If we denote as before the strength of a vortex by m, these results may be written

We have seen above that the strength of each vortex is constant with regard to the time. Hence (36) express that the point whose co-ordinates are

is fixed throughout the motion. This point, which coincides with the centre of inertia of a mass distributed over the plane xy with the surface-density, may be called the 'centre' of the system of vortices, and the straight line parallel to z of which it is the projection may be called the 'axis' of the system.

139. We proceed to discuss some particular cases.

(a) First, let us suppose that we have only one vortex-filament present, and let Ţ have the same sign throughout its infinitely small section. Its centre, as just defined, will lie either within the substance of the filament, or at all events infinitely close to it. Since this centre remains at rest, the filament as a whole will be stationary, though its parts may experience relative motions, and its centre will not necessarily lie always in the same element of fluid. Any particle at a finite distance r from the centre of the filament will describe a circle about the latter as axis, with constant velocity m The region external to the filament is doubly-connected; and the circulation in any (simple) circuit embracing the filament is 2m. The irrotational motion of the fluid external to the filament is the same as in Art. 35.

(b) Next suppose that we have two vortices, of strengths m,, m2, respectively. Let A, B be their centres, O the centre of the system. The motion of each filament is entirely due to the other filament, and is therefore always perpendicular to АВ. Hence the two filaments remain always at the same distance from one another, and rotate with uniform angular velocity about O, which is fixed. This angular velocity is easily

found; we have only to divide the velocity of A (say), viz.

m2 by the distance AO, where

п.АВ

If m1, m, be of the same sign, i.e. if the directions of rotation in the two filaments be the same, O lies between A and B; but if the directions be of opposite signs, O lies in AB, or BA, produced.

If =m1 that A, B move with uniform velocity m

O is at infinity; in this case it is easily seen

π. AB perpendicular to AB, which remains fixed in direction. The motion external to the filaments at any instant is given by the formulæ of Chapter IV, Example 3.

The motion at all points of the plane bisecting AB at right angles is tangential to that plane. We may therefore suppose this plane to form a fixed rigid boundary of the fluid on either side of it; and so obtain the solution of the case where we have a single rectilinear vortex in the neighbourhood of a fixed plane wall to which it is parallel. The filament moves parallel where d is the distance of

to the plane with the velocity the vortex from the wall.

m

2πα

In the last case [m ̧=-m,] the stream-lines are all circles. We can hence derive the solution of the case where we have a single vortex-filament in a mass of fluid which is bounded, either internally or externally, by a fixed circular cylinder. Thus, in

the figure, let DPE be the section of the cylinder, A the position

of the vortex (supposed in this case external), and let B be the 'image' of A with respect to the circle DPE, viz. C being the centre, let

CB. CA=c2,

where c is the radius of the circle. If P be any point on the circle, we have

so that the circle occupies the position of a stream-line due to a pair of vortices, whose strengths are equal and opposite in sign, situated at A, B in an unlimited mass of fluid. Since the motion of the vortex A would then be perpendicular to AB, it is plain that all the conditions of the problems are satisfied if we suppose A to describe a circle about the axis of the cylinder with uniform velocity

m

=

m. CA

T. ABT (CA'— c3) '

In the same way a single vortex of strength m, situated within a fixed circular cylinder, say at B, would describe a circle with m. CB π (c2 — CB2) *

uniform velocity

(c) If we have four parallel rectilinear vortices whose centres form a rectangle ABB'A', the strengths being m for the vortices A', B, and -m for the vortices A, B', it is evident that the centres will always form a rectangle. Further, if the various rotations have the directions indicated in the figure, we see that Fig. 12.

B

the effect of the presence of the pair A', B' on A, B is to separate them, and at the same time to diminish their velocity perpendicular to the line joining them. The planes which bisect AB,

AA' at right angles may (either or both) be taken as fixed rigid boundaries. We thus get the case where a pair of vortices, of equal and opposite strengths, move towards (or from) a plane wall, or where a single vortex moves in the angle between two perpendicular walls.

For other interesting cases of motion of rectilinear vortices we refer to a paper by Professor Greenhill*.

140.

2. Circular Vortices.

Next let us take the case where all the vortices present in the fluid (supposed unlimited as before) are circular, having the axis of x as a common axis. Let denote the distance of any point P from this axis, 9 the angle which makes with the plane xy, v the velocity in the direction of ☎, and w the angular velocity of the element at P. It is evident that u, v, w are functions of x and only, and that the axis of w is perpendicular to x, . We have then

If we make these substitutions, writing added for the volumeelement, in (30), and perform the integration with respect to D, we obtain

The second and third of equations (31) are satisfied identically; the first gives

ffa v w dx da

=

0.

If we denote by m the strength wdxd of the vortex whose co-ordinates are ∞, ∞, these results may be written

where the summations embrace all the vortices present in the fluid. If in these equations we suppose x, always to refer to the same vortex, we may write

Since m is constant for the same vortex, the equation (39) is at once integrable with respect to t, whence

may be called the 'mean radius' of the vortex-rings. The equation (40) shews that this mean radius is constant throughout the motion.

If we introduce in addition a magnitude x, such that

.....

.(42), it is plain that the position of the circle whose co-ordinates are depends only on the strengths and the configuration of the vortices, and not on the position of the origin of co-ordinates. This circle may be called the 'circular axis' of the whole system of vortex-rings. It remains constant in radius; and its motion parallel to x is obtained by differentiating (42), viz. we have

where we have added to the right-hand side a term which

vanishes in virtue of (40).

141. The formulæ (11) become, on making the substitutions (37),

L=0