The value of a function which satisfies Laplace's equation within a closed surface is determined by the values at the surface. If two surface distributions be superposed, the value at any internal point is the sum of those due to the two surface distributions considered separately. By means of this principle a very simple proof may be given of the known theorem that the value of the potential at the centre of a sphere is the mean of those distributed over the surface.

On account of the symmetry it is clear that the central value would not be affected by any rotation of the sphere, to which the surface values are supposed to be rigidly attached.

Thus, if we conceive the sphere to be turned into n different positions taken at random and the resulting surface distributions to be superposed, we obtain a new surface distribution, whose mean value is n times greater than before, determining a central value which is also n times greater than that due to the original distribution. When n is made infinite, the surface distribution becomes constant, in which case the central value is the same as the surface value. From this it follows that in the original state of things the central value was the mean of the surface values.

Intermittent Sounds.

In the Philosophical Magazine for May 1875, Prof. A. M. Mayer describes some experiments on this subject, made by rotating a perforated cardboard disk between a resonator and a vibrating fork. “When the disk is stationary with one of its openings opposite the mouth of the resonator, it is evident that the ear will experience a simple sonorous sensation when a tuning-fork is brought near the mouth of the resonator. On revolving the perforated disk, two additional or secondary sounds appear—one slightly above, the other slightly below the pitch of the fork. An increasing velocity of rotation causes the two secondary sounds to diverge yet further from the note of the beating fork, until, on reaching a certain velocity, the two secondary sounds become separated from each other by a major sixth, while at the same moment a resultant sound appears, formed by the union of the sound of the fork with the upper and lower of the secondary sounds. This resultant is the lower second octave of the note given by the fork. On further increasing the velocity of rotation of the disk, the two secondary sounds and the resultant disappear, and the ear experiences only the sensation of the simple sound produced by the fork, whose beats at this stage of the experiment have blended into a smooth continuous sensation”.

The theory of waves in a uniform canal of rectangular section, in the case when the length of the wave is great in comparison with the depth of the canal and when the maximum height of the wave is small in comparison with the same quantity, was given long ago by Lagrange, and is now well known. A wave of any form, subject to the above conditions, is propagated unchanged, and with the velocity which would be acquired by a heavy body in falling through half the depth of the canal. The velocity of propagation here referred to is of course relative to the undisturbed water. If we attribute to the water in the canal a velocity equal and opposite to that of the wave, the wave-form, having the same relative velocity as before, is now fixed in space, and the problem becomes one of steady motion. It is under this aspect that I propose at present to consider the question; and we will therefore suppose that water is flowing along a tube, whose section undergoes a temporary and gradual alteration in consequence of a change in the vertical dimension of the tube. The principal question will be how far the pressure at the upper surface can be made constant by a suitable adjustment of the velocity of flow to the force of gravity.

That the two causes which tend to produce variation of pressure at the upper surface act in opposition to each other is at once evident. If there were no gravity, the pressure would vary on account of the alteration in the velocity of the fluid.

In a former paper I have shown that, of the various hypotheses which might be made to explain the diminished velocity of light in transparent matter, only one can be reconciled with the observed laws regulating the intensity of polarized light scattered in different directions from an assemblage of particles whose diameters do not exceed a small fraction of the wave-length. We are forced to suppose that the difference between media which is the cause of refraction is a dynamical and not a statical difference, that the rigidity or force with which the aether resists distortion is absolutely invariable. In this view there is nothing novel. Fresnel distinctly adopts it in the investigation of his celebrated formulae for the intensities of reflected light; and, what is more important, Green's rigorous mechanical theory of reflection is based on the same assumption. Cauchy also, to whom much of the credit really due to Green has been transferred, starts from the principle of continuity of movement, which asserts that in the passage from one medium to another there is no break in the continuity of the values, either of the displacements or of their differential coefficients. I believe that Cauchy has nowhere explained the ground or significance of his principle; but it is easy to see that to assume the continuity of strain is equivalent to asserting a complete continuity of statical properties, so that, as has been pointed out by Haughton, Cauchy's theory is essentially the same as Green's.

There is no part of hydrodynamics more perplexing to the student than that which treats of the resistance of fluids. According to one school of writers, a body exposed to a stream of perfect fluid would experience no resultant force at all, any augmentation of pressure on its face due to the stream being compensated by equal and opposite pressures on its rear. And indeed it is a rigorous consequence of the usual hypotheses of perfect fluidity and of the continuity of the motion, that the resultant of the fluid pressures reduces to a couple tending to turn the broader face of the body towards the stream. On the other hand, it is well known that in practice an obstacle does experience a force tending to carry it down stream, and of magnitude too great to be the direct effect of friction; while in many of the treatises calculations of resistances are given leading to results depending on the inertia of the fluid without any reference to friction.

It was Helmholtz who first pointed out that there is nothing in the nature of a perfect fluid to forbid a finite slipping between contiguous layers, and that the possibility of such an occurrence is not taken into account in the common mathematical theory, which makes the fluid flow according to the same laws as determine the motion of electricity in uniform conductors. Moreover the electrical law of flow (as it may be called for brevity) would make the velocity infinite at every sharp edge encountered by the fluid; and this would require a negative pressure of infinite magnitude.

In a paper “On some General Theorems relating to Vibrations,” published in the Mathematical Society's Proceedings for 1873 [Art. xxi], I proved a very general reciprocal property of systems capable of vibrating, with or without dissipation, about a position of stable equilibrium. The principle may be shortly, though rather imperfectly, stated thus:—If a periodic force of harmonic type and of given amplitude and period act upon the system at the point P, the resulting displacement at a second point Q will be the same both in amplitude and phase as it would be at the point P were the force to act at Q.

If we suppose the period of the force to be very great, the effects both of dissipation and inertia will ultimately disappear, and the system will be in a condition of what is called moveable equilibrium; that is to say, it will be found at any moment in that configuration in which it would be maintained at rest by the then acting forces, supposed to remain unaltered. The statical theorem to which the general principle then reduces is so extremely simple that it can hardly be supposed to be altogether new; nevertheless it is not to be found in any of the works on mechanics to which I have access, and was not known to the physicists to whom I have mentioned it. In any case, I think, two or three pages may not improperly be devoted to the consideration of it.