failing to consult Austin's original paper, he has lost sight of the facts and has attempted to prove something which arose solely owing to an unfortunate misunderstanding on the part of Prof. Love.

Mr. van der Pol says, "In discussing some experiments by Dr. Austin, Prof. Love makes use of the audibility factor defined as (R+S)/S, where S is the resistance of the shunt and R the telephone resistance." This is true, but unfortunately Prof. Love overlooked the fact that, although Austin gives the resistance of the telephone receiver as 600 ohms, he adds a footnote to the effect that "the inductive resistance of each telephone used in calculating the shunt ratio was 2000 ohms." No one can doubt that by inductive resistance Austin means impedance, so that both Austin and Hogan used the impedance in calculating the audibility factors which constitute the ordinates of the points in figs. 3 & 4 which Prof. Love reproduces from Hogan's paper. Prof. Love's Table IV. based on the simple resistance of the telephone receiver gives results which are quite incomparable with the plotted results of Hogan's observations. It should also be pointed out that Hogan definitely states that in his experiments R is the impedance of the telephone, as correctly quoted by Prof. Love (p. 126). For this reason his adverse criticism of the conclusions reached by Austin and Hogan needs revision.

If Table IV. is recalculated using the value of R employed by Austin, viz. 2000 ohms, the following results are obtained:

The last column shows that there is not the slightest foundation for Prof. Love's statement that "the results of these experiments are recorded by him [Austin] in a table which does not support the conclusion that the current is proportional to the square root of the audibility factor." In view of the experimental difficulties the proportionality, as indicated by the constancy of the values given in the last column, is wonderfully exact and speaks well for the experimental skill of those who carried out the measurements.

It is seen therefore that the peculiar relationship which

Prof. Love found between the received current and the audibility factor was due entirely to this oversight; his results involve the relationship between the impedance and the resistance of the telephone and would vary with different receivers. There can obviously be no simple relation between the antenna current and the audibility factor, unless the latter be calculated in such a way as to give correctly the ratio between the total sound-producing current and that fraction of it which passes through the telephone receiver. It is obvious, moreover, that for large values of (R+S)/S, the wrongly calculated audibility factor will also vary as the square of the antenna current, but that for small values of (R+S)/S, that is, for weak signals, the wrongly calculated audibility factor will vary less rapidly, as is clearly shown in the table given in my paper (Phil. Mag. Jan. p. 133).

This then is the simple explanation of Prof. Love's suggestions which the experiments of Mr. van der Pol were intended to test.

Mr. van der Pol's experiments have simply confirmed the fact that if, when using such detectors, one miscalculates the audibility factor by taking the resistance instead of the impedance of the telephone receiver, the result will have the peculiarity found by Prof. Love. This was obvious, however, without any experimental confirmation.

Mr. van der Pol's remark that "it is by no means clear whether Austin or Hogan employed the true impedance of their telephones, as in their papers no references at all are given how they determined these impedances" is obviously unjust in view of the work of these two experimenters. The table given above indicates, if it does not prove, that the impedance of the receiver has been fairly accurately determined.

no assump

In reply to Mr. van der Pol's statement that " tions as that made by Prof. Howe that the true impedance of the telephone under actual working conditions is equal to four times I gave 3 and 4 as alternatives] the steady resistance has been justified by any experiments," I wish to say that the figures given were not assumptions but measured values at the frequencies quoted. It will be noticed that Austin's receiver had an inductive resistance 3 times the steady current resistance.

With respect to the closing paragraph of the note I think that" the uncertainty attending the constants employed by Austin and Hogan and the difficulty of determining exact values" are by no means so great as Mr. van der Pol imagines; but even were it otherwise, I cannot agree that it

is "better to base the reduction of the observations on known. measurements rather than on assumptions as to the ratio of impedance to resistance," unless the known measurements are of some magnitude on which the observed phenomena depend. In the case in point, no uncertainty as to the exact wave-form can be regarded as a legitimate excuse for neglecting the fact that one is dealing with an alternating or pulsating current.

31st January, 1918.

South Kensington.

Yours truly,

G. W. O. HOWE.

IT

LVIII. A Doctrine on Material Stresses.

By R. F. GWYTHER*.

