This is otherwise evident from Art. 25 (3), which shews that

dφ

is single-valued, and that therefore the cyclic constants of dt cannot alter.

In Examples 5 and 6, Arts. 36, 37, we had instances to which the above result is not applicable; the reason being that in Ex. 5 the external forces have not a potential, whilst in Ex. 6 their potential is itself a cyclic function.

60. Proceeding now, as in Art. 43, to the particular case of an incompressible fluid, we remark that whether $ be many-valued and therefore all the

or not, its first derivatives do, do, do dx' dy' dz'

higher derivatives, are essentially single-valued functions, so that

$ will still satisfy the equation of continuity

where the surface-integration extends over the whole boundary of any portion of the fluid.

In the theorems of Arts. 45 and 46 the spaces to which (11) is applied are simply-connected, so that it is allowable to suppose $ single-valued throughout them even when the region of which they form a part is multiply-connected. On this understanding the theorems in question still hold when is a cyclic function.

The theorem (a) of Art. 47, viz. that & must be constant throughout the interior of any region at every point of which (10) is satisfied if it be constant over the boundary, still holds when the region is multiply-connected. For $, being constant over the boundary, is necessarily single-valued.

The remaining theorems of Art. 47, being based on the assumption that the stream-lines cannot form closed curves, are however no longer exact. We must introduce the additional condition that the circulation is to be zero in each circuit of the region.

The theorems of Art. 48 also call for modification. The proper extension of (3) is as follows:

61. A function & exists which satisfies (10) throughout a given n + 1-ply-connected region, which has any given cyclic constants K1, K2, ... K, corresponding to the n independent non-evanescible circuits capable of being drawn in the region, and which is such that its rate of variation do in the direction

dn

of the normal has

a

given value must dn

at every point of the boundary. These arbitrary values of do

We follow Thomson in marshalling the following physical considerations in support of this theorem. Let us suppose the region occupied by incompressible fluid of unit density enclosed in a perfectly smooth and flexible membrane. Further, let n barriers be drawn, as in Art. 54, so as to reduce the region to a simply-connected one, and let their places be occupied by similar membranes, infinitely thin, and destitute of inertia. The fluid being initially at rest, let each element of the first-mentioned membrane be suddenly moved

αφ

inwards with the given (positive or negative) norinal velocity an' whilst uniform impulsive pressures K1, K2, ..., are applied to the positive sides of the respective barrier-membranes. Some definite motion of the fluid will ensue, characterized by the following properties :

(a) It is irrotational, being generated from rest;

(b) The normal velocity at every point of the original boundary has the assigned value;

(c) The values of the impulsive pressure, and therefore of the velocity-potential, at two adjacent points on opposite sides of a barrier-membrane, differ by the corresponding value of x, which is constant over the barrier;

(d) The motion on one side of a barrier is continuous with that on the other.

To prove the last statement we remark, first, that the velocities normal to the barrier at two adjacent points on opposite sides of it are the same, being each equal to the normal velocity of the adjacent portion of the membrane. Again, if P, Q be two consecutive points on a barrier, and if the corresponding values of be on one side P, Q, and on the other 'p, φ'q, we have, by (c)

Hence the tangential velocities at two adjacent points on opposite sides of a barrier also agree. If then we suppose the barriermembranes to be liquefied immediately after the impulse, we obtain a state of irrotational motion satisfying the conditions stated at the head of this article*.

2

62. It is easy to shew analytically that the said conditions completely determine 4, save as to an additive constant. For, if possible, let there be two functions 1, ф2 each satisfying the conditions. Since 1, 2 have the same cyclic constants, $ = $1-Фа is a single-valued function, which moreover satisfies (10) throughout the region, and makes = 0 at every point of the boundary. Hence Art. 47 (3) applies, and shews that is constant.

do dn

Hence the irrotational motion throughout an n + 1-ply-connected space is determinate when we know the value of the normal velocity at every point of the boundary, and also the value of the circulation in each of the n independent circuits which can be drawn in that space.

The following theorem, which now replaces that of Art. 52, is proved in like manner.

The irrotational motion through an n + 1-ply-connected region extending to infinity, but limited internally by one or more closed surfaces, is made fully determinate by the following conditions:

(a) The normal velocity has a prescribed value at every point of the internal boundary;

(b) The circulations in the n independent circuits of the region have prescribed values; and

(c) The velocity vanishes at an infinite distance from the internal boundary.

If, for instance, we have an anchor-ring moving in an infinite mass of liquid which is at rest at infinity, the irrotational motion of the fluid at any instant is determinate when we know the motion of the ring (and therefore the velocity of every element of its surface normal to itself), and also the value of the circulation in any circuit embracing it.

63. The theory of multiple continuity seems to have been first developed by Riemann *, for spaces of two dimensions, à propos of his researches on the theory of functions of a complex variable, in which connection also cyclic functions satisfying the equation

through multiply-connected regions present themselves.

The bearing of the theory on Hydrodynamics, and the existence in certain cases of many-valued velocity-potentials were first pointed out by Helmholtzt. The subject of cyclic irrotational motion in multiply-connected regions was afterwards taken up and fully investigated by Sir W. Thomson in his paper on vortex-motion already referred to.

Green's Theorem.

64. In treatises on Electrostatics, &c., many important properties of the potential are usually proved by means of a certain theorem due to Green. Of these the most interesting from our present point of view have been already given; but as the theorem in question leads to a useful expression for the kinetic energy in any case of irrotational motion, we give the following proof of it.

Let u, v, w, φ be any functions which are finite, continuous, and single-valued at all points of a connected region S completely bounded by one or more closed surfaces; let ds be an element of any one of these surfaces, l, m, n the direction-cosines of the normal to it drawn inwards. We shall prove that

where the double-integral is taken over the whole boundary of S, and the triple-integral throughout its interior.

If we conceive a series of surfaces drawn so as to divide Sinto any number of separate parts, the integral

taken over the boundary of S, is equal to the sum of the similar integrals taken each over the whole boundary of one of these parts. For, for every element do of a dividing surface, we have, in the integrals corresponding to the parts lying on the two sides of this surface, elements (lu + mv + nw) do, and (l'u+m'v+n'w) do, respectively. But the normals to which l, m, n, l', m', n' refer being drawn inwards in each case, we have l' = − 1, m' = -m, n' =-n; so that in forming the sum of the integrals spoken of the elements due to the dividing surfaces disappear, and we have left only those due to the original boundary of S.

Now let us suppose the dividing surfaces to consist of three infinite series of planes parallel to yz, zx, xy, respectively. Let x, y, z be the co-ordinates of the centre of one of the rectangular spaces thus formed, dx, dy, dz the lengths of its edges, and let us calculate the value of (18) taken over the boundary of this space. As in Art. 8 the part of the integral due to the yz-face nearest the origin is

d. Du dx) dydz, and that due to the opposite face