These formulae relate of course to the region x > 0*.

→ The formula for occurs in Basset, Hydrodynamics, t. ii. p. 93. See also Nagaoka, l.c.

It was shewn in Art. 150 that the value of 4 is that due to a system of double sources distributed with uniform density κ over the interior of the circle. The values of and for a uniform distribution of simple sources over the same area have been given in Art. 102 (11). The above formulae (18) and (19) can thence be derived by differentiating with respect to x, and adjusting the constant factor*.

162. The energy of any system of circular vortices having the axis of x as a common axis, is

by a partial integration, the integrated terms vanishing at the limits. We have here used to denote the strength wdxdw of an elementary vortexfilament.

Again the formula (7) of Art. 153 becomes†

The impulse of the system obviously reduces to a force along Ox. By Art. 152 (6),

P = PSS(yzn) dx dy dz =пρ [[w2w dx dw= πρΣκω. If we introduce the symbols x。, w。 defined by the equations

......

(3)

.these determine a circle whose position evidently depends on the strengths and the configuration of the vortices, and not on the position of the origin on the axis of symmetry. It may be called the 'circular axis' of the whole system of vortex-rings.

Since

is constant for each vortex, the constancy of the impulse shews, by (3) and (4), that the circular axis remains constant in radius. To find its motion parallel to x, we have, from (4),

* Other expressions for and can be obtained in terms of zonal spherical harmonics. Thus the value of is given in Thomson and Tait, Art. 546; and that of can be deduced by the formulae (11), (12) of Art. 95 ante. The elliptic-integral forms are however the most useful for purposes of interpretation.

† At any point in the plane z=0 we have y = w, ¿=0, n=0, 5=w, v=v; the rest follows by symmetry.

where the added term vanishes, since Σkwv0 on account of the constancy

of the mean radius (wo).

=

163. Let us now consider, in particular, the case of an isolated vortexring the dimensions of whose cross-section are small compared with the radius (wo). It has been shewn that

where r1, 2 are defined by Art. 161 (6). For points (x, w) in or near the substance of the vortex, the ratio r1/r, is small, and the modulus (A) of the elliptic integrals is accordingly nearly equal to unity. We then have

approximately*, where X' denotes the complementary modulus, viz.

(2)

(3)

Hence at points within the substance of the vortex the value of is of the order κ, log (wo/e), where e is a small linear magnitude comparable with the dimensions of the section. The velocities at such points, depending (Art. 94) on the differential coefficients of 4, will be of the order ê/ε.

We can now estimate the magnitude of the velocity dx/dt of translation of the vortex-ring. By Art. 162 (1), T is of the order pr2, log (wo/e), and v is, as we have seen, of the order «/<; whilst x xo is of course of the order e. Hence the second term on the right-hand side of the formula (6) of the preceding Art. is, in the present case, small compared with the first, and the velocity of translation of the ring is of the order κ/w。.log (w。/e), and approximately constant.

An isolated vortex-ring moves then, without sensible change of size, parallel to its rectilinear axis with nearly constant velocity. This velocity is small compared with that of the fluid in the immediate neighbourhood of the circular axis, but may be greater or less than /o, the velocity of the fluid at the centre of the ring, with which it agrees in direction.

For the case of a circular section more definite results can be obtained as follows. If we neglect the variations of and w over the section, the formulae (1) and (2) give

* See Cayley, Elliptic Functions, Arts. 72, 77; and Maxwell, l.c.

or, if we introduce polar co-ordinates (s, x) in the plane of the section,

and this definite integral is known to be equal to 2π log s', or 2π log s, according as s'≥s. Hence, for points within the section,

The only variable part of this is the term wws2; this shews that to our order of approximation the stream-lines within the section are concentric circles, the velocity at a distance s from the centre being ws.

In our present notation, where « denotes the strength of the whole vortex, this is equal to Wolf. Hence the formula for the velocity of translation of the vortex becomes*

The vortex-ring carries with it a certain body of irrotationally moving fluid in its career; cf. Art. 155, 2°. According to the formula (7) the velocity of translation of the vortex will be equal to the velocity of the fluid at its centre when wo/a =86, about. The accompanying mass will be ring-shaped or not, according as a exceeds or falls short of this critical value.

The ratio of the fluid velocity at the periphery of the vortex to the velocity at the centre of the ring is 2waш。/к, or w ̧/ña. For a=100, this is equal to 32, about.

164. If we have any number of circular vortex-rings, coaxal or not, the motion of any one of these may be conceived as made up of two parts, one due to the ring itself, the other due to the influence of the remaining rings. The preceding considerations shew that the second part is insignificant compared with the first, except when two or more rings approach within a very small distance of one another. Hence each ring will move, without

* This result was given without proof by Sir W. Thomson in an appendix to a translation of Helmholtz' paper, Phil. Mag. (4), t. xxxiii. p. 511 (1867) [Papers, t. iv. p. 67]. It was verified by Hicks, Phil. Trans. A, t. clxxvi. p. 756 (1885); see also Gray, "Notes on Hydrodynamics," Phil. Mag. (6), t. xxviii. p. 13 (1914).

sensible change of shape or size, with nearly uniform velocity in the direction of its rectilinear axis, until it passes within a short distance of a second ring.

A general notion of the result of the encounter of two rings may, in particular cases, be gathered from the result given in Art. 149 (3). Thus, let us suppose that we have two circular vortices having the same rectilinear axis. If the sense of the rotation be the same for both, the two rings will advance, on the whole, in the same direction. One effect of their mutual influence will be to increase the radius of the one in front, and to contract the radius of the one in the rear. If the radius of the one in front become larger than that of the one in the rear, the motion of the former ring will be retarded, and that of the latter accelerated. Hence if the conditions as to relative size and strength of the two rings be favourable, it may happen that the second ring will overtake and pass through the first. The parts played by the two rings will then be reversed; the one which is now in the rear will in turn overtake and pass through the other, and so on, the rings alternately passing one through the other*.

If the rotations be opposite, and such that the rings approach one another, the mutual influence will be to enlarge the radius of each. If the two rings be moreover equal in size and strength, the velocity of approach will continually diminish. In this case the motion at all points of the plane which is parallel to the two rings, and half-way between them, is tangential to this plane. We may therefore, if we please, regard the plane as a fixed boundary to the fluid on either side, and so obtain the case of a single vortex-ring moving directly towards a fixed rigid wall.

The foregoing remarks are taken from Helmholtz' paper. He adds, in conclusion, that the mutual influence of vortex-rings may easily be studied experimentally in the case of the (roughly) semicircular rings produced by drawing rapidly the point of a spoon for a short space through the surface of a liquid, the spots where the vortex-filaments meet the surface being marked by dimples. (Cf. Art. 27.) The method of experimental illustration by means of smoke-rings† is too well-known to need description here. A beautiful variation of the experiment consists in forming the rings in water, the substance of the vortices being coloured.

The motion of a vortex-ring in a fluid limited (whether internally or externally) by a fixed spherical surface, in the case where the rectilinear axis of the ring passes through

* The corresponding case in two dimensions was worked out and illustrated graphically by Gröbli, l.c. ante p. 218; see also Love, “On the Motion of Paired Vortices with a Common Axis," Proc. Lond. Math. Soc. t. xxv. p. 185 (1894).

Reusch. "Ueber Ringbildung der Flüssigkeiten," Pogg. Ann. t. cx. (1860); Tait, Recent Advances in Physical Science, London, 1876, c. xii.

Reynolds, "On the Resistance encountered by Vortex Rings &c.," Brit. Ass. Rep. 1876; Nature, t. xiv. p. 477.