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tion of vis viva, but by reasou of the fact that what vis viva is left will be more or less counteracted by the presence of the negative term due to the tensions.
It is clear, therefore, that the conservation of vis viva cannot be relied on as holding in cases of the intersection of waves traversing continuous masses, and that what Dr. Helmholtz bas offered to our attention as a universal law of nature is completely contradicted by fact in cases such as those we have been considering
If we advert to the original derivation of the equation of vis viva from the principle of virtual velocities, if we reflect how completely the arbitrary displacement of the points of application of the respective forces forms the characteristic feature of the principle of virtual velocities, and bear in mind that the transition from that principle to the equation of vis viva is effected simply by the substitution of the actual for the arbitrary mo tions—if we keep in view these various considerations, it can hardly be matter of surprise that, in treating of the internal metions of continuous masses it should have come to be considered that the only terms depending on the internal actions which need be taken into account in forming the equation of vis viva consist of pairs of equal and opposite forces multiplied by a common displacement or common velocity (that, namely, of the common point of application of the forces), and consequently that no such terms will appear in that equation.
That this mode of considering the subject, however specious, must be entirely erroneous, sufficiently appears, if I mistake not, from what has preceded; but it is of the highest importance that the mode as well as the fact of the fallacy should be clearly apprehended; and to this part of the subject I propose now to address myself.
Let Pu Po be the pressures, and vj, v, the particle-velocities at the time t at the surfaces the ordinates of whose positions of rest are respectively x + dx and x + 2dx ; P-1, P-2, V-1, 0-2
the corresponding quantities for the surfaces as to which x— dx and x — 2dx are the ordinates of the positions of rest.
The equation of motion of the first element may be put under the form
dt multiplying which by vdt we get
0=Dv dt dx +P, vdt.— pvdt.
dt The corresponding equations for the elements in contact with
the first will be
dt dx +Pav, dt-Piv, dt,
dvi 0=Dv, di
do- dt dx +pv-dt-p-10-dt. Hence, corresponding to the different elements of the entire wave, we shall have a series of equations, such as
dv-. dt dr+p_10_2dt – P-90_,dt,
dv.–2 dt dx +pv_, dt-p_10_,dt,
&c. = &c. Taking the sum of these, observing that the first and last terms will vanish, the particle-velocity at the extremities being either zero or infinitely small, we get
dv dt dx
dx dt dt
dp (since р
the first term on the right-hand =podx
dx' side of which equation vanishes when the integration is extended over the entire wave, the velocity vanishing at the extremities of the wave).
Integrating (@) with respect to t, we get, as before, the equation of vis viva; this mode of deriving which enables us to see the error committed in supposing that, when the equation is formed for a continuous mass, the equal and opposite forces acting at the common boundary of two elements will be multiplied by the same velocity, viz. that of the bounding surface.
It is clear that we must treat the motion of each element exactly as if it were collected at its centre of gravity—that the velocities by which we multiply the forces acting upon it are not the velocities of their actual points of application, but that of their hypothetical point of application, viz. the centre of gravity of the element—in other words, that the velocity by which we multiply must be the average velocity prevailing throughout the element; from which it follows that the force p, for example, when regarded as acting on the element originally considered, must be multiplied by the velocity v; but when it is regarded as acting on the element immediately in the rear of the former,
dv it must be multiplied by v-, or v- dx; so that, instead of
dx those equal and opposite forces disappearing from the equation
dv of vis viva, they will give rise to a residuum, viz. dx ; and the
dx sum of these small terms taken for the entire wave will in general be a finite quantity.
It can hardly be necessary that I should point out the bearing which the foregoing conclusions have upon the applicability, when compressible bodies are dealt with, of the principle of virtual velocities. 6 New Square, Lincoln's Inn,
March 3, 1875.
XLIV. On the Flow of Electricity in a uniform plane conducting
Surface.- Part I. By G. CAREY FOSTER, F.R.S., and OLIVER J. LODGE*
[With a Plate.) 1. THE THE objects of the following paper are:—first, to show that
the most important laws relating to the flow of electri. city in a plane conducting sheet of uniform conductivity can be established by mathematical considerations of such a simple kind that there is no reason why they should not be introduced into ordinary teacbing, as well as into the more complete elementary text-books of electricity; and in the second place, to describe methods that may conveniently be employed for testing
* Read before the Physical Society, February 27, 1875. Communicated by the Society.
experimentally the theoretical conclusions, together with some results obtained by these methods.
