and by taking careful account of the temperature of the condenser and its temperature coefficient, there will be very little uncertainty in the value of the capacity. The question remains, however, as to what effect the absorption in the condenser produces on the measured value of the inductance when used in Anderson's method. The effect of absorption is to cause the current to lag a little behind its phase in a perfect condenser. That is, it is in advance of the electromotive force by a little less than 90°. We give below a theoretical investigation of this question, and also

a Wien: Ann. d. Phys., 44, p. 711; 1891. Dolezalek: Ann. d. Phys., 12, p. 1153; 1903.