absolutely no friction. I do not expect that there would be this wide difference; but still the actual motion would probably not agree so well with the theoretical, as in those cases in which the forward and backward oscillations are alike. By the theoretical motion is of course meant that which would be obtained from the common theory, in which friction is not taken into

account.

It appears from experiments on pendulums that the effect of the internal friction in air and other gases is greater than might have been anticipated. In Dubuat's experiments on spheres oscillating in air the spheres were large, and the alteration in the time of oscillation due to the resistance of the air, as determined by his experiments, agrees very nearly with the result obtained from the common theory. Other philosophers, however, having operated on smaller spheres, have found a considerable discrepancy, which is so much the greater as the sphere employed is smaller. It appears, moreover, from the experiments of Colonel Sabine, that the resistance depends materially upon the nature of the gas. Thus it is much greater, in proportion to the density, in hydrogen than in air.

NOTE REFERRED TO AT P. 37.

[It may be noticed that two of Helmholtz's fundamental propositions respecting vortex motion follow immediately from Cauchy's integrals; or rather, two propositions the same as those of Helmholtz merely generalized so as to include elastic fluids follow from Cauchy's equations similarly generalized.

On substituting in (7) for R the expression given by (8), and introducing the notation of angular velocities, as in (11), equations (7) become

but (b) are the differential equations of the system of vortex lines at the time 0, and (c), as being of the form

as

are the differential equations of the loci of the particles at the time t which at the time 0 formed the vortex lines respectively. But when we further take account of the values of P, Q, R, exhibited in (c), we see that (c) are also the differential equations of the system of vortex lines at the time t. Therefore the same loci of particles which at one moment are vortex lines remain vortex lines throughout the motion.

0

Let be the resultant angular velocity at the time 0 of a particle P, which at the time t is at P, and has for its angular velocity; let ds drawn from P. be an element of the vortex line at time 0 passing through Po, and ds the element of the vortex line passing through P at the time t which consists of the same set of particles. Then each member of equations (b) is equal to ds, and each member of equations (c) equal to ds/2. Hence we get from any one of equations (a)

Let A, be the area of a perpendicular section, at P,, of a vortex thread containing the vortex line passing through P, at the time 0, a vortex thread meaning the portion of fluid contained within an elementary tube made up of vortex lines; then by what precedes the same set of particles will at the time t constitute a vortex thread passing through P; let A be a perpendicular section of it passing through P at the time t, and draw two other perpendicular sections passing respectively through the other extremities of the elements ds, and ds. Then if we suppose, as we are at liberty to do, that the linear dimensions of A, are indefinitely small compared with the length ds,, we see at once that the elements of volume comprised between the tube and

0

the pair of sections at the time 0 and at the time t respectively contain ultimately the same particles, and therefore

or the angular velocity of any given particle varies inversely as the area of a perpendicular section through it of the vortex thread to which it belongs, and that, whether the fluid be incompressible or elastic.

When these results are deduced from Cauchy's integrals, the state of the fluid at any time is compared directly with its state at any other time; in Helmholtz's method the state at the time t is compared with the state at the time t+dt, and so on step by step.

A remaining proposition of Helmholtz's, that along a vortex line the angular velocity varies at any given time inversely as the perpendicular section of the vortex thread, has no immediate relation to Cauchy's integrals, inasmuch as it relates to a comparison of the state of the fluid at different points at the same moment. It may however be convenient to the reader that the demonstration, which is very brief, should be reproduced here.

where the integration extends over any arbitrary portion of the fluid. This equation gives

[[ w'dydz + [[ w"dzdx + [[w" dxdy = 0,

where the double integrals extend over the surface of the space in question. The latter equation again becomes by a well-known transformation

where dS is an element of the surface of the space, and the

S. II.

4

angle between the instantaneous axis and the normal to the surface drawn outwards.

Let now the space considered be the portion of a vortex thread comprised between any two perpendicular sections, of which let A and A' denote the areas. All along the side of the tube = 90°, and at the two ends = 180° and = 0°, respectively, and therefore if ′ denotes the angular velocity at the second extremity of the portion of the vortex thread considered

which proves the theorem.]

NA = N'A',

[From the Philosophical Magazine, Vol. XXXIII., p. 349 (November, 1848.)]

ON A DIFFICULTY IN THE THEORY OF SOUND.

THE theoretical determination of the velocity of sound has recently been the occasion of a discussion between Professor Challis and the Astronomer Royal. It is not my intention to enter into the controversy, but merely to consider a very remarkable difficulty which Professor Challis has noticed in connexion with a known first integral of the accurate equations of motion for the case of plane waves.

The difficulty alluded to is to be found at page 496 of the preceding volume of this Magazine*. In what follows I shall use Professor Challis's notation.

*

[The following quotation will suffice to put the reader in possession of the apparent contradiction discovered by Professor Challis. It should be stated that the investigation relates to plane waves, propagated in the direction of z, and that the pressure is supposed to vary as the density.

"The function ƒ being quite arbitrary, we may give it a particular form. Let, therefore,

This equation shows that at any time t, we shall have w=0 at points on the axis of z, for which

At the same time w will have the valuem at points of the axis for which