out the hinder (left hand) half of the wave, and negative throughout the front half; and as the velocity in A B C is throughout positive, it follows that throughout the hinder half of A B C we dp dy shall have positive, while throughout the front half of ABC dx dt the same function will be negative. For the entire wave, therefore, it is clear, from the symmetrical form of vibration which we have ascribed to it, that the sum of the tension will be zero*. If we now consider the wave abc taken singly, we shall have dp throughout its hinder (right hand) half negative, while dp dx dx throughout its front half will be positive; and the velocity throughout a b c being negative, we shall have positive dp dy dx dt throughout the hinder half of the wave, and negative throughout the front half. Thus, in the case of either wave taken separately, the positive and negative parts of the term in the equation of vis viva depending on the tensions will counterbalance each other, and the tensions will wholly disappear from that equation. But when the waves are superposed, or interfere, as, for instance, in the manner represented in the figure, this mutual balance of the opposing terms will cease to exist, as I shall now proceed to show. The figure is supposed to represent the state of things occurring after the period of complete occultation, when the front half of each wave has entirely emerged, and while the hinder halves have in part emerged, but as to the remainder are still in the state of superposition. Draw p P M perpendicular to c A C, and on the left of D, point of intersection of the curves A B C, a b c. the When the waves are separate and non-interferent, a stratum of the hinder half of either wave of the undisturbed breadth da, in the case of the one corresponding to PM, in the other to pm, would give rise to a positive term in the equation of vis viva. But when these elementary portions of the two waves become superposed, although the condensation of the combination will be the sum of the condensations of the two elements taken sepa rately, yet, inasmuch as the values of for the separate elements dx have opposite signs, the sign of d for the combination may be dx *Though less obvious, the same is equally true whatever be the form of vibration, provided that the wave is such as to be transmitted without undergoing change in its length, or form of vibration. Phil. Mag. S. 4. Vol. 49. No. 326. May 1875. 2 D dp dx positive or negative according to circumstances. If the sign of in the combination be positive, since it will be multiplied in the equation of vis viva by a negative velocity (for the velocity in the combination will be the difference of the velocities of the components, and the velocity in abc is here predominant), we dp dy shall have at this point negative; so that this portion of dx dt the wave, instead of aiding to balance the negative tensions prevailing throughout the front halves of the two waves, as its components would have done if the two waves had continued separate, will, so to speak, go over to the side of the latter. On the other hand, if at this point of the combined disturbance dp be negative, its value will be less in amount, irrespective of da dp sign, than what it would have been for the wave a b c taken separately, at the same time that, as before, it will be multiplied by a negative velocity, but a velocity which will be less than the velocity with which would be multiplied if we were dealing with the wave abc separately. Hence, though at this point the element of the combined disturbance would, as in the case of each of its components taken separately, tend to counteract the negative tensions of the emerged front halves of the two waves, yet it would do this in a less degree than one only of those components would do when taken separately (the wave a be to wit), and therefore, à fortiori, in a less degree than both. If we had drawn p P M on the right side of D, we should have arrived at precisely the same conclusion, though in a slightly different manner. It thus appears that, while the negative tensions at the time represented in the figure are precisely the same as when the waves were separate, the positive portion of the tensions will be diminished in amount, so that they will no longer counterbalance the former. On the whole, therefore, the sum of the tensions, instead of being zero, will give rise to a negative term of finite magnitude in the equation of vis viva. III. If it should be supposed that the fact of the sum of the tensions not vanishing in the equation of vis viva may afford a possible source of compensation for the loss of vis viva which it has been shown may arise from the interference of waves, the foregoing investigation will suffice to show the fallacy of this view. For, in the case of motion above considered it is evident that the destruction of the vis viva may be accompanied by the development of a negative term due to the tensions; in which case there will be a loss of energy, not only through the destruc tion of vis viva, but by reason 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 has 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 motions-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. P2 Let P1, P2 be the pressures, and v,, 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, v_2 the V—1, corresponding quantities for the surfaces as to which x-da 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 Hence, corresponding to the different elements of the entire wave, we shall have a series of equations, such as 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 0 =Σ (Dvdt dx) +Σp(v_1—v) dt =2 (Dvd dt dx) +Σp (v-ddx-v)de dt the integration being extended over the entire wave, or (since Р dx =pv- 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 (a) 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, it must be multiplied by v-1 or v- dx; so that, instead of dv dx those equal and opposite forces disappearing from the equation dv dx; and the of vis viva, they will give rise to a residuum, viz. Р 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*. 1. THE [With a Plate.] HE objects of the following paper are:-first, to show that the most important laws relating to the flow of electricity 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 teaching, 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. |