It would be somewhat pedantic to point out that the argument as outlined here is applicable to tensors of any rank provided they are alternating. Indeed, it may not be too much to say that alternating tensors suffice for physics; the underlying reason therefore being that the elements of content, i. e. area, volume, etc., are described by means of Jacobian determinants, which are in their very nature alternating. The obvious rejoinder to such a remark is a reference to the contravariant stress-tensor X", which is, as we have remarked above, symmetric. However, the stresstensor should really be regarded in its triply-labelled form Xp, the doubly-labelled form X being artificial. To say that X32X23, for instance, is to say that X13=X813, or, equivalently, that X123+X133=0, since the stress-tensor is alternating in its covariant labels. Expressed in tensor form, this says that the result of contracting the triplylabelled stress-tensor is the zero-tensor, i. e.

Xpa = 0.

A similar remark would apply to the symmetrical tensor Gr., which, in Einstein's theory, vanishes in space at points free from matter. This may be presented as a tensor of rank four, Gimn, which is alternating in the covariant labels. The condition of symmetry is

Gima = 0.

LXXV. A General Theorem on Screened Impedances. By RAYMOND M. WILMOTTE, B.A. (of the National Physical Laboratory)*.

SUMMARY.

FOR accurate measurements, especially at high frequencies, it is necessary that all impedances and apparatus used should be surrounded by metal screens kept at definite potentials. By this means, the leakage and capacity effects to earth can be rendered perfectly definite, and the apparatus electrically independent of the surrounding objects. Even when this is done, there is some ambiguity regarding the value of the impedance of the apparatus. The impedance is

* Communicated by the Author.

the ratio of the potential difference across the apparatus to the current. Now, both the potential difference and the current depend on the potential of the screen; moreover, the current may be measured at various points. One of the most usual values of the impedance is obtained when the screen is kept at the same potential as the terminal where the current is being measured, but is not directly connected to that terminal. On this definition of impedance it would be natural to expect that there would be two possible values, depending on which terminal is kept at the potential of the screen. In this paper the curious result is obtained that these two values are equal, and certain limitations are considered. This general result is of considerable importance, as it makes it unnecessary to connect the apparatus in any special way, so long as the screen is kept at the same potential as the terminal at which the current is measured.

Introduction.

NAPACITY effects to earth have brought about the universal use of metal screens surrounding any apparatus which is to be used for accurate measurements, especially at

high frequencies. The only case in which it is legitimate to do without metal screens is when the impedance of the apparatus is small.

Generally, then, an impedance has three terminals-one at each end, and one on the screen. Let these be A, B, and O respectively, as shown in fig. 1.

The value of the impedance is the ratio of the potential difference between A and B to the current. This will depend on the potential at which the shield O is kept relatively to A or B, and the point at which the current is measured. The screen S is usually connected to some point P in the circuit. This point P is not necessarily directly connected to A or B, but is instrumental in fixing the potential of O relative to A or B. For any setting

of the potential of the screen O relative to A, there will be two possible values of the impedance, namely,

If the screen O is given the same setting of potential but relative to B instead of to A, there should be another two values for the impedance:

It is proved in the next paragraph that the only setting of the potential of O which will make two of these values equal in the general case is when the potential of O is made equal to that of A or B, then

For any special case, of course, some of the values of the impedance can be made equal by suitable settings of the potential of the screen; for instance, if there is absolute symmetry about the centre of the impedance, it will also be found that

In this particular case, then, the values of the impedance can be be reduced to two.

The importance of the equality (1) in the general case is that the impedance of any piece of apparatus can be given without stating which terminal should be kept at the potential of the screen. The potential of the screen must be kept at the potential of one of the terminals A or B, and the screen must not be directly connected to that terminal. This latter condition is necessary, for if the point P coincides with A, the current measured would not be I, but Ip.

The condition, that the screen should be kept at the potential of one of the terminals while there is no direct connexion between the two, happens to be a very frequent condition of bridge measurements where a Wagner earth is used. It would be convenient, therefore, to standardize this particular value of the impedance so that only when other values are required should it be necessary to state the conditions of measurement.

Two other values, sometimes required, occur when the screen is directly connected to one of the terminals. That is, P coincides with either A or B. If P coincides with A, the current IB should be measured and not Ip. The reason for this is that, owing to the comparatively large size of the screen, appreciable current may flow via the earth to the screen from other apparatus in the circuit, and the current Ip will depend not only on the potential difference between A and B, but also on the circuit arrangement. On the other hand, the current ip is independent of the circuit, being equal to (is+io).

The three common values of an impedance can be represented diagrammatically by a network of three impedances, as shown in fig. 2. This method of representation may be useful in some cases (e. g., a condenser) in order to gain a physical conception why the different values exist; but, in general,

Fig. 2.

B

0000000

this is unnecessary, since in any case the complete data of the impedance of a piece of apparatus used in such a way that the screen is kept by some means or other at the potential of one of the terminals, must consist of at least three values.

The above results have been stated for a special case in a previous paper by L. Hartshorn and the author *, but no proof was given and the result was not generalized, nor were the limitations considered.

Proof.

Consider the impedance made of a network of admittances interconnecting the points 1, 2, 3, ... n, such that Y,, is the admittance joining the points r and s. The points 1 and n correspond to the terminals of the apparatus. Let a current Let V1, I enter at the point 1 and I, leave at the point n.

*L. Hartshorn & R. M. Wilmotte, "Note on Shielded Non-inductive Resistances," J. Sci. Inst. iv. pp. 33-37 (1926).

V2, V3, ..., Vn be the potentials of the junction points of the admittances. Let the screen be denoted by the point 0.

Since the the total current entering a junction point is zero, we have

I1 = (Vo−V1) Yo1 + (V2−V1)Y 12+(V3−V1)Y13+

...

+(Vn-V1)Yin +VnY1n+VeYo1.

+VnY2n+VoY02

n

−I2 = V1Y1n+V2Y2n+V3Ygn + ... — V2Ÿ (Yrn) + VoYon

n

0

Eliminating V2, V3, ..., Vn-1 from the first (n-1) equations,

we obtain

It should be noted that both and V, are symmetrical with respect to 1 and n, that is, 1 and n can be interchanged without changing their value. This symmetry does not