T is intended to provide a reasoned basis for a theory of stresses, applicable in the first case to a body which satisfies the geometrical conditions for acting as a Rigid Body (not being in a state of const aint), but which shall be capable of extension to Elastic Bodies or to other approximations to Natural Bodies, without introducing the fiction of "undisturbed" condition in which the body is assumed to be free from stress. The doctrine, for the development of which the several stages are indicated below, is the outcome of a series of papers on the "Specification of Stress," published by the Manchester Literary and Philosophical Society†, but it cannot be described as the motive of the papers, since it has formulated itself during the progress of the series.

an

The body contemplated is not supposed to be crystalline, fibrous, or annealed, and not to be subject to any special conditions either locally or at the surface. The body may, in the first instance, be regarded as either at rest or in motion under the geometrical conditions which define rigidity, and any modification of those geometrical conditions, such as elastic modifications, are to be deduced from the stresses in the initial instance, and from such definitions as may prove necessary in the sequel.

The stages by which the theory is developed are stated below. No analytical expressions are used, but reference is made to such analytical expressions at all the stages. This is inevitable, since the question is, at this stage, an analytical question.

* Communicated by the Author.

Manchester Memoirs, No. 10, vol. lvi. (1912), No. 5, vol. lvii. (1913), No. 5, vol. lviii. (1914), No. 14, vol. 1x. (1916), No. 1, vol lxii. (1917).

(1) The nine elements of mechanical stress obey wellknown laws of resolution. By introducing the notion of infinitesimal rotations of the coordinate axes about their own positions, we can, for our present purposes, replace these relations by the three differential-operators, acting upon the elements of stress, the determination of which follows from the laws of their resolution; that is, these operators are based on mechanical considerations.

(2) If we now consider any arbitrary vector (which we shall speak of conveniently as a virtual or potential displacement), we may look upon the nine first differential coefficients of its components as being replaced by the nine elements of virtual or potential strain (including among them the three rotations).

These nine virtual strains may now be affected by the same infinitesimal rotation of the axes as is employed in (1), and the three consequent differential-operators * acting upon the elements of strain may be deduced; in this case on geometrical grounds. It will be noticed that the forms of the two sets of operators are similar, and may easily be made identical.

(3) I shall now introduce the fundamental assumption on which the theory is based; that the elements of a material stress are functions of the first differential coefficients of the components of some vector quantity; in other words, functions of some set of nine virtual strains.

(4) If we now turn to the two sets of differential-operators concerning elements of stress and of virtual strain, we are able, in consequence of assumption (3), to express each element of stress in terms of the nine elements of virtual strain; or, conversely, each element of strain in terms of the nine elements of stress, by means of sets of simple partial differential equations.

In obtaining these solutions constants will be regarded as uniform and isotropic, and, consequently, we shall exclude, among other things, the possibility of a crystalline structure. We shall also suppose the body not to be in a state of constraint, or otherwise that that state has been eased. I shall, further, limit the solutions to relations of a linear form. From this it will follow that the relations between the elements of stress and of virtual strain agree, in form, with the corresponding relations familiar to us in the Theory of Elasticity.

(5) Supposing the elements of virtual strain to be expressed

Manchester Memoirs, No. 3, vol. ix. (1895), No. 1, vol. lxii. (1917).

in terms of the elements of stress, we may eliminate the former by differentiation, and obtain a set of relations involving only the elements of stress (stress-relations).

(6) I shall now suppose the set of three mechanical stressequations to be introduced which correspond to the state of rest or motion of the body as a rigid body. With these equations I shall suppose the stress-relations of the preceding paragraph to be combined. From this combination we may obtain Stress-Equations, by which we may regard the elements of stress to be defined: as far as they are capable of being defined so long as the surface-traction conditions have not been considered.

(7) Up to this point I have been contemplating such cases as a cube of metal on a rough inclined plane, or a connecting-rod moving in a prescribed manner, the geometrical conditions for rigidity being preserved. The stresses are not to be regarded as undefined, but as being determinate.

(8) The Theory of Elasticity is now introduced by the definition of an Elastic Body as a body such that the strain, hitherto considered as a virtual or potential strain, is the actual strain experienced by the parts of the body. The stresses being definite, the strains become definite, and the displacement may be deduced.

The acceptance of the principles involved in this doctrine. would have the effect of removing the subject of stresses from its accepted place in the Theory of Elasticity, and of making it an integral part of the Statics and Dynamics of a Rigid Body. Apart from the introduction of the cases of motion of the body, this would appear, at first sight, to be merely a matter of exposition of the Elastic Theory. But a further consequence would be that the determination of the elastic strains from the Rigid Body Stresses would be only the first stage in the Theory of Elasticity. A closer approximation to the values of the stresses would follow from the estimated alteration of the surface-traction conditions consequent on the displaced condition of the surface of the body, and the subject would become one of continued approximations on the lines of certain other subjects of Mathematical Physics, and an opportunity would be provided for a theory of permanent set and of rupture.

Lymm, Cheshire.