where the integration extends over any closed surface lying wholly in the fluid. Applying this to the closed surface formed by two cross-sections of a vortex-tube, and the portion of the tube intercepted between them, we find w,o,w,,, where w1, w, denote the angular velocities at the sections σ,,,, respectively.

=

2

Thomson's proof shews that the theorem is true even when , n, are discontinuous (in which case there may be an abrupt bend at some point of a vortex), provided only that u, v, w are continuous.

An important consequence of the above theorem is that a vortex-line cannot begin or end at any point in the interior of the fluid. Any vortex-lines which exist must either form closed curves, or else traverse the fluid, beginning and ending on its boundaries. Compare Art. 44.

The theorem (6) of Art: 40 may now be enunciated as follows: The circulation in any circuit is equal to twice the sum of the strengths of all the vortices which it embraces.

128. The motion of the fluid occupying any simply-connected region is determinate when we know the values of the expansion (0, say,), and of the component angular velocities §, n, 3 at every point of the region, and the value of the normal velocity (~, say,) at every point of the boundary.

If possible, let there be two sets of values, u1, v1, w1, and U, V, w, of the component velocities, each satisfying the above conditions; viz. each set satisfying the differential equations

will satisfy (3), (4), and (5) with 0, §, n, 5, λ put each = 0; that is to say u', v', w' are the components of the irrotational motion of an incompressible fluid occupying a simply-connected region whose boundary is at rest. Hence (Art. 47) these quantities all vanish; and there is only one possible motion satisfying the given conditions.

The above theorem-an extension of one given in Art. 49-is equally true when the region extends to infinity, and (5) is replaced by the condition that the fluid is there at rest.

129. If, in the last-mentioned case, all the vortices present are within a finite distance of the origin, the complete determination of u, v, w in terms of 0, E, n, can be effected, as follows*.

2N =

==

25........(9).

▼2L=-2§‚ ▼3M=-2n, ▼3N

Now (7) and (9) are satisfied by making P,

L, M, N equal to

the potentials of distributions of matter whose densities at the

point (x, y, z) are

*

See Stokes, Camb. Trans. Vol. 1x. (1849), and Helmholtz, Crelle, t. Lv. (1858).

denote the values of these

where the accents attached to 0, §, n,

quantities at the point (x', y', z') and r stands for the distance

{(x − x')2 + (y — y')2 + (z — z′)2}3.

The integrations are supposed to include all parts of space at which 0, &, n, have values different from zero.

We must now examine whether the above values of L, M, N

by (17), Art. 64. The volume-integral vanishes by (1), and the surface-integral vanishes because by hypothesis we have §, n, =0 at all points of the (infinite) surface over which it is taken. Hence (8) is satisfied, and the values (6) of u, v, w satisfy (3) and (4). They also evidently vanish at infinity.

The above results hold even when 0, &, n, are discontinuous functions, provided only that u, v, w be continuous. As regards this is obvious; but a discontinuity in §, n, § will necessitate a modification in (12). Let us suppose that as we cross a certain surface the values of §, n, change abruptly, and let us dis

tinguish the value on the two sides by suffixes.

Two cases

present themselves; the vortex-lines may be tangential to Σ on both sides, or they may cross the surface, experiencing there an abrupt change of direction. In the first case we have

l§ ̧ + mn ̧ + n51 = l§2+ mn2+n52 = 0 ..................................(13)

at Σ; and in the second we have

l§1 + mn, + n§ ̧ = l§, + mn2 + n§2.........................(14). In fact, if dΣ be a section of a vortex, taken parallel and infinitely close to Σ on one side of it, the product (l§,+mn,+n§1) dΣ measures the strength of the vortex, which is (Art. 127) the same on both sides of E. Now in (12) the region through which the triple integration extends is divided by the surfaces Σ into a certain number of distinct portions. For each of these, taken by itself, the equality of the second and third members of (12) holds ; and if we add the results thus obtained, we see that to make (10) true for the region taken as a whole we must add to the third member terms of the form

due to the two sides of each of the surfaces Σ. The relations (13) and (14) shew however that these terms all vanish, so that (8) is still satisfied.

130. Let us examine the result obtained in Art: 129; and let us suppose first that the fluid is incompressible, so that = 0, and therefore P=0. Denoting by Su; Sv, Sw the portions of u, v, w arising from the element dx'dy'dz' in the integrals (11), we find

It appears from the form of these expressions that the resultant

of du, dv, dw is perpendicular to the plane containing the direction of the vortex-line at (x, y, z) and the line r, and also that its sense is that in which the point (x, y, z) would move if it were rigidly attached to a body rotating with the fluid element at (x, y, z). The magnitude of the resultant is

(by an elementary formula of solid geometry), where x is the angle which makes with the direction of the vortex-line at (x', y', z').

A relation of exactly the same form as that here developed obtains between the magnetic force and the electric currents in any electro-magnetic field. If we suppose a system of electric currents arranged in exactly the same manner as the vortex-filaments, the components of the current at (x, y, z) being , n,, the components of the magnetic force at (x, y, z) due to these currents will be u, v, w.

In the general case (ie. when 0 is not everywhere zero) we must add to the values of u, v, w obtained by integrating (15) dP dP dP

the terms

da' dy" dz, respectively, where P has the value (10). These are the components of the force at (x, y, z) produced by a distribution of imaginary magnetic matter with density 0.

131. Let us revise the investigation of Art. 129 with a view to adapting it to the case where the region occupied by the fluid is not infinite, but is limited by surfaces at which the value of the normal velocity λ is given. The equations to be satisfied by u, v, w are (3), (4) and (5). The integrals (10) and (11) being supposed to refer to this limited region, the surface-integral in the last member of (12) will not in general vanish unless all the vortices present form closed filaments lying wholly in the region. If on the other hand the vortex-lines traverse the region, beginning and ending on the boundary, we may suppose them continued outside the region, or along its surface, in such a manner that they form closed curves. We thus obtain a larger region in which all the vortex-filaments are closed, and if we now suppose the integrals in (10) and (11) to refer to this extended region, the surface-inte