The general subject treated of in this paper has attracted the attention of a considerable number of mathematicians and physicists. The earliest published investigation relating to it was contained in a reinarkable memoir by Kirchhoff
, which appeared in Poggendorff's Annalen in 1845 (vol. lxiv. p. 497). In this paper Kirchhoff established the general mathematical theory of the flow of electricity in an unlimited uniformly conductingsheet, and in a limited sheet with a circular boundary, with so much completeness as to leave little for others to do beyond working out the application to special cases of the general principles he laid down, or finding other methods of establishing the conclusions he deduced from them.
We cannot better indicate the general plan of Kirchhoff's investigation than by quoting the following account of it from a paper by Professor W. Robertson Smith*, to which we shall have to make further reference immediately :-"By an application of Ohm's law, he [Kirchhoff] expressed analytically the condition to be satisfied by v [the potential]. When the electricity enters and issues by a number of individual points, he found (apparently by trial) that an integral of the form § (a logr), where ri, rz, &c. are the distances of the point (, y) from the successive points of entrance and issue, satisfied the conditions when the plate is infinite. For a finite plate, it is necessary that the boundary of the plate should be orthogonal to the curves
(a logr)=const. (3) He was thus led to form the orthogonal curves whose equation he gives in the form
E(Q[r, R])=const., (4) where [r, R] is the angle between r and a fixed line R. These equations he applies to the case of a circular plate, completely determining the curves when there is one exit and one entrance point in the circumference, and showing that in any case a proper number of subsidiary points would make the equipotential lines determined by (3) cut the circumference at right angles. Kirchhoff's paper is throughout properly busied with the function v, and the stream-lines are only dealt with incidentally. There is no attempt to give a physical meaning to the equation (4).” To this we have only to add that Kirchhoff proved the accuracy of his theoretical deductions by determining experimentally the form and distribution of the equipotential lines on a circular disk with two electrodes on the edge, as well as (Pogg. Ann.
* Proc. Edinb. Roy. Soc. 1869–70, pp. 79–99.
vol. lxvii. p. 344, 1846) the strength of the current at various parts of the disk; and that, from the expression for the difference of potential between the electrodes, he deduced by Ohm's law the resistance of a circular disk with two small electrodes anywhere upon it. In order to test experimentally the value thus obtained, he seems to have devised independently the arrangement now commonly known as Wheatstone's bridge; but, owing to the smallness of the resistance to be measured, he was unable to obtain satisfactory results.
Soon after the publication of Kirchhoff's paper, Smaasen * gave an investigation of the flow of electricity in a plane conducting sheet, in which he takes account, in determining the potential, of the electricity given off to the air, and deduces the resistance of an infinite sheet with two small circular electrodes by a process which, though longer than that employed by Kirchhoff, may be regarded as more direct. It consists in finding the resistance of the space between two lines of flow at an infinitely small distance apart, and then extending by integration the expression thus obtained so as to make it apply to the unlimited sheet. In a subsequent papert Smaasen determined by an analogous process the resistance of a conducting sphere, or of an unlimited conductor of three dimensions f. Smaasen's treatment of the subject is, like Kirchhoff's, based chiefly upon the mode of distribution of the potential; the only investigation we are acquainted with which deals specially with lines of flow is contained in the paper by Professor W. Robertson Smith from which we have already quoted. The starting-point adopted by Professor Smith is the same as that from which we have set out in the following communication; and, indeed, we found, after making some progress in our own work, that several of our demonstratious (which we at first thought were new) had been already given by him, while all the chief conclusions were, as we have said above, implicitly contained in Kirchhoff's original memoir. Consequently, although the present paper contains a few minor results which, so far as we know, have not been pointed out explicitly before, we do not claim for its contents any essential novelty; and our only reason for venturing to publish it is that
* Pogg. Ann. vol. Ixix. p. 161. † Pogg. Ann. vol. Ixxji.
435. | About the same time, the same subject was taken up by Ridolfi of Florence (Il Cimento, An. V. 1847, May-June), whose paper, however, we only know from the references of Beetz (in Dove's Repertorium, vol. viii. p. 147) and Poggendorff (Pogg. Ann. vol. Ixxii. p. 449). No reference to this paper is given either in the Royal Society's Catalogue of Scientific Papers, or in the carefully compiled “ Bibliographie" of the mathematical theory of the voltaic pile in Verdet's Conférences de Physique (Euvres de Verdet, vol. iv. p. 351).